2017 the History of Philosophical and Formal Logic From Aristotle to Tarski
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Te History o Philosophical and Formal Logic
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Also available rom Bloomsbury Te Bloomsbury Companion to Aristotle, edited by Claudia Baracchi Te Bloomsbury Companion to Bertrand Russell, edited by Russell Wahl Te Bloomsbury Companion to Philosophical Logic, edited by Leon Horsten and Richard Pettigrew Philosophical Logic, George Englebretsen and Charles Sayward
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Te History o Philosophical and Formal Logic From Aristotle to arski Edited by Alex Malpass and Marianna Antonutti Marori
Bloomsbury Academic An imprint o Bloomsbury Publishing Plc LONDON • OXFORD • NEW YORK • NEW DELHI
• SYDNEY
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First published 2017 © Alex Malpass, Marianna Antonutti Marfori and Contributors, 2017 Alex Malpass and Marianna Antonutti Marfori have asserted their right under the Copyright, Designs and Patents Act, 1988, to be identified as the Editors of this work. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers. No responsibility for loss caused to any individual or organization acting on or refraining from action as a result of the material in this publication can be accepted by Bloomsbury or the editors. British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library. ISBN :
HB : 978-1-4725-1350-2 ePDF : 978-1-4725-0525-5 ePub: 978-1-4725-0717-4
Library of Congress Cataloging-in-Publication Data
Names: Malpass, Alex, editor. | Marfori, Marianna Antonutti, editor. Title: The history of philosophical and formal logic from Aristotle to Tarski / edited by Alex Malpass and Marianna Antonutti Marfori. Description: London, UK; New York, NY, USA: Bloomsbury Academic, an imprint of Bloomsbury Publishing, Plc, [2017] | Includes bibliographical references and index. (hb) | ISBN 9781472505255 Identifiers: LCCN 2016039911| ISBN 9781472513502 (epdf) Subjects: LCSH: Logic--History. Classification: LCC BC15 .H57 2017 | DDC 160--dc23 LC record available at https://lccn.loc.gov/2016039911 Cover design: Catherine Wood Cover image © Dylan Griffin Typeset by RefineCatch Limited, Bungay, Suffolk
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Contents Preace
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Introduction Part I
1
Te Origins o Formal Logic
1
Aristotle’s Logic Adriane Rini
2
Stoic Logic
3
Medieval Logic
Katerina Ierodiakonou Sara L. Uckelman
29 51 71
Part II Te Early Modern Period
4
Leibniz
5
Bolzano
6
Boole
Part III
Jaap Maat
101
Jönne Kriener Giulia erzian
121 143
Mathematical Logic
7
C.S. Peirce Peter Øhrstrøm
165
8
Frege Walter B. Pedriali
183
9
Peano and Russell
229
Alexander Bird
10 Hilbert Curtis Franks Part IV
243
wentieth-Century Logic
11 Gödel
P.D. Welch
269
12 arski Benedict Eastaugh
293
Index
315
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Preace
Students o introductory logic courses usually encounter the undamental concepts o ormal logic – notions like valid argument, consistency, truth in a model, soundness, completeness, and so on – or the first time through those courses. However, these concepts are ofen presented in an ahistorical way, which can make them seem highly abstract, inaccessible, and set in stone, rather than the result o many centuries o conceptual development and refinement. Understanding how the current shape o these concepts is the result o their historical development can make them appear less abstract and artificial, and thus more comprehensible. Furthermore, logicians o previous centuries understood undamental logical questions and concepts differently than we typically do now. Te richness o their ideas and approaches to logic have an intrinsic interest, but they may also provide a valuable source or philosophicallyminded students who seek to challenge the currently dominant approach to philosophical and ormal logic. Te History of Philosophical and Formal Logic: From Aristotle to arskiis a partial history o these undamental questions and concepts, beginning with the work o Aristotle, the ounder o the discipline, and ending with that o a modern giant, Alred arski. Each chapter has been written especially or this collection and ocuses on the work o a particular figure or school, with the first part (‘Te Origins o Formal Logic’) concentrating on Ancient Greek logic (Aristotle and the Stoics), together with the logical work o the scholastics in the Middle Ages. In the second part (‘Te Early Modern Period’) we see how Gottried Leibniz’s mathematizing approach to logic in the 17th century was ollowed by nineteenth century thinkers with a similar cast o mind, Bernard Bolzano and George Boole. Tey can be seen as initiators o the mathematical revolution in logic that took place in the late nineteenth and early twentieth centuries. Te third part, ‘Mathematical Logic’, is concerned with figures whose work exemplifies that revolution: C.S. Peirce, Gottlob Frege, Giuseppe Peano, Bertrand Russell, and David Hilbert. Te final part, ‘wentieth-century Logic’, assesses the work o two vii
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Preface
key figures in the development o contemporary mathematical logic, Kurt Gödel and Alred arski, who together shaped much o the course o logic in the twentieth century. Each o the aorementioned figures was surrounded by other logicians whose achievements, and their importance or the development o logic, we were unortunately unable to explore more extensively in this book. Te History of Philosophical and Formal Logic was conceived as a way or undergraduate students with little training ormal logic to discover rootsor o logical concepts. However, we also hopeinthat it provides a startingthe point anyone with an interest in this wonderul discipline to become acquainted with its history. By producing an introductory book whose chapters span the entire history o the discipline, we aim to show how the elements o the standard undergraduate logic curriculum took their current orm only relatively recently, emerging rom a long development by many hands whose interests and motivations varied widely. Moreover, by shedding light on a ew topics that have not been given as much attention in standard histories o logic, such as the contributions o Leibniz and Bolzano, as well as the diagrammatic logics o C.S. Peirce, we hope to play a part in encouraging a broader view o the history o logic, which as a discipline surely has much to gain in being more inclusive o other figures and traditions. We only regret not having been able to contribute more in this regard ourselves. Tis book was born rom a series o lectures on the history o logic at the University o Bristol, organised by Alex Malpass. Aimed at undergraduates in philosophy as well as members o other departments and aculties, the lectures were given by philosophers and logicians at Bristol, together with guest lectures rom researchers at other institutions . In order to bring the content o these lectures to a wider audience,the speakers in the lecture series were invited by Alex Malpass to contribute to this book. In the end, the chapters ‘Bolzano’ by Jönne Kriener, ‘Boole’ by Giulia erzian, ‘Peano and Russell’ by Alexander Bird, and ‘arski’ by Benedict Eastaugh were written by speakers in the lecture series, while the other chapters were contributed by international experts invited by the editors. Marianna Antonutti Marori joined the project in 2014 to complete the book. Te writing o the Introduction reflects the respective expertise o the two editors: the first hal, rom Aristotle to Boole, was written by Alex Malpass, while the second hal, rom Peirce to arski, was written by Marianna Antonutti Marori. Marianna Antonutti Marori Paris and Munich March 2017
Introduction Put simply, logic deals with the inerence o one thing rom another, such as the inerence o a conclusion rom the premises o an argument. An example o this would be the ollowing: Te emission o certain chemicals that harm the ozone layer is likely to cause global warming, and petrol-consuming vehicles emit these chemicals in large quantities; thereore, global warming is likely.
Premises can either ‘succeed’ or ‘ail’ at ensuring that the conclusion holds, and success means that whenever the premises are true, the conclusion is either also true or at least more likely to be true. In the previous example, the premises about the harmulness o certain emissions and the act that petrol-based vehicles emit these chemicals in large quantities supports the conclusion that global warming is likely. I the premises are indeed successul in establishing the conclusion, then we call the resulting inerence valid. Logic can be thought o as the study, or perhaps even the ‘science’, o valid inerence. In this book we will be looking at the core o this science o inerence, by which we mean deductive logic. In a valid deductive argument, the truth o the premises ensures the truth o the conclusion, such as in the ollowing amous argument: All men are mortal, Socrates is a man; thereore, Socrates is mortal.
In this example, one cannot grant the truth o the premises without also granting that Socrates is mortal – the conclusion ollows deductively rom the premises. Deductive logic is only one branch o logic, however, and there is also inductive logic. Generally, inductive logic covers arguments where the conclusion is not established conclusively rom the premises, but the likelihood o its being true is supported by the premises. Consider the ollowing argument: All observed swans have been white; thereore, the next swan to be observed will be white.
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2
Introduction
o illustrate the pitalls o inductive arguments, Bertrand Russell proposed the ollowing example: a turkey which has been ed every single day until Christmas Eve most likely thinks that it will be ed the ollowing day too, but, unortunately or the turkey, this is not the case. Inductive arguments are extremely useul when conducting science, and according to some philosophers inductive reasoning is characteristic o the scientific methodology. However, we will not be concerning ourselves with inductive logic, and will be ocusing specifically on deductive logic. Like most subjects in philosophy, the general notion o logical inerence is amiliar rom everyday lie, and is ofen used in casual conversational settings (especially i you happen to be a Vulcan on Star rek); but when we reflect on it, the phenomena that once seemed so obvious can come to take on a new light and begin to seem mysterious. Consider the curious act that deductive logic is at the same time both entirely intangible and also incredibly powerul. It is intangible in the sense that it is a completely abstract phenomenon (or instance, one cannot see logic, or measure it with experiments), and yet it is powerul because, in a sense, its laws come with a orce stronger than any known to physics, because the conclusions o sound deductive arguments are known to be true with absolute certainty (Wittgenstein once reerred to this phenomenon as the ‘hardness o the logical must’,PI #437). Mysterious though this notion might seem to be, over the past 150 years logicians have made staggering advances. It is no exaggeration to say that we live in a golden age or logic, which has seen oundational developments in pure logic that have led to practical consequences ollowing on rom them, primarily the establishment o computer programming, but also the ormalization o various branches o the arts and sciences (consider the use o game theory in economics, or example). In this book we will look at the ascinating and highly complicated story o the history o logic, rom the first tentative reflections o the ancient Greeks to the development o ormal languages and mathematical models amiliar to the twenty-first-century philosopher, mathematician and computer scientist. Roughly speaking, the development o logic in Europe has had three ‘golden ages’: the Greek period, rom around 350 to 200 BCE ; the high medieval period, rom around 1100 to 1400; and the modern period, rom around 1850 up to the present day. Tese classifications are, o course, generalizations that do not capture the ull reality o logical developments by lone pioneers (such as Leibniz) and in other cultures (such as in the Indian, Chinese, and Arabic and Islamic worlds). For the purposes o this book, we will be ocusing on deductive logic,
Introduction
3
and its development in Europe. Tis is not to say that developments elsewhere are not o value – quite the contrary. Te work o logicians in the Islamic world and the influence o their work on logicians in Western Europe during the High Middle Ages, in particular, is a ascinating topic that has recently gained much attention rom scholars, but that or limitations o space will not be treated in this book. first golden logic happened within like the space o a (384–322 ew generations in Te Ancient Greece,age andoincluded great figures Aristotle BCE), Diodorus Cronus (c. 340–280 BCE ) and Chrysippus (279–206 BCE ). Tese early logicians managed to make huge progress, with very little in the way o genuine intellectual predecessors. Tere were orerunners to this period who used logical inerences, particularly in ethics and metaphysics, but the ormalization and reflection on the general principles o validity seems to have been lacking. For instance, Zeno o Elea ( c. 490–430 BC E) is well known to have offered various paradoxes, o motion and o length etc., in an effort to deend the metaphysical doctrines o his master, Parmenides (c. 50 0 BCE ). Zeno essentially argued that motion is an illusion, because it leads to impossible consequences. Tus, he used the argument orm: I p, then q; but not-q (because it is impossible); thereore, not-p
Zeno obviously had a good intuitive grasp o logical inerence, and probably had the eeling that he was arguing in a ‘correct’ manner. However, simply arguing logically is not sufficient to make one a logician. What is required in addition is an appreciation o the notion o valid inerence in abstraction rom any particular argument; and this, it seems, did not properly arrive beore Aristotle. From the beginning o the Greek golden period we get many anticipations o later developments. From Aristotle we get a wonderully worked-out system o ‘term-logic’, known as the ‘syllogism’, which was a precursor to the firstorder logic, or ‘predicate calculus’, and which became easily the most influential contribution in logic or two thousand years. From the Stoics we get the first ormulations o elements o propositional logic, with definitions o connectives given via truth-values, but we also see many other interesting ormulations o logical problems, such as proposals or the nature o implication that are very similar to some contemporary theories on subjunctive and indicative conditionals. Much o the best work orm the Stoic period, notably the work o Chrysippus, has been lost to history, but we know indirectly that he wrote much on logic generally, and specifically about paradoxes, including the Liar and the Sorites. It was or a very long time thought that the Aristotelian and Stoic
Introduction
4
conceptions o logic were at odds with one another, beore their reconciliation in the modern era under the umbrella o first-order logic. Unlike the contributions o Chrysippus, Aristotle’s work on logic was very widely read, especially in Europe rom the twelfh century onwards, and over the course o history it has exerted an enormous influence on how philosophers have viewed logic, giving up its dominance over the discipline only in the modern Tis influence so great that Józe Bocheński wrote in period. the introduction to hiswas monumental A History o Formal (1902–1995) Logic (1961) that the word ‘logic’ can actually be defined in relation to Aristotle: [W]e find that there is one thinker who so distinctly marked out the basic problems o [logic] that all later western inquirers trace their descent rom him: Aristotle. Admittedly, in the course o centuries very many o these inquirers – among them even his principal pupil and successor Teophrastus – have altered Aristotelian positions and replaced them with others. But the essential problematic o their work was, so ar as we know, in constant dependence in one way or another on that o Aristotle’sOrganon. Consequently we shall denote as ‘logic’ primarily those problems which have developed rom that problematic. Bocheński 1961: 2
Even i ew would be happy with Bocheński’s definition today, Aristotle’s influence on the history o logic is undeniable. Given its importance, we shall now say a little about Aristotle’s main logical system, known as the ‘categorical syllogism’ (or more see Chapter 1). As stated above, Aristotle’s syllogistic logic was primarily concerned with ‘general terms’, which are words like ‘Greeks’ or ‘mortals’, as opposed to singular terms, such as ‘Socrates’ or‘Plato’. General terms generally pick out a group o objects according to their shared possession o a property; or example the term ‘Greeks’ picks out all those people who have the property o having been born in Greece, etc. According to Aristotle, each basic declarative sentence has a subject–predicate structure; so there is something that the sentence is about (the subject), and we say something about it (the predicate). For example, in the sentence ‘Greeks are mortal’, the term ‘Greeks’ is the subject, and ‘mortal’ is the predicate. Lastly, we also need to say something about the quantity in which the predicate term applies to the subject term. Tis can come in our grades:
1. 2. 3. 4.
Every Greek is mortal Some Greek is mortal Greek is mortal Not every Greek No is mortal
Universal Affirmative) Particular Affirmative) ( Particular Negative) ( Universal Negative) ( (
Introduction
5
According to Aristotle, the syllogism is a tightly defined type o argument, in which there are two premises and a conclusion, each o which is a term-based quantified subject-predicate sentence, like 1 to 4 above, and which share three terms between them. Consider the ollowing example o a syllogism,
Premise 1: Premise 2:
Every Athenian is Greek Every Greek is mortal
Conclusion:
Every Athenian is mortal
In this argument, there are three terms (‘Athenian’, ‘Greek’ and ‘mortal’). Te terms are reerred to as minor, middle and major, depending on their position in the argument. Te minor term is that which is the subject o the conclusion (in the above example, this would be the term ‘Athenian’), the major is the predicate o the conclusion (i.e. ‘mortal’), and the middle term is that which is shared by both o the premises (i.e. ‘Greek’). Te premises each have one o the minor or major terms, and are consequently labelled as the minor premise and the major premise. For example, premise 1 is the minor premise, as, in addition to the middle term (‘Greek’), it eatures the minor term (‘Athenian’); premise 2 is accordingly the major premise. One o Aristotle’s main innovations in the course o developing the syllogism, and one o the reasons why we date the beginnings o logic to his work, was the use o letters to stand in or the terms. Tis simple technique provided a useul way o abstracting away rom the linguistic and conceptual detail o the terms used, and allowed one to grasp the proper logical orm o the underlying argument. Tus, the above argument is really just a particular instantiation o an argument orm, later known as Barbara, which looks like this (you can find out why and how Aristotelian syllogisms were given names like Barbara in Chapter 1): Every α is β Every β is γ Every α is γ Tis syllogism is just the first o ourteen that Aristotle categorizes or us in the Prior Analytics. Tese are arranged in three ‘figures’, which are distinguished by the position in the premises o the shared middle term. Aristotle takes the first two syllogisms o the first figure, which includes Barbara, to be ‘perect’ or selevidently valid. Using some intuitive conversion rules or sentences (like: rom ‘Some α is β’ we can derive ‘Someβ is α’), Aristotle shows a method according to which one can prove that each o the other twelve syllogisms is reducible to the
6
Introduction
first two, and thus that each syllogism is correspondingly valid. Although it was not presented as such at the time, his theory o syllogisms then can be viewed as a system, akin to that o Euclid’s (c. 300 BCE ) geometry, in which the two perect syllogisms are the axioms, and the others are theorems, provable via the axioms and the rules o inerence he advocates. Tus, Aristotle’s achievement was not just the outlines o some general thoughts about logic, as one might expect rom atheory. pioneering theoretician, butachievement a ully developed andlater proto-prooTe sheer scale o the seems logical to havesystem hampered attempts at developing logic, as there seemed to be little prospect o improving on Aristotle’s example. Indeed, we see Kant (1724–1804) (who seems to be unaware o the achievements o the medieval logicians) in the preace to the second edition o the Critique o Pure Reasonsaying the ollowing: Tat logic rom the earliest times has ollowed this sure path may be seen rom the act that since Aristotle it has not had to retrace a single step, unless we care to count as improvements the removal o some needless subtleties or the clearer exposition o its recognized teaching, eatures which concern the elegance rather than the certainty o the science. It is remarkable also that to the present day this logic has not been able to advance a step, and is thus to all appearance a closed and completed body o doctrine. Kant, Critique o Pure Reason(1781), B viii, p. 17
Te act that Aristotle’s achievement was so great actually had the unexpected consequence o holding back logical progress. It turns out that there are in act a number o additional requirements that Aristotle built in to the theory o the syllogism which i dropped would have allowed a much more general approach to be developed, but which were not ully appreciated or many centuries due to the compelling nature o the system as it stands. When a logical system was developed that could work without these restrictions, modern logic was born. One o these eatures o the logic which were later dropped in the modern period was the inerence rom ‘All A is B’ to ‘Some A is B’. Tis effectively meant that no term is empty, i.e. no terms like ‘Martian’ eature in the theory o the syllogism. Also, no term reers to everything. So the terms are always o greater extension than the least possible thing (nothing), and lesser than the greatest possible thing (everything). Tese requirements were later dropped, in the development o the modern logic by people like Charles Sanders Peirce (1839–1914) and Gottlob Frege (1848–1925). However, the personal achievement o Aristotle was so great that it took around two thousand years or these developments to take place.
Introduction
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Te end o this Greek golden age coincided more or less with the decline o Macedonian dominance and the rise o the newly emergent Roman power. It seems that, or whatever reason, the study o logic was not something that flourished under the Roman regime. Consequently, we find ew real innovations in logic in the ollowing centuries, which were more typified by commentaries on previous works, such as those o Alexander o Aphrodisias ( fl. 200 CE) and Porphyry CE ), rather than genuine developments. with so many o(234–305 the intellectual achievements o thetechnical ancient world, the study As o logic gradually aded into obscurity in Europe with the decline o the Western Empire by about 450 CE . Boethius’ (480–526 CE) translation o Aristotle’sOrganon in the final years o his lie represents almost the last reserve o Western knowledge o Aristotelian logic, and soon even this was lost to the learned men o Europe. While learning in the West declined, the study o logic was kept alive in the thriving Islamic world under the Abbasid Caliphate (750–1258), and reached great heights o sophistication with notable contributors including Al-Farabi (872–950), Avicenna (980–1037) and Averroes (1126–1198). From the start o the twelfh century onwards, Europe became increasingly exposed to the Arabic world, in part due to the Crusades, and with this exposure came a rediscovery o the ancient Greek texts on logic, especially those o Aristotle, along with the benefit o the sophisticated Arabic commentaries. As these texts circulated around Europe we find the beginning o our second golden age o logic, rom about 1100 to 1400, the high medieval or ‘scholastic’ period, with notable contributors including Peter Abelard (1079–1142), William o Sherwood (1190–1249), Peter o Spain (c. 1250), William o Ockham (1295–1349) and Paul o Venice (1369–1429). Curiously, during this period there was a combination both innovative progress and reverence or ancient authority, most particularly that o Aristotle. Te progress came in the orm o greater understanding o quantification, reerence and logical consequence, a more intricate treatment o semantics, and developments in modal and temporal logic. Unortunately, this high point or European logic was itsel not to last. Te precise reasons or this decline are very complicated, and still hotly debated by contemporary historians o logic. Various actors contributed to the decline o medieval logic, including the ollowing. Firstly, as we saw ollowing the decline o Greek logic, the period o great novel development in the medieval period was ollowed by a period o commentaries and comparisons. Tere were many logicians working in this period, and each had their own particular way o using technical terms and o distinguishing between concepts. It is tempting to think that the task o simply
8
Introduction
comprehending and compartmentalizing the state o the art o medieval logic was such an arduous task that it became all that the later generations could manage. Secondly, there is an argument that socio-economic actors led to changes in the education system o the universities, which in turn led to a declining ocus on logic. Tis has been eloquently outlined by Lisa Jardine: In the early decades o the sixteenth century, across northern Europe, we find the introductory arts course being adapted to meet the requirements o an influx o students rom the proessional classes. Within this arts course, with its humanist predilection or Greek and Latin eloquence, and legal and ethical instruction, there was an acknowledged need or some rigorous underpinning o instruction in ‘clear thinking’. But the meticulous introduction to ormal logic and semantic theory provided by the scholastic programme came to look increasingly unsuitable or this purpose. Jardine 1982: 805
Te idea is that these new ‘business classes’ needed to learn practical knowledge rom their education, and the dusty logical and semantic distinctions o the scholastics no longer fitted the bill. Another closely related argument is that medieval logic was swept aside by the rise o humanism, and its ocus on rhetoric over logic. One figure who is notable as a passionate advocate o humanism and critic o scholasticism is Lorenzo Valla (1407–1457). In particular, Valla championed two major and distinctively humanist ideas that clashed with the medieval outlook. Firstly, he held that the great Roman authors o antiquity, such as Cicero (104–43 BCE) and Quintilian (35–100 CE), were the final authorities when deciding a point about the correct use o Latin, which is the language that discussions o philosophy and logic were conducted in during the medieval period. In contrast to the linguistic recommendations that might ollow rom an abstract consideration o Aristotle’s logic, Valla urged that the authority o Latin grammar ought to flow rom the period when it was used as a living language. So, i a certain usage was urged by Aristotle, but a contrary one was urged by Cicero, then we ought to go with Cicero. In addition to this, Valla maintained that the most important part o argument is persuasion, rather than certain proo. One can see that in practical matters, perhaps pertaining to legal cases, it is persuasion, rather than proo, that is practised. Tus, in general, one can say that, in the pedagogical priorities o the humanist period, logic gave way to rhetoric. By the time we get to Peter Ramus (1515–1572) we get a particularly drastic rejection
Introduction
9
o the Aristotelian authority which was so characteristic o the scholastic period, as Ramus is supposed to have chosen the topic o his master’s thesis to be that ‘everything Aristotle taught was alse’ (Kneale and Kneale [1962]: 301). While the study o pure logic may have declined in the early scientific period, there were a great many advances in mathematics, in particular in algebra and geometry. By the end o the sixteenth century European mathematicians had recovered manythe o Arabic the works romdevelopments. antiquity thatTis hadperiod been lost, and largely improved upon algebraic o mathematical progress arguably culminated with the creation o the infinitesimal calculus o Leibniz (1646–1716) and Newton (1642–1727). Leibniz himsel was an outstanding mind who worked in many fields and, against the intellectual tide o his day, he conducted a solitary programme o innovative research in ormal logic. Although the main body o Leibniz’s works on logic remained relatively unknown until the turn o the twentieth century, in retrospect he is hugely significant or articulating a completely new vision o the subject – one that stressed the use o the recently developed mathematical and algebraic methods. In classical antiquity, logic was conceived o as a tool to be used in oratory and argumentation, as exemplified by Plato’s Socratic dialogues. Although early logicians such as Aristotle and Chrysippus did reflect on rules o inerence and the ormalization o arguments, they too saw logic first and oremost as a tool to be used in philosophical debate. Tis general conception o logic was also maintained by the medieval logicians, as evidenced by the distinctively medieval notion o the obligationes, which was an attempt to produce rules or public philosophical debate (see Chapter 3). In contrast to this, Leibniz had a vision o logic as a purely symbolic and algebraic discipline, divorced entirely rom the all too human roles o respondent and interlocutor. According to the new vision, logic would be a subject matter o the highest importance in itsel, not just a tool to help someone win an argument. Leibniz’s motivations or pursuing an algebraic treatment o logic were deeply rooted in his philosophical views. He conceived o a mathematically reconstructed logic as a medium that would reflect the nature o thought more aithully than the comparatively clumsy linguistic approach o the medieval logicians. Te idea was that this new logic, being symbolic rather than linguistic, would actually improve the way that we think, by laying out the real elements o thought with perect clarity, and would thereore be a crucial key in advancing science generally. In his ambition, Leibniz saw the possibility or a universal language o thought based on logic; a lingua philosophica (or ‘philosophical language’), which is something that would be recognizable to any student o analytic philosophy
10
Introduction
today. Despite the details o Leibniz’s logical innovations remaining largely out o sight in the Royal Library o Hanover until the turn o the twentieth century, the general character o his vision managed to exert an influence on a number o early pioneers in the modern age o logic. For instance, Leibniz’s ideal o a perect language o thought heavily influenced two o the most important logicians o the early modern period; George Boole (1815–1864) and Gottlob Frege. Tis influence reflected titles o their best-known works on logic – (which Boole’s 1854 workis Te Laws inothe Tought and Frege’s 1879 work Begriffsschrif means ‘concept-script’). Although Boole and Frege differed significantly in their approaches, both were trying to realize the mathematical vision o logic that Leibniz had envisaged. Both Boole and another important British logician, Augustus De Morgan (1806–1871), published major works on logic in the year 1847 – Boole’s Mathematical Analysis o Logic, and De Morgan’sFormal Logic. It is rom the publication o these works that we shall date the third, and current, golden age o logic – the modern period. Boole’s work was more systematic than De Morgan’s, and exerted greater influence on later thinkers. In order to give an idea o Boole’s main innovations, we shall say a little about his work now. Boole’s primary concern was to give an algebraic account o the laws o logic, as ound in Aristotle’s categorical syllogisms. However, his developments quickly went beyond this rather limited goal. As we have seen, the categorical syllogism dealt with the logic o general terms, such as ‘Greeks’ and ‘mortal’, and Boole ollowed this lead by using the more algebraic symbols x, y, z etc. to stand or terms, but he decided to find new ways o presenting the logical relations between the terms themselves. Boole used the combination o xy (the ‘multiplication’ o x and y) in much the same way that natural language combines predicates in expressions like ‘green cars’. Te result is the intersection o the green things and the car things (only those objects that are in both classes). Tis allowed a very simple equation to represent the universal affirmative ‘All x are y’: in Boole’s notation it becomes xy = x. Te class o things that are xy coincides with the class o things that are x, because all o the things that are x are also y. One o Boole’s chie innovations came was his use o the symbols ‘1’ or the ‘universe’, or class o everything, and ‘0’ or the empty class. Using this, Boole could symbolize the notion o the things which are not-x, the ‘complement’ o x, as 1–x (i.e. everything minus the x things), which allowed him to ormulate an alternative expression or the universal affirmative as x(1–y) = 0 (i.e. the set which is the intersection o the xs and the not-ys is empty). Also, we have the translation o the universal negative, ‘No x is y’, as simplyxy = 0. Te translation
Introduction
11
o the particular statements was more contentious. His proposed translation o ‘Some x is y’ was v = xy, where the auxiliary letter ‘v’ is used to mean ‘some’. He could then translate ‘Not every x is y’ by the equation v = x(1–y). Boole’s algebraic approach to logic was urther developed by Charles Sanders Peirce (1839–1914), beginning in the late 1860s. Amongst Peirce’s many contributions, one o the most striking is his abandonment o the Aristotelian dictum no even term is empty.no Onunicorns Peirce’s account, the statement ‘All unicorns are white’ isthat true, though exist. Tis change legitimated some inerences, and invalidated others: ‘Some unicorn is white’ could no longer be inerred rom the premise A ‘ ll unicorns are white’, but ‘All unicorns are coloured’ could be inerred rom the premises ‘All unicorns are white’ and ‘All white things are coloured’. Peirce was a polymath whose work in logic embraced an innovative range o diagrammatic proo systems, the Existential Graphs. Within these, the system o Alpha Graphs corresponds to what we now know as propositional logic, and his Beta Graphs to first-order predicate logic. Te Existential Graphs were largely ignored at the time, but more recent interest in the expressive power and pedagogical advantages o diagrammatical reasoning has produced a renewed ocus on this aspect o Peirce’s work. While less developed, his complex system o Gamma Graphs reflected Peirce’s desire to extend logic beyond what we now call classical logic, and incorporate modal eatures such as notions o temporality and possibility. His analysis was to prove influential on A.N. Prior (1914–1969), an important figure in the development o intensional logics that treated modal notions such as possibility, necessity and tense. Between 1870 and 1883, Peirce also created what we now know as the calculus o relations. Te calculus o relations is the ormal study o the kinds o relationships that hold between objects, such as ‘larger than’ or ‘parent o’, and not just properties that hold o single objects, such as ‘is black’ or ‘is a horse’. Te operations o intersection, union and complement which appeared in Boole’s system can be extended to the relational setting, but Peirce wenturther, including other operations on relations such as composition. Consider the relations ‘is the mother o’ and ‘is a parent o’: these compose to produce the relation ‘is a grandmother o’, since given any three peoplex, y and z, i x is the mother o y and y is a parent o z, then x is the grandmother o z. Te calculus o relations was also the setting or a urther Peircian innovation, namely the introduction o quantifiers in something like their modern orm. Quantifiers are a linguistic device which allow one to speciy how many things o a certain sort satisy some property; in natural language they include locutions
12
Introduction
such as ‘every’, ‘there exists’, ‘most’ and ‘none’. Quantifiers allow us to make general statements, such as those we can reason about syllogistically, but the quantifiers introduced by Peirce are a more flexible and expressively and deductively powerul device. Tey solve the ollowing problem that De Morgan raised or Aristotle’s logic. Given that a horse is an animal, it ollows that the head o a horse is the head o an animal. Aristotelian syllogistic logic does not allow us to make this inerence, and thus does notother capture this do seemingly correct and simple inerence. Peirce’s quantifiers, on the hand, allow this inerence. Extending logic with quantifiers was a major advance, and as sometimes happens with such discoveries, it was made independently by different researchers. Peirce’s initial introduction o the existential and universal quantifiers, in the guise o unrestricted unions and intersections o relations, took place in his 1880 paper ‘On the Algebra o Logic’, but a ull presentation o his quantification theory only appeared in 1885 in ‘On the Algebra o Logic: A Contribution to the Philosophy o Notations’. Six years earlier, Gottlob Frege had also introduced quantifiers in his Begriffsschrif, and although the discovery should be credited to both logicians, Frege’s logical work includes other distinctively modern eatures that do not appear in Peirce, and which we shall discuss in due course. Aristotelian quantifier phrases take two predicates as arguments: they have the orm ‘All A are B’, ‘Some A is B’, ‘No A is B’ or ‘Not all A are B’. Fregean or Peircian quantifiers are unary and introduce a bound variable: a variable standing or an object o the domain o quantification, o which properties can then be predicated. In Peirce’s system there are two kinds o quantifier: the existential quantifier ∃, and the universal quantifier ∀. Tanks to his treatment o negation, Frege only needed the universal quantifier: he could define the existential quantifier by combining negation with universal quantification. In modern notation, the statement ‘All A are B’ can be ormalized using a universal quantifier and the connective or the conditional, as ollows: ∀x(Ax → Bx). Inormally this reads ‘For every object x, i x has the property A, then x has the property B’. Suppose that we take the unary predicate H to denote the property o being a horse; the unary predicate A to denote the property o being an animal; and the binary predicate C to denote the property o being the head o something. We can then ormalize the statement ‘Every horse is an animal’ as ∀x(Hx → Ax), and the statement ‘Te head o a horse is the head o an animal’ as ∀x[∃y(Hx ∧ Cyx) → ∃y(Ax ∧ Cyx)]. Inormally we can read this second sentence as asserting that or every object x, i there is some y such that x is a horse and y is the head o x, then there is some y such that y is the head o an animal (namely x). Te
Introduction
13
inerence rom the ormer to the latter is a valid one in the predicate calculus, and proving it is a good exercise or a beginning student o logic. Te logic that Frege presents in the Begriffsschrif has a spe cial place in history, as it draws together in a systematic way several important elements o ormal systems as we now conceive o them. One way to think o Frege’s system is as a predicate calculus in the modern sense: a ormal language containing quantifiers, logical predicates both and relational ones), togetherconnectives with logicaland axioms and laws(including o inerence. Teunary system o the Begriffsschrif included nine logical axioms and one law o inerence, namely modus ponens: rom a conditional statement A → B and the antecedent A, iner B. It is this packaging o a ormal language, axioms and laws o inerence together as a coherent whole that makes Frege’s work a landmark contribution to logic. Nevertheless, one should not read too much into the of-made claim that Frege is the ather o modern logic, since his system differed in important ways rom those that were to ollow. o begin with, Frege’s logic wassecond-order: there were not only quantifiers ranging over the objects o the domain, but secondorder quantifiers which ranged over unctions on the domain. For example, i the first-order domain consists o students, then the second-order domain consists o groups o students, such as all those enrolled in a philosophy degree, or all students with names o one syllable. Moreover, Frege took a ‘unctions first’ approach, casting properties and relations in terms o unctions, which he called concepts: properties were nothing but unctions rom objects to truth- values. A predicate like ‘is happy’ was understood by Frege as a unction that mapped to ‘the rue’ all the objects which are happy, and to ‘the False’ all the objects which are not happy. Finally, Frege’s notation was, as Russell put it, ‘cumbrous’ and ‘difficult to employ in practice’ (Russell 1903: 501). Frege’s work is also important in another sense, namely that its intertwining o logic and the oundations o mathematics represents the continuation by ormal means the programme o rigorization o the calculus (and mathematics more generally) pursued by Bolzano, Weierstraß and Cantor. InDie Grundlagen der Arithmetik (1884) and the two-volume Grundgesetze der Arithmetik (1892, 1903), Frege developed a oundation or mathematics based – at least in Frege’s own view – entirely on principles which were logical in nature. Te system o the Grundgesetze included the powerul Rule o Substitution, which is equivalent to a comprehension principle or Frege’s second-order logic, namely a scheme asserting that every set definable by a ormula in the language exists. It also included the inamous Basic Law V, which in modern terminology can be defined more or less as ollows: or all unctions X and Y, the extension o X (the
14
Introduction
collection o all objects with the property o being X) is equal to the extension o Y i, and only i, X = Y. Bertrand Russell (1872–1970) realized that Basic Law V implied the contradiction we now know as Russell’s paradox. Te ‘extension o’ operator could be used to orm new concepts. Tis means that the ormation o the ollowing concept is legitimate within the system: the concept o all concepts whose extensions are not members o themselves. Call this concept R, and call its extension E. Does all so under the concept o R? it does, then does satisy the definition o R,Eand E cannot all under theI concept o R.EBut i Enot does not all under the concept o R, then it satisfies the definition, and so E must all under the concept o R. We thus have a contradiction either way. Russell’s paradox was seemingly already known to Cantor, as the same phenomenon appears in set theory, and indeed is closely related to Cantor’s theorem itsel. While Russell’s paradox was a shock to Frege, it has proved immensely ruitul or logic and the oundations o mathematics: the development o set theory and type theory, and indeed our entire understanding o the philosophy o mathematics have been shaped by responses to Frege and Russell’s paradox. wo o these responses have been particularly influential, reflecting as they do two different ways o restricting logic and the oundations o mathematics to avoid Russell’s paradox. Te first o these is the theory o types, as developed primarily in the Principia Mathematica o Russell and Alred North Whitehead (1861– 1947). Russell and Whitehead’s ramified theory o types is highly complex and we will not attempt to explain it here; instead we shall sketch the ideas behind a simpler type theory that suffices to illustrate the point at hand. Te theory o types guards against paradox by imposing a hierarchy on the universe o objects defined in the theory. At the lowest level, 0, is the type o individuals. At level 1 we have the type o classes o individuals – or example, i the individuals are natural numbers, then the class o even numbers is a type 1 object. At level 2 we have classes o classes o individuals, and so on. Tis restriction blocks Russell’s paradox because there is no way to orm the class o all classes that do not contain themselves: each class can be ormed only o objects o lower type, while the definition o the Russell class R ranges over all classes. Te second response to Russell’s paradox was the axiomatic theory o sets developed by Ernst Zermelo (1871–1953). Zermelo’s solution to the paradox was to restrict the comprehension principle in two ways, giving rise to his Aussonderungsaxiom (Separation Axiom). Te first restriction was that new collections ormed via this axiom could only be subsets o existing sets: given a set X already proved to exist, the set o all members o X satisying a given property could then be ormed. Secondly, only ‘definite’ properties were allowed
Introduction
15
to define sets. Te paradoxical Russell class was thus legislated out o existence. Modern axiomatic set theories are typically based on Zermelo’s system, and can thus prove that there is no set containing all and only those sets that do not contain themselves. Zermelo did give some guidance on what the ‘definite’ properties were supposed to be, but the notion remained vague. Hermann Weyl (1885–1955) and Toral Skolem (1887–1963) both proposed that the criterion o definiteness replaced by theinnotion o oarder one-place predicate ormula in first-order logic,bei.e. any ormula the firstlanguage o set theory with only one ree variable is allowed in the Separation Axiom scheme. It is Skolem’s proposal, and the first-order theory o sets that resulted, that is now the dominant approach both in set theory and in the oundations o mathematics more generally. Te off-putting nature o Frege’s notation meant that the field was clear or a system o logical notation that mathematicians would be more eager to adopt. Just such a system was provided by Giuseppe Peano (1858–1932), who provided, amongst other things, the symbols ∩ or intersection, ∪ or union, and ∈ or set membership. He also introduced the backwards ‘E’ symbol∃ or existential quantification (the upside-down ‘A’ symbol∀ or the universal quantifier was introduced much later), and the symbol ∼ or negation. Te rotated ‘C’ (or ‘consequentia’) that he used to denote the material implication was adapted by Russell or Principia Mathematica as the now-amiliar horseshoe symbol ⊃. Peano’s notational improvements were important because they allowed or the more widespread adoption o a logic that took on board Frege’s deeper innovations. Te algebraic tradition o De Morgan, Peirce and Ernst Schröder (1841–1902) remained important in the early twentieth century, through its influence on the area we now know as model theory, and in particular in the work o Marshall Stone (1903–1989) on Boolean algebras, and Leopold Löwenheim (1878–1957) and Skolem on the model theory o first-order logic – that is, logic where the quantifiers range only over the objects o the domain, and not over unctions on the domain, as in second-order logic. First-order logic was properly isolated as a system o central importance some years later, in the work o Gödel and arski, but the first-order ragment o the calculus o relations – mentioned briefly in Schröder’s Vorlesungen über die Algebra der Logik– was interesting enough to inspire Löwenheim’s work on his eponymous theorem. o understand Löwenheim and Skolem’s contributions, we must return to the nineteenth century and the work o German mathematician Georg Cantor (1845–1918). In the mid-1870s, Cantor created the area o mathematics known
16
Introduction
as set theory, and proved a number o key results about infinite sets (or collections), the most important o which is known as Cantor’s theorem: that there are different sizes o infinite sets; that every set is strictly smaller than its powerset (the set o all subsets o a given set); and in particular that the set o all real numbers is bigger than the set o all natural numbers. Te notion o size at play here is known as cardinality, and we say that two sets have the same cardinality just in case a one-to-one them. For example, the there set o exists all natural numberscorrespondence and the set o between all even numbers have the same cardinality, since we can construct a unction which takes every natural number to an even number: (0) = 0, (1) = 2, (2) = 4, and in general (n) = 2n. Sets which are either finite or have the same cardinality as the natural numbers are called countable. Sets which are infinite but have a greater cardinality than the natural numbers (or example, the real numbers) are called uncountable. One o the undamental insights o the algebraic approach to logic was that logical symbols are subject to interpretation, as symbols in algebra are (or example, the geometric interpretation o complex numbers). Tis leads naturally to the idea o a model: an interpretation o the vocabulary o a logical system which satisfies its axioms. We thereore come to the notion oBoolean a algebra: any model o Boole’s axioms o logic. Such algebras consist o a set o atoms, the elements o the algebra; distinguished elements 0 and 1; and conjunction (intersection or multiplication), disjunction (union or addition) and complementation (negation or subtraction) operations on the set o elements. Te study o Boolean algebras was begun by Marshall Stone, who proved a number o important results, including the representation theorem that bears his name: every Boolean algebra is isomorphic to (that is, has the same structure as) a Boolean algebra o sets. We can trace the birth o model theory to Löwenheim’s paper ‘Über Möglichkeiten im Relativkalkül’. Working within Schröder’s version o the calculus o relations, Löwenheim showed that i a first-order sentence has a model, then it has a countable model. Te Norwegian mathematician Toral Skolem improved Löwenheim’s proo in a paper o 1920, and in doing so generalized the result rom single sentences to (possibly infinite) sets o sentences. Te result is important because it shows that first-order logic is not, in general, able to fix the cardinalities o its models: there can, or example, be models o the theory o real numbers which are countable, or models o set theory where sets that the theory holds to be uncountable are actually (rom the external, model-theoretic perspective) countable. Te completeness theorem or
Introduction
17
first-order logic, later proved by Gödel in his 1929 PhD thesis, is an easy corollary o Skolem’s work rom the early 1920s, but this was not understood until later. In order or model theory to grow into a mature discipline, a urther ingredient was needed, namely a precise understanding o the central notion o a model. Tis was supplied by Alred arski (1901–1983), whose landmark 1933 paper ‘On the Concept o ruth in Formalized Languages’ contained a definition o the satisaction relation. Tis was subsequently to arski create the o truth in a first-order structure published inupdated 1956 by anddefinition his student Robert Vaught, and which is very close to the standard textbook definition used today. Te essential idea is that a structure M o a language L consists o a set D o elements – the domain, over which the quantifiers range – together with interpretations or the nonlogical symbols o the language such as constants, relation symbols and unction symbols. A relation o satisaction, or ‘truth-in-M’, can then be defined recursively. I every sentence o a theory in the language L is true in M, we say that M is a model o . Similar definitions can be constructed or other logics, and while the model theory o first-order logic is still a major part o the discipline, logicians also study models o other logics such as modal logic and infinitary logic. Algebraic logic as an area o study in its own right was revived some decades later by arski, an admirer o Peirce and Schröder. In his 1941 paper ‘On the Calculus o Relations’, he presented two axiomatizations o the calculus o relations. arski was also responsible or a broader reawakening o interest in algebraic logic, and published a number o landmarks in the field including the two books Ordinal Algebras and Cardinal Algebras, as well as developing the notion o a cylindric algebra. Cylindric algebras stand in the same relation to first-order logic with equality that Boolean algebras do to propositional logic. In act, they are Boolean algebras, but equipped with additional operations that model quantification and equality. We can also see elements o the algebraic approach in the contributions o David Hilbert (1862–1943). Hilbert was one o the leading mathematicians o his era, and his contributions to logic were conceptual and methodological, as well as technical. We can identiy at least three separate and substantial achievements: his pioneering o the axiomatic method through his investigations into the oundations o geometry; his programme or the oundation o mathematics on a finitist basis, which bore ruit through the creation o proo theory, and also through the limitative results o Gödel; and his ormulation o the Entscheidungsproblem (decision problem), whose solution led to the ounding o computability theory and computer science.
18
Introduction
Te roots o the axiomatic method lie in geometry, with the work o the ancient Greek mathematicians, as collated and extended by Euclid in the Elements. It is thereore appropriate that it is in geometry, and specifically in Hilbert’s work on the oundations o geometry, that we find the first modern exemplar o this method. During the nineteenth century, figures like Gauss, Bolyai, Lobachevsky and Riemann created non-Euclidean geometry: geometrical spaces obeyed all oa Euclid’s geometrical postulates save questions the parallel postulatethat . Hilbert provided general axiomatic ramework in which o consistency and independence o geometric principles could be addressed. A central plank in this programme was the idea that once a geometrical system had been axiomatized as a collection o statements in a fixed ormal language, the geometrical vocabulary could be entirely reinterpreted: models o geometrical axioms no longer had to conorm to spatial intuitions, but were ree to take any orm at all, provided that they satisfied the axioms. In this spirit, Hilbert is reputed to have said that instead o points, straight lines and planes, one must be able to say tables, chairs and beer mugs. By using the theory o real numbers to construct models o geometry, Hilbert proved the axioms o geometry to be consistent relative to the theory o real numbers – that is, analysis. But then analysis itsel needed to be axiomatized, and proved consistent. While Hilbert developed an axiomatization o analysis in 1900, the consistency o analysis proved to be a difficult problem, partly because the set-theoretic path to the oundations o analysis pursued by Richard Dedekind (1831–1916) seemed to rest on assumptions just as questionable as simply assuming the consistency o analysis directly. Hilbert thereore determined to pursue a direct consistency proo, firstly or the more basic theory o arithmetic (that is, the natural numbers) beore moving on to analysis. In his consistency proos or geometry, Hilbert reinterpreted the language in order to produce models, but in the search or a consistency proo or arithmetic, Hilbert proposed leaving the arithmetical theory uninterpreted, so that the axioms and ormal derivations could be treated as mathematical objects in their own right. Tis was a ground-breaking idea: ormulas could be conceived o simply as finite sequences o symbols, not endowed with any intrinsic meaning, while ormal proos were ‘concrete’ objects which could be grasped intuitively, namely finite sequences o sentences consisting only o axioms and sentences derived by fixed rules o inerence rom sentences earlier in the proo. Reasoning about these simple, symbolic objects was – so Hilbert argued – obviously secure, and could provide a metatheoretic standpoint rom which to prove the consistency o arithmetic, analysis and all the rest o mathematics. Tings did
Introduction
19
not pan out that way, but Hilbert’s finitism has been influential nonetheless, both in the direction o the discipline o proo theory he created through his research into the consistency o arithmetic, and as a oundational stance in the philosophy o mathematics. Hilbert’s lectures on mathematical logic rom 1917 to 1922 are justly amous, and his assistant Wilhelm Ackermann (1896–1962) helped him to turn them into a book, published 1928 as Grundzüge andother later der theoretischen Logik translated into English asinPrinciples o Mathematical Logic. Amongst many things, the book contained the first complete presentation o the system o firstorder predicate calculus. It also spelled out a question Hilbert had been interested in or over twenty years: ‘Is it possible to determine whether or not a given statement pertaining to a field o knowledge is a consequence o the axioms?’ Tis question was known as the Entscheidungsproblem, or decision problem, and a restricted orm o it can be stated with more precision as ollows: is there a mechanical method that can determine whether or not a ormula in a first order language L is logically entailed by a theory in the language L? Tere are really two problems here: one conceptual, the other technical. Te conceptual problem is to determine what a ‘mechanical method’ is, and to give a precise mathematical account o it; the technical problem to determine whether or not such a method or deciding first-order validity exists. Tese problems ound a solution in the work o Alan uring (1912–1954) and Alonzo Church (1903–1995) in the 1930s. Another major part o Hilbert’s work in this period was his programme o finding a finitistically-acceptable consistency proo or arithmetic. Tis was later thrown into disarray by the discoveries o Kurt Gödel (1906–1978), to which we now turn. In his 1929 doctoral thesis at the University o Vienna, Gödel proved the completeness theorem or first-order logic, which states that every sentence which is universally valid – that is, which comes out true regardless o which domain the variables are taken to range over, and whatever interpretation is given to the relation and unction symbols o the language – is in act provable. Te completeness theorem, however, was to be ollowed by a result which, while at least as proound, was ar more shocking: Gödel’sincompleteness theorems, proved in 1930 and published in 1931. Te first incompleteness theorem states that or any recursively axiomatized ormal system containing elementary arithmetic, i the system is consistent then there is a sentence P in the language o the system such that neither P nor ∼P are provable in the system. Tis amounts to saying that no ormal axiomatic theory that meets certain minimal requirements can capture the totality o true
20
Introduction
mathematical statements. Tesecond incompleteness theoremstates that or any recursively axiomatized ormal system containing elementary arithmetic, i the system is consistent then the consistency o the system is not provable in the system itsel. Note that, despite the conusing similarity between the names o these two properties, there is a crucial difference between the completeness o a deductive system (which is proved in the completeness theorem), and the (which shownor notevery to hold in inerence the incompleteness completeness o aormer set o axioms theorems). Te property holdsiswhen valid there is a proo veriying that inerence in the deductive system in question, while the latter is true when or every statement P in the language o a set o axioms, either P or ∼P is a logical consequence o those axioms. It is this latter property that Gödel showed did not hold or consistent theories that contain enough arithmetic. In proving the incompleteness theorems, Gödel developed novel proo techniques which have since become basic tools o mathematical logic. One such technique is Gödel coding (also called arithmetization or Gödel numbering), a method or representing – i.e. encoding – the syntax o ormal theories in the language o arithmetic. Tis allowed Gödel to represent statements about a given ormal system within the system itsel. For example,coding allows one to express, within the ormal system o first-order Peano Arithmetic, the statement that Peano Arithmetic is consistent. Te basic idea isthe ollowing: the codingfixes a mapping between natural numbers and the statements o a language in a way that always allows one to recover an expression rom its Gödel code, and vice versa. Te inormation encoded in digital computers by series ozeros and ones constitutes a amiliar instance o a similar kind o coding. It was the possibility o representing statements about ormal systems within the systems themselves that allowed Gödel to construct statements which were (provably!) unprovable in those systems. Gödel’s incompleteness theorems had proound consequences or logic and the oundations o mathematics. Te second incompleteness theorem, in particular, was a devastating blow to Hilbert’s programme. It meant that there could be no finitary system o arithmetic capable o proving its own consistency, let alone that o analysis or set theory. However, it also triggered many positive developments, including amongst others the development o ordinal proo theory under Hilbert’s student Gerhard Gentzen (1909–1945), and the programme o relative consistency proos in set theory. Afer many decades, the incompleteness theorems have ound their place as one o the building blocks o mathematical logic: the field as it exists would be inconceivable without them, and they continue to inspire new research to this day.
Introduction
21
Furthermore, the incompleteness theorems have also had an influence outside the bounds o logic. Gödel thought that the ollowing disjunctive thesis, now known as Gödel’s disjunction, was a consequence o incompleteness: either the human mathematical mind exceeds the power o a computer, or there are unknowable mathematical truths (or both). Tis idea has been the subject o much philosophical discussion and popular interest, as demonstrated by the appearance o Roger Penrose’s bookShadows o the Mind on the New York imes bestseller list. During his lietime, Gödel made other significant contributions to many areas o logic, set theory and philosophy o mathematics, but most o his important results in logic were proved during his time in Vienna in the 1930s. Due to the worsening political climate and a sequence o events in his personal lie, Gödel moved to the United States in 1940, where he took up a position at the Institute or Advanced Study at Princeton. During the decade rom 1930 to 1940 Gödel had already paid two visits to Princeton (in 1933–4 and in 1935), where a number o conditions created uniquely avourable grounds or major advancements in mathematics and logic, and almost every logician who made significant and long-lasting contributions to logic passed through Princeton at some point in this period. Many notable personalities such as Albert Einstein, Hermann Weyl and John von Neumann were attracted to Princeton as a result o the increasing power gained by the Nazi Party in Germany and Austria. Tey joined an already outstanding group o mathematicians and physicists. Among these, a key figure or mathematical logic was Alonzo Church, who received his PhD at Princeton in 1927 and became Assistant Proessor there in 1929. In the first years o the 1930s the logicians Stephen C. Kleene (1909–1994) and J. Barkley Rosser (1907–1989) were Church’s doctoral students. John von Neumann introduced Church’s group to the incompleteness theorems, in a lecture given at Princeton in 1931 on Gödel’s recent work whose proound consequences he was the first to understand. ogether with Alan uring and Emil Post (1897–1954), Church and Kleene contributed to the establishment o a branch o mathematical logic now known as computability theory (or recursion theory). Hilbert’s collaborator Paul Bernays (1888–1977) was also in Princeton in 1935–6 as a visiting scholar, and ruitully interacted with Church, Kleene and uring. What ollows is a brie reconstruction o the main events at Princeton that helped shape the modern orm o mathematical logic. Between 1929 and 1931 Church developed the intensional ormal system o λ-calculus. Tis was intended to be a oundation or mathematics that avoided the known paradoxes in a different way than Russell’s type theory. Te system
22
Introduction
captured the notion o effective calculabilityon which Church had been ocusing since at least 1934. While the λ-calculus was less than successul as a oundation or mathematics, it was extremely influential in the development o semantics or programming languages and it is still widely researched in theoretical computer science. In an address to the American Mathematical Society in April 1935, Church first proposedcould the idea that ‘the somewhat oclasses effective calculability’ be characterized in termsvague o twointuitive preciselynotion defined o unctions: the notion o a recursive unction o positive integers, or a λ-definable unction o positive integers. Tis thesis first appeared in print in Church’s 1936 paper ‘An Unsolvable Problem o Elementary Number Teory’, and came to be known as ‘Church’s Tesis’ since Kleene (1952) reerred to it by this name. Te notion o a recursive unction had already been employed by Gödel, who refined it using a key suggestion rom Jacques Herbrand (1908–1931) in 1931, the same year in which the proo o the incompleteness theorems was published. In act, the notion o recursiveness plays a undamental role in the proo o the incompleteness theorems. It is needed in order to ormulate a precise criterion or the decidability o the proo relation, i.e. the requirement that ‘x codes a proo o y’ can be verified or alsified by applying a mechanical method in a finite number o steps. Tis means that given a set o ormulas and an arbitrary number (e.g. the Gödel code o a proo), there is a method that ‘decides’ whether the number belongs to the set (i.e. whether the number in question is the Gödel code or a correct derivation). In this case, we say that the set o ormulas is ‘decidable’ or ‘recursive’. Another key property o a set o ormulas in this context is the property o being recursively enumerable, which belongs to sets whose elements we can effectively generate by means o a mechanical method. Tis means that we have instructions that tell us how to list, in a mechanical way, the elements o the set. However, i a set is recursively enumerable but not recursive, we have no general mechanical method or determining whether or not a number is, or is not, a member o that set. It is a consequence o the incompleteness theorems that the set o arithmetical statements provable rom the axioms o Peano Arithmetic is recursively enumerable, but not recursive: any ormal system that is powerul enough to express basic arithmetic is incomplete, in the sense that it is always possible to find a sentence expressible in the language o the system that is true, but not provable within the system in question. Te intuitive notion o a definite method or solving problems had been used throughout the history o mathematics, but it became clear to Gödel that in order to obtain general undecidability and
Introduction
23
incompleteness results, a precise mathematical definition o this notion was necessary. In 1936 and 1937, Church and uring independently published their own accounts o the notion o computability. Te importance o such an advance is demonstrated by the act that, thanks to a mathematically precise definition o the notion o computability, both Church and uring were finally able to solve Hilbert’s Teir strategy unctions was to provide model o Entscheidungsproblem computation in whichinallthe thenegative. effectively computable can bea represented, and then show that no such unction could decide the set o validities o first-order logic. As mentioned earlier, in ‘An Unsolvable Problem o Elementary Number Teory’, Church argued that the notion o effective calculability could be defined in terms o the notion o recursive unction or λ-definable unction (the case or the converse – that every recursive or λ-definable unction is effectively computable – was much easier to make). At the time o his 1935 address, however, Church was still not completely convinced that effective calculability could be identified with λ-definability and gave his first ormulation o the thesis in terms o recursive unctions, even though Gödel himsel was unsatisfied with the idea. Church later wrote that his proposal to identiy the notions o effective calculability and recursiveness arose in response to a question rom Gödel. Church, Kleene and Rosser quickly realized that the notions o λ-definable unction and recursive unction were equivalent, and the proo o this equivalence was published by Kleene in 1936. Church thought that this would strengthen the argument or thinking that these two classes o unctions constituted a general characterization o the intuitive notion o effective calculability. In a two-page paper also published in 1936, Church then provided a negative solution to the Entscheidungsproblem, thus proving what is now known asChurch’s theorem. At the same time, in April 1936, Alan uring – at the time a newly elected ellow o King’s College, Cambridge – completed a paper in which he developed his own account o effective computability, entirely independently rom Church. Afer being shown Church’s work, uring realized that his characterization o effective computability was equivalent to Church’s (the proo o the equivalence was added to the paper as an appendix prior to publication). uring then contacted Church, and went to Princeton as a visiting graduate student in the autumn o 1936, eventually staying – on von Neumann’s advice – or a urther year to complete a PhD under Church’s supervision. uring’s paper ‘On Computable Numbers, with an Application to the Entscheidungsproblem’ eventually appeared in print at the end o 1936. In it,
Introduction
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uring characterized the notion o computation as a mechanical procedure, namely a procedure which manipulates finite strings o symbols (the inputs) according to finitely given sets o rules (the program or algorithm) through a series o steps to obtain a result (the output). Te novelty o uring’s approach was to define computation in terms o what an idealized human agent could in principle compute in a purely mechanical ashion; that is, ollowing instructions in a stepwiseprocedure manner without the aidtoo or ingenuity. Te concept o mechanical is then shown beinsight equivalent to the concept o a uring machine, a machine which is at any time in one o a finite number o states and whose behaviour is ully determined by the rules which govern the transitions between states. Te thesis that every effectively computable unction is uring computable (that is, computable by a uring machine) now goes under the name o uring’s thesis. uring went on to define the notion o aUniversal uring Machine as a uring machine that can take as its input the rules o another uring machine, and execute them. Modern digital computers are, in some sense, concrete instances o this idea: they can run any program (i.e. algorithm) that is given to them as input, and thereby emulate the running o any other computer. uring’s analysis has had a major influence on the development o logic and the philosophy o mathematics. Tis is shown by the reactions o uring’s contemporaries. Church wrote in a 1937 review o uring’s paper that It is . . . immediately clear that computability, so defined, can be identified with (especially, is no less general than) the notion o effectiveness as it appears in certain mathematical problems (various orms o the Entscheidungsproblem, various problems to find complete sets o invariants in topology, group theory, etc., and in general any problem which concerns the discovery o an algorithm). Church 1937: 43
Gödel said in 1946 lecture that It seems to me that [the] importance [o uring’s analysis o computability] is largely due to the act that with this concept one has or the first time succeeded in giving an absolute definition o an interesting epistemological notion, i.e. one not depending on the ormalism chosen. Gödel 1990: 150
He also said in 1964 that only uring’s work provided ‘a precise and unquestionably adequate definition o the general concept o ormal system’, as a mechanical procedure or generating theorems rom a given set o axioms. Moreover, uring’s conceptual analysis o effective computability, and its definition in terms
Introduction
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o uring computability, marked the beginning o both theoretical computer science and artificial intelligence. His work currently finds applications in many scientific domains ranging rom physics to biology and cognitive science.
Reerences Bocheński, J. M. 1( 961), A History o Formal Logic, Notre Dame: University o Notre Dame Press. English edition translated and edited by Ivo Tomas . Church, Alonzo (1937), ‘Review: A. M. uring, On Computable Numbers, with an Application to the Entscheidungsproblem’, Journal o Symbolic Logic (2): 42–3. Gödel, Kurt (1986), ‘Postscriptum (1964) to “On undecidable propositions o ormal mathematical systems”’, in Solomon Feerman, John W. Dawson,Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay and Jean van Heijenoort (eds),Kurt Gödel: Collected Works Vol. ,Ipages 346–72, Oxord: Oxord University Press. Gödel, Kurt (1990), ‘Remarks beore the Princeton Bicentennial Conerence on Problems in Mathematics’, in Solomon Feerman, John W. Dawson,Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay and Jean van Heijenoort (eds),Kurt Gödel: Collected Works Vol. II, pp. 150–3. Oxord: Oxord University Press. Jardine, Lisa (1982), ‘Humanism and the teaching o logic’, in Norman Kretzmann, Anthony Kenny and Jan Pinborg (eds),Te Cambridge History o Later Medieval Philosophy, pp. 797–807, Cambridge: Cambridge University Press. Kant, Immanuel (1999), Critique o Pure Reason, English edition translated and edited by Paul Guyer and Allen W. Wood , Cambridge: Cambridge University Press. Kneale, William and Kneale, Martha (1962), Te Development o Logic, Oxord: Clarendon Press, Oxord. Kleene, Stephen C. (1952), Introduction to Metamathematics, Amsterdam: North Holland. Russell, Bertrand (1903), Te Principles o Mathematics, Cambridge: Cambridge University Press. uring, A.M. (1937), ‘On Computable Numbers, With an Application to the Entscheidungsproblem’, Proceedings o the London Mathematical Society Series2, 442, 230–65. Wittgenstein, Ludwig (2009), Philosophical Investigations. English translation by G.E.M. Anscombe with revisions by Peter Hacker and Joachim Schulte , German text on acing pages, ourth edition, Chichester: Wiley-Blackwell.
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Part I
Te Origins o Formal Logic
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1
Aristotle’s Logic Adriane Rini
1 Te syllogism Aristotle is generally credited with the invention o logic. More than two thousand years ago, he noticed that good, persuasive arguments have certain kinds o shapes, or structures. Logic was born when he began to study those structures, which he called syllogisms, identiying underlying patterns o ordinary human reasoning and then devising simple methods which can be used to determine whether someone is reasoning correctly. Aristotle was aware that this was a momentous discovery, and he himsel thought that all scientific reasoning could be reduced to syllogistic arguments. Tere are hints that Aristotle saw his syllogistic logic as providing useul practice or participants in ancient debating contests, contests which he himsel would have encountered as a student in Plato’s Academy. Tere are also hints that Aristotle might have envisioned his syllogistic logic as providing a way to catalogue acts about science. But Aristotle never fleshes out any such details. His Prior Analytics, the text in which he presents his syllogistic, ocuses on the mechanics o the syllogistic ar more than on its interpretation, and in act the comparative lack o interpretive detail has sometimes led scholars to suppose that what we have today o Aristotle’s logic is his own sometimes scrappy lecture notes or his students’ notes, and not a polished, finished work. But in the end these are o course only guesses. In modern times, the main scholarly interest in Aristotle’s ancient logic has ocused increasingly on the proper interpretation o the mechanical methods which Aristotle invented and on their relation to his wider philosophy. Scholars today describe Aristotle’s invention as a theory o inerence or deduction. Aristotle’s own explanation is amous:
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30 (De)
Te History o Philosophical and Formal Logic
A deduction [a syllogism] is an argument in which, certain things being supposed, something else different rom the things supposed ollows o necessity because o their being so. Prior Analytics 24a18–20
Suppose, or example, that a riend tells us that (* )
‘i all birds fly, and i all seagulls are birds, then it must be the case that all seagulls fly.’
In reasoning according to (*) our riend is trying to reason deductively. But in the particular case o (*) our riend’s reasoning involves some conusion about the acts – acts about birds. Certainly many and perhaps even most birds fly, but not all o them do. So, it is not true that ‘all birds fly’. Aristotle saw that one could separate general patterns o deductive reasoning rom the particular subject matter that one reasons about. And once these are separated then the patterns themselves can be studied independently o their actual subject matter. Tere is nothing wrong with the structure o ( *). I (*) is not persuasive it is only because it involves a actual error, a actual errorabout birds. For it is a actual matter that there flightless birdsthe such as or thealsity New o Zealand kiwi. such And because thisfly’. is a actualare matter it affects truth a statement as ‘all birds In opics 105, Aristotle describes philosophical inquiry in terms o three possible sub-fields: he argues that any philosophical question will belong either to natural science or to logic or to ethics. His point seems to be that we should regard each o these separate fields as having different subject matter. Tis distinction helps to explain the difference between what is good about *() and what is not so good about (*), since it lets us say that it is a matter o natural science whether or not it is in act true that all birds fly. For example, it is a matter o biology, or o animal behaviour, but it is not a matter o logic. Te distinction between the subject matter o natural science and the subject matter o logic also makes it easier to distinguish the patterns o good reasoning rom the particular science that one reasons about. Tese patterns o reasoning are themselves the subject matter o logic, and by concentrating on what is general to the patterns, we can in effect move beyond matters o act. Te natural sciences are about discovering the truth and alsity o our claims about the world – such as discovering the alsity o the claim that all birds fly. But a helpul way o understanding(De) is as saying that logic takes us beyond the matters o truth and alsity o any natural science – the realm o logic concerns what happens ‘o necessity’. In order to describe the patterns themselves, Aristotle developed a vocabulary with which to distinguish the various components o a syllogism.
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Furthermore, there is a long tradition o representing syllogistic reasoning in three-line arguments – and representing syllogisms in this way makes it easier to identiy the separate logical components. Consider, or example, the ollowing: (1) All birds fly All seagulls are birds All seagulls fly (1) is composed o two premises, ‘all birds fly’ and ‘all seagulls are birds’, together with a conclusion, ‘all seagulls fly’. Te two premises are the ‘certain things being supposed’ that are described in (De) . From this we can see that (De) tells us that in syllogizing according to (1) we suppose that ‘all birds fly’ and ‘all seagulls are birds’ are both true. Te supposition that both o the premises are true leads us to the truth, as well, o ‘something else different’ rom the premises – that is, the supposition that the premises are true leads us to the truth o the conclusion. Aristotle notices that in an argument such as (1) when we assume the truth o the premises, then the truth o the conclusion ‘ollows o necessity’. He does not say merely that the conclusionis also true. His claim is stronger. In a syllogism i the premises are true then the conclusion cannot ail to be true. Or, another way o saying the same thing: i the premises are true then the conclusion has to be true as well. Tis basic insight is at the heart o Aristotle’s discovery o logic. (It is at the heart o modern logic too, since what Aristotle is describing is what today we call logical validity.) On the other hand, i the conclusion can be alse even when the premises are true, then, according to Aristotle, there is no syllogism.1 In act Aristotle’s syllogistic is very limited. He only counts as a syllogism an argument which has exactly two premises and one conclusion.
2 A lesson rom Plato: Names and verbs While syllogisms are composed o premises and conclusions, these premises and conclusions are themselves built out o still more basic components. Plato, who was Aristotle’s teacher, initiated the study o these more basic components, taking them to be eatures o language. In doing so, Plato laid the oundation or a link between human reasoning and human language, a link which strongly persists in modern philosophy. Plato explains his motivation in the dialogue called the Sophist:
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Te History o Philosophical and Formal Logic Te signs we use in speech to signiy being are surely o two kinds . . . One kind called ‘names’, the other ‘verbs’ . . . By ‘verb’ we mean an expression which is applied to actions . . . And by ‘name’ the spoken sign applied to what perorms these actions . . . Now a statement never consists solely o names spoken in succession, nor yet verbs apart rom names. Sophist 261e262b
Plato, here, introduces the idea that a simple statement (something which can be true or alse) is itsel built out o two basic components: name (onoma) and verb (rhema). And Plato seems, at least in this passage, to take it as obvious that names and verbs are different kinds o things: one picks out the agent; the other picks out the agent’s actions. Aristotle was amiliar with this passage rom Plato’s Sophist and clearly carried rom it the lesson that meaningul statements must involve the combination o specific linguistic elements, and this is reflected in Aristotle’s own consideration inCategories o ‘things that are said’: O things that are said, some involve combination [literally, ‘interweaving’] while others are said without combination. Examples o those involving combination are ‘man runs’, ‘man wins’; and o those without combination ‘man’, ‘ox’, ‘runs’, ‘wins’.
Categories 1a16
You do not succeed in communicating anything true (or alse) i you merely utter a noun (‘man’ or ‘ox’), or merely a verb (‘runs’ or‘wins’). Nouns alone do not produce truth or alsehood. Verbs alone do not either. In thisCategories passage, Aristotle offers two examples o ordinary, meaningul statements – ‘man runs’, ‘man wins’ – each o which is the result o our ‘combining’ nouns and verbs and each o which does communicate either a truth or a alsehood. Simple sentences such as these are called affirmatives because they affirm something (here, running or winning) o a subject (man). And o course, such sentences can be true or alse, depending on whether the world is as the sentence says. As we saw in Section 1, Aristotle regards such actual knowledge about how the world is as the subject matter o natural science. Not only does Aristotle inherit rom Plato the distinction between name and verb, but in Aristotle’s De Interpretatione this distinction takes on what appears to be a syntactic importance according to which the name and the verb ulfil specific grammatical roles within a simple statement: A name (onoma) is a spoken sound significant by convention, without time, none o whose parts is significant in separation. De Interpretatione 16a19
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A verb (rhema) is what additionally signifies time, no part o it being significant separately; and it is a sign o things said o something else. De Int 16b6
Tese passages create some difficulty about the project right away: is Aristotle’s syllogistic logic about things in the world (such as men and birds), or is it about the language we use to describe those things (such as our words ‘man’ and ‘bird’)? It is hard to say. InDe Int, Aristotle’s writing is careul: names and verbs ‘signiy’, they unction as ‘signs o things’. On the ace o it, this would seem to suggest that he thinks o logic as a study which concerns linguistic statements which are themselves made up o names and verbs. But even i Aristotle starts out this way, it becomes difficult to sustain this view, particularly in later, more sophisticated parts o the syllogistic. Tis lack o clarity gives rise to a tension that runs through Aristotle’s syllogistic. Modern scholars ofen describe the difficulty as a ailure to distinguish between using a word and mentioning (talking about) a word. As we shall see in later sections, questions about the precise nature o the roles o the clearly linguistic items o name and o verb become a oundational issue within the history o the development o logic. Sometimes, in ancient Greek, as in modern English, grammar requires more than the simple concatenation o nouns and verbs. Tat is, we do not always produce grammatical sentences by stringing nouns and verbs together. Sometimes the nouns and verbs themselves are linked explicitly by a copula, a orm o the verb ‘to be’. In English, or example, we do not say ‘cows animals’ – English requires the introduction o a copula to give the grammatical statement ‘cows are animals’. Aristotle recognizes this eature o Greek and allows an explicit copula when needed, but scholars disagree about how much importance to attach to the copula. (See or example Geach 1972, Smith 1989, Patterson 1995, Charles 2000.)
3 Affirmation and denial As noted above, our ability to make an affirmation provides us with a way o describing the world, and such descriptions themselves will be either true or alse. But Aristotle also noticed that ‘it must be possible to deny whatever anyone has affirmed, and to affirm what anyone has denied’ (De Int 17a31). Every affirmation has an opposite denial, and every denial has an opposite affirmation. Furthermore, in just the same way that an affirmation can be either true or alse,
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Aristotle recognizes that so too can a denial be either true or alse. When two statements ‘affirm and deny the same thing o the same thing’ they are opposites and they stand in contradiction. It is perhaps obvious that even the most basic human communication requires that affirmation must be tractable and meaningul. But Aristotle’s analysis in De Int makes denial tractable and meaningul in just the way that affirmation is. o take an example, ‘all birds fly’ is an affirmation, ‘some birds which do not isfly’alse, is a denial, o these two is example statements it iswhile the affi rmation and theand denial which true – though o course there will also be examples in which the denial is alse and the affirmative true (e.g. ‘some men are pale’; ‘no men are pale’). By considering affirmation and denial together as the two components o any contradiction Aristotle avoids problems which worried his predecessors. Plato, and Parmenides beore him, sought to ground philosophy in what is, but then struggled to provide an analysis o what is not. I what is ‘exists’, doeswhat is not ‘not exist’? How can we talk meaningully about what is not? I truth reflects what is, then is alsity about what is not? Te need or an answer led earlier philosophers to approach the notion o alsity with caution. Following Aristotle’s instructions inDe Int, denial is not a problem and the analysis is straightorward: we can construct the ollowing illustration.
Such a diagram is called a square o opposition. It provides a way to represent various relations that hold between propositions. For example, when two propositions are such that one must be true when the other is alse, and vice versa, Aristotle calls them contradictories. Reading diagonally: ‘every man is pale’ and ‘some man is not pale’ are contradictories; and ‘some man is pale’ and ‘no man is pale’ are contradictories. Tat is to say, the negation o ‘every man is pale’ is ‘some man is not pale’, and vice versa; and the negation o ‘some man is pale’ is ‘no man is pale’, and vice versa. Te horizontal arrows represent other relations which can hold between propositions.
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When two propositions cannot both be true and cannot both be alse, Aristotle calls them contraries. So, reading lef-to-right or right-to-lef: ‘every man is pale’ and ‘no man is pale’ are contraries. When two propositions cannot both be alse, they are called subcontraries. Again, reading lef-to-right or right-to-lef, this time along the bottom o the diagram: ‘some man is pale’ and ‘some man is not pale’ are subcontraries – that is, at least one o them must be true. Te centrality o affirmation and denial to Aristotle’s logic becomes obvious in the opening lines o Prior Analytics. When Aristotle introduces his system o syllogistic he explains that, at the most basic level, ‘a premise is a sentence affirming or denying something o something’ (An Pr 24a16). (In Aristotle’s system, a conclusion, too, will always beeither an affirmation or a denial.)
4 Categorical propositions All o the example statements in the previous sections involve some subject matter. In the same way that ‘all birds fly’ – the first premise in (1) – isabout birds, the statement ‘some man is not pale’ is about men. Tis quality o aboutness guides us to ocus on the subject matter at hand, so that we might then assess the statements themselves or truth or alsity. But in studying logic and in distinguishing it rom natural science, Aristotle takes a deliberate step away rom any specific scientific subject matter. His driving concern in the syllogistic is with the structure o argument, and such structure is easier to appreciate when it is general and not bound to any specific subject matter. So since the science which guides our assessments o truth and alsity o simple statements is subjectspecific, Aristotle wants to consider categorical statements generally and without regard to their specific content. He achieves this by using term variables – e.g., A, B and C – in place o ordinary language terms such as ‘birds’, ‘fly’, ‘seagulls’, ‘man’ and ‘pale’. He begins by putting the variable letter A in place o the subject term, and the variable letter B in place o the predicate term. By this method the premise in (1), ‘all birds fly’, becomes ‘all A are B’.
Aristotle’s own example statement ‘every man is pale’ has exactly the same underlying structure as ‘all birds fly’. One can see that this is so by taking Aristotle’s
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‘every man is pale’ and replacing the subject ‘man’ with the variable A, and replacing the predicate ‘pale’with the variable B. We get ‘every A is B’, which is the same as ‘all A are B’.
Whether we say ‘every man is pale’ or ‘all men are pale’, we are making a universal affirmative statement about a subject ‘man’ and a predicate ‘pale’ – and Aristotle himsel recognizes similarly equivalent expressions in his Greek. He treats them just as different ways o making the very same statement. In early chapters o the Prior AnalyticsAristotle usually describes the predicate B as belonging (or not belonging) to the subject A, so, or example, we find affirmations and denials (i.e. affirmative and privative propositions) described as ollows: Affirmative ‘B belongs to all A’ ‘B belongs to some A’
Privative ‘B belongs to none o the As’ ‘B does not belong to some A’
Te ‘belongs to’ expression reflects some o Aristotle’s background science where belonging captures a certain priority o scientific terms. According to this background science, or example, man belongs to animal, but man does not belong to plant. Trough the chapters o Book 1 o Prior Analytics, Aristotle ocuses increasingly closely on details that have more to do with his logic than with his background science, and as this happens his language shifs and the ‘belongs to’ expression becomes less common. So, instead o talking about how B ‘belongs to’ A, Aristotle begins to say simply that ‘A is B’. Later inAn Pr , when Aristotle is explaining points o special logical complexity, even this new ‘A is B’ expression sometimes gets still urther abbreviated to the very cryptic expression ‘the AB’, where Aristotle leaves it to his reader to fill in the missing details. What are these urther details? As Aristotle’s own examples in the square o opposition indicate, contraries and contradictories involve more than attaching subject and predicate, or name and verb. Constructing a ull square o opposition requires that we add more: we must be able to signiy both affirmation and denial and we must be able to indicate quantity – i.e. to indicate whether we are talking about ‘all’ or ‘some’ o a subject. When a premise says, or example, that all ‘ A are B’, or that every ‘ A is B’, or that ‘none o the As are B’, Aristotle calls the premise auniversal. A premise
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which says that ‘some A is B’, or thatsome ‘ A is not B’, he calls aparticular (An Pr 24a17). In considering affirmations and denials which are themselves either universal or particular, Aristotle’s notion o what counts as a premise is restricted to a small class o simple propositions, and scholars traditionally call this restricted class o premises categorical propositions. (Te subject and predicate o a categorical proposition name things within the Aristotelian categories. See especially Chapter 4.) Te ourollowing kinds o convention, categorical propositions are Categories, requently abbreviated according to the where A is the subject and B is the predicate: (A) (E) (I) (O)
universal affirmative universal privative particular affirmative particular privative
B belongs to every A BaA B belongs to no A BeA B belongs to some A BiA B does not belong to some A BoA
Every syllogistic premise (and conclusion) iseither an A-type, E-type, I-type or Otype proposition. Tis notational convention is not Aristotle’s own but a medieval contribution which modern scholars still use today. In BaA, the lower case ‘a’ indicates that the premise is an A-type, or universal affirmative; the upper case ‘A’ indicates the subject term. Te logical relations o contrariety and contradiction that stand between the various categorical propositions can be expressed using a more general square o opposition than the one inSection 3, above:
Reading lef-to-right or right-to-lef: A propositions and E propositions are contraries; and I and O are subcontraries. Reading diagonally: A and O are contradictories; and I and E are contradictories. So, the negation o an A proposition is its contradictory, i.e. an O proposition, and vice versa; and the
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negation o an E proposition is its contradictory, i.e. an I proposition, and vice versa. Tese contradictories get put to work when Aristotle develops methods or proving syllogisms in Prior Analytics.2
5 Further syllogisms Aristotle takes pairs o A, E, I and O propositions which share exactly one term in common and then asks whether rom some given premise pair a conclusion relating the remaining two terms must ollow o necessity. For this to happen, the terms must fit one o three possible patterns: Te term in common to the premises might be the subject o one premise and the predicate o the other. Te term in common to the premises might be the predicate o both premises. Or the term in common might be the subject o both premises. Aristotle calls these possible combinations figures. Where A and B and C are variables or terms, we can represent the three figures as ollows: First figure
Second figure
Tird figure
predicate-subject A-B B-C A-C
predicate-subject A-B A-C B-C
predicate-subject A-C B-C A-B
Aristotle describes the three figures in An Pr Book A, chapters 4–6, and he considers possible combinations o A, E, I and O premises in each figure. Te term which occurs in each o the two premises Aristotle calls the middle term. Te middle term drops out and does not appear in the conclusion. Aristotle calls the other two terms the extremes – the extremes are the only terms which appear in the conclusion. (Scholars have noticed a link between Aristotle’s labels or the various terms and the labels ancient geometers used in their study o proportions or harmony. Any line-segment, or example, can be divided into its extreme and its mean ratios. See Euclid Book IV, Definition 3.) By approaching his logic in this way Aristotle places some noticeable limits on the system he develops. First, his description o the figures appears to be based solely on possible premise combinations. He gets three figures because there are only three possible combinations given that each premise must contain two terms and that the two premises must share one term in common. He also limits his system by looking or a conclusion o a particular orm: in the first figure he wants a conclusion specifically relating a C subject to an A predicate; in
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the second figure the conclusion relates a C subject to a B predicate; and in the third figure, a B subject to an A predicate. Tis method rules out any other figures, but it is easy to see that there are, nonetheless, other possibilities, and Aristotle could have approached the system differently. He himsel recognizes this in An Pr 29a21–27, where he briefly notes the possibility o a ourth figure. able 1 lists, using all o their the syllogisms establishes Prior Analytics traditionalAristotle (medieval) names.3 in Chapters A4–6 o
able 1.1 Non-modal syllogisms First figure
Second figure
Tird figure
Barbara A belongs to every B B belongs to every C A belongs to every C
Cesare A belongs to no B A belongs to every C B belongs to no C
Darapti A belongs to every C B belongs to every C A belongs to some B
Celarent A belongs to no B B belongs to every C A belongs to no C
Camestres A belongs to every B A belongs to no C B belongs to no C
Felapton A belongs to no C B belongs to every C A does not belong to some B
Darii A belongs to every B B belongs to some C A belongs to some C
Festino A belongs to no B A belongs to some C B does not belong to some C
Datisi A belongs to every C B belongs to some C A belongs to some B
Ferio A belongs to no B B belongs to some C A does not belong to some C
Baroco A belongs to every B A does not belong to some C B does not belong to some C
Disamis A belongs to some C B belongs to every C A belongs to some B Bocardo A does not belong to some C belongs to every B does not belong to C some B A Ferison A belongs to no C B belongs to some C A does not belong to some B
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Te syllogisms in able 1.1 are called non-modal syllogisms – they involve premises which merely assert belonging or not belonging. For this reason the syllogisms in able 1.1 are ofen called assertoric syllogisms. Scholars use these labels to distinguish these syllogisms rom ones which do more than merely assert.4 Consider a syllogism with the same structure as (1) but with all true premises: (2) All birds are animals All seagulls are birds All seagulls are animals Notice that (2) is an example o a syllogism in the first figure. In act, it is an example o the syllogism called Barbara. Each o the categorical statements in (2) is an A-type categorical proposition. Aristotle thinks that first-figure syllogisms with A-type premises and an A-type conclusion involve the kind o basic reasoning that all rational humans can perorm – even i they themselves are not conscious that such reasoning is ‘syllogizing in the first figure rom universal affirmative premises’. Rational beings employ syllogistic reasoning all the time without actually reflecting on how they are reasoning, but it is the logician’s job to investigate how reasoning itsel works. When we have an example o Barbara, i we suppose that the premises are true, then we are guaranteed that we will never have a alse conclusion relating a C subject to an A predicate. (Te crucial difference between examples (1) and (2) is that in (2) the premises are both true, but in (1) they are not – however, in syllogizing according to Barbara we have to suppose the premises are true.) A tension arises between what the syllogistic in An Pr requires and what the combination o name and verb in De Int requires. Te De Int passages, cited above, make clear that Aristotle regards the combination o name and verb as providing the underlying structure o any meaningul proposition. In an I proposition o the orm ‘B belongs to some A’, there is a special syntactic or grammatical role or each o A and B: simply, A serves as the subject and B as the predicate. Likewise or the other types o propositions. But in An Pr, as soon as Aristotle describes his three figures, a problem begins to become obvious. Te separate syntactic roles played by name and verb turn out not to be preserved in the syllogistic figures: in the first figure the B term is the subject o one premise but the very same B term is the predicate o the other premise. In the second figure the B term is the subject o one premise but the predicate o the conclusion. In the third figure the B term is the predicate o one premise but the subject o
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the conclusion. So the figures themselves do not respect the earlier syntactic roles, and so the name/verb distinction which Aristotle inherited rom Plato is thereore no longer a good fit. Aristotle notices this, and he clearly chooses to be guided by the demands o his logic, rather than by the demands o the syntax outlined in Plato’s Sophist. Eventually Aristotle steps away rom Plato’s name/ verb distinction, preerring instead to describe onoma and rhema using the more neutral label (horos ). Tis move away rom Plato represents move towards the point o ‘term’ view o a logician because it allows Aristotle to stepa urther away rom what any particular names and verbs might mean, and allows him to ocus instead on how general terms themselves, represented by variables, eature in patterns o reasoning. Tis becomes particularly important in a method which Aristotle calls conversion. Conversion involves the transposition o subject and predicate. Consider something which is white – perhaps a flower, or a cloud, or a piece o paper. Aristotle’s own example, in Posterior Analytics 1.22, is o a white log. I it is true that (2) some white thing is a log, then you know that it must also be true that (3) some log is white. And more generally, i it is true that (2 ) some A is B, then you know as well that (3 ) some B is A.
Aristotle explains that i we are given any true I-type premise with subject A and predicate B, then we also have another true I-type premise with the subject and predicate transposed, so that the A subject becomes the predicate, and the B predicate becomes the subject o the new proposition. Scholars call this I-conversion. E-type premises convert in the same way: that is, i you are given that ‘no A are B’ is true, then you also have ‘no B are A’.5 Tis is called E-conversion. Aristotle notices that he can transpose subject and predicate terms in A-type premises also, but that these A-conversions do not work exactly the same as I-conversions and E-conversions. Consider a subject such as ‘horse’. It is true that all horses are animals. It does not ollow rom this that all animals are horses, but it does ollow that some animals are horses. And this generalizes: I ‘all A are B’, then ‘some B are A.’ Aristotle calls such A-conversion accidental conversion. It takes us rom an A-type premise to an I-type premise – i.e. rom an affirmative universal to an
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affirmative particular proposition. (Tere is no O-conversion because you cannot guarantee rom an O-type premise with an A subject and B predicate anything about a premise with a B subject and an A predicate.) One question which many twentieth-century commentators raise is: what happens to Aristotle’s A-conversion i there are not any As? Tat is, what i the term A is ‘empty’? Elsewhere in his philosophical works Aristotle does sometimes consider or non-existent things. His example o stag, a ‘goat-stag’ (Posteriorterms Analytics 92b5–7), something thatamous is part goat and is part and so something which, definitely, does not exist. But in his syllogistic logic he does not consider what happens i our terms are empty, and we have no direct textual evidence about his thoughts on how empty terms are in his logic. wentiethcentury scholars, however, raise this question because modern logic does have to deal with empty terms, and this difference marks another way in which Aristotle’s system is limited. It is narrower in scope than modern logic. Tis difference means that it is most straightorward to approach Aristotle’s logic as a system which is about things in the world, or, as explained in Section 1, above, about what natural sciences study and discover. Tis way, any issues about empty syllogistic terms do not arise. I-conversion, E-conversion, and A-conversion serve as special rules which Aristotle uses in order to construct proos or second and third figure syllogisms. His most important proo method involves putting his conversion principles to work. Tis is easiest to see in an example. Aristotle offers a proo o the secondfigure syllogism which has as its premises ‘no B is A’ and ‘all C are A’. He wants to know whether these premises guarantee a conclusion linking a C subject to a B predicate. Here is how Aristotle proves the second-figure syllogism which the medievals called Cesare: (1) (2)
No B is A All C are A
premise premise
(3) (4)
No A is B No C is B
E-conversion, (1) Celarent, (2)(3)
First, Aristotle converts the E-type premise (1) ‘no B is A’ to get (3) ‘no A is B’. (3) together with premise (2) are in act the premises o the first-figure syllogism known as Celarent (with A in place o B, and B in place o A). So, conversion lets us turn a second-figure schema into a first-figure schema, or as Aristotle describes the method, conversion ‘brings us back’ to the first figure. Since Celarent is a syllogism and since we can bring Cesare back to the first figure via conversion, then Cesare must also be a syllogism. Te main difference, here,
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between Cesare and Celarent is that the validity o the first-figure Celarent is immediate and obvious, whereas the validity o second-figure Cesare is not immediate but needs to be brought out.6 It is brought out by using conversion to turn Cesare into Celarent, thus guaranteeing that we can syllogize rom (1) and (2) to a conclusion (4). Here is how a proo in the third figure works. ake the syllogism known as Datisi, whose premises are given as ollows: (1) All C are A premise (2) Some C is B premise (3) Some B is C I-Conversion, (2) (4) Some B is A Darii, (1)(3) In the proo o Datisi, the I-type premise (2) ‘some B is C’ gets converted to (3) ‘some C is B’. (1) and (3) are the premises o the first-figure syllogism Darii. So, again, a first-figure syllogism together with conversion guarantees that the thirdfigure Datisi (1)(2)(4) is a syllogism. In each o the syllogisms in able 1.1, a conclusion does ollow o necessity rom a premise pair. As noted above, Aristotle argues that in the first figure it is ‘obvious’ that the conclusion ollows rom the premises in the sense that we can grasp them immediately, without the need or any proo. Te syllogisms o the first figure, thereore, unction asaxioms which ground the proos o syllogisms in the other figures. Since the first figure is obvious, i we can bring a secondfigure or a third-figure premise pair ‘back to the first’ by converting premises, then, even i our proo is not itsel immediate and obvious, it is nonetheless something to be relied upon, something that guarantees second- and thirdfigure syllogisms.
6 Counterexamples: ‘Tere is no syllogism’ Te syllogisms in able 1.1, however, do not include all possible premise combinations. Tere are other premise pairs rom which no conclusion can be guaranteed. When a conclusion does not ollow rom a given premise pair, Aristotle offers instructions or constructing counterexamples, showing how in such cases the truth o any conclusion cannot be guaranteed. When there is a counterexample, Aristotle says ‘there is no syllogism’ – or, as modern logicians would say, there is not a valid schema. Here is an example o how Aristotle constructs a counterexample, in An Pr 26a4–9:
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Te History o Philosophical and Formal Logic [N]othing necessary results in virtue o these things being so. For it is possible or the first extreme [the A term] to belong to all as well as to none o the last [the C term]. Consequently, neither a particular nor a universal deduction becomes necessary; and, since nothing is necessary because o these, there will not be a deduction. erms or belonging are animal, man, horse; or belonging to none, animal, man, stone.
Aristotle to show there is no this syllogism rom that the first-figure premise pair ‘all B wants are A’ and ‘no Cthat are B’ . He does by showing rom true premises, first, we do not get a privative conclusion. Second, we do not get an affirmative conclusion. He gives two sets o terms or the variables A, B and C, where the terms make the premises true. His first set o terms are ‘terms or belonging’ which show that the conclusion cannot be privative. With animal, man and horse as A, B and C, our premises are ‘all men are animals’ and ‘no horses are men’. We cannot have either an E-type conclusion ‘no horse is an animal’ or an O-type conclusion ‘some horse is not an animal’ because all horses are by nature animals – nothing can be a horse without also being an animal. But these E and O statements are the only possibilities or a privative conclusion, so we cannot have a privative conclusion. With the second set o terms – animal, man, stone – our premises are ‘all men are animals’ and ‘no stones are animals’. But, in this case, we cannot have an A-type conclusion ‘all stones are animals’ or an I-type conclusion ‘some stones are animals’ – because both the A and I propositions are clearly alse. A and I and E and O propositions exhaust all the possible orms o a conclusion – universal and particular, affirmative and privative. Since our terms show that none o these is in act available, we thereore have a proo which establishes that there is no syllogism rom the initial premise pair. Some scholars describe the premise pair as inconcludent. Aristotle is thorough about his claims about which premise combinations yield syllogisms. He does not take it or granted that there is a syllogism or is not a syllogism – rather, in the Prior Analytics, he goes step-by-step through each possible premise pair, offering either a proo or a counterexample. While his counterexamples are, o course, perectly adequate, Aristotle does not say much about how he initially devises them or how he expects his readers to do so. Tis led early twentieth-century scholars sometimes to worry that Aristotle’s approach must thereore bead hoc, the result o mere trial and error, and not a part o a ormal system. Tis o course is only conjecture, but Aristotle’s counterexamples remain one o the least studied areas in the syllogistic.
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7 Modal syllogistic As noted in Section 5, there is more to the syllogistic than just the ourteen syllogisms in able 1.1 together with the counterexamples against the inconcludent premise pairs. In Chapters A.8–22 o An Pr, Aristotle extends his syllogistic to investigate patterns o reasoning involving modal claims about necessity possibility. It is not he ishis interested patternsand o reasoningand involving necessity and surprising possibility,that because general in scientific metaphysical project involves studying what cannot-be-otherwise. What can be otherwise is, according to Aristotle, not the stuff o real science. In spite o this obvious link between science and the modal notions o necessity and possibility, and in spite o the near universal praise or the invention o the syllogistic, Aristotle’s modal syllogistic has not been well received. Tere are many reasons why scholars have struggled to interpret the modal syllogistic – too many, in act, to cover in such a chapter as this, so that the present discussion will only concentrate on what has been perceived to be one o the most serious problems or Aristotle’s modal syllogistic. Te most amous o Aristotle’s modal syllogisms is a version o Barbara with one necessary premise and one premise which merely asserts: (3) A belongs o necessity to every B B belongs to every C A belongs o necessity to every C Tis is usually known as Barbara NXN , where N indicates a proposition involving necessity, and X indicates a proposition which merely asserts (nonmodally). Aristotle argues that i we suppose such premises then we are guaranteed the truth o the conclusion. But taking Barbara NXN as a syllogism seems to commit us to one way o interpreting Aristotle’s propositions about necessity. Tat is, (3) makes sense i the necessity is understood as attaching to the predicate A term, so that we might paraphrase the modal premise as ‘every B is a necessary A’, where being a necessary A is a special kind o modalized predicate. Using the modern logical symbol ‘’ to indicate necessity, then the premise about necessity in (3) has a structure which can be represented as ollows: (i) Every B is A Medieval logicians called this kind o necessity de re (the necessity holds o the thing) because it attaches to the predicate term itsel. On the ace o it, de re
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necessity seems to be precisely what is needed i (3) Barbara NXN is, as Aristotle insists, a (valid) syllogism. We can then represent Barbara NXN as ollows: (4) Every B is A Every C is B Every C is A Most o themake proos modal syllogistic, theflip-flop proos in assertoric syllogistic, useinothe conversion principles like which thethe order o the subject and predicate terms, and this flip-flopping now has to account or the necessity. A problem arises, here, because the modal conversion principles do not work well using de re necessity. Simple A-conversion takes us rom ‘every B is A’ to ‘some A is B.’ So, A-conversion involving necessity (ofen called NA conversion) should take us rom ‘every B is A’ to ‘some A is B’ Tis conversion is illegitimate and is easy to alsiy with a counterexample: let A be animal and B be bachelor. Ten the NA-conversion would have to take us rom Every bachelor is a necessary animal to Some animals are necessary bachelors. Tis de re conversion clearly does not work, because no animal is a necessary bachelor. So, the modal NA-conversion seems to require a different analysis altogether. We might try, then, to interpret an A-type premise about necessity in a different way – or example, we might suppose that, rather than the structure represented by (i), it has the ollowing structure: (ii) (B belongs to every A) where what is necessary is the proposition ‘B belongs to every A’. Te medievals called the proposition ‘B belongs to every A’ a dictum, and so called the necessity in (ii) necessity de dicto. (It helps to think o the necessity here as being a necessary connection between two predicates neither o which need apply by necessity to a thing.) As an example, the NA-conversion would take us rom, say,
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By necessity every spouse is married to By necessity some married things are spouses. Tis de dicto modal conversion is valid. But de dicto necessity does not make sense o Barbara NXN . By necessity every spouse is married. Every Wellingtonian is a spouse By necessity every Wellingtonian is married. Let’s suppose that in act everyone who lives in Wellington happens to be a spouse, but o course even in such a case no one is a spouse by necessity. Since the modal syllogisms seem to demand one interpretation and the conversion rules another, it is not immediately obvious what Aristotle himsel means. Tere is no agreement among scholars about the correct way to handle this problem, and that is why the modal syllogistic is such a controversial part o Aristotle’s logic.
8 Appendix: Te medieval mnemonics Te names o the syllogisms ‘Barbara’,‘Celarent’, etc., which are ound in able 1.1, encode instructions or Aristotle’s proos. Tis system o mnemonics was not due to Aristotle but was rather a medieval invention. Te codes work as ollows: ●
●
Te vowel letters A, E, I, and O, introduced in Section 4, are used to indicate the quantity and quality o the propositions in the square o opposition and to indicate the quantity and quality o propositions in the syllogisms. Te vowels tell whether the propositions are about ‘all’ or about ‘some’ o a subject, and whether the propositions are affirmative or privative. Te name Barbara, with its three As, identifies the syllogism which contains three universal affirmative propositions, or A-type propositions. So, any instance o Barbara has two A-type premises and an A-type conclusion. Te name Cesare describes a syllogism in which the first premise is an E-type, the second is an A-type, and the conclusion is an E-type proposition. Te first letter o each name – always a B, C, D, or F – tells us which o the first figure syllogisms is used in the proo; or example, as we saw in
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●
Section 5, above, Aristotle bases the proo o Cesare on the first-figure syllogism Celarent. Te initial letter o Cesare is a reminder o that act. O the consonants within the names, both ‘S’ and ‘P’ indicate the need or conversion. ‘S’ afer an E or an I indicates that the E or I premise must be ‘converted’ in order to complete the proo. For example, Datisi has an S occurring afer an I, so we know the I premise gets converted. Te need or A-conversion is indicated by arequires ‘P’ occurring in a name an A. proo o Darapti, or example, conversion o theafer second A Aristotle’s premise: (1) (2) (3) (4)
A belongs to every C B belongs to every C C belongs to some B A-Conversion (2) A belongs to some B Darii (1)(3)
Notes 1 In modern logic, the difference between ‘having a syllogism’and ‘not having a syllogism’ is described as the difference between having avalid argument and an invalid one. Aristotle, however, has no separate term to describe an invalid argument. 2 In more sophisticated sections o the syllogistic, Aristotle requently uses proos which require that we suppose the contradictory o a given proposition. Te relations established by the square o opposition provide the ramework required or such proos. One amous but controversial example comes atAn Pr A.15, 34a2–3, in what Aristotle calls proo ‘through an impossibility’. It is what modern logicians call a reductio proo. (See Chapter 13 o Rini 2011 or a discussion o Aristotle’s proo.) 3 Te appendix in section 8 o this chapter explains how to decode the medieval names o the syllogisms. 4 In Aristotle’s philosophy generally, he is not especially interested inmere assertion o belonging – or in his own science he is mainly interested in studying what belongs essentially and what possibly belongs. Tese interests are reflected in his logic also. Necessity and possibility are known as modal notions, and Aristotle develops his syllogistic to accommodate these in what is called the modal syllogistic. Section 7 o this present chapter includes a brie introduction to some o the interpretive issues that arise in the case o the modals. 5 Or, another way o saying the same thing: the E-type proposition‘B belongs to no A’ converts to another E-type proposition A ‘ belongs to no B’. 6 What makes the first-figure syllogisms obvious and easy to grasp? Aristotle calls them ‘perect’ or ‘complete’ deductions. Tese labels are technical terms in Aristotle’s
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philosophy and have attracted much attention rom scholars. See or example the discussions in Patzig 1968 (43–87) and in Corcoran 1974.
Reerences Primary Barnes, J. (ed.) (1984), Te Complete Works o Aristotle, Te Revised Oxord ranslation, Princeton: Princeton University Press (Bollingen Series) R. Smith, R. (1989), Aristotle: Prior Analytics; translated, with introduction, notes, and commentary, Indianapolis: Hackett Publishing Company. Aristotle’s Prior and Posterior Analytics. A Revised ext with Ross, W. D. 1949), ( Introduction and Commentary, (Reprinted with corrections 1957, 1965.) Oxord: Clarendon Press. odhunter, I. (1933) Euclid: Elements, London: Dent. redennick, H. (1938), Te Organon, I: Te Categories, On Interpretation, Prior Analytics. Cambridge, Massachusetts: Harvard University Press (Loeb Classical Library). White, N. (1993), Plato: Sophist. ranslated, with introductionand notes, Indianapolis: Hackett Publishing Company.
Secondary Charles, D. (2000), Aristotle on Meaning and Essence. Oxord: Clarendon Press. Corcoran, J. (1974), A ‘ ristotle’s Natural Deduction System’, inAncient Logic and Its Modern Interpretations, Dordrecht: D. Reidel, 85–131. Geach, P. 1( 972), Logic Matters, Oxord: Basil Blackwell. Patterson, R. (1995), Aristotle’s Modal Logic. Essence and Entailment in the Organon. Cambridge: Cambridge University Press. Patzig, G. (1968), Aristotle’s Teory o the Syllogism, ranslated by Jonathan Barnes. Dordrecht: D. Reidel. Rini, A. (2011), Aristotle’s Modal Proos, Prior Analytics A8–22 in Predicate Logic, Dordrecht: Springer.
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2
Stoic Logic Katerina Ierodiakonou
1 Stoic logic in context ogether with the Epicureans and the Sceptics, the Stoics dominated philosophy during the Hellenistic period, i.e. during the three centuries afer Aristotle’s death in 322 B CE . Tey amously claimed that philosophy has three parts that are closely interrelated: physics, ethics and logic. Given their belie in the rationality o nature, they treated logic as inseparable rom the other parts o philosophy; whereas the end o physics is knowing the world, and that o ethics is living in accordance with the natural order, logic aims at distinguishing the true rom the alse and thus makes it possible to find out the truths o the other parts o philosophy. In other words, the purpose o logic, according to the Stoics, was the establishment o a true and stable understanding o the cosmos that was supposed to be essential to human beings, i they were to live a well-reasoned and happy lie. o show the special role o logic in the interrelation between the three parts o philosophy, the Stoics compared logic to the shell o an egg, to the surrounding wall o a ertile field, to the ortification o a city, or to the bones and sinews o a living being (e.g. Diogenes Laertius 7.40 = LS 26B; Sextus Empiricus, Against the Mathematicians 7.19 = L S 26D). Te Stoics did not use the term ‘logic’ ( logikē) as we do nowadays. Logic or them was the study o logos, that is, the study o reason as expressed in all orms o articulate speech; it was meant to help people ask and answer questions methodically, argue correctly, clariy ambiguous statements, solve paradoxes. Tus understood, logic was divided by the Stoics into rhetoric and dialectic: rhetoric was defined as the art o speaking well in the orm o whole, continuous speeches; dialectic, on the other hand, was defined as the art o conducting discussions by means o short questions and answers, though in a much broader sense it was also defined as the science o what is true and what is alse. More 51
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specifically, Stoic dialectic was subdivided into the topics o significations and utterances, or it separately studies what is signified by our utterances and the utterances themselves. Te study o utterances includes purely linguistic and grammatical phenomena; or instance, it includes a physical account o sound appropriately ormed by the speech organs, a discussion o the phonemes or letters o the alphabet, an analysis o the parts o speech, an examination o the criteria good Te study what by is signified covers whatthe getsrelations said by using allorsorts o style. utterances, but o mainly using propositions, between them, the arguments composed o such propositions, and especially their validity (e.g. Diogenes Laertius 7.41–4 = LS 31A; Cicero, On the Orator 2.157–8 = LS 31G). Hence, the modern sense o logic was treated by the Stoics under dialectic, i.e. under the subpart o the logical part o Stoic philosophy, although they also treated under dialectic what we would nowadays call grammar, linguistics, epistemology and philosophy o language (Gourinat 2000). In what ollows, I ocus on the part o Stoic dialectic that corresponds to our modern sense o the term, reerring to it as Stoic logic. Chrysippus, the third head o the Stoic school, developed Stoic logic to its highest level o sophistication. It is reported that he was so renowned in logical matters that people believed that i the gods had logic, it would be no different rom that o Chrysippus (Diogenes Laertius 7.180 = LS 31Q). Te ounder o the Stoic school, Zeno o Citium, and its second head, Cleanthes, were not logicians in the sense that they constructed a ormal logical system, but they both used valid arguments o a considerable degree o complexity; given the rather standardized patterns o their arguments, they must have been aware o the logical orms in virtue o which these arguments were considered as valid (c. Schofield 1983; Ierodiakonou 2002). On the other hand, Zeno’s pupil, Aristo, ervently advocated the view that only ethics should be studied, because physics is beyond us and logic is none o our concern. He claimed that logic is not useul, since people who are skilled in it are no more likely to act well, and he compared logical arguments to spiders’ webs (Diogenes Laertius 7.160–1 = LS 31 N). In general, though, the Stoics studied logic systematically and used it assiduously. Tey were, in act, ofen criticized or being overconcerned with logical orm, or elaborating empty theories, and or ignoring the useul parts o logic such as scientific proos. Diogenes Laertius’ catalogue o Chrysippus’ books lists 130 titles in its section on logic, while its section on ethics contains several titles that suggest a logical content; moreover, other ancient sources offer a ew supplementary titles o Chrysippus’ logical treatises. But although Chrysippus was undoubtedly the authoritative figure in Stoic logic, there seem to have been
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divergences within the Stoa over logic, so that it is reasonable to suppose that, at least in minor ways, Stoic logic changed in the course o its long career (e.g. Sextus Empiricus, Against the Mathematicians 8.443 = L S 36C). o better understand the emergence o Stoic logic, it is useul to look into the logical background out o which it historically developed (c. Barnes 1999b). Tere are two philosophical schools that may have influenced Stoic philosophers in their logical endeavours: Aristotleor and his ollowers, namely Teophrastus and Eudemus; and second,first, the Megaric Dialectical school, namely Diodorus Cronus and Philo the Logician. Tere is no need to search or the influence o Epicurus, since he explicitly rejected logic, though later Epicureans did show some interest in such things as the truth conditions o conditional propositions. And there is no evidence that the philosophers in Plato’s Academy, who at the time adhered to the sceptical stance, attempted to advance logic. Te established view in the nineteenth century was that Stoic logic should be considered as a mere supplement to Aristotle’s logical theory; or Stoic logic, so it was alleged, does nothing more than either copy Aristotelian syllogistic or develop it in a vacuous and ormal way. It is only since about the middle o the twentieth century, afer the important advances in ormal logic, that it has become obvious how Stoic logic essentially differs rom Aristotle’s (c. Łukasiewicz 1935; Kneale and Kneale 1962). It has even been suggested that the Stoics could not have been influenced by Aristotle, since his logical treatises were not available to them and only recovered in the first century BCE (c. Sandbach 1985). Tis view, however, has been extremely controversial. Afer all, even i the Stoics were amiliar with Aristotle’s works, there can still be no doubt concerning the srcinality o their logical system. For although it is true that Teophrastus and Eudemus published treatises on what they called ‘syllogisms based on a hypothesis’, which Aristotle had promised to write about in hisPrior Analytics but never did, and these syllogisms have a great deal in common with the types o arguments discussed by the Stoics, there is no evidence that the Peripatetic logicians anticipated the outstanding eature o Stoic logic; that is, the construction o a logical system to prove the validity o a whole class o arguments o a different kind than those Aristotle ocused on in his syllogistic (c. Barnes 1999b). As ar as the Megaric or Dialectical school is concerned, the historical connections between the philosophers o this school and the Stoics are well documented. Zeno knew both Diodorus’ and Philo’s works well, and Chrysippus wrote treatises in which he criticized their logical theses. Although Diodorus and Philo are usually presented as mainly occupied with the study o logical
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puzzles or paradoxes, it is clear that they also put orward srcinal views about, or instance, logical modalities and the truth conditions o conditional propositions (see below). But they never came close to ormulating a logical calculus as elaborate and sophisticated as that o the Stoics (c. Bobzien 1999).
2 Te Stoic logical system Te reconstruction o the Stoic logical system in all its details is quite problematic, and this is or the ollowing reasons: First, the logical treatises o the Stoics themselves are lost. Second, most o our ancient sources or Stoic logic are rather hostile and late; or instance, the Platonist Plutarch (c. 45–120 CE) or the sceptic Sextus Empiricus (c. 160–210 CE ). Tird, some o the available material is composed by ancient authors whose competence in the intricacies o logic is questionable; or instance, the doxographer Diogenes Laertius (c. third century CE ). Nevertheless, on the basis o what has survived, modern scholars have provided us with a airly reliable understanding o the undamental eatures o Stoic logic, which show the logical ingenuity o Chrysippus and o the other Stoic logicians (c. Mates 1953; Frede 1974a; Bobzien 1999, 2003; Ierodiakonou 2006).
2.1 Sayables and assertibles Te main characteristic o Stoic logic is that the inerences it studies are about relations between items that have the structure o propositions. Whereas Aristotle ocused his attention on inerences that involve relations between terms, and thus introduced a logical system similar to what we nowadays call ‘predicate logic’, Stoic logic marks the beginning o the socalled ‘propositional logic’ (c. Frede 1974b; Barnes 1999a). o say, though, that Stoic logic is propositional may be somewhat misleading; or, to start with, the Stoics have quite a different understanding o what a proposition is, or to use their own term, o what an axiōma, i.e. an assertible, is. o ully grasp the Stoic notion o an axiōma, we first need to get some idea about another basic Stoic notion, namely the notion o a lekton, since axiōmata are defined by the Stoics primarily as lekta (c. Frede 1994). Te Greek term ‘lekton’ is derived rom the verb legein, i.e. ‘to say’, and hence it is what has been or gets said, or something that can be said, i.e. a sayable. In act, the Stoics distinguished between what can be said, by uttering or using an expression, and
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the expression itsel that we utter or use in saying something. For instance, they distinguished between the expression ‘Socrates is walking’, and what gets said by using this expression, namely that Socrates is walking. Tus the kind o item that gets said by using the appropriate expression in the appropriate way, the Stoics called a lekton. Te Stoics also talked about a lekton as the state o affairs signified, i.e. the signification (sēmainomenon), distinguishing it rom the utterance, i.e. the signifier (sēmainon ), placed and rom the external object whichorthe is about. lektonsentences, Tereore, lekta are items between mere vocal sounds written on the one hand, and the objects in the world, on the other (Sextus Empiricus, Against the Mathematicians 8.11–12 = LS 33B). Very roughly speaking, lekta or sayables are the underlying meanings in everything we say, as well as in everything we think; or lekta were also defined by the Stoics as what subsists in accordance with a rational impression, that is, as the content o our thoughts (Sextus Empiricus, Against the Mathematicians 8.70 = L S 33C). But not everything that gets thought gets said, and not everything that can be said gets thought. Tere are indeed many things that never get thought or said, although they are there to be thought or said. In other words, Stoic sayables are not mind-dependent items; at the same time, though, they certainly do not exist in the way bodies exist in the world. Te Stoics stressed that sayables are incorporeal like void, place and time (Sextus Empiricus, Against the Mathematicians 10.218 = LS 27D), and in order to characterize their mode o being they introduced the notion o subsistence (huphistanai) as opposed to that o existence (einai). Reality, they claimed, is not just constituted by bodies, but also by predicates and propositions true about bodies. Hence, sayables are given in Stoic ontology some status, namely the status not o bodies but o incorporeal somethings (tina). Te Stoics divided sayables into complete and incomplete (Diogenes Laertius 7.63 = LS 33F). Incomplete sayables include predicates, or instance what is meant by the expression ‘is walking’, or it is simply a thing to say about something. On the other hand, questions, oaths, invocations, addresses, commands and curses, are all complete sayables. Te most important kind o complete sayables are the axiōmata, i.e. the assertibles. An assertible is mainly defined by the act that it is the kind o complete sayable that when it gets said one is asserting something, and it differs rom other kinds o complete sayables by the property o being true or alse (e.g. Diogenes Laertius 7.65 = LS 34A; Sextus Empiricus, Against the Mathematicians 8.74 = LS 34B). Since they constitute a particular class o sayables, assertibles do not exist as bodies do, but they are said to subsist. Moreover, i an assertible is alse it is said to merely subsist ( huphistanai), but i
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it is true it is a act and can also be said to be present ( huparchein). In this sense true assertibles correspond to the world’s having certain eatures, and they are available to be thought and expressed whether anyone is thinking about them or not. On the other hand, since alse assertibles are said to subsist, the philosophical question o how alse statements and thoughts are possible gets a reasonable answer; alse assertibles are the contradictories o acts, and as such they may be saidSo, to Stoic have some ontological status. assertibles and propositions, as we nowadays conceive them, share common characteristics. For instance, they are expressed by complete indicative or declarative sentences, they are either true or alse, and they are incorporeal. But there are also differences between them. For instance, whereas propositions are timelessly true or alse, Stoic assertibles are asserted at a particular time and have a particular tense; that is to say, an assertible can in principle change its truth-value without ceasing to be the same assertible. For example, the assertible ‘I Dion is alive, Dion will be alive’ is not true at all times, or there will be a time when the antecedent will be true and the consequent alse, and thus the conditional will be alse (Simplicius, Commentary on Aristotle’s Physics 1299.36– 1300.10 = LS 37K). Furthermore, since Stoic assertibles include token reflexive elements, e.g. ‘this’ or ‘I’, they may cease to exist and presumably also, though this is not clearly stated, begin to exist at definite times; or a Stoic assertible requires that its subject exists, otherwise it is said to be destroyed. For example, the assertible ‘Tis man is dead’ is destroyed at Dion’s death, i ‘this man’ reers to Dion (Alexander o Aphrodisias, Commentary on Aristotle’s Prior Analytics 177.25–178.1= LS 38F). Assertibles are divided into simple and non-simple assertibles (e.g. Sextus Empiricus, Against the Mathematicians 8.93–8 = LS 34H). Simple assertibles are those which are not composed either o a repeated assertible or o several assertibles; they are subdivided into:
(i) Definite; e.g. ‘Tis one is walking’. (ii) Indefinite; e.g.‘Someone is walking’. (iii) Intermediate; e.g. ‘It is day’ or ‘Socrates is walking’. In addition, the Stoics classified among simple assertibles three different kinds o negative assertibles (e.g. Diogenes Laertius 7.69–70 = LS 34K):
(i) Negations; e.g.‘Not: it is day’. (ii) Denials; e.g. ‘No one walks’. (iii) Privatives; e.g. ‘Tis man is unkind’.
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An interesting special case o a negation is what the Stoics called ‘super-negation’ or, as we would say, ‘double negation’; or example, ‘Not: not: it is day’. Tis is still a simple assertible and its truth conditions are the same as those or ‘It is day’. Note that the scope o the negative particle is, according to the Stoics, the entire assertible, which means that an assertible, or instance, o the orm ‘It is not day’ is treated as affirmative and not as negative. Hence, the negative particle ‘not’ was not thethe Stoic logicians a connective; connectives partsconsidered o speech,by and Stoic negativeasparticle does not do that. bind together Non-simple assertibles, on the other hand, are those which are composed either o a repeated assertible or o several assertibles which are combined by one or more connectives. Te main types o non-simple assertibles studied by the Stoics are the ollowing (e.g. Diogenes Laertius 7.71–4 = LS 35A):
(i) A conjunctive assertible is one which is conjoined by the conjunctive connective ‘both . . . and . . .’; e.g. ‘Both it is day and it is light’. A conjunctive assertible is true when all its conjuncts are true. (ii) A disjunctive assertible is one which is disjoined by the disjunctive connective ‘either . . . or . . .’; e.g. ‘Either it is day or it is night’. Te Stoics understand the disjunctive relation as exhaustive and exclusive; that is to say, a disjunction is true when one and only one disjunct is true. (iii) A conditional assertible is one linked by the conditional connective ‘i’; e.g. ‘I it is day, it is light’. A conditional, according to the Stoics, is true when there is a ‘connection’ (sunartēsis) between the antecedent and the consequent, i.e. when the contradictory o its consequent conflicts with the antecedent; or instance, the conditional ‘I it is day, it is day’ is true, since the contradictory o its consequent ‘Not: it is day’ conflicts with its antecedent ‘It is day’. A conditional is alse when the contradictory o its consequent does not conflict with its antecedent; or instance, the conditional ‘I it is day, I am talking’ is alse, since the contradictory o its consequent ‘Not: I am talking’ does not conflict with its antecedent ‘It is day’. Chrysippus, thereore, assigned to the conditional connective ‘i’ a strong sense, compared to what our ancient sources attribute to Diodorus Cronus and Philo the Logician. Philo claimed that a conditional is true simply when it does not have a true antecedent and a alse consequent, e.g. ‘I it is day, I am talking’. Tis use o the conditional connective ‘i’ is equivalent to what we nowadays call ‘material implication’ and is clearly truth-unctional. Diodorus, on the other hand, advocated that a conditional is true when it neither was nor is able to have
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a true antecedent and a alse consequent. According to this view, the conditional ‘I it is day, I am talking’ is alse, since when it is day but I have allen silent it will have a true antecedent and a alse consequent; but the conditional ‘I there are no partless elements o things, there are partless elements o things’ is true, or it will always have the alse antecedent ‘Tere are partless elements o things’. On the Chrysippean view, however, both the conditional ‘I it is day, I am talking’ and the conditional ‘I there are no partless o things, there between are partless elements o things’ are alse, since there iselements no connection in them the antecedent and the consequent (Sextus Empiricus, Outlines o Pyrrhonism 2.110–13 = L S 35B). Chrysippus’ interpretation o the conditional connective ‘i’ has the disadvantage o rendering at least part o Stoic logic non-truth-unctional. On the other hand, it is able to adequately express intelligible connections in nature and avoid cases that are counter-intuitive, such as the conditionals ‘I it is day, I am talking’ or ‘I there are no partless elements o things, there are partless elements o things’. Similarly, the Stoics’ interest in adequately expressing intelligible connections in nature shows in Chrysippus’ decision not to use the conditional when discussing astrological predictions merely based on empirical observation o the correlations between astral and terrestrial events. For example, it may be that it is not the case both that Fabius was born at the rising o the dog-star and that Fabius will die at sea. Chrysippus would not express this as ‘I Fabius is born at the rising o the dog- star, he will not die at sea’, precisely because he was not convinced that there was a necessary causal connection between being born at that time o the year and dying on dry land. Tis is the reason why Chrysippus preerred in such cases the negated conjunction, i.e. ‘Not: Both Fabius was born at the rising o the dog-star and Fabius will die at sea’ (Cicero, On Fate 12–15 = LS 38E; c. Sedley 1984; Barnes 1985). In addition to conjunctions, disjunctions and conditionals, we find in our ancient sources more kinds o non-simple assertibles, which may have been introduced afer Chrysippus. For instance, a subconditional assertible is one which is joined by the connective ‘since’, and it is true when the antecedent holds and the consequent ollows rom the antecedent; or example, ‘Since it is day, the sun is above the earth’ is true when said in daytime. Also, nonsimple assertibles can be composed o more than two simple assertibles, either because the constituent assertibles are themselves non-simple, or because certain connectives, namely the conjunctive and the disjunctive connective, are two-or-more-place unctions; or example, the Stoics used the conditional ‘I both it is day and the
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sun is above the earth, it is light’, or the three-place disjunction ‘Either wealth is good or it is evil or it is indifferent’.
2.2 Modalities Te Stoics also discussed modal assertibles in their logical system. Although Stoic deal withassertibles assertiblesare o standardly the orm ‘Itclassified is possible it is day’, logic simpledoes andnot nonsimple as that possible, impossible, necessary and non-necessary. Tat is to say, the Stoic logicians regarded modalities as properties o assertibles, just like truth and alsehood. According to their view, thereore, an assertible may in principle change its modal value, since it has it at a time. Stoic modal logic developed out o the debate over the amous Master Argument (kurieuōn logos), presented by Diodorus Cronus (e.g. Epictetus, Discourses 2.19.1–5 = LS 38A; Alexander o Aphrodisias, Commentary on Aristotle’s Prior Analytics 183.34–184.10 = LS 38B; c. Bobzien 1993). According to the Master Argument, the ollowing three propositions mutually conflict: ‘Every past truth is necessary’; ‘Something impossible does not ollow rom something possible’; and ‘Tere is something possible which neither is nor will be true’. Diodorus saw this conflict and exploited the convincingness o the first two propositions to establish the conclusion that ‘Nothing which neither is nor will be true is possible’. It seems that the Stoics made various attempts to rebut Diodorus’ view, reacting to the threat o a weakened orm o logical determinism implied by his account o the possible as ‘that which is or will be’. Chrysippus proposed his own definitions o modalities (Diogenes Laertius 7.75 = LS 38D):
(i) A possible assertible is that which admits o being true, and is not prevented by external actors rom being true, e.g. ‘Dion is alive’. (ii) An impossible assertible is that which does not admit o being true, or admits o being true but is prevented by external actors rom being true, e.g. ‘Te earth flies’. (iii) A necessary assertible is that which is true and does not admit o being alse, or admits o being alse but is prevented by external actors rom being alse, e.g. ‘Virtue is beneficial’. (iv) A non-necessary assertible is that which is capable o being alse, and is not prevented by external actors rom being alse, e.g. ‘Dion is sleeping’.
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(v) A plausible assertible is that which invites assent to it, e.g. ‘I someone gave birth to anything, she is its mother’. (vi) A probable or reasonable assertible is that which has higher chances o being true than alse, e.g. ‘I shall be alive tomorrow’. 2.3 Arguments Te Stoics defined an argument (logos) as a complex or a compound o premises and a conclusion. Te ollowing is a typical Stoic argument (Sextus Empiricus, Outlines o Pyrrhonism 2.135–6 = LS 36B2; Diogenes Laertius 7.76–7 = LS 36A1–3): I it is day, it is light. But it is day. Tereore it is light.
Moreover, the Stoics discussed arguments in terms o their modes ( tropoi), which are the abbreviations o particular arguments; or instance, the mode o the previous argument is: I the first, the second. But the first. Tereore the second.
Te ordinal numbers here stand or assertibles, though they have exactly the same role as the letters o the alphabet in Aristotelian logic which stand or terms. Finally, the Stoics also used the so-called ‘mode-arguments’ (logotropoi), in which the assertibles are given in ull when first occurring, but are then replaced by ordinal numbers, obviously or purposes o simplicity and clarity: I Plato is alive, Plato is breathing. But the first. Tereore the second.
It was the orthodox Stoic view that an argument must have more than one premise, though it seems that some Stoics accepted single-premise arguments, like the ollowing (Sextus Empiricus, Against the Mathematicians 8.443 = LS 36C7; Apuleius, On Interpretation 184.16–23 = LS 36D): You are seeing. Tereore you are alive.
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O arguments some are valid, others invalid. Invalid are those the contradictory o whose conclusion does not conflict with the conjunction o the premises (Diogenes Laertius 7.77 = LS 36A4); or instance, the argument I it is day, it is light. But it is day. Tereore Dion is walking
is invalid, because the contradictory o its conclusion, i.e. ‘Not: Dion is walking’, does not conflict with the conjunction o its premises, i.e. ‘Both i it is day it is light and it is day’. In other words, the validity o an argument depends on the truth o the corresponding conditional ormed rom the conjunction o the premises as antecedent and the conclusion as consequent (Sextus Empiricus, Outlines o Pyrrhonism2.137 = LS 36B3). o take again the previous argument, it is invalid because the corresponding conditional ‘I both i it is day it is light and it is day, Dion is walking’ is alse, at least according to Chrysippus’ truth conditions or conditional assertibles. O valid arguments, some are just called ‘valid’, others‘syllogistic’ (sullogistikoi). Te Stoics defined syllogistic arguments as those which either are what they called ‘indemonstrables’ (anapodeiktoi) or can be reduced to the indemonstrables (Diogenes Laertius 7.78 = LS 36A5). Indemonstrable arguments, or simple syllogisms, are those whose validity is not in need o demonstration, given that it is obvious in itsel (Diogenes Laertius 7.79–81 = LS 36A11–16). Te lists o indemonstrable arguments which are to be ound in our ancient sources vary, but there is no doubt that Chrysippus himsel distinguished five different types o such arguments. Te basic logical orms o the five standard indemonstrables are described and illustrated by examples in the ancient texts as ollows:
(i) A first indemonstrable argument is constructed out o a conditional and its antecedent as premises, and the consequent as conclusion, e.g. I it is day, it is light. But it is day. Tereore it is light.
(ii) A second indemonstrable argument is constructed out o a conditional and the contradictory o its consequent as premises, and the contradictory o its antecedent as conclusion, e.g. I it is day, it is light. But not: it is light. Tereore not: it is day.
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(iii) A third indemonstrable argument is constructed out o a negated conjunction and one o its conjuncts as premises, and the contradictory o the other conjunct as conclusion, e.g. Not: both Plato is dead and Plato is alive. But Plato is dead. Tereore not: Plato is alive.
(iv) A ourth indemonstrable argument is constructed out o a disjunction and one o its disjuncts as premises, and the contradictory o the other disjunct as conclusion, e.g. Either it is day or it is night. It is day. Tereore not: it is night.
(v) A fifh indemonstrable argument is constructed out o a disjunction and the contradictory o one o its disjuncts as premises, and the other disjunct as conclusion, e.g. Either it is day or it is night. Not: it is day. Tereore it is night.
In suggesting this particular list o the five types o indemonstrable arguments, Chrysippus was obviously not trying to introduce the smallest possible number o different types o indemonstrable arguments. Rather, it seems that he included in his list all types o arguments which rely merely on the argumentative orce o the different basic types o connectives known to him. In the case o the ourth and fifh indemonstrables, or instance, they just rely on what it means to use the disjunctive connective, namely to say that i one o the disjuncts holds the contradictory o the other holds too, and i the contradictory o one disjunct holds the other disjunct holds too.
2.4 Analysis o demonstrate the syllogistic validity o any argument whatsoever, the Stoic logicians considered it necessary to reduce it to one or more o the indemonstrable arguments, which are thus regarded as the first principles o the Stoic logical system. Te procedure o reducing non-simple syllogisms to indemonstrable arguments was called by the Stoics ‘analysis’ a( nalusis). o carry out this
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procedure, the Stoic logicians had at least our logical rules, the so-called themata, and in Latin constitutiones or exposita (e.g. Galen, On Hippocrates’ and Plato’s Doctrines 2.3.18–19 = LS 36H; Diogenes Laertius 7.78 = LS 36A5). We only know the first and the third Stoic thema, and it is on the basis o extremely meagre evidence that modern scholars have suggested different reconstructions o the other two. Te first thema is the ollowing (Apuleius, On Interpretation 191.5–10 = LS 36I): I rom two propositions a third is deduced, then rom either one o them together with the contradictory o the conclusion the contradictory o the other is deduced.
Te third thema is the ollowing (Alexander o Aphrodisias, Commentary on Aristotle’s Prior Analytics 278.12–14 = LS 36J): When rom two propositions a third is deduced, and extra propositions are ound rom which one o those two ollows syllogistically, the same conclusion will be deduced rom the other o the two plus the extra propositions rom which that one ollows syllogistically.
As to the second and ourth themata, we try to reconstruct them mainly on the basis o a logical principle, the so-called ‘dialectical theorem’ (dialektikon theōrēma) or ‘synthetic theorem’ (sunthetikon theōrēma), which is most probably Peripatetic and which is supposed to do the same job as the second, third, and ourth themata together (e.g. Sextus Empiricus, Against the Mathematicians 8.231 = LS 36G4): When we have the premisses rom which some conclusion is deducible, we potentially have that conclusion too in these premisses, even i it is not expressly stated.
o get a clearer ideao how we are supposed to apply the Stoicthemata, and thus how Stoic analysis actually unctions, let us ocus on the analysis o the ollowing non-simple syllogism (Sextus Empiricus,Against the Mathematicians 8.235–6 = LS 36G7): I things evident appear alike to all those in like condition and signs are things evident, signs appear alike to all those in like condition. But signs do not appear alike to all those in like condition. And things evident do appear alike to all those in like condition. Tereore signs are not things evident.
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o this purpose, it would be easier to use the mode o the non-simple syllogism: I both the first and the second, the third. But not the third. But also the first. Tereore not the second.
Sextus suggests that this argument can be reduced to two indemonstrables o different types, namely to a second and a third indemonstrable argument, by going through the ollowing two steps:
(i) By combining the first premise, which is a conditional, with the second premise, which is the contradictory o the conditional’s consequent, we get a second indemonstrable which has as its conclusion the contradictory o the conditional’s antecedent: I both the first and the second, the third. But not the third. Tereore not: both the first and the second.
combining the conclusion o this indemonstrable, which is a negated (ii) By conjunction, with the third remaining premise, which affirms one o the two conjuncts, we get a third indemonstrable which has as its conclusion the affirmation o the other conjunct: Not: both the first and the second. But the first. Tereore not: the second.
Hence, the dialectical theorem in this case validates the use o the conclusion o the second indemonstrable, that is to say the use o the negated conjunction, in the construction o the third indemonstrable; or according to this logical rule, the negated conjunction which is deduced rom some o the premises o the argument is implicitly contained in the argument, though it is not expressly stated. And it is obvious that we may similarly use the third thema; or a single application o the third thema on the second and third indemonstrables that we have constructed, could help us deduce the non-simple syllogism whose validity we try to prove. o summarize, Stoic analysis starts with a non-simple syllogism and continues with a series o arguments which are either indemonstrables or arguments directly derived rom the indemonstrables by appropriate application o one o the Stoic themata. Indeed, the Stoic logicians, and in particular Chrysippus, seem
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to have believed that their standard list o five indemonstrables is complete in the sense o containing all that is required or reasoning. It is said, or instance, that every argument is constructed out o these indemonstrables, and that all other arguments are thought to be validated by reerence to them (e.g. Diogenes Laertius 7.79 = LS 36A11). Tereore, we may iner that some claim o completeness was made by the Stoic school, but it is not at all clear what precisely the Stoics’ 1979; Milnedefinition 1995). o completeness was, i they ever offered one (c. Mueller Afer all, the Stoics themselves admitted that we cannot apply the method o analysis to all valid arguments, that is, we cannot reduce all valid arguments to the five indemonstrables by using the our Stoic themata; or, as we have already said, there are arguments in Stoic logic which are just valid, but not syllogistic. It thus seems that, according to the Stoics, the validity o such arguments is guaranteed not by their own analysis, but by their being equivalent to syllogistic arguments. For instance, the so-called ‘subsyllogistic arguments’ (huposullogistikoi logoi) differ rom syllogisms only in that one or more o their constituent assertibles, although being equivalent to those in a syllogism, diverge rom them in their linguistic orm (e.g. Diogenes Laertius 7.78 = LS 36A6): ‘It is day and it is night’ is alse. But it is day. Tereore not: it is night.
In this example, i it were not or the first premise that slightly diverges rom the linguistic orm o a negated conjunction, the argument would have been a third indemonstrable. In general, what emerges rom the Stoic logicians’ treatment o subsyllogistic arguments is that they tried to eliminate unnecessary ambiguities by standardizing language, so that the orm o a sentence would unambiguously determine the type o assertibles expressed by it; or one and the same sentence may express assertibles that belong to different classes, and equally two different sentences may express the same assertible. But i there is some agreement to fix language use in a certain way, it becomes possible to easily discern the logically relevant properties o assertibles and their compounds, by simply examining the linguistic expressions used. Finally, a scientific demonstration or proo ( apodeixis) is, according to the Stoics, a syllogistic argument with true premises which by means o deduction reveals, i.e. gives knowledge o, a non-evident conclusion (e.g. Sextus Empiricus, Outlines o Pyrrhonism 2.140–3 = LS 36B7–11). For instance, the ollowing argument was treated by the Stoics as an example o a proo:
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It is exactly the revelation o this non-evident conclusion by the orce o the premises that constitutes the requirement o a genuine proo, and it is this discovery to which knowledge, according to the Stoics, aspires.
2.5 Invalid arguments and paradoxes In Stoic logic invalid arguments arise in our ways (e.g. Sextus Empiricus, Against the Mathematicians 8.429–34 = LS 36C1–5):
(i) By disconnection, when premises have no connection with one another or with the conclusion, e.g. I it is day, it is light. But wheat is being sold in the market. Tereore it is light.
(ii) By redundancy, when they contain premises which are superfluous or drawing the conclusion, e.g. I it is day, it is light. But it is day. But also virtue benefits. Tereore it is light.
(iii) By being propounded in an invalid schema, e.g. I it is day, it is light. But not: it is day. Tereore not: it is light.
(iv) By deficiency, when they contain premises which are incomplete, e.g. Either wealth is bad, or wealth is good. But not: wealth is bad. Tereore wealth is good.
For the first premise should be ‘Either wealth is bad, or wealth is good, or wealth is indifferent’; and moreover, a premise is missing. Furthermore, there is abundant evidence o the Stoics’ interest in solving the ollowing logical paradoxes (c. Mignucci 1999).
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(i) Te Liar
Various versions o the Liar paradox were known in antiquity, but there is no single text that gives us with certainty the precise ormulation o the argument. A plausible reconstruction reads as ollows: ‘I you say that you are lying, and you say so truly, you are lying, and i you are lying, you are telling the truth.’ Presumably it was Eubulides who invented this paradox in the ourth century, but there is no doubt that it was Chrysippus who more than anyone else in ancient times tried to solve it. Te peculiarity o the case o the Liar seems to be not only that we are unable to find out what the truth o the statement is, but that in this case there is no truth o the matter. Perhaps Chrysippus’ view was that in cases like this the statement is neither true nor alse. However, i this were the case, the solution would put the very notion o an assertible under great pressure and would orce a reconsideration o its definition (Cicero, On Academic Scepticism 2.95 = LS 37H5; Plutarch, On Common Conceptions 1059D–E = LS 37I). (ii) Te Sorites
Te name o the Sorites comes rom the Greek noun ‘sōros’, which means ‘heap’ or ‘pile’. Tis paradox exploits the vagueness o certain predicates such as ‘heap’: Is a single grain o wheat a heap? Te answer is obviously ‘No’. Are two grains a heap? Te answer is again ‘No’. I we continue adding one grain to the previous quantity we never get a heap. Chrysippus is reported to have claimed that this paradox does not pose any real difficulty, because the wise man knows at which moment he should stop replying to questions o the orm ‘Are so-and-so many grains a heap?’ (Cicero, On Academic Scepticism 2.95 = LS 37H3; Galen, On Medical Experience 17.1 = LS 37E3). (iii) Te Veiled Man
According to one version o this paradox, Chrysippus asks someone whether he knows his own ather. Te person replies that he does. Next Chrysippus asks him again what he would have said i a veiled man were to be placed in ront o him and was asked whether he knows him. Te same person replies that he would have said that he doesn’t know him. Chrysippus concludes that, i the veiled man were his ather, the person would have thus admitted that he both knows and does not know his own ather (Lucian, Philosophers or Sale 22 = LS 37L).
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(iv) Te Horned Man
In Diogenes Laertius (7.187) we find the ollowing ormulation o this paradox: ‘I you have not lost something, you have it still. But you have not lost horns. Tereore you still have horns.’ Unortunately, there is no evidence as to the way in which the Stoics tried to solve the last two paradoxes.
3 Conclusion In his Introduction to Logic the ancient physician and philosopher Galen (129– c. 200/216 CE) discussed elements o the Stoic logical system alongside Aristotle’s syllogistic. He even introduced what he considered as a third kind o syllogisms, the ‘relational’ (pros ti) syllogisms; or example, ‘Dio owns hal as much as Teo. Teo owns hal as much as Philo. Tereore Dio owns a quarter o what Philo owns.’ Te act that at the end o his account Galen made reerence to the late Stoic philosopher Posidonius (c. 135–51 B CE ) has led some modern scholars to suggest that it was Posidonius who had previously invented relational syllogistic and urther developed Stoic logic. However, the evidence is extremely scant and Posidonius’ contribution to logic most probably is limited. In act, no ancient logician seriously attempted to explore relational syllogisms or, or that matter, to combine the two major logical systems o antiquity. Te Byzantine scholars who paraphrased or commented on Aristotle’s logical treatises in the middle ages simply added the standard list o Stoic indemonstrable arguments to the Aristotelian syllogistic; in most cases, they did not even attribute it to the Stoics. Regrettably, the synthesis o Aristotle’s predicate logic and the Stoic propositional calculus had to wait or a long time.
Reerences Barnes, J. (1985), ‘Pithana Sunēmmena’, Elenchos 6, 454–67. Barnes, J. (1999a), A ‘ ristotle and Stoic Logic’, in K. Ierodiakonou (ed.),opics in Stoic Philosophy, Oxord: Clarendon Press, 23–53. Barnes, J. (1999b), ‘Logic and Language. Introduction & Logic I. Te Peripatetics ’, in K.
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Algra, J. Barnes, J. Manseld and M. Schofield (eds), Te Cambridge History o Hellenistic Philosophy, Cambridge: Cambridge University Press, 65–83. Bobzien, S. (1993), ‘Chrysippus’ modal logic and its relation to Philo and Diodorus ’, in K. Döring and . Ebert (eds),Dialektiker und Stoiker, Stuttgart: Franz Steiner, 63–84. Bobzien, S. (1999), ‘Logic II . Te Megarics & III.1–7 Te Stoics’, in K. Algra, J. Barnes, J. Manseld and M. Schofield (eds.),Te Cambridge History o Hellenistic Philosophy, Cambridge: Cambridge University Press, 83–157. Bobzien, S. (2003), ‘Logic’, in B. Inwood (ed.),Te Cambridge Companion to the Stoics, Cambridge: Cambridge University Press, 85–123. Frede, M. (1974a), Die stoische Logik, Göttingen: Vandenhoeck and Ruprecht. Frede, M. (1974b), ‘Stoic vs. Aristotelian Syllogistic’, Archiv ür Geschichte der Philosophie 56, 1–32. Frede, M. (1994), ‘Te Stoic Notion o a Lekton’, in S. Everson (ed.),Companions to Ancient Tought (vol. 3): Language, Cambridge: Cambridge University Press, 109–28. Gourinat, J.-B. (2000), La Dialectique des stoïciens, Paris: Vrin. Ierodiakonou, K. (2002), ‘Zeno’s Arguments’, in . Scaltsas and A.S. Mason (eds), Te Philosophy o Zeno, Larnaca: Te Municipality o Larnaca, 81–112. Blackwell Ierodiakonou, K. (2006), ‘Stoic Logic’, in M.L. Gill and P. Pellegrin (eds), Companions: A Companion to Ancient Philosophy, Oxord: Blackwell, 505–29. Kneale, Press.W. and Kneale, M. (1962), Te Development o Logic, Oxord: Oxord University Long, A.A. and Sedley, D.N. (eds. and trans.) (1987), Te Hellenistic Philosophers, (2 vols.), Cambridge: Cambridge University Press (=LS ). Łukasiewicz, J. (1935), ‘Zur Geschichte der Aussagenlogik’, Erkenntnis 5, 111–31. Mates, B. (1953), Stoic Logic. Berkeley: University o Caliornia Press. Mignucci, M. (1999), ‘Logic III .8 Te Stoics’, in K. Algra, J. Barnes, J. Manseld and M. Schofield (eds.),Te Cambridge History o Hellenistic Philosophy, Cambridge: Cambridge University Press, 157–76. Milne, P. 1( 995), ‘On the Completeness o non-Philonian Stoic Logic’, History and Philosophy o Science16, 39–64. Mueller, I. (1979), ‘Te Completeness o Stoic Propositional Logic,’ Notre Dame Journal o Formal Logic 20, 201–15. Sandbach, F.H. (1985), Aristotle and the Stoics, Cambridge: Cambridge Philological Society. Schofield, M. (1983), ‘Te Syllogisms o Zeno o Citium’, Phronesis 28, 31–58. Sedley, D.N. (1984), ‘Te Negated Conjunction in Stoicism’, Elenchos 5, 311–16.
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Medieval Logic Sara L. Uckelman
1 Introduction Many people unamiliar with the history o logic may think o the Middle Ages as a ‘Dark Ages’ in logic, with little development beyond Aristotelian syllogistic and ull o scholastic wrangling ocused on uninteresting details. Tis could not be urther rom the case. Instead, the Middle Ages, especially at the end o the High Middle Ages and into the ourteenth century, was a period o vibrant activity in logic, in many different areas – the (re)birth o propositional logic, the development o interactive and dynamic reasoning, sophisticated semantic theories able to address robust paradoxes, and more. Te period can be characterized by a ocus on the applied aspects o logic, such as how it relates to linguistic problems, and how it is used in interpersonal contexts. o attempt to survey the ull chronological and biographical story o medieval logic rom the end o antiquity until the birth o the Renaissance in the space o a chapter would be an impossible endeavour. Instead, this chapter ocuses on our topics which are in certain respects uniquely and particularly medieval: (1) the analysis o properties o terms, specifically signification and supposition; (2) theories o consequences; (3) the study osophismata and insolubilia; and (4) the mysterious disputationes de obligationibus . We treat each o these in turn, giving short thematic overviews o their rise and development. First, we provide a short historical comment on the introduction o the Aristotelian corpus into the medieval West and the effects that this had on the development o logic in the Middle Ages.
2 Te reception o Aristotle In the second decade o the sixth century, Anicius Manlius Severinus Boethius decided to embark upon the project o translating rom Greek into Latin all o 71
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the Aristotelian and Platonic works that he could obtain, beginning with Aristotle’s logical corpus. In 523, he was sentenced to prison or treason and by his death, around 526, he had completed an admirable percentage o his task, translating Aristotle’s Categories, On Interpretation, the Prior Analytics, the opics, the Sophistical Reutations, and most likely the Posterior Analytics, as well as Porphyry’s Introduction (to the Categories). He also wrote commentaries on the theunique , andsyllogisms his own textbooks on Introduction Categories On Interpretation logic, including ,the littleand treatise on hypothetical (syllogisms with molecular, i.e. noncategorical, premises). Tese translations, the first o Aristotle into Latin, were to shape the path o medieval Western philosophy in a tremendous ashion. Afer Boethius’s death, many o his translations o Aristotle were lost. Only the translations o On Interpretation and the Categories remained. Tese two books, along with Porphyry’s Introduction, ormed the basic logical and grammatical corpus or the next six centuries. It was not until the 1120s that his other translations were rediscovered.1 In the 1150s, James o Venice translated the Posterior Analytics (i Boethius had translated the Posterior Analytics, this translation has never been ound), and also retranslated the Sophistical Reutations. Te newly completed Aristotelian logical corpus was available in western Europe by the latter hal o the twelfh century, though it was not until the birth o the universities in the early thirteenth century that they began to circulate widely (the exception being the Sophistical Reutations, which was seized upon almost immediately). Tese works established their place as canonical texts in logic and natural philosophy, and their effect on the development o these fields was quickly seen. Te material in these new texts provided medieval logicians not only with a stronger logical basis to work rom, but also proved to be a ‘jumping off’ point or many novel – and in some cases radically non-Aristotelian – developments. Aristotle long remained the authority on logic, but though most medieval logicians give the required nod to the Philosopher, they certainly elt ree to explore and develop their own ideas.
3 Properties o terms In the Middle Ages, logic was a linguistic science; it arose rom a desire to understand how language is used (properly) in order to assist in textual exegesis. As a result, rom an early period the study o logic was closely connected to the study o grammar (indeed, these two studies, along with rhetoric, made up the
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trivium, the branch o learning that was the core o the undergraduate’s curriculum in the early universities). William o Sherwood (1190–1249) in his Introduction to Logic (c. 1240) explains that ‘Logic is principally concerned with the syllogism, the understanding o which requires an understanding o the proposition; and, because every proposition is made up o terms, an understanding o the term is necessary’ (William o Sherwood 1966: 21). Hence i we wish to become proficient in logic we first important become masters obeing terms,signification and, more specifically, their properties, themust two most o which and supposition. Te period rom 1175 to 1250 marked the height o what is known as terminist logic, because o its emphasis on terms and their properties. In this section, we discuss the ideas o signification and supposition ound in late terminist writers such as William o Sherwood, Peter o Spain (mid-thirteenth century), and Lambert o Lagny (mid-thirteenth century), which reflect the most mature and settled views on the issues.
3.1 Signification According to Lambert o Lagny in his treatise on theProperties o erms, the signification o a word is the‘concept on which an utterance is imposed by the will o the person instituting the term’ (1988: 104). As such, signification is one o the constitutive parts o a word’s meaning. Four things are requiredor signification: ●
●
●
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A thing A concept o the thing An utterance (which may be mental, spoken, or written) A union o the utterance and the concept (effected by the will’s imposition)
A thing is any extra-mental thing, whether it be an Aristotelian substance (e.g. Socrates the man), an Aristotelian accident (e.g. the whiteness which is in Socrates) or an activity (e.g. the running which Socrates is doing right now). On the medieval view, these extra-mental things are presented to the soul by means o a concept. A term gains signification when it is used in an utterance and the utterance is connected to a concept by the will o the speaker (or thinker or writer). Te concept imposed upon the term in the speaker’s utterance is the signification o the term. Te signification o a sentence is then built out o the signification o its terms, in a compositional ashion. Signification served as the basis or one o the primary divisions o terms recognized by medieval grammarians and logicians, into categorematic and syncategorematic (or significative and consignific ative).A categorematic term is one
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which signifies or has signification on its own, apart rom any other word. Nouns, proper names and verbs are, or the most part, categorematic terms. (Different grammatical orms o the being verbest ‘is’ are an exception). On the other hand, a syncategorematic term is one which does not signiy on its own, but only in conjunction with another word or in the context o a sentence. Now, clearly, these ‘definitions’ are not sufficient to uniquely identiy all words as either categorematic or syncategorematic. a result, most medieval illust rated definitions o categorematic and As syncategorematic terms authors ostensively, andthe different authors highlighted different syncategorematicterms. Logical connectives (‘and’, ‘or’, ‘not’,‘i’) and quantifiers (‘all’, ‘some’, ‘none’) were all recognized as syncategorematic terms, but the list o syncategorematic terms is longer than the list o what are modernly recognized as the so-called ‘logical constants’, and many o them are perhaps surprising. For example, William o Sherwood in hisSyncategoremata (midthirteenth century) 2( 012) discusses the ollowing syncategorematic terms: omnis ‘every’,totum ‘whole’,decem ‘ten’,infinita ‘infinite’,qualislibet ‘o whatever sort’,uterque ‘each’,nullus ‘none’,ni- hil ‘nothing’,neutrum ‘neither’,praeter ‘except’, solus ‘single’,tantum ‘only’, est ‘is’, non ‘not’,necessario ‘necessarily’, contingenter ‘contingently’,incipit ‘begins’,desinit ‘cease’,si ‘i’, nisi ‘i not’,quin ‘without’,et ‘and’ and vel ‘or’. Some o these words, such as infinita, can be used both categoremati cally and syncategorematically, and this act gave rise to certain logical puzzles, or sophismata. I a term has only one signification imposed upon it, then it is univocal; i it has more than one signification imposed on it, then it is equivocal, such as ‘bat’ (the athletic equipment) and ‘bat’ (the flying mammal), in English. Equivocal terms must be distinguished rom each other, in order to avoid the allacy o quaternio terminorum ‘our terms’, a type o equivocation ound in syllogisms. But univocal terms must also be careully attended to, or even though they will always have the same signification no matter the context, these terms can still be used to stand or or reer to different things, depending on the context o the sentence in which they appear. Tis notion o a term ‘standing or’, or reerring to, different objects is called suppositio ‘supposition’. While a (categorematic) term can signiy in isolation rom other terms, it will supposit or things only in the context o a complete sentence, in the same way that syncategorematic terms only have signification in the context o a complete sentence.
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3.2 Supposition Signification gives only an indirect way o reerring to things, via the concepts which are signified. In order to reer directly to things, and not just the concepts they all under, the mechanism o supposition is used. Lambert o Lagny defines the supposition o a term as the ‘acceptance o a term or itsel, or its signified thing, or some suppositum contained under its signified thing, or or more than one suppositum contained under its signified thing’ (1988: 106). For example, the univocal term ‘man’ signifies the concept o man, but it can supposit or the word ‘man’, the concept man, or individual or multiple men, depending on the grammatical context in which it appears. Every author writing on supposition divided supposition into a different number o types, and distinguished them in different ways, but three main divisions were recognized, between simple or natural supposition, material supposition, and ormal or personal supposition. A word that supposits simply stands or what it signifies; or example, in ‘Man is a species’, ‘man’ supposits simply. A word that supposits materially stands or the word itsel; or example, in ‘Man is a three-letter word’, ‘man’ supposits materially. Finally, a word that supposits ormally or personally stands or actual things which all under the concept signified; or example, in ‘A man is running’, ‘man’ supposits personally. Personal supposition is the most important type o supposition, since it allows us to talk about individuals. Both simple and material supposition are not urther subdivided. Personal supposition, the most common type, has many different subtypes. O these, the ones that are most interesting are the types o supposition had by terms which, either by their nature or due to some added word such as a universal quantifier, can apply equally well to more than one thing. Tese two types o supposition are called strong mobile supposition and weak immobile supposition.2 A common term has strong mobile supposition when it is preceded by a universal affirmative or universal negative quantifier (e.g. omnis ‘every’ or nullus ‘no, none’). It has weak immobile supposition ‘when it is interpreted necessarily or more than one suppositum contained under it but not or all, and a descent cannot be made under it’ (Lambert 1988: 112). For example, rom the sentence ‘All men are running’, it is possible to descend to the singular sentences ‘Socrates is running’, ‘Aristotle is running’, ‘Plato is running’, and so on or all (currently existing) men. Similarly, rom the conjunction o the singulars ‘Socrates is running’, ‘Aristotle is running’, ‘Plato is running’, etc., it is possible to ascend to the universal ‘All men are running’. Such ascent and descent is not possible when the universal claim is
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prefixed by a negation, or example‘Not all men are running.’ Likewise, when the subject term has been modified by an exceptive or an exclusive, it is not possible to descend. For example, it is not possible to descend rom ‘Every man except Socrates is running’ to a singular because, or example, ‘Aristotle except Socrates is running’ is ungrammatical. Likewise, rom ‘Only men are running’, we cannot conclude o any individual man that he is running, since the addition o the exclusive term ‘only’ changes the supposit supposition o ‘man’. Te basic mechanism supposition explains how terms in simple present-tensed assertorico sentences. When the tense o a sentence is past or uture, or the sentence has been modified by a modal operator such as ‘necessarily’, ‘is known’, ‘is thought o’, medieval logicians appealed to the notions o ampliationand restriction. Lambert defines restriction as ‘a lessening o the scope o a common term as a consequence o which the common term is interpreted or ewer supposita than its actual supposition requires’ (1988: 134). One way that a common term can be restricted is through the addition o an adjective. For example, we can restrict the supposition o ‘man’ by adding to it the adjective ‘white’; ‘white man’ supposits or ewer things than ‘man’ unmodified. I the modiying word or phrase does not restrict the term’s supposition but rather expands it, then we are dealing with the opposite o restriction, which is called ampliation. Ampliation is an ‘extension o the scope o a common term as a consequence o which the common term can be interpreted or more supposita than its actual supposition requires’ (Lambert 1988: 137). As an example, Lambert offers ‘A man is able to be Antichrist’ (137); in this sentence, ‘man’ is ampliated to stand not only or existing men but or uture men. Ampliation can be caused by the addition o predicates (such as ‘possible’), verbs (‘can’), adverbs (‘potentially’) or participles (‘praised’). Lambert divides ampliation into two types, ampliation ‘by means o supposita’ and ampliation ‘by means o times’. Te ormer is caused by ‘verbs whose corresponding action is related to the subject and said o the subject but is not in the subject – such as ‘can’, ‘is thought’, ‘is praised’ (138). A term ampliated in this way stands or both existent and non-existent things. Ampliation by reason o times is caused by modifiers which ‘cause a term to be extended to all the differences o time’ (138). Examples o this kind o modifier are temporal operators such as ‘always’, modal operators such as ‘necessarily’ and ‘possibly’, and changes in the tense o the verb. Teories o supposition are closely tied to considerations o temporal and alethic modalities (such as ‘necessary’ and ‘possible’) (Uckelman 2013); we cannot go urther into these issues here due to reasons o space.
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4 Teories o consequences Te central notion in logic, both then and now, is the concept o ‘ollowing rom’ or, more technically, ‘logical consequence’. Te question o which propositions ollow rom other propositions, at heart, is what all o logic is about. Tere is a distinct lack o any general theory o logical consequence in Aristotle. Te portion logical consequence is studied in the Aristotelian corpus –othe syllogistic – is but which a raction o the and fielddiscussed and moving rom the study o syllogisms to the study o logical consequence in general is one o the highest accomplishments o the medieval logicians. What do we mean by ‘(logical) consequence’? Modern logic distinguishes different logical relationships between two propositions. Tere is the relationship o implication, which holds between the antecedent and the consequent o a true conditional. Tere is the relationship o entailment, which holds between two propositions when one cannot be true without the other one also being true. And then there is the relationship o inerence or derivation, which is the relationship which arises rom the process o moving rom one proposition (the premise) to another (the conclusion) in an argument. While there are certainly good reasons or distinguishing between these three relationships, these distinctions were not always made in medieval logic. Te Latin word consequentia literally means ‘ollows with’, and it was used indiscriminately to cover implication, inerence and entailment. One consequence o this is that the primary evaluative notion was not validity but ‘goodness’. Whereas a conditional is either true or alse, and not valid or invalid, and an argument is either valid or invalid, and not true or alse, both o these can be bona consequentiae ‘good consequences’ or bene sequiter ‘ollow correctly’. Te word consequentia was introduced into Latin by Boethius as a literal translation o Greekκἀ ύθησις. Some scholars have argued that the roots o theories o logical consequence are to be ound not in the syllogistic but instead in the theory o topical inerences (Green-Pedersen, 1984). As a result, Boethius’s translation o the opics and his commentary on the same were influential. Another important work o his was the short, non-Aristotelian treatise ‘On Hypothetical Syllogisms’. Despite the name, this text does not ocus solely on ‘i . . . then . . .’ statements; rather,Boethius (and others afer him) used ‘hypothetical’ in contrast with ‘categorical’. Any molecular proposition, i.e. one ormed out o other propositions by means o negation, conjunction, disjunction or implication, was considered ‘hypothetical’, and Boethius’s treatise on syllogisms or arguments using such propositions was a first step towards modern propositional logi c.3
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Te study o logical consequence began in earnest in the twelfh century. Te concept proved difficult to properly define and classiy. In the twelfh century, the most sophisticated attempt was produced by Peter Abelard (c. 1079–1142), a brilliant logician who produced not one but two comprehensive theories o logical consequence. Te first, in the Dialectica (c. 1116), ocuses on the treatment o topics and o hypothetical syllogisms, ollowing Boethius and Aristotle. Te other, later, discussion occurs in ais ragment o his commentary on Boethius’s commentary on the opics , which a part o the Logica Ingredientibus (beore 1120). O the two discussions, that in the Dialectica is more complete and clear. Unortunately, it also turns out to be inconsistent Martin ( 2004). Trough the course o the next two centuries, logicians continued to wrestle with the concept o consequence. In the early part o the ourteenth century treatises devoted solely to the notion o consequence begin to appear, with the earliest being written by Walter Burley (c. 1275–1344) and William o Ockham (c. 1285–1347). Tese treatises define different types o consequences (e.g. ormal and material, simple and ‘as o now’, etc.), what it means or a consequence to be good, or or one proposition to ollow rom another, and list rules o inerences which preserve the goodness o a consequence. Tese rules o inerence mirror modern propositional logic very closely. For example, in Burley’sDe puritate artis logicae, which appeared in both longer and shorter versions, the ollowing rules all appear, where P, Q and R are all atomic propositions (Boh 1982: 312–13; Burley 1951):4 • • • • • •
P → Q ¬ (P ∧ ¬Q) P → Q (Q → R) → (P → R) P → Q ( R → P ) → ( R → Q) P → Q, Q → R, . . ., → U P → U P → Q, (P ∧Q) → R P → R P → Q, (Q ∧ R) → S (P ∧ R) → S s
• P → Q ¬Q → ¬P • ¬ (P ∧ Q) ¬P ∨ ¬Q • ¬ (P ∨ Q) ¬P ∧ ¬Q Te ourteenth century saw the gradual codification o two main views on the nature o logical consequence, the English and the continental, with Robert Fland (c. 1350), John o Holland (1360s), Richard Lavenham (d. 1399) and Ralph Strode (d. 1387) as canonical examples on one side o the channel and Jean Buridan (c. 1300–c. 1358), Albert o Saxony (1316–1390) and Marsilius o Inghen (1340–1396) on the other side.
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4.1 Te English tradition Te English tradition (which was taken up in Italy at the end o the ourteenth century and into the fifeenth (Maierù 1983) is characterized by an overtly epistemic definition o (ormal) consequence in terms o the containment o the consequent in the antecedent. For example, Strode gives the ollowing definition: A consequence said to be ormally valid is one o which i it is understood to be as is adequately signified through the antecedent then it is understood to be just as is adequately signified through the consequent. For i someone understands you to be a man then he understands you to be an animal. 1493
Tis echoes ideas ound in Abelard’s views on consequences, which stress a tight connection between the antecedent and consequent, or between the premises and conclusion. It is not sufficient that a consequent be merely ‘accidentally’ truth-preserving; there must be something more that binds the propositions together.5 Such a definition certainly entails necessary truth-preservation, but it goes beyond it. In stressing the epistemic aspects o consequence, Strode, and others in the English tradition, are emphasizing one o the hallmarks o the medieval approach to logic, namely the emphasis on the epistemic context o logic and the idea that logic is an applied science which must be evaluated in the context o its applications.
4.2 Te continental tradition In contrast, the defining marks o validity in the continental tradition are modality and signification. On this side o the channel, Jean Buridan’sractatus de consequentiis (1330s) (Hubien 1976) provides a canonical example. Te first chapter o this treatise is devoted to the definition o consequence. Buridan begins by presenting a general definition which he then revises on the basis o objections and counterexamples. Te first general definition is: Many people say that o two propositions, the one which cannot be true while the other is not true is the antecedent, and the one which cannot not be true while the other is true is the consequent, so that every proposition is antecedent to another proposition when it cannot be true without the other being true. Hubien 1976: 21
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However innocuous this definition might seem, it is problematic or Buridan, and others who ollow suit, or whom the relationships o antecedent and consequent are not between propositions in the modern sense o the term – abstract entities which are necessarily existing – but rather between specific tokens o propositions, spoken, written or mental, which only have truth- values when they exist, and do not otherwise. But proposition tokens have specific properties which interere with this definition. Buridan points out that Every man is running, thereore some man is running
(1)
is a valid consequence, but it does not satisy the definition given, because it is possible or the antecedent to be true without the consequent, i someone ormed ‘Every man is running’ without orming ‘Some man is running’, in which case it would be possible or the ormer to be true without the latter. His second revision is to supplement the definition with the ollowing clause ‘when they are ormed together’ (Hubien 1976: 21), but even this is not sufficient, or consider the ollowing: No proposition is negative, thereore no donkey runs.
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On the second definition, this would be a consequence, since there is no circumstance under which the antecedent is true, so there is no circumstance under which the antecedent is true without the consequent. Te problem with the sentence is that its contrapositive: Some donkey runs, thereore some proposition is negative.
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is not a valid consequence, and Buridan wishes to maintain contraposition as a sound rule o inerence (Hubien 1976: 22). Te final revision that he gives does away with reerence to truth-value altogether, and is defined in terms o signification: Some proposition is antecedent to another which is related to it in such a way that it is impossible or things to be in whatever way the first signifies them to be without their being in whatever way the other signifies them to be, when these propositions are ormed together.6
Te problem with a proposition such as ‘No proposition is negative’ is that it is a sel-reuting proposition; it cannot be ormed without its very ormation making it alse. Sel-reuting propositions cause problems or theories o truth and consequence, and turn up as central players in treatises on insolubilia and sophismata, to which we turn next.
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5 Insolubilia and sophismata Te Sophistici Elenchiwas one o the first o the new Aristotelian works translated in the middle o the twelfh century to gain a wide readership (Dod 1982: 69). Te study o allacies and sophistical reasoning held the same draw in the Middle Ages as it did in ancient Greece and in modern times: it is not sufficient to know how taught viabethe syllogism in the ); in Priorone’ Analytics ordertotoreason win in properly a dispute,(as oneismust also able to recognize when s opponent is reasoning improperly. Tis gave rise to the study o insolubiliae ‘insoluble sentences’ and sophismata ‘sophisms’. Te medieval genre o sophismata literature developed in the twelfh century and was firmly established in both grammatical and logical contexts by the end o that century (Ashworth 1992: 518). In the context o logic, a sophisma or insoluble is a problematic sentence, a sentence whose analysis either leads to an apparent contradiction, or or which two equally plausible analyses can be given, one or its truth and one or its alsity.7 reatises on sophisms generally ollowed a similar ramework:
1. Te sophism is stated, sometimes along with a casus, a hypothesis about how the world is, or extra inormation about how the sophism should be analysed. 2. An argument or its truth and an argument or its alsity are presented. 3. Tere is a claim about the truth- value o the sophisma. 4. Te apparent contradiction is resolved by explaining why the arguments supporting the opposite solution are wrong. Te result is a sentence which has a definite truth- value under the casus (i one is given). Many sophisms dealt with paradoxes that arise rom logical predicates, such as truth, necessity, validity, etc. Medieval logicians recognized the importance o the task o providing non-trivial and non-ad hoc resolutions to these insolubles and sophisms. Te most productive era in the theory o insolubles was rom 1320 to 1350. During this period, many treatises on insolubilia and sophismata were written, and discussions o insoluble sentences appeared in other, nondedicated works, too. Some o the most important authors writing on the topic during this period include Tomas Bradwardine (c. 1295–1349), Richard Kilvington (d. 1361), Roger Swyneshed (d. 1365), William Heytesbury ( c. 1310– 1372), John Wycli (c. 1330–1384) and Peter o Ailly (1351–1420). In the remainder o this section, we discuss (1) the liar paradox and related insolubles,
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(2) sophisms relating to validity and logical consequence, and (3) other types o sophisms and insolubles.
5.1 Te liar paradox Te most amous insoluble is the liar paradox: Tis sentence is alse.
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Te earliest known medieval ormulation o the paradox is in Adam o Balsham’s Ars disserendi (1132). However, Adam ‘says nothing whatever to indicate that he was aware o the very special problems they pose, that they were current topics o philosophical discussion in his day, or how one might go about trying to answer those questions’ (Spade 1987: 25). It was not until the later part o the twelfh century that the problematic aspects o the liar sentence (and related sentences) were taken up in earnest. Over the course o the next two centuries, many attempts to solve the paradox were provided. Tese solutions can be divided into the ollowing five amilies: (1) classification under the allacy
secundum quid et simpliciter; (2) transcasus theories, (3) distinguishing between the actus exercitus ‘exercised act’ and the actus significatus ‘signified act’; (4) restrictio theories; and (5) cassatio theories. Secundum quid et simpliciter
Tis is Aristotle’s solution. In Chapter 25 o theSophistical Reutations, Aristotle makes a distinction between ‘arguments which depend upon an expression that is valid o a particular thing, or in a particular respect, or place, or manner, or relation, and not valid absolutely’ (Aristotle 2013), that is, between expressions which are valid secundum quid ‘according to something’ and those which are valid simpliciter ‘simply’ (or ‘absolutely’). He goes on to say: Is it possible or the same man at the same time to be a keeper and a breaker o his oath? . . . I a man keeps his oath in this particular instance or in this particular respect, is he bound also to be a keeper o oaths absolutely, but he who swears that he will break his oath, and then breaks it, keeps this particular oath only; he is not a keeper o his oath . . . Te argument is similar, also, as regards the problem whether the same man can at the same time say what is both alse and true: but it appears to be a troublesome question because it is not easy to see in which o the two connexions the word ‘absolutely’ is to be rendered – with ‘true’ or with ‘alse’. Tere is, however,nothing to prevent it rom being alse absolutely, though
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true in some particular respect or relation, i.e. being true in some things, though not ‘true’ absolutely. Aristotle 2013
How this solves the paradox is not entirely clear; on this view, the liar sentence can apparently be solved both ways. Many medieval logicians who adopted the Aristotelian reply argued that the liar is alse simpliciter, and true secundum quid; however, it was lef unspecified with respect to what quid it is true. ranscasus
Te Latin word transcasus has no straightorward translation into English. It is a literal translation o Greek μεταπτωσις ‘a change, transerring’. Conceptually, it is related to the Stoic notion o μεταπιπτωντα, rom the same root, which are propositions whose truth-value changes over time (Spade and Read 2009: 37). In transcasus, it is not that the truth-value o the liar sentence changes, but rather, what the sentence reers to (and hence how its truth-value should be evaluated). On such solutions, when someone says ‘I am speaking a alsehood’, the sentence is not sel-reerential but instead reers to what that person said immediately prior. I he did not say anything beore, then the liar sentence is just alse. Te actus exercitus and the actus significatus
Tis solution takes advantage o the act that the liar sentence (as usually ormulated in medieval treatises) involves assertion: I say, ‘I am saying something alse’, or Plato says, ‘Plato is saying something alse’, or similar. When such an assertion is made, it is possible to distinguish between what the speaker says he is doing (signified act) and what he is actually doing (exercised act). Tis view, which is not well understood, is espoused by Johannes Duns Scotus in his Questiones super libro elenchorum(c. 1295), who says that the exercised act o the liar is ‘speaking the truth’ and the signified act o the liar is ‘speaking a alsehood’ (Duns Scotus 1958). Because the liar sentence expresses something which is not true, it is alse. Restrictio
Restriction solutions are the most straightorward: by restricting the allowed grammatical/syntactic orms to disbar sel-reerential sentences, it is possible to rule the liar paradox as without truth- value because it is sel-reerential
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(ungrammatical). Spade (1975) discusses seventy-one different texts dealing with the liar paradox. Fourteen o these texts espouse some type o restriction theory, either explicitly or implicitly. Tese include a number o mid- to lateourteenth-century anonymous treatises, as well as treatises by well-known logicians such as Walter Burley, in hisInsolubilia (beore 1320) (31) and William o Ockham, in his Summa logicae (1324–1327) (Ockham 1974) and his ractatus 1328)a (Ockham 1979). ranging rom very weak, super libros elenchorum Restriction solutions(beore exist across broad spectrum, orbidding only a small amount o sel-reerence, to very strong, orbidding all sel-reerence. On such strong restriction theories, not only does it turn out that the liar has no truth-value, but so also such insolubles as the linked liars: Plato: What Socrates says is alse Socrates: What Plato says is alse
(5) (6)
as well as seemingly non-paradoxical sentences which just happen to be selreerential, such as: Tis sentence has five words.
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Cassatio
Te Latin word cassatio means to make null or void, or to cancel. On the view o the cassators, when you are uttering an insoluble, you are saying nothing; the paradoxicality o the sentence cancels out any meaning it might have had. Tereore an insoluble like the liar has the same truth- value as the empty utterance: none. Tis solution was avoured in the early period, and died out by the 1220s, though it continued to be mentioned in later catalogues o types o solutions (Spade and Read 2009, §2.5). In addition to the liar and the linked liars, other liar-like insolubles were also considered. For example, suppose that Plato promises to give everyone who tells the truth a penny. Socrates then announces, ‘You won’t give me a penny’. Or similarly, Plato is guarding a bridge, and will let only those who tell the truth cross; anyone who tells a lie will be thrown into the water. Socrates approaches and says, ‘You will throw me rom the bridge.’ Both o these present the same problems or analysis as the liar paradox, though it is clear that they cannot be solved in similar ways (or example, restriction strategies make no sense here, since there is no sel- or cross-reerence).8
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5.2 Paradoxes o validity Earlier we mentioned some o the problems that arise in the analysis o propositions such as ‘No proposition is negative’. In Chapter 8 o hisSophismata, Buridan (2001) considers the related inerences ‘Every proposition is affirmative, thereore no proposition is negative’ and ‘No proposition is negative, thereore some proposition is negative’. Tese inerences are problematic, because, on the one hand, the antecedent is either the contrary or the contradictory o the consequent, so any time it is true the consequent will have to be alse, according to the rules in the Square o Opposition; and on the other hand, ‘No proposition is negative’ is itsel a negative proposition, and any time that it exists, some negative proposition will exist, and thus some proposition will be negative. Nevertheless, it is not impossible that there be no negative propositions; as Buridan points out, ‘Every proposition is affirmative’ would be true i God annihilated all negatives, and then the consequent [o the first inerence] would not be true, or it would not be’ (2001: 953). Tis analysis leads Buridan to make an interesting distinction: He concludes that such a proposition ‘is possible, although it cannot be true’ (956); that is, he distinguishes between being ‘possible’ and being ‘possibly-true’; a proposition can be one without being the other. Other paradoxes o validity include sel-reerential propositions,but unlike the liar sentence they involve logical predicates other than truth. For example,PseudoScotus offers a counterexample to Buridan’s definition o logical con sequence in terms o necessary truth-preservation (c. §4.2 above) (Yrjönsuuri 2001): God exists, hence this argument is invalid.
(8)
Under standard medieval metaphysical and ontological assumptions, ‘God exists’ is a necessary truth.9 Suppose, then that (8) is valid. Te antecedent is not only true, but necessarily true. However, i the argument is valid, then the consequent is alse, since it asserts that the argument is invalid. But then the argument is not truth-preserving, and so cannot be valid. But i the argument is invalid, then it is necessarily invalid, and as a result the consequent is necessary. But a necessary truth ollowing rom a necessary truth is necessarily a valid inerence, and thus (8) is valid. Similar paradoxes can be levelled against Buridan’s final definition, in terms o signification, too. For example, the definition is adequate or the counterexample that it was designed to obviate, but would have problems dealing with the ollowing inerence:
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Te consequent o (9) does not signiy as is the case, thereore, the consequent o (9) does not signiy as is the case.
(9)
While this precise example is not ound, similar ideas are treated by Roger Swyneshed in his Insolubilia (c. 1330–1335) (Spade 1979).
5.3 Other classes o insolubilia and sophismata In this section we briefly catalogue other common types o insolubilia and sophismata. 1. Sophisms which arise rom exponibilia ‘exponible [terms]’. An exponible term is one whose analysis requires breaking the sentence in which it appears down into a collection o sentences, each o which are simpler in orm. A common example o a pair o exponible terms are incipit ‘be gins’ and desinit ‘ceases’. Tere are two ways that a sentence such as ‘Socrates begins to be white’ can be analysed: Socrates is not white at time t, and tis the last moment at which he is not white.
(10)
Socrates is not white at time t, and tris the first moment at which he is white.
(11)
Conusing the two ways that such sentences can be expounded can result in sophisms. Such sophisms are discussed by William Heytesbury in his Regulae solvendi sophismata (1335) (Heytesbury 1979; Wilson 1956) and Richard Kilvington in hisSophismata (beore 1325) (Kilvington 1990). 2. Sophisms which arise rom conusing the syncategorematic and categorematic uses o terms. Te most common example is infinita est finita, where infinita can be interpreted either categorematically or syncategorematically. Tis sentence is difficult to translate into English without losing the ambiguity; when infinita is used categorematically, it is taken as a substantive noun, ‘the infinite’, and then the sentence says that ‘the infinite is finite’, which is alse. But syncategorematically, it is taken adjectivally, and means that ‘infinite are the finite’, i.e. ‘there are infinitely many finite things’, which is true, or example numbers. Illicit shifs between the categorematic reading and the syncategorematic reading can lead to paralogisms and sophisms. Heytesbury discusses this in his treatise De sensu composito et diviso(beore 1335) (Heytesbury 1988).
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3. Sophisms which arise rom inelicities o presupposition and supposition. A typical example o these is ‘Socrates promises Plato a horse’ (or in some cases, a penny), and yet or any given horse (or penny), it is not the case that Socrates has promised Plato this horse. 4. Sophisms which arise rom the re- imposition o terms, stipulating that they signiy things other than their ordinary signification. We see examples o these in the next section. Finally, we would be remiss in not mentioning (5) sophisms which illustrate that the more things change, the more things stay the same, or, more precisely, that medieval humour isn’t all that different rom modern humour, and that is the class o sophisms whose conclusion is u es asinus ‘You are an ass’. One example o such is the paralogism: ‘Tis donkey is yours, this donkey is a ather, thereore this donkey is your ather’, and i your ather is a donkey, then you are one as well.
6 Obligational disputations Te final area o medieval logic that we cover in this chapter is the most peculiar and the most unamiliar. While theories o meaning and reerence, systems o logical consequence, and the study o paradoxes and sophistical reasoning are all part and parcel o the modern study o logic and philosophy o language, there is no such counterpart or the uniquely medieval genre o disputationes de obligationibus. Te earliest treatises on these disputations are anonymous, and date rom the first decades o the thirteenth century. In the ollowing two centuries, scores o treatises on obligations were written, including ones by William o Sherwood, Nicholas o Paris fl(. 1250), Walter Burley, Roger Swyneshed, Richard Kilvington, William o Ockham, Albert oSaxony, John o Wesel (1340/50s), Robert Fland, John o Holland, Richard Brinkley fl(. 1365–1370), Richard Lavenham, Ralph Strode, Peter o Ailly, Peter o Candia (late ourteenth century), Peter o Mantua (d. 1399), Paul o Venicec(. 1369–1429) and Paul o Pergola (d. 1455). So what are these mysterious disputations, and why are they mysterious? An obligatio is a dispute between two players, an Opponent and a Respondent. Te Opponent puts orward propositions (one at a time, or in collections, depending on different authors’ rules), and the Respondent is obliged (hence the name) to ollow certain rules in his responses to these propositions. Tese rules depend on the type o disputation; we give an example o one type below in §6.1 (which the reader can consult now to have a sense o what we are talking about).
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Early authors distinguish six types, or species, o obligationes: (1) positio ‘positing’, (2)depositio ‘de-positing’ (a type o denial), (3) dubitatio ‘doubting’, (4) petitio ‘petition’, (5)impositio ‘imposition’ and (6) sit verum ‘let it be true’. Later authors argued that some o these types could be derived rom the others and so reduced the number o species, generally to three (Uckelman 2012, §4). Tey are mysterious because their background and their purpose is unclear. Early texts allude to Aristotle’s threeold division o disputations Book(Burley VII I, 1988). Chapter 4 o the opics (c. e.g. Walter Burley’s De obligationibus (c. in 1302) Tere are indications in other texts that show that the medieval authors were interested in developing obligationes in the tradition o disputations as described in the opics – not only in motivating the genre but also in discussions concerning what type o disputations obligationes are (Uckelman 2012, §2). Modern scholars have advanced many hypotheses about the purpose o these disputations. In our opinion, no single answer is going to tell the whole story. It is clear – certain texts tell us so explicitly (de Rijk 1975) – that obligationes were used as training exercises or students. Tat there is a close connection between obligationes and insolubilia-literature is also clear given the use o obligationes -language in treatises on insolubles and sophisms (Kilvington 1990; Martin 2001). While the idea that obligations were developed as a type o counteractual reasoning is not in general tenable, it can be justified in some specific contexts (Uckelman 2015). Te general procedure ollowed in the disputations did not vary drastically rom author to author or type to type. Te Respondent had three (in some cases, our) possible responses: concede, deny or doubt (some authors also allowed him to draw distinctions in the case o ambiguous propositions).Which response was the correct response depended, in part, on whether the proposition was relevant (or pertinent) or irrelevant (or impertinent). In the tradition o Walter Burley, which came to be termed theresponsio vetus‘the old response’, a proposition was defined as relevant i it, or its negation, ollowed rom the conjunction o all the propositions conceded along with the negations o all denied. On this definition, the set o relevant sentences potentially changed with each step othe disputation. Such a definition can also be ound in the works o William o Sherwood, Ralph Strode and Peter o Candia. Tis dynamic conception o relevance resulted in a number o consequences that later authors, particularly Roger Swyneshed, ound problematic. Swyneshed, in what came to be termed theresponsio nova ‘new response’, redefined relevance in hisObligationes(1330×1335) (Spade 1977) into a static notion, where a proposition is relevant i it or its negation ollows rom the positum (the first proposition o the disputation), and is irrelevant otherwise. It is clear that on this definition, whether a proposition is relevant or not does not
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change with the course o the disputation. Swyneshed was ollowed in this redefinition by Robert Fland, Richard Lavenham and John o Wesel, among others. Regardless o which definition o relevance was used, the ollowing general rules were accepted by everyone: ●
●
A relevant proposition should be conceded i it ollows, and denied i its negation ollows. An irrelevant proposition should be conceded i it is (known to be) true, denied i it is (known to be) alse, and doubted i it is neither (known to be) true nor (known to be) alse.
6.1 Positio
Positio is the crown jewel o the obligationes regalia. It is the most prominently discussed, by both medieval and modern authors. Positio can be divided into multiple types. Te first division is into possible and impossible positio; both divisions are urther divided as to whether the proposition is simple or complex, and then urther as to whether the complex propositions are ormed by conjunction (‘conjoined positio’) or disjunction (‘indeterminate positio’). In any o these types o positio, it is also possible that a urther stipulation is added, in which case the positio is called ‘dependent’ (Burley 1988: 378). Most texts ocus their discussion on possible positio.
Figure 3.1 A simple example
Te rules given above are, in slightly simplified orm, Burley’s rules or positio. Tese rules exhaustively cover all o the possibilities that the Respondent may ace in the course o the disputation; how these rules play out in the context o actual disputations is made clear in Burley’s examples, which are typical o thirteenth-century developments. Due to issues o space, we cannot ollow this up with a discussion o the later ourteenth-century developments, but instead direct the reader to Uckelman (2012), §§4.2, 4.3. We consider an examplepositio which appears, with slight variation, in many thirteenth-century treatises. It is airly simple but illustrates Burley’s rules nicely.
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Suppose that ϕ does not imply ¬ψ and ϕ is known to be alse; or example, let ϕ be ‘Te capital o England is Paris’ and ψ is ‘It is raining’. Sinceϕ is satisfiable (because i it were not, then it would imply ¬ψ), the Respondent should admit it when it is put orward as a positum. In the second round, the Opponent asserts ¬ϕ ∨ ψ. Now, eitherϕ implies ψ, in which case the proposition ollows rom the positum and hence the Respondent should concede it, or ψ is independent o ϕ, and proposition is irrelevant. Intrue that irrelevant case, we know that sinceshould ϕ is alse, ¬ϕ ishence true, the so the disjunction is true, and propositions be conceded. But then, ϕ ollows rom the positum along with something correctly conceded, and hence when the Opponent asserts ψ, the Respondent must concede it too. Tis example shows how, given a positum which is alse, but not contradictory, the Opponent can orce the Respondent to concede any other proposition consistent with it. Te act that this is possible is one o Swyneshed’s primary motivations or revising the standard rules. Further ormal properties, ollowing rom the assumption o a consistentpositum (which is the definition o possible positio), include that no disputation requires the Respondent to concedeϕ in one round and to concede¬ϕ in another round (or to concedeϕ in one round and to deny it in another); the set o ormulas conceded, along with the negations o those denied, will always be a consistent set; yet, ti may be that the Respondent has to give different answers to the same propositions put orward at different times.
6.2 Depositio
Depositio is just like positio, except that the Respondent is obliged to deny or reject the initial proposition (the depositum). A depositio with depositum ϕ will be completely symmetric to a positio with ¬ϕ as the positum. Nevertheless, early treatises on obligationes, such as that by Nicholas o Paris which dates rom c. 1230–1250, still treat depositio at some length. 6.3 Dubitatio In dubitatio, the Respondent must doubt the statement that the Opponent puts orward (called the dubitatum). While dubitatio was discussed in thirteenthcentury texts, ofen at some length, later authors (both later medieval and modern authors) call dubitatio a trivial variant o positio, and thus spend little time discussing it. For example, Paul o Venice (1988) reduces dubitatio to positio (in much the same way that he, and others, reducesdepositio to dubitatio);
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Swyneshed, Lavenham (Spade 1978), John o Wesel (1996), Richard Brinkley (Spade and Wilson, 1995) and John o Holland (1985) do not mention dubitatio at all. However, such a trivializing view odubitatio ails to recognize the higherorder aspects o the disputation, the mixing o both knowledge and truth, which result in a significantly more difficult type o disputation. Just as positio is only interesting when the positum is alse, dubitatio is only interesting when the truthvalueRybalko, o the dubitatum is known (whether is truethe or complexity alse) (Uckelman, Maat and orthcoming ; Uckelman 2012).it Tus, o this type o disputation partly arises rom the interaction between knowledge, truth and the obligations o the Respondent, as the Respondent in many cases is required to respond dubio ‘I doubt it’ to propositions that he actually knows. A second cause o complexity in dubitatio is the act that the rules, unlike those or positio, are not deterministic. For example, Nicholas o Paris’s rules or dubitatio (Braakhuis 1998: 72–6) include the ollowing: ●
●
●
●
Just as in positio a positum put orward in the orm o the positum, and everything convertible to it in the time o positing is to be conceded and its opposite and things convertible with it is to be denied and just as in
depositio a depositum put orward in the orm o the depositum, with its convertibles, must be denied and its opposite with things convertible with it must be conceded; so in dubitatio or a dubitatum put orward in the orm o dubitatum and or its convertibles and moreover or the opposite o the dubitatum with its convertibles must be answered ‘prove!’ (223). For everything irrelevant to the dubitatum the response must be according to its quality. For everything antecedent to the dubitatum the response must be ‘alse’ or ‘prove!’ and never ‘true’ (224). For everything consequent to the dubitatum it is possible to reply ‘it is true’ or ‘prove’ and never ‘it is alse’ (224).
Whereas there is always a unique correct response or Respondent in positio (in both the responsio antiqua and nova), here, the rules give Respondent a range o choices. Tis non-determinacy means that there is a plurality o ways that Respondent may act, and still be disputing according to the rules, a eature which no other version o obligatio has. However, this eature odubitatio seems not to have been noticed by later authors who insisted that dubitatio could be reduced to positio. Nicholas’s dubitatio has similar ormal properties to positio. Provided that the dubitatum is neither a contradiction nor a tautology, it can be proved that the
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Respondent can win the disputation playing by Nicholas’s rules ordubitatio: that is, there is never any case where he will be orced either to concede or to deny the dubitatum (Uckelman 2011, Teorem 24).
6.4 Impositio/Institutio/Appellatio Te obligation involved in impositio, also called institutio‘institution’ orappellatio ‘appellation’, unctions in a relevantly different manner rom the obligation in positio, depositio or dubitatio. Whereas in these latter three, the Respondent’s obligation involves how he is to respond to the obligatum, impositio involves the redefinition (re-imposition) o certain terms or phrases.10 Impositio can take place in conjunction with any o positio, depositio and dubitatio; that is, once a new imposition is introduced, then the Respondent may also urther be obliged to concede, deny, or doubt the initialobligatum o the disputation. Sometimes the imposition is simple and straightorward: I impose that ‘a’ signifies precisely that God exists . . . I impose that this term ‘man’ may be converted with this word ‘donkey’, or I impose that this proposition ‘God exists’ signifies precisely that man is donkey. Spade 1978, ¶¶2, 21
In the first example, ‘a’ is being instituted as the name o a proposition that signifies that God exists; likewise in the third example, the phrase ‘God exists’ is instituted as the name o a proposition signiying that man is donkey; thus any time that ‘God exists’ is asserted in a disputation, it must be understood as meaning ‘Man is donkey’. In the second example, the institution is not at the level o propositions but at the level o words; it changes the meaning o the term ‘man’ so that it no longer means ‘man’ but instead means ‘donkey’. Simple impositions like these are relatively easy; the only skill they require beyond the skills needed or positio is the skill to remember the new imposition o the term or proposition. However, much more complicated examples can be provided, such as the ollowing (also due to Lavenham): I impose that in every alse proposition in which ‘a’ is put down that it signifies only ‘man’ and that in every true proposition in which ‘a’ is put down that it signifies only ‘donkey’, and that in every doubtul proposition in which ‘a’ is put down that it signifies alternately with this disjunction ‘man or non man’. Spade 1978, ¶24
Now suppose that the proposition ‘Man is a’ is put orward. Te proposition is either true, alse or o unknown truth- value (doubtul). Suppose it is true. Ten,
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it means ‘Man is a donkey’, which is impossible; hence, contrary to supposition, the proposition is in act alse. But i it is alse, it means ‘Man is man’ – but this is true! Tus, i it cannot be true or alse, then it must be doubtul. But i it is doubtul, then it means ‘Man is man or not man’, which is true, and hence not doubtul! No matter which assumption is made about the value o the proposition, the Respondent is led into contradiction.
6.5 Petitio In petitio, the Opponent asks (petitions) the Respondent to respond in a certain way, or example by conceding or denying the initial proposition.Petitio is rarely treated at any length, because, as a number o authors (Nicholas o Paris (Braakhuis 1998: 183), Marsilius o Inghen (1489), Peter o Mantua (Strobino 2009), Paul o Venice (1988: 38–9) argue, petitio can be reduced to positio. Tus, rom the disputational point o view, there are little more than cosmetic differences between positio and petitio.
6.6 Rei veritas / sit verum Te sixth type, sit verum or rei veritas ‘the truth o things’, is rarely discussed by the medieval authors, and sometimes not even explicitly defined. As a result, it is difficult to give a precise explanation or characterization o this type. Many discussions o sit verum ocus on epistemic aspects o the disputation (Stump 1982: 320). For instance, Paul o Venice gives the ollowing example osit verum: ‘Let it be true that you know that you are replying’ (1988: 45). Nicholas o Paris also gives an example o a rei veritas that cannot be sustained which is couched in epistemic terms (Braakhuis 1998: 166, 233). However, one cannot generalize too broadly rom this, as other examples show more in common with counteractual reasoning than epistemic reasoning (Uckelman 2015).
7 Conclusion In this chapter we have given an overview o medieval logic which we hope is sufficient to show that the Middle Ages were, in the history o logic, not a period o darkness and crudeness, but rather one, particularly during the thirteenth and ourteenth centuries, o new insights into the nature o language and inerence.
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Tese insights, building on Aristotle’s Organon but going ar beyond him, provided a oundation or the ormal education o centuries o young men, regardless o whether they intended to continue their studies in philosophy. Tere is much that we have not been able to address in great detail in this chapter, and still more that we have not touched on at all. (For example, we have almost completely omitted developments o the syllogistic, both assertoric and modal, as well as theNevertheless, interesting and deal o with uture contingents.) whatcomplex we havequestion shown is o thathow the to impact logic and the study o it in the Middle Ages thereore cannot be dismissed out o hand. Modern study o medieval logic is still, to a large extent, in its early stages, and decades to come will continue to prove the importance and sophistication o the medieval logicians.
Notes 1 Dod notes that ‘how and where these translations .. . were ound is not known’ (1982: 46), nor indeed how and where they were lost in the first place. 2 Te mobility reerred to in their names is a reerence to the possibility (or lack thereo) o ‘descending to’ or ‘ascending rom’ singulars; see examples below. 3 It is, however, not modern propositional logic, since the notion o implication or conditional which he uses is neither the modern material conditional nor strict implication, but is instead based on the idea o subjects and predicates either containing each other or being repugnant to (contradictory with) each other. As a result, the logical theory in ‘On Hypothetical Syllogisms’ validates many theorems which are not acceptable with either material or strict implication. 4 Note that the medieval logiciansdid not use symbolic notation (with the exception o using letters to stand or arbitrary terms in discussions o syllogistic). Nevertheless, these are accurate and aithul representations o the rules which are ound in Burley. 5 In this respect, the medieval approach to logic shares many methodological characteristics with logicians who pursue relevance logic projects nowadays. 6 Buridan is not entirely content with this definition either,but this has to do with his theory o truth, rather than any problem with the definition itsel, and we do not have the space to go into these problems here. But c. Hubien1976) ( and Klima (2004). 7 In this section, we group insolubilia and sophismata in our discussion, even though historians o logic will sometimes try to biurcate the two (c. e.g Spade and Read 2009; Pironet and Spruyt 2011). Both o the terms are somewhat wider in scope than modern ‘paradox’, which implies some sort o logical contradiction. Not all insolubles are in act unsolvable; rather, they are so namedbecause they are difficult to solve. And not all sophisms involve the use o sophistical (i.e. allacious) reasoning.
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8 Both o these examples are adapted rom Peter o Ailly’sInsolubilia (Ailly 1980). 9 In act, when it shows up in logical examples, it is almost always being used asa ‘generic’ logical truth, rather likep V ¬p. Nothing important turns on the act that this proposition is about God. 10 For example, the anonymous author oObligationes Parisiensesnotes that ‘Institutio is divided into certain institutio, and uncertain or obscure institutio, or example i the name ‘Marcus’ is fixed that it might be a name o Socrates or Plato, but you would not know o which’ (de Rijk 1975: 28) and Lavenham definesimpositio as an ‘obligation by means o which a term or proposition is assigned a [new] signification’.
Reerences Ailly, Peter o (1980), Peter o Ailly: Concepts and Insolubles, ed. and trans. Paul V. Spade , Berlin: Springer. Aristotle (2013), On Sophistical Reutations, trans. W.A. Pickard-Cambridge , Adelaid: University o Adelaide Library. Web edition, https://ebooks.adelaide.edu.au/a/ aristotle/sophistical/. Ashworth, E. Jennier (1992), ‘New light on medieval philosophy: Te Sophismata o Richard Kilvington’, Dialogue, 31:517–21. Boh, Ivan (1982), ‘Consequences’, in Norman Kretzmann,Anthony Kenny and Jan Pinborg, eds, Cambridge History o Later Medieval Philosophy, pp. 300–15, Cambridge: Cambridge University Press. Braakhuis, H.A.G (1998), ‘Obligations in early thirteenth century Paris: the Obligationes o Nicholas o Paris (?)’, Vivarium, 36(2):152–233. Buridan, Jean (2001), ‘Sophismata’, trans. Gyula Klima, in Summulae de Dialectica, New Haven: Yale University Press. Burley, Walter (1951), De Puritate Artis Logicae, ed. P. Boehner,St Bonaventure, NY: Franciscan Institute; E. Nauwelaerts. Burley, Walter (1988), Obligations(selections), in Norman Kretzmann and Eleonore Stump , eds, Te Cambridge ranslations o MedievalPhilosophical exts, vol. 1: Logic and the Philosophy o Language, pp. 369–412 , Cambridge: Cambridge University Press . de Rijk, L.M. (1975), ‘Some thirteenth century tracts on the game o obligation II’, Vivarium, 13(1):22–54. Dod, Bernard G. (1982), A ‘ ristoteles latinus’, in Norman Kretzmann,Anthony Kenny, and Jan Pinborg, eds, Cambridge History o Later Medieval Philosophy, pp. 45–79, Cambridge: Cambridge University Press. Duns Scotus, John (1958). ‘Quaestiones super libro elenchorum’, in Luke Wadding, ed., Opera Omnia, vol. 1, pp. 268–9, Hildesheim: Georg Olms. Green-Pedersen, Neils Jørgen (1984), Te radition o the opics in the Middle Ages. Te Commentaries on Aristotle’s and Boethius’s ‘opics’ , Investigations in Logic, Ontology, and the Philosophy o Language. Munich: Philosophia Verlag.
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Heytesbury, William (1979), On Insoluble Sentences:Chapter One o His Rules or Solving Sophisms, ed. and trans Paul V. Spade , oronto: Pontifical Institute o Medieval Studies. Heytesbury, William (1988), ‘Te compounded and divided senses’, in N. Kretzmann and E. Stump, eds and trans., Cambridge ranslations o Medieval Philosophical exts, vol. 1: Logic and the Philosophy o Language, pp. 413–34, Cambridge: Cambridge University Press. Holland, John o (1985), Four racts on Logic, ed. E.P. Bos,Nijmegen: Ingenium Publishers. Hubien, Hubert (ed.) (1976), Iohannis Buridani: ractatus de Consequentiis, vol. XVI o Philosophes Médiévaux, Louvain: Publications Universitaires. Kilvington, Richard (1990), Te Sophismata o Richard Kilvington, ed. N. Kretzmann and B.E. Kretzmann, Cambridge: Cambridge University Press. Klima, Gyula (2004), ‘Consequences o a close, token-based semantics: Te case o John Buridan’, History and Philosophy o Logic, 25:95–110. Laigny Lambert o (o Auxerre) (1988), ‘Properties o terms’, in N. Kretzmann and E. Stump, eds and trans., Cambridge ranslations o Medieval Philosophical exts, vol. 1: Logic and the Philosophy o Language, pp. 102–62, Cambridge: Cambridge University Press. Maierù, Alonso (1983), English Logic in Italy in the 14th and 15th Centuries , Naples: Bibliopolis. Marsilius o Inghen (1489), Obligationes, Paris: Erroneously published under the name Pierre d’Ailly. Martin, Christopher J. (2001), ‘Obligations and liars’, in M. Yrjönsuuri, ed., Medieval Formal Logic, pp. 63–94, Dordrecht: Kluwer Academic Publishers. Martin, Christopher J. (2004), ‘Logic’, in J. Brower and K. Giloy , eds, Cambridge Companion to Abelard, pp. 158–99, Cambridge: Cambridge University Press. Ockham, William o (1974), ‘Summa logicae’, in Gedeon Gál et al., eds, Opera Philosophica, vol. I. St Bonaventure, NY: Franciscan Institute. Ockham, William o (1979), Expositio Super Libros Elenchorum, vol. 3 o Opera Philosophica, ed. F. Del Punta, St Bonaventure, NY: Franciscan Institute. Pironet, Fabienne and Joke Spruyt (2011), ‘Sophismata’, in E.N. Zalta, ed.,Stanord Encyclopedia o Philosophy, Winter edition, Stanord: Stanord University Press. Roure, Marie-Louise Roure (1970), ‘La problématique des propositions insolubles au XIII e siècle et au début du XIVe, suivie de l’édition des traités de W. Shyreswood, W. Burleigh et T. Bradwardine. Archives d’Histoire Doctrinale et Littérature du Moyen Âge, 37:262–84. Sherwood, William o (1966), Introduction to Logic, trans. Norman Ktetzmann, Minneapolis: University o Minnesota Press. Sherwood, William o (2012), Syncategoremata, ed. and trans C. Kann and R. Kirchhoff, Hamburg: Meiner. Spade, Paul V. 1975), ( ‘Te mediaeval liar: A catalogue o the Insolubilia-literature’, Subsidia Mediaevalia, 5.
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Spade, Paul V. 1977), ( ‘Roger Swyneshed’s Obligationes: Edition and comments’, Archives d’Histoire Doctrinale et Littéraire du Moyen Âge, 44:243–85. Spade, Paul V. 1( 978), ‘Richard Lavenham’sObligationes: Edition and comments’, Rivista Critica di Storia della Filosofia, 33:224–41. Spade, Paul V. 1979), ( ‘Roger Swyneshed’s insolubilia: edition and comments’, Archives d’histoire doctrinale et littéraire du Moyen Âge, 46:177–220. Spade, Paul V. 1( 987), ‘Five early theories in the medieval Insolubilia-literature’,
Vivarium, 25:24–46. Spade, Paul V. and Stephen Read 2009), ( ‘Insolubles’, in E.N. Zalta, ed., Stanord Encyclopedia o Philosophy. Winter edition, Stanord: Stanord University Press, http://plato.stanord.edu/archives/win2009/entries/insolubles/. Spade, Paul V. and Gordon A. Wilson, eds (1995), ‘Richard Brinkley’s Obligationes: A Late Fourteenth Century reatise on the Logic o Disputation ’, vol. 43 oBeiträge zur Geschichte der Philosophie und Teologie des Mittelalters, neue Folge . Muünster: Aschendorff. Strobino, Riccardo (2009), ‘Concedere, negare, dubitare: Peter o Mantua’s reatise on Obligations’, PhD thesis, Scuola Normale Superiore, Pisa. Strodus, Radolphus (1493), Consequentiae cum Commento Alexandri Sermonetae et Declarationibus Getnai de Tienis, Venice: Bonetus Locatellus. Stump, Eleonore (1982), ‘Obligations: From the beginning to the early ourteenth century’, Norman Kretzmann , Anthony Kenny Cambridge: and Jan Pinborg , eds, Cambridge History oinLater Medieval Philosophy , pp. 315–34, Cambridge University Press. Uckelman, Sara L. (2011), ‘Deceit and indeeasible knowledge: Te case o Dubitatio’, Journal o Applied Non-Classical Logics, 21, nos. 3/4:503–19. Uckelman, Sara L. (2012), ‘Interactive logic in the Middle Ages’, Logic and Logical Philosophy, 21(3):439–71. Uckelman, Sara L. (2013), A ‘ quantified temporal logic or ampliation and restriction ’, Vivarium, 51(1):485–510. Uckelman, Sara L. (2015), ‘Sit Verum Obligationesand counteractual reasoning’, Vivarium, 53(1):90–113. Uckelman, Sara L., Jaap Maat and Katherina Rybalko (orthcoming), ‘Te art o doubting in Obligationes Parisienses’, in C. Kann, C. Rode and S.L. Uckelman, eds, Modern Views o Medieval Logic, Leuven: Peeters. Venice, Paul o (1988), Logica Magna Part II Fascicule 8, ed. E.J. Ashworth, Oxord: Oxord University Press. Wesel, John o (1996), ‘Tree questions by John o Wesel onobligationes and insolubilia’, ed. with an introduction and notes by Paul Vincent Spade , http://pvspade.com/ Logic/docs/wesel.pd. Wilson, Curtis (1956), William Heytesbury:Medieval Logic and the Rise o Mathematical Physics, Madison: University o Wisconsin Press. Yrjönsuuri, M. (ed.) (2001), Medieval Formal Logic. Dordrecht: Kluwer Academic Publishers.
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Te Early Modern Period
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Leibniz Jaap Maat
1 Introduction Gottried Wilhelm Leibniz (1646–1716) was a polymath who made significant contributions to a number o disciplines. In the history o mathematics, Leibniz is primarily remembered as one o the inventors o the infinitesimal calculus. In the history o philosophy, he occupies a canonical place as one o the great seventeenth-century rationalists. In the history o logic, his importance is uncontested, although the evaluation o his achievements has differed considerably rom one historian o logic to another. In the present chapter, a necessarily somewhat sketchy overview o Leibniz’s work on logic is provided. Afer considering his place in the history o logic, his efforts to systematize syllogistics are discussed. Next, a central idea that guided his work is described, namely the ideal o a philosophical language based upon an alphabet o human thoughts. Finally, several projects closely related to this ideal are examined in some detail: the arithmetization o logical inerence, the creation o logical calculi, and rational grammar.
2 Leibniz as a logician In their classic history o logic, the Kneales called Leibniz ‘one o the greatest logicians o all time’, while criticizing his work so severely that they admitted this epithet might seem surprising (Kneale and Kneale 1962: 336). In their view, Leibniz had gone wrong on crucial points, but his greatness as a logician was apparent in that he was the first to devise an abstract calculus that was open to several interpretations. Te Kneales were echoing an evaluation o Leibniz’s work first put orward by Couturat, who published a comprehensive and still 101
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valuable account o Leibniz’s logic in 1901. Couturat likewise praised Leibniz or his logical ingenuity, while finding ault with many o his results: Leibniz wrongly preerred the intensional view o the proposition over the extensional one (more on this below), and ailed to see that subalternation, that is, a type o inerence rom universal to particular propositions (e.g. ‘all A are B, thereore some A are B’) is invalid. Tese mistakes, in Couturat’s view, were due to Leibniz’s excessive admiration orsuch Aristotle, which prevented him rom developing the insights that later logicians as Boole were able to ormulate. In the course o the twentieth century, several historians have evaluated Leibniz’s work in logic much more positively (e.g. Rescher 1954, Kauppi 1960). Recently, Lenzen (2004a, 2004b) has argued most orceully or the sophistication o Leibniz’s logical work, and claimed that 160 years beore Boole, Leibniz had worked out a ull ‘Boolean’ algebra o sets. Tere are several reasons why historians have reached such diverse conclusions not only about the merits but also about the content o Leibniz’s logical work. One o these is that writers like Couturat and the Kneales were perhaps too prone to use more recent theories and achievements as a yardstick against which Leibniz’s work was measured. Another reason is the rather chaotic state o Leibniz’s papers. He published very little on logic, but wrote a large number o texts on the subject. Very ew o these were ready, or even intended, or publication. Many papers are filled with tentative notes; others are longer pieces in which various lines o investigation are pursued. ogether, these papers provide a treasure trove ull o intriguing ideas and analyses, but it ofen requires quite some interpretative work to distil unambiguous viewpoints and results rom them. Tus, it could happen that whereas some commentators ound evidence o a logician vainly struggling to ree himsel rom traditional misconceptions, others ound striking results that had previously been ascribed to logicians working in later centuries. Apart rom the question to what extent modern insights and results are present in Leibniz’s logical papers, it may be asked how, i at all, Leibniz’s work has contributed to the development o modern logic. It is uncontroversial that Leibniz was an eminent logician, but it has ofen been claimed that this act has gone unnoticed or centuries. As Parkinson remarked, Leibniz was the first symbolic logician, but ‘he cannot be called the ounder o symbolic logic’, because most o his writings were only published afer symbolic logic had been independently established by others in the nineteenth century (Leibniz 1966: lix). In act, when Couturat edited his collection o Leibniz’s unpublished logical papers in 1903, he was convinced that he had unearthed entirely unknown
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aspects o Leibniz’s work on logic. Although Couturat had good reasons to believe this, it is certain that most logicians working in the nineteenth century, with the exception o Boole, were amiliar with Leibniz’s work through an edition by Erdmann (1840), which contained a limited sample o logical papers. Frege, or one, knew these papers very well, and his ‘Begriffschif’ (1879) derived some o its inspiration rom Leibniz’s ideal o a philosophical language. Yet, according to (2009), Leibniz’s o Peckhaus mathematical logic as such.work did not exert any influence on the emergence In order to understand the place o Leibniz in the history o logic, it is o course insufficient to look only at later developments, o which Leibniz could have no knowledge. It will also be necessary to put his work in the context o his own time, and compare it with that o contemporaries and predecessors. A general point about the context in which Leibniz worked is that the seventeenth century was not the most stimulating period or a logician to be working in. Logic was probably a more widely known subject than it is today, because it ormed a standard ingredient o every educated man’s knowledge. extbooks on logic prolierated in the sixteenth and seventeenth centuries, but they all rehearsed roughly the same body o traditional learning, which was increasingly regarded as useless. Some o the most influential philosophers o the period, such as Bacon, Descartes and Locke, explicitly denounced the logic o the schools as obsolete, and even detrimental to clear thinking and scientific progress. Although Leibniz was not the only one to deend logic against such attacks, his view o the value o logic was quite uncommon. Against Locke’s contention that little would be lost i Aristotelian logic were abolished, he maintained that the invention o the syllogistic orm was among the most important ones ever made by the human mind, because it included an ‘art o inallibility’ (Nouveaux Essais IV, xvii, § 4). What Leibniz perceived was that algebra, geometry and logic shared a characteristic that he considered to be o the greatest value, namely that they used ormal patterns o reasoning that enabled one to draw conclusions concerning all the particular instances covered by their equations and theorems, and that this could be done in mechanical ashion once the patterns were established. Logic was the most general discipline in this regard, ‘a kind o universal mathematics’. Syllogistic orm, in Leibniz’s view, was only one among many orms o logical inerence. Convinced that the elaboration and expansion o these orms would help advance science enormously, he pursued his ideal o mathematizing logic throughout his lie. As Leibniz’s view o logic was shared by ew i any o his contemporaries, he worked mostly in isolation. Te seventeenth
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century was a period in which the prestige o mathematics increased, with many new developments taking place in the field, natural science being put on a mathematical ooting, and philosophers such as Spinoza expounding their systems in geometrical ashion. But Leibniz was probably the only person in his time who envisaged, and worked towards, a close connection between mathematics and logic.
3 Systematizing syllogistics Although Leibniz viewed syllogisms as only part o the grander logic he envisaged, he devoted considerable time and effort to investigating syllogistic theory. Te first results o this are ound in one o the ew pieces on logic he ever published, in certain passages o A Dissertation on the Art o Combinations (DAC , 1666, G IV 27–104), written as a student thesis when he was nineteen years old. Tis tract was organized around a large number o ‘uses’ or applications o the mathematical theory o combinations and permutations. Although the uses pertained to various fields, ranging rom law to geometry and logic, Leibniz described all these uses as instances o ‘the logic o invention’. It is in one o the uses o this combinatorial logic o invention that a systematic treatment o syllogistics is ound. Te problem Leibniz set himsel was to determine the number o valid types o syllogisms. In answering it, he proceeded by steps, employing combinatorics and traditional rules or the syllogism in turn. Te actual procedure he used was not the most simple, and a short description o it may serve to give an impression o Leibniz’s tract on combinations. Leibniz started out by distinguishing our possible quantities a proposition may have, namely universal, particular, singular and indefinite. He combined these in groups o three, as in each syllogism three propositions occur. Next, he sorted out which o the sixty-our resulting combinations may give rise to a valid syllogism, using such rules as ‘rom pure particulars nothing ollows’, which lef thirty-two ‘useul’ moods. Subsequently, he combined these with the useul moods with respect to the two qualities, affirmative (A) and negative (N), o which there are only three, namely AAA (both premises and the conclusion affirmative), NAN, and ANN , resulting in thirty-two times three, which equals ninety-six useul moods. He then applied rules that apply to specific figures. O the ninety-six useul moods, eight turned out to be valid in none o the our figures. Tis lef eighty-eight useul moods, which Leibniz urther reduced by giving up the distinction o our quantities he had started out with. He now
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equated singular propositions to universal ones, and indefinite propositions to particular ones, so that ultimately only universal and particular propositions remained. Tis rather roundabout procedure resulted in the identification o twenty-our types o valid syllogisms, neatly divided into six types in each o the our figures. Tat combinatorics has a role to play in determining the number o valid syllogisms was asprocedure obvious incan Leibniz’s time is now. Forlogic example, a similar, though different be ound, in as theitPort Royal (Arnauld and Nicole 1662, part 3, ch. 4).However, there was no consensus on the exact number o valid syllogisms when Leibniz wrote this. Contemporary authors held different views, some wishing to exclude subalternate moods such as Barbari and Celaront (Arnauld and Nicole part 3, ch. 3, rule 2, corollary 4), and others denying that the ourth figure should be regarded as a genuine figure at all (Sanderson 1672 part 3, ch. 4; Wallis 1687, part 3, ch. 9). Leibniz insisted, also in later writings, that the ourth figure is as legitimate as the other three. Leibniz returned to syllogistics several times in later years, since he was interested in providing it with a solid theoretical oundation. Among his papers is one that is devoted to the so-called reduction o syllogistic moods. It was a topic already treated by Aristotle, who was concerned to show that all valid syllogistic orms are reducible to the our ‘perect’ syllogisms o the first figure that were aferwards labeled Barbara, Celarent, Darii and Ferio, whose validity he assumed to be sel-evident. Leibniz proposed a similar way o proving the validity o syllogistic orms, but systematizing the procedure. He first showed that the our perect syllogisms derive their certainty rom an axiom o ‘no less geometrical certainty than i it were said that that which contains a whole contains a part o the whole’ (P 106), and which was known rom scholastic times as ‘the dictum de omni et nullo’. It says that whatever is affirmed or denied o the members o a class is also affirmed or denied o the members o a subclass o that class. Now this is what is expressed, as ar as the affirmative part goes, by Barbara and Darii, or the whole or part o a subclass respectively, and similarly in the negative case by Celarent and Ferio. As a next step, Leibniz proved subalternation by means o Darii, and the ‘identical’ statement ‘some A is A’, assumed to be sel-evidently true, as ollows: ‘Every A is B, some A is A,thereore, some A is B.’ He proved the negative case in a similar way by means o Ferio: ‘No A is B, some A is A, thereore some A is not B’. Once subalternation was proved, two urther moods o the first figure could be derived: Barbari and Celaro, in which a particular conclusion replaces, by subalternation, the universal ones o Barbara and Celarent, respectively.
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Having thus established six moods in the first figure, Leibniz derived rom each o these a valid mood in the second, and a valid mood in the third figure, using a single principle, which he called ‘regress’. It was traditionally also known as ‘reductio per impossibile’, and was used by Aristotle. It works by assuming that the conclusion o a valid syllogism is alse and demonstrating that on this assumption one o the premises must be alse as well. Tus, assuming that the conclusion syllogism in be Barbara is alse andtheitsalse major premise isby true, it ollows thatothea minor must alse. Replacing propositions their respective contradictories and interchanging the minor and the conclusion results in a valid syllogism o the second figure, Baroco: rom ‘Every C is D, every B is C, thereore every B is D’ (Barbara) results ‘Every C is D, some B is not D (contradictory o the conclusion), thereore some B is not C (contradictory o the minor) (Baroco). Analogously, a syllogism o the third figure Bocardo is derived by assuming the conclusion and the major premise, rather than the minor, o a syllogism in Barbara to be alse. Tis procedure applied to each o the six moods o the first figure, so that all six valid moods in the second, and all six valid moods in the third figure could be derived. Leibniz’s point in proposing this procedure was to show that a uniorm method could be used or deriving all the valid moods o the second and third figure. Furthermore, he maintained that it was the best method o proo, as it was synthetic rather than analytic, which meant that it contained ‘the method by which they could have been discovered’ (P 110). However, the tidy systematicity o the procedure did not extend to the moods o the ourth figure, the derivation o which requires the principle o conversion (e.g. ‘No A is B, thereore no B is A’) that Leibniz had been able to avoid so ar. In sum, Leibniz systematized the metatheory o syllogistics, taking the dictum de omni et nullo as an axiom, and using identical propositions, subalternation, the method o regress, which as he noted presupposes the principle o contradiction, and finally conversion as urther principles to prove the validity o syllogistic moods in all our figures. All these principles were traditional, except the use o ‘identical’ propositions, which Leibniz employed in proving principles such as subalternation and conversion that were usually taken or granted without proo. Tis use o identical propositions was, as Leibniz noted, invented by Peter Ramus (Couturat 1901: 8; G IV 55). A urther example o Leibniz’s efforts to investigate the theoretical basis o syllogistics is ound in a paper entitled ‘A Mathematics o Reason’ (P 95–104). Again, he used insights that had been developed by logicians working beore him, but aiming at a more rigorous treatment. In this case, it was the theory o
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the distribution o terms that Leibniz took as a starting point. Medieval logicians had observed that in categorical propositions some terms are ‘distributed’, and others are ‘undistributed’, meaning that only a distributed term applies to every individual belonging to the class denoted by the term. Tus, in a universal affirmative proposition such as ‘All horses are animals’ the term ‘horse’ is distributed, whereas the term ‘animal’ is not, since the proposition is about every individual not about every individual Onpropositions the basis o and this criterion, ithorse, can bebut established that subject terms inanimal. universal predicate terms in negative propositions are distributed. Te distribution o terms lay at the basis o several rules that could be used as a test or the validity o a syllogism; or example, in every valid syllogism, the middle term should be distributed in at least one o the premises. In ‘A Mathematics o Reason’, Leibniz called distributed terms ‘universal’ and undistributed terms ‘particular’. He discussed the traditional rules concerning distribution, providing a justification or them: or example, i the middle term is particular (i.e. undistributed) in both premises, nothing can be concluded rom the premises, because there is no guarantee that the same individuals are denoted by both occurrences o the middle term in each premise. He also enumerated and explained a series o other rules and observations, such as ‘i the conclusion is a universal affirmative, the syllogism must be in the first figure’, and ‘in the second figure, the major proposition is universal and the conclusion negative’, all o which he could justiy on the basis o principles and corollaries he had proved first. It is clear, however, that, just as with the reduction o syllogistic moods, Leibniz was working within a traditional ramework, and putting orward results that or the most part were already known. By contrast, a third example o Leibniz’s concern with syllogistics constituted a distinctive novelty. In a urther attempt to assimilate logic to geometry, he employed diagrams o various sorts in representing the our traditional types o proposition. At first, he used circles to represent the subject and predicate terms, and made the way they did or did not overlap indicate the quality and quantity o the proposition. Similar circles are usually called Venn diagrams today, but it was Leibniz who introduced their use as a graphical representation o terms and their interrelations. A second type o diagram that Leibniz devised consisted o parallel horizontal lines representing terms, with their overlap in a vertical direction indicating how the terms were related. For example, a universal affirmative proposition was represented by a horizontal line symbolizing the predicate term, while the subject term was symbolized by a shorter parallel line drawn under it and nowhere extending beyond the longer line. Tus, it could be
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read off rom the diagram that the subject term and the predicate term coincided in part. By adding a third line, an entire syllogism could be represented by this means, and the validity o an inerence rom the premises was made apparent by the resulting diagram.
4 Combinatorics, the alphabet o human thoughts and the philosophical language Leibniz’s tract on combinations contained discussion o a urther logical topic that he himsel considered so important that he mentioned it in retrospect as one o the main reasons why this otherwise immature work was still valuable. Tis topic was ‘the alphabet o human thoughts’ – an idea that was linked to a line o thinking that can be ound with numerous philosophers rom Plato to Descartes. It entailed, roughly, that everything we know and think can be broken down into smaller units. Just as sentences are composed o words, and words consist o letters or speech sounds, so is the thought expressed by a sentence composed o ideas. Ideas themselves can be analysed into components, which in their turn can be analysed. But analysis cannot go on or ever; it must stop where it reaches primitives: simple notions that have no component parts. Leibniz assumed that a class o such ideas exists, and this is what he called ‘the alphabet o human thoughts’. He made it a guiding principle or many o his logical investigations. He derived the idea rom his study o logic textbooks when he was still a boy. He noticed that such textbooks were usually organized according to increasing levels o complexity: they started with the treatment o terms, presenting them as belonging to a hierarchical scheme o categories called genus and species, proceeded to propositions, i.e. combinations o terms, and concluded with syllogisms, i.e. combinations o propositions. But it did not seem to be transparent how the levels were connected; in particular, it was unclear how complex terms, i.e. propositions, were connected to syllogisms, since propositions were not divided into categories as simple terms were. On urther thought, he realized that the connection between the level o terms and the level o propositions was unclear as well, and that the categories o simple terms also needed revision. Tus, in the tract on combinations he proposed to reorganize the theory o terms altogether, using combinatorial principles or a new arrangement, so that terms were classified on the basis o the number o terms o which they are composed. Tere would be a first class o primitive notions, a second class
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containing concepts resulting rom combinations o two primitives, a third class o concepts in which three primitives are involved, and so on. Supposing that such a classification is given, it would become possible to produce a list o all the terms that could be a predicate in a true proposition with the given term as subject, as well as a list o terms which could be the subject o a true proposition with the given term as predicate. o illustrate this, let us arbitrarily assume that there are only primitive notions: a, b, c, d. Ten there are our classes in all, containing the our ollowing concepts: class I: a, b, c, d class II : ab, ac, ad, bc, bd, cd class III : abc, abd, acd, bcd class IV: abcd For every complex term, it can be determined which terms can be predicated o it so as to orm a true, universal proposition. For example, the term ‘ab’ has ‘a’, ‘b’ and ‘ab’ as such predicates, or ‘ab is a’, ‘ab is b’, and ‘ab is ab’ are all true propositions. I the same term ‘ab’ occurs as a predicate, the combination with the terms ‘ab’, ‘abc’, ‘abd’, and ‘abcd’ in subject position yields a true proposition, as or example ‘abc is ab’ is true. Leibniz also gave rules or determining the subjects and predicates o a given term in particular and negative propositions. On analysis, these rules turn out to be inconsistent (Kauppi 1960: 143–4; Maat 2004: 281ff.). Leibniz probably saw this eventually; he did not pursue this particular idea o computing the number o subjects and predicates in later writings. But he did not give up the general principles on which it relied. Tese principles are, first, that all concepts result ultimately rom the combination o simpler concepts and that the entire edifice o knowledge is based upon, and reducible to a set o primitive unanalysable concepts, the alphabet o human thoughts. Secondly, it is assumed that in a true affirmative proposition the predicate is contained in the subject, as shown in the examples ‘ab is a’ and ‘abc is ab’. Tis latter principle is what is called the intensional view o the proposition. On this view, a proposition expresses a relation between concepts, such that in a true proposition ‘All A is B’ the concept o the subject A contains the concept o the predicate B. According to the alternative, extensional, view, a proposition expresses a relation between classes or sets; a proposition such as ‘All As are Bs’ is true i the class denoted by B contains the class denoted by A. Leibniz preerred the intensional view, or several reasons. First, it fitted quite well with the classification o concepts according to combinatorial complexity. Tus, the truth o the proposition ‘man is rational’ can be proved by
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replacing the term ‘man’ by its definition, consisting o supposedly simpler concepts, namely ‘rational animal’. Afer this substitution, the proposition reads ‘rational animal is rational’, which shows on the surace that the concept o the predicate ‘rational’ is identical to part o the concept o the subject ‘man’. Tis type o proo relied on two principles that Leibniz later ormulated explicitly in a number o papers. First, the law o substitutivity, according to which two terms or expressions the same theyo can be substituted affecting truth. are Secondly, the i‘law identity’, expressedor as the ‘A isother A’, orwithout in some equivalent orm, which became an indispensable axiom in his logical calculi. A second reason why Leibniz preerred the intensional view is that he did not want logic to be dependent on the existence o individuals. As he put it in a paper o 1679, he rather considered ‘universal concepts, i.e. ideas, and their combinations’, apparently because in this way he was dealing with propositions that can be true or alse regardless o whether individuals exist to which the concepts it contains are applicable. A choice between the intensional and the extensional view o the proposition was not o particular importance in Leibniz’s view, because he believed that the two were ultimately equivalent, and that results obtained within the intensional perspective could be expressed in extensional terms, and vice versa, by ‘some kind o inversion’ (P 20). He noted that the extension and intension o terms are systematically related: the more individuals a term is applicable to, i.e. the greater its extension, the ewer parts its concept contains, i.e. the smaller its intension. For example, the term ‘animal’ has greater extension but smaller intension than the term ‘man’. Evaluating the proposition ‘every man is an animal’ can be done in two equivalent ways: either by checking whether the intension o ‘animal’ is contained in the intension o ‘man’, or by checking whether the extension o ‘man’ is contained in the extension o ‘animal’. Tis relationship between the intension and extension o terms has been called the ‘law o reciprocity’. It may seem a questionable principle, because it entails that i two terms have the same intension, they also have the same extension. Now this seems to be alse or numerous pairs o terms, such as terms that are coextensive (have the same extension) but differ in meaning (have different intension) such as ‘creature with a heart’ and ‘creature with a kidney’. However, Leibniz wished to consider not just individuals that in act exist, but all possible individuals, assuming that i two terms differ in intension, there would be a possible individual to which only one o the two terms applies. Conversely, i such an individual is impossible, this proves that the two terms have the same intension afer all.
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Leibniz’s views concerning knowledge (as ultimately derived rom simple notions) and truth (as consisting in a relation between concepts and provable through analysis by means o definitions) were intimately connected with a scheme he first sketched in his tract on combinations, and which he pursued or the rest o his lie. Tis was the construction o an artificial language that he believed would have extremely beneficial effects: it would be a reliable tool or thinking, so that by its meanso. human knowledge and science could be enhanced at a pace previously unheard Quite a ew seventeenth-century philosophers were preoccupied with searching or means to advance science. Tey usually sought the solution in developing new methods o research. Leibniz, by contrast, believed that the construction o a new symbolism was crucial or the improvement o science. He was not altogether unique in this respect, as schemes or artificial or ‘philosophical’ (i.e., roughly, scientific) languages were widespread in the period, with their authors ofen claiming ar-reaching advantages or their inventions. wo o the most ully developed schemes o this kind were created in England, by Dalgarno (1661) and Wilkins (1668), who claimed that the languages they had constructed were more logical and more philosophical than existing languages. Leibniz studied their schemes careully, and used the vocabularies and grammar o their languages in preliminary studies or his own project. Nevertheless, he was convinced that neither Dalgarno nor Wilkins had perceived what a truly logical and philosophical language could accomplish. Such a language would not only be a means or communication, and not just an instrument or the accurate representation o knowledge, both o which characteristics were realized to some extent by the English artificial languages, but it would first o all be a tool or making new discoveries, and or checking the correctness o inerences. Tis logical language amously made it possible to decide controversies by translating conflicting opinions into it; or this purpose, participants in a debate would sit down together and say ‘let’s calculate’ (G VII 200). Tus, Leibniz’s scheme included the construction o an encyclopedia encompassing all existing knowledge, as well as a ully general logical calculus that would comprise all sorts o logical inerences, including but not limited to syllogistic ones. Leibniz worked towards the achievement o these goals throughout his lie. He produced numerous lists o definitions, which were apparently intended or use in the encyclopedia. He also drafed many versions o a logical calculus, and he wrote a number o papers on rational grammar, investigating how existing languages express logical relationships. Te next two sections provide a brie description o some o the logical calculi, and o rational grammar, respectively.
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5 Te logical calculi Leibniz drafed outlines o logical calculi in a considerable number o papers, written over a long period o time. He ofen started aresh, and obtained results that sometimes differed rom, and sometimes overlapped with previous ones. By the term ‘calculus’ he meant various things; he used the term to reer to a coherent series theorems, also to a method o providing an arithmetical modelooraxioms logicaland inerences. Webut will discuss examples o both types o calculus, starting with the latter.
5.1 Te arithmetization o logical inerence In a series o papers written in 1679 (P 17–32), Leibniz explored the possibility o interpreting the terms o categorical propositions by means o numbers. Te purpose o this was to reduce logical inerences to arithmetical calculations, so that all reasoning would have the same perspicuity and certainty as arithmetic. In a first attempt, Leibniz equated the combination o terms into more complex ones with multiplication; or example, just as the term ‘man’ results rom combining the terms ‘rational’ and animal’, so does multiplication o two and three make six. By assigning a unique, ‘characteristic’ number to each term in such a way that the number o a complex term is equal to the product o the component terms, determining the truth o a proposition would come down to a simple calculation: in the example just given the truth o ‘man is rational’ would appear rom the act that six, or ‘man’, is divisible by three, or ‘rational’. Tis shows that ‘rational’ is part o the concept ‘man’, that the predicate is contained in the subject, and hence that the proposition is true. Te assignment o characteristic numbers to terms would also have to be such that i o two terms neither contains the other, their respective numbers are not divisible: the truth o the proposition ‘no ape is a man’ would be clear, or example, i the number or ‘man’ would be six, and that or ‘ape’ would be ten. Leibniz perceived that this system could not work, or it seemed impossible to account or particular affirmative (PA) propositions. He tried several methods o providing a model or these. First, he assumed it would be sufficient or a PA proposition to be true i either the number o the subject term were divisible by the number o the predicate term (making the universal affirmative (UA), and hence also the PA true) or the number o the predicate term were divisible by the subject term. On urther thought, this did not seem satisactory, probably because only terms related as genus and species could be accounted or
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in this way. Tus, he tried a second method, requiring that or a PA proposition ‘some A is B’ to be true, there should be numbers r and v such that the product o r and the number or A (call this A) equals the product o v and the number or B (call this B): rA = vB. Tis turned out to be even less satisactory, as on this model all affirmative particular propositions are true (choose B or r and A or v), and consequently all universal negative (UN) propositions are alse, rendering theLeibniz model useless or Leibniz’s purposes (Glashoff 2002:in174). soon devised a more sophisticated system, which each term was assigned an ordered pair o natural numbers, such that the numbers were prime with respect to each other. Although each number was preceded by a plus sign or a minus sign, the system did not rely on a distinction between positive and negative numbers; the signs unctioned merely as indicators o the first and second member o a pair. For a UA proposition to be true it was now required (and sufficient) that the first number o the subject term was divisible by the first number o the predicate term, and the second number o the subject term was divisible by the second number o the predicate term. On this model, a UN proposition is true i, and only i, either the first number o the subject and the second number o the predicate or the second number o the subject and the first number o the predicate have a common divisor. For example, ‘No pious man is unhappy’ can be shown to be true i the numbers or ‘pious man’ and ‘unhappy’ are +10 −3 and +5 −14, respectively, because 10 and 14 have 2 as a common divisor. Te condition or PA propositions, which are contradictories o UN propositions, ollows rom this: they are true i no such common divisor exists. Similarly, PN propositions are true i the condition or the truth o the corresponding UA proposition is not ulfilled. With this model, all the problems that were connected with earlier attempts had been solved. In the 1950s, Łukasiewicz showed that Leibniz’s arithmetical interpretation o syllogistic logic is sound in that it satisfies Lukasiewicz’s axioms (1957: 126–129). Leibniz intended this interpretation also to work or modal and hypothetical syllogisms, but he did not proceed to investigate this urther. He knew that the arithmetical calculus was only a modest step towards his ultimate goal, the philosophical language. Tis was because the calculus showed that i number pairs were chosen according to certain rules, it would be possible to test the validity o inerences by means o a calculation. But the philosophical language could not be achieved until the characteristic number pairs or each term had been definitively established – and that was a colossal task he had not even begun to undertake. Recently, Leibniz’s calculus has inspired logicians to do urther
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research on arithmetical interpretations o syllogistic logic (Glashoff 2002; van Rooij 2014).
5.2 Te calculus in the General Inquiries In a lengthy piece called ‘General Inquiries about the Analysis o Concepts and o ruths’ P 47–87),Tus, Leibniz treated a several numberways o subjects, which he clearly saw as(1686, all connected. he examined o classiying types o terms, sought definitions o the terms ‘true’ and ‘alse’, and tried to find an account o the meaning o the terms ‘exist’ and ‘possible’. In the course o these investigations, he ormulated a series o principles, which together were meant to constitute a universal logical calculus. Similar attempts at establishing such principles are ound in a large number o other texts. From the 1980s onward, Lenzen has produced a number o analyses o these texts, showing that one can extract several logical calculi o diverse expressive power rom them. Leibniz’s ‘General Inquiries’ thus turns out to contain, i supplemented with a ew principles expressed elsewhere, a system that Lenzen has called L1, which he has shown to be an algebra o concepts that is isomorphic to a Boolean algebra o sets. In this chapter, we can only give an impression o what this looks like, relying on Lenzen 2004a (ch. 3) and Lenzen 2004b. Te calculus that Lenzen has called L1 consists o the ollowing main elements. Te lef-most column contains names o the elements given by Lenzen, and the second column Leibniz’s ormulation. Te third column states the corresponding set-theoretical interpretation, where (A) denotes the set o possible individuals that the unction assigns to the concept letter A. Te ourth column contains Lenzen’s notation.
Elements o L1
Leibniz
Set-theoretical Interpretation
Lenz en’s Notation
Identity Containment Converse containment Conjunction Negation Possibility
A = B, coincidunt A et B A est B, A continet B A inest ipsi B
φ(A) = φ(B)
A=B AB A ιB
AB , A + B, A ⊕ B Non-A A est ens, A est res, A est possibile
φ(A) ∩ φ(B)
φ(A) ⊆ φ(B) φ(B) ⊆ φ(A)
φ(A) φ(A) ≠ ∅
AB A P(A)
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Using these elements, Leibniz ormulated a series o principles, which can all be interpreted as stating basic set-theoretical laws. Te ollowing is a sample o these principles (in Lenzen’s notation, I(A)’ ‘ symbolizes impossibility or inconsistency o A): Laws o L1
Leibniz
Lenzen’s Notation
(1) (2) (3)
B is B I A is B and B is C, A will be C Tat A contains B and A contains C is the same as that A contains BC Not-not-A = A I I say ‘A not-B is not’, this is the same as i I were to say ‘A contains B’ I A contains B and A is true, B is also true
AA AB∧BC→AC A BC ↔ A B ∧ A C
(4) (5) (6)
A=A I (AB) ↔ A B A B ∧ P(A) → P(B)
As Lenzen has shown (2004a, ch. 3), the principles (1) to (6) together provide a complete axiomatization o Boolean algebra. For discussion o Lenzen’s results, see Burkhardt 2009.
6 Rational grammar Leibniz’s efforts in developing a ormal calculus were or the most part driven by his ideal o a philosophical language, o which such a calculus would orm a central element. In constructing his calculi, he usually took the traditional orms o the categorical proposition as a starting point. But he realized rom an early stage onward – there is a hint o this already in the tract on combinations – that the patterns studied in traditional logic, even i not only categorical but also hypothetical, disjunctive and modal propositions were included, did not exhaust the possible orms o logically valid inerence. As he put it in a piece called ‘A Plan or a New Encyclopedia’ (1679, A VI 4 A, 344): ‘very requently there occur inerences in logic that are to be proved not on the basis o logical principles, but on the basis o grammatical principles, that is, on the basis o the signification o inflections and particles.’ In order to identiy and analyse the grammatical principles involved, a thorough investigation o natural languages was called or, since these languages consist o the ‘symbols that human kind in general uses in speaking and also in thinking’. Rather than rejecting existing languages as
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deective and illogical, Leibniz thus undertook to investigate them with a view to extracting the logical principles they embody. Leibniz devoted a airly large number o papers to rational grammar, none o which were published during his lietime, and most o which were only published in ull recently (A VI 4, 1999); some o them are still unpublished. In conducting his studies o language in the context o rational grammar, Leibniz understandably took theory as a himsel starting what point.unction Tus, hethey investigated the ‘partstraditional o speech’grammatical or word classes, asking have in the expression o thought. A first result o this was that most word classes seemed superfluous. Verbs, or example, proved to be composed o a single necessary element, the copula ‘is’, and a noun, either adjective or substantive, or participle. Instead o ‘I love’ one can equivalently say ‘I am loving’. In the latter expression, the verb ‘love’ has been resolved into the copula ‘am’, and the participle ‘loving’, which is the predicate o the proposition. Tis analysis o the verb was quite common among grammarians at the time, especially those concerned with philosophical or rational grammar, such as Wilkins and the Port Royal grammar. urning to substantives, Leibniz ound that they can be resolved into a combination o the single substantive ‘thing, being’ (res, ens) with an adjective o some sort: or example, ‘gold’ is ‘a yellow, heavy, malleable thing’. Although adverbs are less easily reduced to other types o expressions, Leibniz observed that they ofen unction in the same way as adjectives do. Tis led him to conclude that ‘everything in discourse can be resolved into the noun substantive Being or Ting, the copula or the substantive verb is, adjective nouns, and ormal particles.’ It is tempting to view this drastic simplification o the grammar o natural languages as an attempt to squeeze all sentence types into the mould o the subject–predicate proposition, so that they would become more manageable or both traditional logic and the logic that Leibniz was developing. Although there is some justification or this view, the importance o the last item on Leibniz’s short list o indispensable types o expressions should not be overlooked, namely the ‘ormal particles’. Tese included prepositions, conjunctions, certain adverbs, and pronouns. Words o this type were called ‘ormal’, as opposed to the ‘material’ words such as adjectives, as they signiy operations, relations and inerences rather than concepts or things and their properties. Again, Leibniz was ollowing a long-standing tradition in using the distinction between material and ormal words. Te specific approach he took to the analysis o ormal particles, however, sets his rational grammar apart rom his contemporaries and most o his predecessors (except or some medieval grammatical treatises), as his analysis
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was primarily aimed at determining the logical characteristics o these particles, such as the inerential patterns they signal. Unlike words belonging to the other word classes, the particles required individual treatment in rational grammar. Tus, Leibniz devoted considerable effort to the description o the meaning o a large number o prepositions, (grammatical) conjunctions and pronouns, usually rom the perspective o logical inerence. example, the647). preposition ‘with’ he noted: ‘i A is with B, it ollows that BFor is with A’ (Aabout VI 4 A, About ‘i . . . then’ he observed: ‘I I say I L is true it ollows that M is true, the meaning is, that it cannot be simultaneously supposed that L is true and M is alse. And this is the true Analysis o I and o It ollows’ (A VI 4 A, 656). Leibniz examined not only the conjunctions that were amiliar candidates or logical analysis, such as ‘i . . . then’, but also words like ‘although’ and‘nevertheless’: Although the teacher is diligent, yet the student is ignorant. Te sense o this is: the teacher is diligent, rom which it seems to ollow that the student is not ignorant, but the conclusion is alse, because the student is ignorant . . . It is clear rom this that ‘although’ and ‘yet’ involve some relation to the mind, or that they are reflexive.
A VI 4 A, 656
Te observation that some words, or, as he noted elsewhere, notions, or propositions, or utterances, are ‘reflexive’ occurs repeatedly in Leibniz’s papers. Reflexivity in this sense is one o the characteristics o natural languages that showed the need or an expansion o logic. For the contexts created by ‘reflexive’ expressions are ones in which the law o substitutivity does not seem to hold. A word requently used by Leibniz to indicate that such a context is present is ‘quatenus’ –‘in so ar as’. It is used in the ollowing example, in which substitutivity is blocked: the expressions ‘Peter’ (proper name o the apostle) and ‘the apostle who denied Christ’ can be substituted or each other, unless they occur in the context created by the reflexive word ‘in so ar as’: the sentence ‘Peter, in so ar as he was the apostle who denied Christ, sinned’, cannot, without affecting the truth o the sentence, be replaced by: ‘Peter, in so ar as he was Peter, sinned’ (A VI 4 A, 552). Reflexivity proved to be a pervasive aspect o natural languages, as it is present in all expressions containing grammatical cases (such as genitive and accusative). Cases other than the nominative were called ‘oblique’ cases; Leibniz accordingly called combinations o terms in which such cases are involved ‘oblique’. For example, in the expression ‘manus hominis’, ‘the hand o a man’, the term ‘man’
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enters obliquely into the expression, which is indicated by the genitive case. Such an expression is clearly different rom ‘direct’ combinations o terms, such as ‘rational animal’, because unlike the latter it cannot be treated as a conjunction o the terms ‘hand’ and‘man’. Put differently, what‘the hand o a man’ reers to is not the intersection o the set o all hands and the set o all men. Such combinations o terms thus had to be accounted or in a way different rom what either syllogistics Leibniz’s own logical calculi such as the one called L1 by Lenzen (c. section 5orabove) could provide. Leibniz’s rational grammar, unsurprisingly, remained uncompleted.
Reerences Works by Leibniz Gerhardt, C[arl] I[mmanuel], ed. (1875–1890). Die philosophischen Schrifen von Gottried Wilhelm Leibniz. Berlin: Weidemann. Abbreviated as ‘G’. Reerences are to volume and page number. Leibniz, G.W. (1839/40). God. Guil. Leibnitii opera philosophica quae extant Latina Gallica Germanica omnia, 2 vols, J.E. Erdmann (ed.), Berlin: Eichler. Leibniz, Gottried Wilhelm (1923ff). Sämtliche Schrifen und Briee. Hrsg. v. der Deutschen Akademie der Wissenschafen zu Berlin. Darmstadt: Reichl (1950ff. Berlin: Akademie Verlag). Abbreviated as ‘A’. Reerences are to series, volume and page number. Leibniz, Gottried Wilhelm (1966). Logical Papers. A selection translated and edited by G.H.R. Parkinson. Oxord: Clarendon Press. Abbreviated as ‘P’. Leibniz, G.W.Nouveaux essais sur l’entendement humain . ranslated in Remnant, Peter and Jonathan Bennett (1996). New Essays on Human Understanding, Cambridge: Cambridge University Press.
Other works Arnauld, Antoine and Pierre Nicole (1970[1662]). La Logique ou l’Art de Penser. Paris: Flammarion. Burkhardt, Hans (2009). ‘Essay Review’ [o Lenzen 2004a]. History and Philosophy o Logic, 30, 293–9. Couturat, Louis (1901). La Logique de Leibniz, d’après des documents inédits . Paris: Alcan.
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Couturat, Louis (1903). Opuscules and ragments inédits de Leibniz. Extraits des manuscrits de la Bibliothèque royale de Hanovre.Paris: Alcan. Reprinted by G. Olms Verlagsbuchhandlung (1961). Dalgarno, George (1661). Ars Signorum, vulgo character universalis et lingua philosophica. London: J. Hayes. [Reprinted 1968, Menston: Te Scolar Press, and edited with translation in Cram and Maat 2001: 137–289]. Glashoff, Klaus (2002). ‘On Leibniz’s Characteristic Numbers’. Studia Leibnitiana, Bd 34, H 2, pp. 161–84. Kauppi, Raili (1960). Über die Leibnizsche Logik. Helsinki: Societas Philosophica. Kneale, William and Martha Kneale (1962). Te Development o Logic. Oxord: Clarendon Press. Lenzen, Wolgang (2004a). Calculus Universalis. Studien zur Logik von G.W. Leibniz . Paderborn: Mentis. Lenzen, Wolgang (2004b). ‘Leibniz’s Logic’. In: Gabbay, Dov M. and John Woods (eds.), Handbook o the History o Logic, Volume 3. Elsevier, pp. 1–83. Lukasiewicz, Jan (1957). Aristotle’s Syllogistic rom the Standpoint o Modern Formal Logic. Oxord: Clarendon Press. Maat, Jaap (2004). Philosophical Languages in the Seventeenth Century:Dalgarno, Wilkins, Leibniz. Dordrecht / Boston / London: Kluwer. Peckhaus, Volkero(2009). ‘Leibniz’s Influence on 19th Edward CenturyN. Logic ’, Te Stanord Encyclopedia Philosophy (Spring 2014 Edition), Zalta (ed.),http://plato. stanord.edu/archives/spr2014/entries/leibniz-logic-influence/ Rescher, Nicholas (1954). ‘Leibniz’s Interpretation o his Logical Calculi’.Te Journal o Symbolic Logic 19(1): 1–13. Rooij, Robert van (2014). ‘Leibnizian Intensional Semantics or Syllogistic Reasoning’. In: Ciuni, R et al. (eds.),Recent rends in Philosophical Logic, Springer, pp. 179–94. Sanderson, Robert (1672 [1615]). Logicae Artis Compendium. Oxord. Wallis, John (1687). Institutio Logicae. Oxord. Wilkins, John (1668). An Essay towards a Real Character and a Philosophical Language. London: Samuel Gellibrand and John Martyn.
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Bolzano Jönne Kriener
1 Introduction Tis chapter presents core elements o the logic developed by the Austrian mathematician and philosopher Bernard Bolzano during the first decades o the nineteenth century.* For Bolzano, logic deals with scientific reasoning quite generally. A science or him is an ordered body o true propositions. Accordingly, I will begin by explaining Bolzano’s notion o proposition. When we engage in science, our reasoning crucially involves the derivation o some propositions rom others. Bolzano’s most advanced innovation in logic is his theory o deducibility (Ableitbarkeit). Famously, it anticipates some aspects o the modern concept o logical consequence. Finally we deal with a more demanding, and less well understood, way in which Bolzano took scientific truths to be ordered: his notion o grounding (Abolge). Grounding is central to Bolzano’s thinking about science, and thus an important part o Bolzano’s logic. O course, this chapter is not a comprehensive presentation o Bolzano’s logic. It is not intended to be one. I would urge you to at least have a look at some o the scholarly work I list in the bibliography, such as theStanord Encyclopedia entries on Bolzano (Morscher 2012; Sebestik 2011). Bolzano’s main work Wissenschafslehre(1837) will be reerred to as ‘WL’. Bolzano anticipated several insights o modern logic. I will ocus on these innovations, and mark them clearly. Bolzano wrote in German. His technical terms have been translated into English differently by different authors. I will ollow the very useul handbook entry by Rusnock and George (2004). Beore we go into details, however, let me emphasize one point. Despite being two centuries old and dealing with deep and difficult matters, Bolzano’s work is a pleasure to read. Bolzano is a careul and modest thinker, and his writing is 121
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remarkably clear and well structured. Do give it a try yoursel and have a look at, or instance, WL §147.
2 Lie Bernard Bolzano was born in 1781. At the age o 15, he entered university to study physics, mathematics and philosophy. In addition, rom 1800 onwards he was a student o theology. Only our years later Bolzano was appointed proessor o religious instruction. His chair had been created by the Austrian emperor to fight revolutionary tendencies. Treatened by Napoleon’s revolutionary armies, the Habsburg monarch eared that its own subjects could want to liberate themselves rom his absolutist reign. However, Bolzano’s teaching did not comply with the Kaiser’s intentions. Here is a quote rom one o his lectures. Tere will be a time when the thousandold distinctions o rank among men that cause so much harm will be reduced to their proper level, when each will treat the other like a brother. Tere will be a time when constitutions will be introduced that are not subject to the same abuse as the present one. Bolzano 1852: 19
In 1819 he was dismissed, orbidden to publish and orced to leave Prague. Friends gave him shelter at their rural estate. Tere, during the 1820s and 1830s, he created most o his logical and philosophical work. Eventually he was allowed to move back to Prague, and in 1842 even to become director o the Bohemian Academy o Sciences. Bolzano died in 1848.
3 Propositions and ideas As you have read in previous chapters, traditional logic starts out rom terms or ideas, and understands judgements and propositions in terms o them. Bolzano reverses this order: the basis o Bolzano’s logic is his theory o propositions. Recall that in present-day mainstream philosophy, a proposition is what is said by an indicative sentence. For example, the sentence‘Snow is white’expresses the proposition that snow is white. It is sae to understand Bolzano as working with a close relation o this notion. Importantly, or Bolzano a proposition is not located in space-time. In this precise sense, he takes propositions to be abstract entities. In particular, a proposition is not a mental entity. Tus, Bolzano reuses psychologism about logic, hal a ce ntury beore Frege and Husserl (see Chapter 8).
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Innovation
Logic is not about judgements or thoughts as mental events, but about abstract propositions. Instead o attempting an explicit definition o ‘proposition’, Bolzano characterizes its meaning by a number o postulates.
1. 2. 3. 4. 5.
A proposition is either true or alse. Propositions are abstract. Judgements have propositions as their content. Interpreted sentences express propositions. wo sentences o different structure may express the same proposition.
As compact as this characterization is, it ully suffices or Bolzano’s purpose o developing a general theory o logic. Beore we move on, it is worth noting the novel character o the method itsel, that Bolzano uses to convey his concept o proposition. He explains his technique as ollows. We set out various sentences in which the concept . . . appears in such combinations that no other concept could be thought in its place i these sentences were to express something reasonable. By considering and comparing these sentences, the reader will gather by himsel the meaning o our sign. Bolzano 1981, §9 Innovation
Bolzano deploys, and reflects on, the method o implicit definition.
3.1 Ideas Given this basic theory o proposition, Bolzano proceeds to define an idea as the component o a proposition. An idea is anything that can be part o a proposition . . . without being itsel a proposition. WL §48
Bolzano assumes that there are simple ideas, rom which all other, complex ideas are built up. Bolzanian complex ideas are structured: it matters how they are built up rom their simple parts. For example, the idea o being allowed not to talkis distinct rom the idea o not being allowed to talk(WL §92). More generally, the
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idea i is the idea j just in case i and j are built up in the same way rom the same simple ideas. Bolzano emphasizes the distinction between an idea, which is an abstract entity, and its objects. Objects may be spatio-temporal, such as what alls under the idea o a table, or may not, such as the prime numbers. Every idea i has an extension (WL §66), the collection o its objects. Some ideas any (WL object, or Teir example the ideaisoempty a golden roundlack square §67). extension .1 mountain, or the idea o a An intuition is a simple idea with exactly one object, such as the idea expressed by successully using, in a specific context, the demonstrative ‘this’. Aconcept is an idea that is not an intuition nor contains any. Bolzano’s prime example is the idea o being something, which is a concept because firstly it has not one but infinitely many objects, and secondly is simple and thereore does not contain any intuition (WL§ 73). All other ideas are called mixed. Te distinction between intuitions and concepts allows or an analogous classification o propositions. A proposition that does not contain, directly nor indirectly, any intuition, Bolzano calls a conceptual proposition. Every proposition that contains some intuition is called empirical. Te next section will give examples. According to Bolzano, every proposition is composed o ideas, which themselves are composed o simple ideas. However, it is crucial to his approach that the concept o a proposition is undamental, and that the notion o an idea is defined in terms o it. Bolzano gives an identity criterion or propositions: A is B i and only i A and B are composed o the same simple ideas, in the same manner. More precisely, every proposition immediately contains three ideas, o three distinct kinds.
1. One subject idea, 2. one predicate idea and 3. the simple idea o having. Tus, every proposition is o the orm
i has j or ideas i and j. In particular, this means that Bolzano does not have the modern notion o a logical connective. Every proposition is o the traditional subject-predicate orm. For example, the proposition that
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(1) either snow is white or snow is green. is analysed as ollows (WL§166): Te idea o a true proposition among the propositions that snow has whiteness and the proposition that snow has greenness, has non-emptiness.
Tis part o Bolzano’s work is still in the grip o traditional logic. However, when in the next section we turn to his logic proper, we will find that Bolzano in act was able to analyse propositions in a much more flexible way than his official statements about their orm suggest. Exercise 5.1
Consider the ollowing technical terms. ‘intuitions’, ‘empirical’, ‘simple’, ‘mixed’, ‘propositions’, ‘connectives’, ‘ideas’, ‘concepts’, ‘complex’, ‘conceptual’, ‘judgement’
Some play a role in Bolzano’s theory o propositions. Put those terms at their right place in Figure 5.1. ruth is an abstract idea that applies to propositions. Since we identiy propositions by their structure, and truth is an idea o its own, the proposition A is not identical with the proposition that A is true.
Figure 5.1 (Exercise) Te key technical terms o Bolzano’s theory o propositions,
and how they connect
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3.2 A toy Bolzanian science Let us amiliarize ourselves with Bolzano’s theory o propositions by looking at a toy model. I will work with a basic science o amily relations (‘BF’). It contains propositions such as that every sister has a sibling, or the proposition that Fredo and Michael are brothers. In act, or simplicity, let us assume that there are only the Corleones (as known rom Te Godather 1–3). Tat is, let us assume that as Bolzanian intuitions (recall: ideas that have exactly one object) we have the intuitions o Vito Corleone, Michael and the others.2 Let the idea o being emale, as well as being male, also be intuitions. Tese are all the intuitions we have in our toy Bolzanian science BF. By way o simpleconcepts, we have the concepts o parenthood, marriage and the concept o sharing. Let us also treat as simple the concepts o negation, conjunction and existential quantification as well as the other concepts o firstorder logic. Putting together these simple concepts we obtain complex ones, such as x and y being siblings (sharing a parent),x being a cousin or a parent oy and so on. Now, we can distinguish between conceptual and empirical BF-propositions. For example, the proposition mentioned above, that Fredo and Michael are siblings, is empirical because it contains intuitions, such as the idea o Fredo. Exercise 5.2
1. What mixed ideas do we have? Give three examples. 2. Give a conceptual proposition o the basic science o amily relations.
4 Variation Bolzano’s conception o logic is much more inclusive than how we nowadays understand the subject. In this section, I turn to that part o Bolzano’s work which corresponds to our modern, more restricted notion o logic. I turn to Bolzano’s theory o consequence. (2) Obama is in Berlin today. Te sentence (2) expresses a true proposition on 19 June 2013, but a alsehood on 19 July. Does this contradict Bolzano’s view o truth as an idea, i.e. something independent o space and time? Afer all, i the sentence (2) expresses the same
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proposition on both days, then the truth value o this proposition would depend on time. However, Bolzano replies, (2) does not express the same proposition at different times. Te reason is that ‘today’ expresses distinct ideas on different days and thereore, when it seems to us that the proposition expressed by an utterance o (1) changes its truth value over time, we really consider o this proposition, with different ideas substituted or the idea o variants 19 June 2013. Tis astute observation provides the basis or Bolzano’s variation logic. Given a proposition, we could merely inquire whether it is true or alse. But some very remarkable properties o propositions can be discovered i, in addition, we consider the truth values o all those propositions which can be generated rom it, i we take some o its constituent ideas as variable and replace them by any other ideas whatever. WL §147 Innovation
Bolzano develops and makes great use o an unprecedented method o variation, which anticipates to some extent modern model-theoretic semantics. For a proposition A and ideas i, j, let A(j/i) be the proposition obtained rom A by substituting j or every occurrence o i. Call A(j/i) an ‘i-variant’ o A. Note that we do not assume A to contain i. I it does not, A(j/i) is just A. For example, the proposition that Anthony is Francesca’s brother is a cousin-variant o the truth that Anthony is Francesca’s cousin. Indeed, it is a alse such variant, just as the proposition that he is her niece. Figure 5.2 gives one way o visualizing the replacement o the idea is cousin o, that is part o the proposition that Anthony is cousin o Francesca, by one or the other idea. As indicated in the quote above, Bolzano allows not merely one but several ideas i, j, k, . . . together to be considered as variable.
4.1 Validity It is not the case that every idea is a suitable substitution or any other. For example, the idea o being a sibling is not a good substitution or the idea o Vito Corleone. Every idea comes with its own range o variation: those ideas
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Figure 5.2 Variation
that may be substituted or it. Tis puts constraints on what variants we consider or a given proposition. An i-variant A(j/i) is relevant i and only i:3
1. Te subject idea o A(j/i) has objects, 2. i i ≠ j then j and i do not have the same objects, and 3. j is in the range o variation o i. Definition
A proposition A is valid (invalid) with respect to ideas i, j, . . . iff every relevanti, j, . . . -variant oA is true (alse). For example, the proposition that Michael’s brother is male is valid with respect to the idea o Michael. Te variation o ideas provides Bolzano with a sophisticated tool to identiy ormal properties o a given proposition. It allows him to ormulate a rich and very general theory o these properties. In the remainder o this section, we will get to know its most important parts. Beore we do so, let me note that the power
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o Bolzano’s variation method stems rom the act that he allows any idea to be varied. Tis effectively liberates his logic rom the restrictions o the traditional subject-predicate analysis even though, officially, he clings to this received doctrine. Definition
A is logically valid i A is valid with respect to all non-logical ideas in A. Which ideas are logical? Bolzano gives examples ( and, or, not, some) but admits that there may be distinct notions o logicality, each with its own range o logically valid propositions. Note that modern logic, too, knows such a sense o relativity. o give a amous example, arski acknowledged that validity (in the modern sense o truth in all models) is relative to a set o logical constants (arski 1956: 418.). Our toy science BF comprises the ideas o first-order logic, such as conjunction (and) and existential quantification (some). Let these be the logical ideas o our toy model. Exercise 5.3
Give two logically valid propositions, one containing the idea o Fredo Corleone, the other containing the idea o being a emale cousin.
4.2 Compatibility We may consider variants o several propositions at once. Tis allows us to distinguish ways in which propositions relate to one another. Bolzano speaks o collections o propositions. His relevant use o this term is well viewed as anticipating how logicians today have come to use the term ‘plurality’, as a convenient but4inessential shorthand orplural reerencealways to sometothings (Yi 2005; Linnebo 2012). Like Bolzano, I will assume a collection be non-empty. Definition
A, B, . . . are compatible with respect to some ideas i, j . . . (more simply:i, j, . . .-compatible) just in case there are some ideas i’, j’, . . . such thatA(i’/i, j’/j, . . .) is true, B(i’/i, j’/j, . . .) is true, . . . . Note that the ideas i, j, . . . need not occur in each o A, B, . . . Since we do not assume the variant o a proposition A to be distinct rom A, in particular every
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true proposition is a true variant o itsel. Hence, whenever we have some true propositions, they are compatible. At the same time, however, a true proposition may be compatible with a alsehood. For example, the truth that Anthony is cousin o Francesca and the alse proposition that Sonny is cousin o Vito are compatible with respect to the ideas o Francesca and Vito – since it is a true Francesca–Vito-variant o the latter that Anthony is cousin o Vincenzo (note that variation with respectproposition to Francesca does not have any effect). Bolzano’s compatibility resembles the modern semantic concept o satisfiability. However, Bolzano’s notion o having true variants is not having a true model. Importantly, it is not sentences o some specified language that have variants, but propositions which exist independently o all language and interpretation. Exercise 5.4
Give three BF -propositions compatible with respect to the idea o being a sister, and two propositions incompatible with respect to the idea o Michael.
4.3 Deducibility Next, Bolzano considers special cases o compatibility (WL §155). Sometimes, it is not only that, say, propositionsA and B each give rise to a truth i the idea j is substituted or the idea, i, but in act every true i-variant o A corresponds to a true i-variant o B. Every way o generating a true proposition rom A by replacing i is a way o turning B into a truth, too. Tis is Bolzano’s concept o consequence, or deducibility. Definition
A is deducible rom B, C, . . ., with respect to some ideas i, j, . . . (‘i, j, . . .-deducible’) just in case A, B, C, . . . arei, j, . . .-compatible, and or allk, l, . . .B(k/i, l/j), C(k/i, l/j), . . . are true only iA(k/i, l/j) is.5 For example, the proposition that Francesca is emale is deducible rom the proposition that every sister is emale, with respect to the idea o being emale (WL §155.20). At the same time, the alse proposition that Francesca is male is being-male-deducible rom the alsehood that every sister is male.
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Exercise 5.5
Give an example to show that, with respect to the idea o Carlo, it is not deducible that Carlo is male, rom the proposition that Fredo is Carlo’s uncle.
A is logically deducible rom B just in case A is deducible rom B with respect to to all non-logical ideas in A and B. As with logical validity, the concept o logical deducibility is relative to a notion o logicality. I continue to identiy the logical ideas o our basic science o amily relations BF with the ideas expressed by the connectives, quantifiers and identity. Tus, the proposition that Francesca is emale is logically deducible rom the proposition that every sister is emale together with the proposition that Francesca is a sister. Exercise 5.6
Let us extend our basic science o amily relations by the idea o admiration. Show that the proposition that Sonny admires Vito is logically deducible rom the ollowing propositions.
1. 2. 3. 4.
Francesca admires Michael. Sonny does not admire Vito i and only i Vincenzo admires his ather. Vincenzo admires Sonny only i Kathryn does admire Michael. No two sisters admire the same person.
How does Bolzano’s notion o logical deducibility relate to the modern concept o classical logical consequence? Firstly, both deducibility and consequence are relations between sentences o ormal languages; deducibility is a relation between abstract, non-linguistic entities. Secondly, Bolzano’s concept o deducibility applies at least to some cases o material consequence. I gave an example: the proposition that Francesca is emale is deducible rom the proposition that every sister is emale. Finally, since Bolzano requires premises and conclusion to be logically compatible, i.e. consistent, deducibility is more restrictive than classical consequence. For one, it is not the case that everything can be derived rom inconsistent premises. In act, nothing can be derived rom such assumptions.6
4.4 Probability Given Bolzano’s theory o propositions, the variation o ideas is a powerul tool or logical inquiry. Bolzano discusses several other applications o it. I will ocus
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on his theory o probability (WL §161). For propositions, A, B, . . ., and ideasi, j, . . . let |A, B, . . .|i, j, . . . denote the number o true i, j, . . .-variants o.A, B,. . . . Definition
Let A, B, . . . be some compatible propositions, and leti, j, . . . be some ideas. Let M be a proposition not among but i, j, . . .-compatible with them. |M, A, B, . . . . divided by | |. i,. j,.,. .with . . i,| j, i,. . .jis, .called A,the B, .ideas respect to . . . the probability o M conditional on A, B,
o give an example, let us extend our basic science o amily relations by the idea o someone being older than someone else.7 Ten, the probability o Michael being older than Francesca conditional on Michael being older than some cousin o hers is 8/17. I the probability o M conditional on A, B, . . . (with respect to i, j, . . .) is greater than ½, Bolzano points out, we are entitled to take M to be true i A, B, . . . are W ( L §161.1). Tus, Bolzano uses his concept o probability to define a notion o valid inductive reasoning. I M is i, j, . . .-deducible rom A, B, . . . then M’s i, j . . .-probability conditional on A, B, . . . is 1 W ( L §161.3). In this precise sense, Bolzano’s inductive logic includes his deductive logic as a special case. Assume that we have distinguished a collection o ideas as logical. Definition
Te logical probability o M conditional on A, B, . . . is its probability with respect to all non-logical ideas. As beore, in the examples I will treat all and only connectives, quantifiers and identity as logical ideas. Bolzano’s concept o logical conditional probability satisfies, modulo his concept o logical deducibility, many o the axioms o the modern concept o conditional probability,or support unctions.8 For example, we have that iA and B are mutually logically deducible then or everyM, its logical probability conditional onA is its logical probability conditional on B (WL §161.4). Tus, Bolzano’s work on conditional probability anticipates to some extent modern inductive logic. Innovation
Using his method o variation, Bolzano extends his deductive logic by a ormal logic o probability. A special case o this concept anticipates modern systems o conditional probability.
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Note, however, that what is assigned probability in Bolzano’s system is not a belie, but a proposition whose existence does not depend on any subject. Further, the probability o a proposition is determined by whether or not certain other propositions are true, which itsel is an entirely objective matter. Consequently, Bolzano’s concept o probability is thoroughly objective. As such, unlike modern inductive logic it does not lend itsel to applications in epistemology.9
5 Bolzano’s theory o grounding 5.1 Grounding Bolzano’s logic as developed so ar applies equally to true as to alse propositions. However, Bolzano has more to offer: a special system or truths. rue propositions are ordered by what Bolzano calls the relation o Abolge. Let me translate it as ‘grounding’. Bolzano motivates his theory o grounding rom examples o the ollowing kind (WL §198). (3) It is warmer in Palermo than in New York. (4) Te thermometer stands higher in Palermo than inNew York. Both propositions are true. However, it is the truth o (3) that explains (4) and not vice versa. Te truth o (3) grounds the truth o (4). Tis relation o grounding stands out rom Bolzano’s system in that it is not defined in terms o variation. In particular, the act that (3) grounds (4) and not vice versa cannot be captured by deducibility: (3) can be derived rom (4). Tereore, a stronger concept is needed: (3) grounds (4). For a long time, interpreters have ound this part o Bolzano’s work ‘obscure’ (Berg 1962: 151). Nothing in a modern logic textbook corresponds to Bolzanian grounding. Nonetheless, the concept has a long and venerable tradition. Bolzano connects with Aristotle’s distinction betweenwhy-proos and mere that-proos (Aristotle 2006: 1051b; Betti 2010). Te truth that it is warmer in Palermo than in New York iswhy the thermometer stands higher in Palermo than in New York. Generally, the grounds o A is why A. Moreover, very recently ormal systems o grounding have been developed, prominently by Kit Fine, that are well viewed as resonating Bolzano’s concept o Abolge (Schnieder 2011; Fine 2012a; Correia 2014). Bolzano gives urther examples (WL §§ 162.1, 201). (5) Te proposition that the angles o a triangle add up to 180 degrees grounds the proposition that the angles o a quadrangle add up to 360 degrees.
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(6) Te proposition that in an isosceles triangle opposite angles are identical grounds the proposition that in an equilateral triangle all angles are identical. (7) Te proposition that God is perect grounds that the actual world is the best o all worlds. I A grounds B then it is the case that B because it is the case that A. Bolzano’s grounding is a concept o objective explanation. However, it mustnot be conflated with epistemic notions, such as justification. For one,just as deducibility, grounding concerns how propositions, that do not have spatio-temporal location, are ordered independently o any subject. For another, justification suffers rom the same shortcoming as deducibility, in that it does not respect the asymmetry between the truths (3) and (4). I you know that the thermometer stands higher in Palermo than in New York, then you are justified in believing that it is warmer in Palermo than in New York. Bolzano discusses whether grounding can be defined in terms o deducibility, and possibly other notions (WL §200); his conclusion is that such a definition is not available. Tereore, Bolzano introduces grounding as a primitive concept and characterizes it by a system o principles, analogously to how he characterized his notion o proposition.
5.2 Principles o grounding Grounding is a relation between single or pluralities o propositions. I will use Greek capital letters (those, such as ‘Γ’ or ‘Δ’, which differ typographically rom capital Roman letters) as variables ranging over pluralities o propositions. Note that Bolzano assumes grounds to be always finite collections o propositions (WL §199). I use the symbol ‘’ or grounding such that Γ ‘ A’ reads: the propositions Γ ground the proposition A. What stands on the lef-hand side o ‘’ is called the grounds, and that which is grounded, and stands on the right-hand side o ‘’, is called theirconsequence. Officially, the consequence may also be a plurality o propositions. However,in practice Bolzano mostly uses the relation between several groundsΓ and a single consequence A. Tereore, many scholars have taken this to be the basic notion o grounding, and defined the relation ‘Γ Δ’, with multiple consequences, in terms o it (Berg 1962). I will ollow them, and present Bolzano’s principles o grounding as or a relation between a single consequence and its possibly many grounds. Te first principle says that only truths stand in the relation o grounding.
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Factivity
I A0, A1, . . . B then A0, A1, . . . andB . . . are true (WL §203). For example, (7) is a case o grounding only i the actual world really is the best o all worlds. Generally, only true propositions stand in the relation o grounding. urning to our toy science o the Corleones, we thereore know that (8) Every sister is male. does not ground (9) Francesca is male. Te ground o A is why it is the case that A; it explains the truth that A. Te sense o explanation at work here is objective and exhaustive. Tis allows us to draw two conclusions about the ormal properties o grounding. Firstly, what grounds a proposition does not involve this proposition itsel, neither directly or indirectly.10 Non-Circularity
Tere is no chain A0, . . .,An such that or every i < n, Ai is among some Γ such that Γ Ai + 1, and A0 = An. (WL §§204, 218) Secondly, the grounds o A are unique. Uniqueness I Γ A and Δ A then Γ = Δ. (WL §206)
On the one hand, this implies that i A is grounded in Γ, then it is not grounded in Γ together with arbitrary other truths. Tus, it is ensured that every truth among Γ matters or the truth A.11 On the other hand, the principle o uniqueness means that the truths that a proposition is grounded in, are its complete grounds. Tis captures our pre-theoretic idea o grounding as a relation o exhaustive explanation. However, I do not include among Bolzano’s ormal principles o grounding that it is unique on its right-hand side. Tat is, I do not assume that i Γ A and Γ B then then A = B. Tese principles describe the relation o grounding ormally. For example, rom (Uniqueness) we know that i the truth that (10) Vito is Michael’s male parent. grounds that
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(11) Michael is son o Vito. then it is not the case that (11) is grounded in the truth that (12) Sonny, Fredo and Michael are Vito’s sons. However, we would like to know more. Does (10) in act ground (11)? More generally, what cases o grounding are there? Bolzano gives examples (such as (6) to (7)), but not many general principles. One such principle, however, is that every truth A grounds the proposition that A is true (WL §205.1). By the same principle we have that the proposition that A is true itsel grounds the proposition that it is true that A is true. Recall that the proposition A is not identical with the proposition that A is true. Hence, uniqueness ensures it not to be the case that A grounds the proposition that it is true that A is true. Generally, grounding is the notion o complete, immediate objective explanation. From it, partial such explanation is defined easily (WL §198): Γ partially ground A i there are some Δ such that Γ are among Δ, and Δ ground A.
5.3 Ascension Bolzano does not stop at the relation between a truth and its immediate grounds. He analyses the order that it imposes on a collection o true propositions (WL §216). I someone starting rom a given truth M asks or its ground, and i finding this in . . . the truths [A, B, C . . .] he continues to ask or the . . . grounds, which . . . these have, and keeps doing so as long as grounds can be given: then I call this ascension rom consequence to grounds. Definition
I, ascending rom M to its grounds, we arrive at some truth A, then M is said to depend on A, and A is called an ‘auxiliary truth’ orM. Exercise 5.7
Explain that Bolzano’s concept o dependence is partial mediate grounding. One worry may be raised. Te way Bolzano conveys his idea o ascension, and dependence, stands in tension with his declared view that, firstly, propositions are not subject to time or space (p. 000 above), and secondly, they stand in the
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relation o ground and consequence also independently o time and space. In the quote above, however, Bolzano uses spatial-temporal language. Indeed, the word ‘ascension’ suggests an upwards movement. However, this is mere metaphor. o characterize dependence in a less metaphorical way, it helps to ormulate ascension as agame.
Definition Let the ascension game G(M) or a true proposition M be played as ollows.
Player 1 starts by playing the true proposition A. 2 responds by playing the propositions A, B, . . . such thatA, B, . . . M. In response, 1 chooses one o A, B, . . ., and so on.One player wins i the other cannot make a move. I a run continues indefinitely, 1 loses. Such games can then be described in purely mathematical terms. For example, each play o G(M) corresponds to a tree o true propositions whose root is M and every other node o which represents an auxiliary truth o M (see Figure 5.3). Tus, we can say that M depends on A i and only i A is represented by a node o such a tree. I say ‘represents’ and not ‘is’, since one and the same truth may be played several times during a play o G(M). Bolzano himsel proposes this characterization o dependence in terms o trees (WL §220). Innovation
Having set up a sophisticated system o extensional logic, Bolzano realizes its restrictions and characterizes a more fine-grained relation o objective explanation. It anticipates, indeed partly inspired, recent work in metaphysics (Fine 2001; Rosen 2010; Fine 2012b).
Figure 5.3 Ascension rom M to its auxiliary truths
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6 Conclusion Bolzano’s work on logic was ahead o his time. In this chapter, we have got to know his core innovations.
1. Logic is not about judgements or thoughts as mental events, but about objective, abstract propositions. 2. Bolzano deploys, and reflects on, the method o implicit definition. 3. Bolzano develops and makes great use o an unprecedented method o variation, which anticipates to some extent modern semantics. 4. Using his method o variation, Bolzano extends his deductive logic by a ormal logic o probability. A special case o this concept anticipates modern systems o conditional probability. 5. Having set up a sophisticated system o logic, Bolzano realizes its restrictions and characterizes a more fine-grained relation o objective explanation: grounding.
Notes * I thank Antje Rumberg orher constructivecriticism andmany helpul comments. Tis work was supported by the European Research Council and the University o Oslo. 1 Note that the idea o a round square necessarily lacks an object. 2 For a comprehensive list, see https://en.wikipedia.org/wiki/Corleone_amily. 3 Tis way o bundling tacit assumptionso Bolzano’s is inspired by Künne 1998, to whom the term ‘relevant’is due, too. 4 However, Bolzano uses ‘collection’ also in other ways; see Simons (1997). 5 o improve readability, I suppress some o the remarkablegenerality o Bolzano’s logic. As a matter o act, Bolzano allows or derivations not only rom multiple premises, but also derivations o multiple conclusions. 6 o this extent, Bolzano’s logical deducibility maybe viewed as anticipatingrelevance logics (Read 1988). 7 o fix matters, let us say one Corleone is older thananother i and only i the ormer is o an earlier generation than the latter or the ormer is above the latter in the list at https://en.wikipedia.org/wiki/Corleone_amily. 8 See e.g. Hawthorne (2012, §2.2). 9 Under the label o ormal epistemology, recently much work has been done that connects the logic o probability with epistemological questions.
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10 Tis ormulation o non-circularity uses the notion o a grounding chain rom Rumberg (2013). 11 o use a technical term, grounding isnon-monotone.
Reerences Aristotle (2006), Metaphysics Book Θ, ed. Stephen Makin, Oxord: Clarendon Press. Berg, Jan (1962), Bolzano’s Logic. Stockholm, Almqvist & Wiksell. Betti, Arianna (2010), ‘Explanation in Metaphysics and Bolzano’s Teory o Ground and Consequence’, Logique & Analyse 211: 281–316. Bolzano, Bernard (1837), Wissenschafslehre, Sulzbach: Seidel. Bolzano, Bernard (1852), ‘Homily on the first Sunday o Advent, 1810’, in ErbauungsredenVol. 4, Prague–Vienna: Houchhandlung bei Wilhelm Braumüller, 1852, p. 19. Bolzano, Bernard (1981), Von der Mathematischen Lehrart, ed. Jan Berg, Stuttgart–Bad Cannstatt: Fromann-Holzboog. Correia, Fabrice (2014), ‘Logical Grounds’, Te Review o Symbolic Logic 7(1): 31–59. Fine, Kit (2001), ‘Te Question o Realism’, Philosophers’ Imprint1: 1–30. Fine, Kit (2012a), ‘Te Pure Logic o Ground,’ Te Review o Symbolic Logic5: 1–25. Fine, Kit (2012b), ‘Guide to Ground’, in Fabrice Correia and Benjamin Schnieder (eds), Metaphysical Grounding: Understanding the Structure o Reality, pp. 37–80, Cambridge, New York and Melbourne: Cambridge University Press. Hawthorne, James (2012), ‘Inductive Logic’, in Edward N. Zalta, ed., Te Stanord Encyclopedia o Philosophy, Stanord: Stanord University Press. Künne, W. 1( 998), ‘Bernard Bolzano,’ in Routledge Encyclopedia o Philosophy, pp. 824–8. London and New York: Routledge. Linnebo, Øystein (2012), ‘Plural Quantification’, in Edward N. Zalta, ed., Te Stanord Encyclopedia o Philosophy, Stanord: Stanord University Press. Morscher, Edgar (2012), ‘Bernard Bolzano’, in Edward N. Zalta, ed., Te Stanord Encyclopedia o Philosophy, Stanord: Stanord University Press. Read, Stephen (1988), Relevant Logic: A Philosophical Examination o Inerence , Oxord: Basil Blackwell. Rosen, Gideon (2010), ‘Metaphysical Dependence: Grounding and Reduction’, in. Bob Hale and Aviv Homan (eds), Modality: Metaphysics, Logic and Epistemology, Oxord: Oxord University Press. Rumberg, Antje (2013), ‘Bolzano’s Concept o Grounding (Abolge) Against the Background o Normal Proos’, Te Review o Symbolic Logic 6(3): 424–59. Rusnock, Paul and Rol George 2004), ( ‘Bolzano as Logician’, in Dov M. Gabbay and John Hayden Woods (eds), Rise o Modern Logic: From Leibniz to Frege , Amsterdam: Elsevier.
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Schnieder, Benjamin (2011), A ‘ Logic or Because’, Te Review o Symbolic Logic 4 (September). Sebestik, Jan (2011), ‘Bolzano’s Logic’, in Edward N. Zalta, ed., Te Stanord Encyclopedia o Philosophy, Stanord: Stanord University Press. Simons, Peter (1997), ‘Bolzano on Collections’, in Mark Siebel Wolgang Künne and Mark extor (eds),Bolzano and Analytic Philosophy, pp. 87–108, Amsterdam: Rodopi. arski, Alred (1956), ‘On the Concept o Logical Consequence’, in Logic, Semantics, Metamathematics, pp. 409–20, Indianapolis: Hackett Publishing Company. Yi, Byeong-Uk (2005), ‘Te Logic and Meaning o Plurals. Part 1’, Journal o Philosophical Logic34 (Oct./Dec.): 459–506.
Solutions to exercises Exercise 5.1 Consider the ollowing technical terms: ‘intuitions’, ‘empirical’, ‘simple’, ‘mixed’, ‘propositions’, ‘connectives’, ‘ideas’, ‘concepts’, ‘complex’, ‘conceptual’, ‘judgement’. Some play a role in Bolzano’s theory o propositions. Put those terms at their right place in Figure 5.1.
Exercise 5.2
1. What mixed ideas do we have? Give three examples. 2. Give a conceptual proposition o the basic science o amily relations.
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Solution
1. Te idea o being a son o Vito Corleone; (b) Te idea o being a sister o Sonny Corleone; (c) Te idea o being a married to Michael Corleone. 2. All sisters are emale. Exercise 5.3 Give two logically valid propositions, one containing the idea o Fredo Corleone, the other containing the idea o being a emale cousin. Solution
1. Either Fredo Corleone has a sister or not. 2. I every emale cousin is a sibling, then everything is a sibling i everything is a emale cousin. Exercise 5.4 Give three BF -propositions compatible with respect to the idea o being sisters, and two propositions incompatible with respect to the idea o Michael. Solution
Tree propositions compatible with respect to the idea o being sisters:
1. Fredo and Michael are sisters. 2. Francesca and Kathryn are sisters. 3. Vincenzo and Francesca are cousins. o see that these propositions are compatible, note that the ollowing are all true
4. Fredo and Michael share a grandmother. 5. Francesca and Kathryn share a grandmother. 6. Vincenzo and Francesca are cousins. wo propositions incompatible with respect to the idea o Michael:
7. Michael and Sonny are sisters. 8. Michael is male.
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Exercise 5.5 Give an example to show that,with respect to the idea o Charles, it is not derivable that Charles is male, rom the proposition thatRichard is Charles’ uncle. Solution
It is true that Fredo is Francesca’s uncle, but alse that Francesca is male.
Exercise 5.6 Let us extend our basic science o amily relations by the ideas o admiration. Show that the proposition that Sonny admires Vito is logically derivable rom the ollowing propositions.
1. 2. 3. 4.
Francesca admires Michael. Sonny does not admire Vito i and only i Vincenzo admires his ather. Vincenzo admires Sonny only i Kathryn does admire Michael. No two sisters admire the same person.
Solution
Dependence is (the inverse o) partial mediate grounding, because i M depends on A, then A is among the truths that ground M, or that ground some truth that M depends on.
6
Boole Giulia erzian
1 Lie and works George Boole (1815–1864) was born in Lincoln, England, where he also spent more than hal his lietime working as a teacher and headmaster. From an early age, his ather encouraged him to cultivate his interests in modern and ancient languages, and in mathematics. Boole senior was also ‘evidently more o a good companion than a good breadwinner. [He] was a shoemaker whose real passion was being a devoted dilettante in the realm o science and technology, one who enjoyed participating in the Lincoln Mechanics’ Institution; this was essentially a community social club promoting reading, discussions, and lectures regarding science’ (Agazzi 1974). Boole lef school at the age o sixteen in order to support his amily, but also took it upon himsel to continue and urther his education. Tus while he was working as an assistant he taught himsel several modern languages and above all mathematics. Boole opened his own school only three years later, spending the ollowing fifeen as a schoolmaster in Lincoln. In 1849 Boole was appointed proessor o mathematics at Queen’s College in Cork, becoming the first to occupy this chair. A short time later he met Mary Everest: then a student o Boole’s, Everest later became a mathematician in her own right. Te two married and had five daughters together. Everest shared Boole’s passion or teaching as well as or mathematics; and indeed she is best known or her work and remarkably progressive ideas on the pedagogy o mathematics.1 Boole remained in Cork until his premature death at the age o orty-nine. Different sources offer slightly varying accounts o how this came about. By piecing them together, it emerges that on a winter day Boole ‘walked rom his residence to the College, a distance o two miles, in a drenching rain, and lectured in wet clothes’ (MacFarlane 1916: 32). By the time he got home later 143
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that day, he already had a ever which quickly developed into pneumonia. But this was the 1860s, when tragically little was known about lung inections; which explains why Mary Boole, armed only with the very best intentions, attempted to heal her husband by dousing him in ice-cold water. Boole died soon aferwards.
1.1 Boole’s accomplishments Boole was a mathematician by training, and during the course o his prolific career he contributed numerous articles to mathematics journals. But Boole’s main interest was logic, and his most significant contributions lie within this field. Boole’s key innovation in logic consisted in the systematic application o mathematical methods to the discipline. Indeed, according to philosopher and historian o logic John Corcoran, George Boole is one o the greatest logicians o all time and . . . can be, and ofen is, regarded as the ounder o mathematical logic. Corcoran 2003: 284
More specifically, Boole’s most prominent contribution to this field o inquiry is the so-called algebra o logic. Boole effectively gave lie to what became a well-established tradition in logic, which continued to flourish long afer his death. What is more, Boole’s work in logic played a key role in the transition rom the Aristotelian tradition in logic to the discipline we know and study today. Boole’s most important work is contained in two monographs: MAL Te Mathematical Analysis o Logic(1847) LO An Investigation o the Laws o Tought, on which are ounded the
Teories o Logic and Probabilities(1854) Boole wrote MAL in response to a controversy between mathematician Augustus De Morgan (1806–1871) and philosopher Sir William Hamilton (1788–1856). De Morgan and Hamilton both demanded credit or the same idea, namely ‘quantiying the predicate’: this was the proposal to add the orms ‘All/Some/No As are All/Some/No Bs’ to the standard orms o the syllogism. Boole’s MAL, an eighty-two-page-long pamphlet, already contained many o the key ideas and results later expounded and expanded upon in his magnum opus, LO.2 Incidentally, De Morgan’sFormal Logic was published in the same year as MAL ; according to some sources, they even came out o press on the very same day. Nevertheless, and although there is a certain amount o overlap between the
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two monographs, Boole’s received ar greater recognition than De Morgan’s. De Morgan himsel later came to recognize the higher merit o Boole’s work, as transpires very clearly in the ollowing passage rom Alexander MacFarlane’s Lectures on en Mathematicians o the Nineteenth Century(1916: 32): De Morgan was the man best qualified to judge o the value o Boole’s work in the field o logic; and he gave it generous praise and help. In writing to the Dublin Hamilton he said, ‘I shall be glad to see his work ( Laws o Tought) out, or he has, I think, got hold o the true connection o algebra and logic.’ At another time he wrote to the same as ollows: ‘All metaphysicians except you and I and Boole consider mathematics as our books o Euclid and algebra up to quadratic equations.’ We might iner that these three contemporary mathematicians who were likewise philosophers would orm a triangle o riends. But it was not so; Hamilton was a riend o De Morgan, and De Morgan a riend o Boole; but the relation o riendship, although convertible, is not necessarily transitive. Hamilton met De Morgan only once in his lie, Boole on the other hand with comparative requency; yet he had a voluminous correspondence with the ormer extending over 20 years, but almost no correspondence with the latter.
Compared to both MAL and De Morgan’s Formal Logic, however, LO received ar greater attention rom logicians and mathematicians alike; or while MAL already contains the seeds o Boole’s main innovations, it is in LO that they are articulated and developed in ull depth. In view o the purpose o this chapter – to provide an introduction to Boole’s logic that is both comprehensive and accessible – we ollow the literature by ocusing mainly on LO.
2 Te laws o thought As illustrated in earlier chapters o this anthology, during the late eighteenth and the nineteenth centuries both logic and philosophical logic had made some significant advances, most notably by the hands o Leibniz and Bolzano. Yet Aristotelian logic still dominated scholarly textbooks and articles in Boole’s time. It was in this context that Boole almost single-handedly set the stage or Gottlob Frege’s revolution and the transition to modern logic. Boole was not just measuring himsel against a century-strong tradition; the backdrop to his education and early work was the psychologist school in philosophy – the same that Frege was to criticize so amously and vehemently only a ew decades later.
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Psychologism has it that logic is really just a branch o psychology. Tus, logic is primarily a descriptive activity, the main purpose o which is to correctly understand and represent the structure and modes o our reasoning. On this view, any normative question about how we ought to reason should thereore deer to (and is thus dependent upon) the descriptive account. Tis is in stark contrast with the modern conception o logic. Te latter is largely predicated on the assumption o logic is that itoharbours uncovering the principles o correct reasoning;that andthe or goal precisely this reason a natural expectation that the descriptive question (How do we reason?) and the normative question (How ought we to reason?) would receive rather different answers. And yet things were very different just over a century ago. Boole’s assessment o psychologism was somewhat more measured than (or instance) Frege’s; nevertheless, ultimately Boole distanced himsel rom the view altogether. However, Nicla Vassallo (1997) makes the interesting observation that Boole only did so afer an initial period during which he was openly quite sympathetic to the view. Te transition (i such it was) seems to have occurred some time between the publication o MAL – where signs o at least a weak psychologistic leaning can still be discerned – and that o LO – where the view is more clearly and explicitly endorsed that logic is first and oremost a normative enterprise. In LO, Boole clearly envisions the purpose o logic as that o representing each and every component o correct or valid reasoning; he is also quick to see the limitations o the Aristotelian system in this respect, and the need to replace (or enrich) it with a much more comprehensive set o principles. Tis, in a nutshell, is what LO sets out to achieve: Te design o the ollowing treatise is to investigate the undamental laws o those operations o the mind by which reasoning is perormed; to give expression to them in the symbolical language o a Calculus, and upon this oundation to establish the science o Logic and construct its method; to make that method itsel the basis o a general method or the application o the mathematical doctrine o Probabilities; and, finally, to collect rom the various elements o truth brought to view in the course o these inquiries some probable intimations concerning the nature and constitution o the human mind. Boole 1854: 1
Notice that in the above passage Boole is rehearsing the main steps in the construction o a ormal theory. oday the concept ‘ormal theory’ is a amiliar
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one, so much so that this specific passage may strike readers as rather unremarkable. But place it in historical context and its significance becomes immediately apparent: aside rom one or two exceptions3 Boole’s LO contains one o the very first approximations o a ormal theory in the modern sense. Tis act alone would give sufficient reason to consider Boole to be a key figure in the history o logic (and philosophical logic); and at the very least it clearly identifies himInaslight a precursor o what o hasFrege just and beenarski. said, we may begin to identiy two principal elements o Boole’s contribution to logic: first, the resort to a ormal theory (and associated ormal language); secondly the creation and adoption o a rigorous method to analyse and construct arguments within such a theory. Te next two subsections provide an overview o each o these in turn.
2.1 Boole’s system Te first task in the construction o a ormal theory is to set up a ormal language or the theory. Te ‘symbolical language o a Calculus’ eatured in LO comprises the ollowing elements: ●
●
●
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terms x, y, z . . . → interpreted as classes the special class 0 → interpreted as the empty class the special class 1 → interpreted as the universe a relation symbol = → interpreted as identity between classes operation symbols: – addition: x + y = v → interpreted as union o classes multiplication: x · y = w → interpreted as intersection o classes – subtraction: −x = 1 −x → interpreted as taking complements o classes no quantifiers.
–
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Te signs and operation symbols that make up the ormal language just described are all borrowed rom the branch o mathematics known today as abstract algebra: the sub-area o mathematics which studies the properties o abstract
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algebraic structures starting rom a set o equations (which hold o the structures in question).4 O course the discipline o abstract algebra was not defined in the same way (or as precisely) then as it is now; and in Boole’s time it was still customary or mathematicians to reer to this discipline as symbolical algebra (owing to the act that variables were then reerred to as symbols). In modern textbooks it is common find ‘the abstract algebraocontrasted with so-called elementary – the lattertobeing arithmetic indefinite quantities or variables’ (Prattalgebra 2007) amiliar rom secondary school. But the two only began to be properly distinguished around the turn o the last century, when the work o several algebraists o the time – Boole being one o them – led to the inception o abstract algebra in the modern sense. Hilary Putnam writes in this regard: Te program – the program o the ‘Symbolic School’ o British analysts – is today out o date, but in a certain sense it was the bridge between traditional analysis (real and complex analysis) and modern abstract algebra . . . Boole was quite conscious o the idea o misinterpretation, o the idea o using a mathematical system as an algorithm, transorming signs purely mechanically without any reliance on meanings. In Boole’ connection logic, this very293–4). important idea appears on the opening pages o s [LOwith ] (Putnam 1982:
It is important to note rom our historical perspective that Boole still had one oot in the old conception o algebra. Tus, or instance, he took afer the algebraists o the time in treating the symbols o his algebra as coextensional with the objects symbolized. In Boole’s case, the symbols were thought o as roughly akin to mental acts, as well as being used to represent and reer to classes: Let us then suppose that the universe o our discourse is the actual universe, so that words are to be used in the ull extent o their meaning, and let us consider the two mental operations implied by the words ‘white’and ‘men’. Te word ‘men’ implies the operation o selecting in thought rom its subject, the universe, all men; and the resulting conception, men, becomes the subject o the next operation. Te operation implied by the word ‘white’ is that o selecting rom its subject, ‘men’, all o that class which are white. Te final resulting conception is that o ‘white men’. Boole 1854: 31–2
It is likely owing to a perceived ambiguity between interpretation and reerent o the symbols o the calculus, in act, that some commentators were later so quick to place Boole in the psychologistic camp. But in this regard it is certainly worth noting that Boole was one o the scarce ew to remain untouched by Frege’s
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accusations o sympathizing with psychologism. On the contrary, in discussing Boole’s work Frege openly expressed his interest in and respect or the ideas developed and put orward in LO. Tis is all the more notable considering the undamental disagreement, at heart, between Frege’s logicist project – which sought to show that arithmetic (at least) could be reduced to pure logic – and Boole’s algebraic approach – which effectively, although not as explicitly, sought to perorm a reduction the opposite Afer introducing theinsymbols o his direction. algebra, Boole goes on to explain that the special ‘universal’ class (represented by the symbol 1) ‘is the only class in which are ound all the individuals that exist in any class’ (1854: 34). Here is Corcoran on Boole’s universal class: Boole’s . . . use o the symbol ‘1’ ound in [LO ], and by translation also the word ‘entity’, marks a milestone in logic . . . In [LO ], ‘1’ indicates not ‘the universe’ [in the sense o ‘most comprehensive class’] but the limited subject matter o the particular discourse in which it is used, what Boole calls ‘the universe o [the] discourse’ which varies rom one discourse-context to another . . . Tis is the first time in the history o the English language that the expression . In this way Boole opens up room or one and ‘universe discourse’ ever in used the same o language to was be used many different interpretations . . . Modern logic . is almost inconceivable without the concept o universe o discourse Corcoran 2003: 273–4; emphases added
We may now go on to present the laws o thought themselves. Te list that Boole finally settled upon comprises six axioms: Law 1 Law 2 Law 3 Law 4
x+y=y+x x·y=y·x x2 = x z · (x + y) = z · x + z · y
Law 5 Law 6
x − y = −y + x z · (x − y) = z · x − z · y
Complementing these axioms are two rules o inerence: Law 7 Law 8
Adding or subtracting equals rom equals gives equals Multiplying equals by equals gives equals
Te principles displayed above bring into sharper ocus Boole’s conception o ‘algebra o logic’ – or ‘mathematical analysis o logic’ – as a kind o applied mathematics. In the displayed ormulas we immediately recognize amiliar
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algebraic properties such as commutativity o addition (Law 1), multiplication (Law 2) and subtraction (Law 5); and distributivity o multiplication over addition (Law 4) and over subtraction (Law 6). Boole was a strongly philosophically minded mathematician and logician, and as such he was sensitive to the importance o providing arguments to support his claims. Tat commutativity is a law o logic, or instance, is argued in the ollowing passage: Now it is perectly apparent that i the operations above described had been perormed in a converse order, the result would have been the same. Whether we begin by orming the conception o men ‘ ’, and then by a second intellectual act limit that conceptionto ‘white men’, or whether we begin by orming the conception o ‘white objects’, and then limit it to such o that class as are ‘men’, is perectly indifferent so ar as the result is concerned . . . And thus the indifference o the order o two successive acts o the aculty o Conception . . . is a general condition o the exercise o that aculty. It is a law o the mind, and it is the real srcin o the law o the literal symbols o Logic which constitutes its ormal expression . . . Boole 1854: 32
O course it is one thing to give an argument, and quite another to give a good argument. We will see in the next section that among the criticisms levelled against Boole is a complaint over the weakness o arguments such as the above. Law 3 departs quite noticeably rom the other laws: in contrast to those, it is not an axiom scheme but an ordinary algebraic equation, the (only) roots o which are x = 0 and x = 1. Partly on this account, it has received considerably more attention in the literature. Boole reers to Law 3 as the ‘special law’ o the algebra o thought, and to its generalized version (below) as the index law:
xn = x5 Why did Boole hail this as a general principle, and conceive o it as belonging among the ‘laws o thought’ ? Boole’s explanation is the ollowing: As the combination o two literal symbols in the orm xy expresses the whole o that class o objects to which the names or qualities represented by x and y are together applicable, it ollows that i the two symbols have exactly the same signification, their combination expresses no more than either o the symbols taken alone would do. In such case [we have]
xx = x. . . . It is evident that the more nearly the actual significations [o x and y] approach to each other, the more nearly does the class o things denoted by the
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combination xy approach to identity with the class denoted by x, as well as with that denoted by y. Boole 1854: 22
Tus, the key to understanding the index law is to think o it not as a principle o algebra but as one o logic (where the latter is o course to be thought o as Boole did). I we place ourselves within Boole’s ramework and adopt his interpretation o the symbols occurring in the equation, then it is a straightorward matter to understand the rationale or Law 3. Recall that the symbol ‘x’ is interpreted as the act o selecting a class o individuals within the universe o discourse (or, more concisely,‘x’ represents a class), and the multiplication symbol (‘·’) is interpreted as intersection (o classes). Ten the logical product x · x = x2 represents the operation o taking the intersection o the class x with x itsel; and this is o course x again. Tus, we have thatx2 = x.6 Boole continues: Te law which it expresses is practically exemplified in language. o say ‘good, good’, in relation to any subject, though a cumbrous and useless pleonasm, is the same as to say ‘good’.are Tus ‘good, good’ men, is equivalent to ‘good’ men. Such repetitions o words indeed sometimes employed to heighten a quality or strengthen an affi rmation . . . But neither in strict reasoning nor in exact discourse is there any just ground or such a practice. Boole 1854: 23
We are now in a position to make a first (partial and provisional) assessment o Boole’s project in LO. Most prominently, it should be clear enough by now that the theory o LO neither coincides with the theory o abstract algebra, nor is construed as such; Boole’s Law 3 alone leaves no doubt about this. Nor is the claim being made that the laws o thought, the laws o logic and the laws o algebra are one and the same thing. Boole’s thesis is more subtle than this, and can be broken down into two components: (i) that algebra can serve as a toolbox and a medium by which to express the laws o logic; and (ii) that the laws o logic, not algebra, embody the laws o thought, i.e. the laws o correct reasoning.
2.2 Boole’s method Having set up a ormal language, the second task envisaged by Boole is ‘upon this oundation to establish the science o Logic and construct its method . . .’ (Boole 1854: 1). Te method developed by Boole to analyse arguments is ofen reerred to as his ‘General Method’ in the literature, and we ollow this usage. We begin by
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presenting the General Method and looking at some o its applications beore embarking on a more critical appraisal o Boole’s work. Boole’s General Method may be reconstructed schematically as comprising the ollowing steps:
1. Convert initial propositions (premises) into algebraic equations. 2. Apply one or more algebraic transormations to the premise equations, ultimately producing conclusion equations. 3. Convert conclusion equations into propositions (conclusions). Rather than merely discussing the General Method in the abstract, it is much more helpul to examine a ew examples ohow it can be put to work (as envisaged by Boole). For a case study that is both natural and historically appropriate, let us consider the categorical propositions o Aristotelian logic. Recall rom Chapter 1 that Aristotelian logic countenances eight types o propositions:
(i) All Ys are Xs (ii) No Ys are Xs (iii) Some Ys are Xs (iv) Some Ys are not- Xs (v) All not- Ys are Xs (vi) No not- Ys are Xs (vii) Some not- Ys are Xs (viii) Some not- Ys are not- Xs Now let us apply the General Method to (i). Since this is a single proposition – as opposed to a multi-line argument – application o the General Method reduces to application o (1): converting the proposition in question into an algebraic equation. But which equation? o work this out, one must examine the internal structure o (i). Te proposition ‘All Ys are Xs’ imposes a specific condition on all things Y and all things X: namely, or the proposition to hold it must be the case that the Xs subsume all o the Ys. Put differently, i ‘X’ and ‘Y’ denote two classes (or subclasses o the universe class), then it must be the case that the Y -class is itsel a subclass o X. In the terminology introduced in Section 2.1, let x, y, v denote X, Y, and the intersection o X and Y, respectively. Te conversion o (i) into algebraic orm then yields the ollowing: All Ys are Xs y = vx ake now the categorical proposition ‘SomeYs are Xs’: in this case the relevant satisaction condition amounts to there being a non-empty intersection o the
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X-class and the Y -class. Let the symbols x, y, z be interpreted as above. Te General Method then produces the ollowing result: Some Ys are Xs xy = v For a final example, let us look at proposition (v). When is the proposition ‘All not- Ys are Xs’ satisfied? When X and Y are disjoint, and everything that is not a
Y is among the Xs. In algebraic orm, this reads: All not- Ys are Xs 1 −y = vx Exercise 1
(a) Convert the remaining five types o propositions rom Aristotelian logic into algebraic orm. (b) Apply Boole’s method to the ollowing argument in propositional orm: (P1) All tigers are mammals. (P2) All mammals are mortal. (C) Tereore, all tigers are mortal.
3 Limitations o Boole’s theory Boole had several critics, both during his lietime and posthumously; in this section we chart some o the main worries raised against LO. Discussion o the more widespread reception o Boole’s work, and more specifically o his legacy to modern logic, will be delayed until Section 4. Here we ocus on two specific charges moved against Boole. Te first concerns the cogency o the arguments presented in LO or the adoption o his General Method; the other, the logical apparatus developed in LO itsel.
3.1 Metatheory It was mentioned earlier on that an important motivation behind Boole’s project was the realization that the Aristotelian ramework, which embodied the orthodoxy at the time, was too narrow in its scope to serve as a satisactory theory o logic. A particularly prominent limitation o the Aristotelian system was its ailure to account or propositions outside o the eight categories, as seen in Chapter 1.
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While Boole was right about the weaknesses in Aristotle’s system, it is also true that a more global comparison o LO with Aristotle’s work is not entirely avourable to Boole. Tere is at least one respect in which Aristotle’s system still surpasses Boole’s, namely its attention toormal rigour. More specifically, it has been noted by some commentators that Boole’s system is neither sound nor complete – nor is there evidence, in LO, that Boole was concerned with veriying whether either oothese metalogical desiderata was satisfied. Granted, the concepts soundness and completeness as we know,define, and prove them today are both more recent than LO, not to mention Aristotle’s Prior Analytics. But the ideas were ar rom novel to logicians working in Boole’s time. According to Corcoran, or instance, where Aristotle had a method o deduction that satisfies the highest modern standards o soundness and completeness, Boole has a semi-ormal method o derivation that is neither sound nor complete. More importantly, Aristotle’s discussions o his goals and his conscientious persistence in their pursuit make o both soundness and completeness properties that a reader could hope, i not expect, to find Aristotle’s logic to have. In contrast, Boole makes it clear that his primary goal was to generate or derive solutions to sets o equations regarded as conditions on unknowns. Te goal o gaplessly deducing conclusions rom sets o propositions regarded as premises is mentioned, but not pursued. Corcoran 2003: 261
Exercise 2
(a) ry to think o two (or more) examples o propositions that cannot be accommodated by the Aristotelian system (or which c. Section 2.2 above). (b) ry to explain in your own words why examples such as these count heavily against the adoption o standard Aristotelian logic.
3.2 Language It is well known that Gottlob Frege (1848–1925), who lived and worked a mere ew decades afer Boole, is almost universally credited with the invention o modern logic. It is also well known that nothing ever happens in a vacuum; and while Frege certainly deserved all the recognition he has received during his lie and ever since, there is some plausibility in the conjecture that Boole’s work on algebraic logic helped set the stage or the Fregean revolution.
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Boole’s work was firmly on the radar by the time Frege began to publish his own work; indeed, there is no doubt that Frege was very well acquainted with LO when he started work on his Begriffsschrif. Evidence to this effect comes rom one o Frege’s earliest unpublished articles (reprinted in Frege 1979), where Boole’s system is discussed on its own merits and in comparison with the theory later developed by Frege himsel. Troughout discussion project. Frege maintains a largely appreciative towards Boole’s this algebraization But he is also (rightly) critical oattitude certain lingering expressive limitations in the theory o LO, which will be mentioned briefly here. For, as Frege notes, Boole had only an inadequate expression or particular judgments such as ‘some 4th roots o 16 are square roots o 4’ . . . and or existential judgments such as ‘there is at least one 4th root o 16’ . . . apparently no expression at all. Frege 1979: 14
Aristotle’s logic is inadequate to account or quantified propositions o either kind; in this instance, however, so is Boole’s. For as we saw earlier in the chapter, Boole’s language o thought does not include quantifier symbols, and a ortiori his General Method is unequipped to treat such propositions. So while Boole’s algebra can accommodate a greater variety o propositions than Aristotle’s logic can, LO also does not reach quite as ar in this direction as one could hope or. Exercise 3
(a) Give two urther examples o quantified propositions which cannot be adequately represented in the language o LO. (b) ry to explain in your own words why these examples are indeed a problem or Boole’s project.
3.3 Method Yet another worry about LO is that Boole offers rather thin arguments in support o the adoption o the algebraic laws as the ‘laws o thought’. Indeed, Boole does little more beyond claiming that the laws may be ‘seen’ to hold by reflecting on the operations perormed by our mind: Sufficient illustration has been given to render maniest the two ollowing positions, viz.: First, that the operations o the mind, by which . . . it combines and modifies the simple ideas o things or qualities, not less than those operations o the
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reason which are exercised upon truths and propositions, are subject to general laws. Secondly, that those laws are mathematical in their orm, and that they are actually developed in the essential laws o human language. Whereore the laws o the symbols o Logic are deducible rom a consideration o the operations o the mind in reasoning. Boole 1854: 32; emphasis added
We may also note that exceptions to the applicability o the algebraic operations are easy to find: ●
●
x + y is defined only i x, y are disjoint x − y is defined only i y is a subclass o x
On the other hand when these conditions do not obtain, the logical operation in question is labelled an uninterpretable. Here is Boole on this very point: Te expression x + y seems indeed uninterpretable, unless it be assumed that the things represented by x and the t hings represented by y are entirely separate; that they embrace no individuals in common. Boole 1854: 66
Te use o just such uninterpretables attracted strong criticism among Boole’s contemporaries, and most notably rom William Stanley Jevons (1835–1882). Jevons, a prominent economist o his time, also cultivated an interest in logic and even published a volume (Jevons 1890) in which he developed a system o logic designed to rival – perhaps, in Jevons’ intentions, to replace – Boole’s LO. But the heart o their dispute is perhaps best captured by certain extracts in the correspondence into which the two entered prior to the publication o Jevons’ monograph. Tus Jevons wrote, or instance: It is surely obvious . . . that x + x is equivalent only to x . . . Proessor Boole’s notation [process o subtraction] is inconsistent with a sel-evident law.
o this, Boole replied that x + x = 0 is equivalent to . . .x = 0; but [. . .] x + x is not equivalent to . . .x.
He continued, o be explicit, I now, however, reply that it is not true that in Logicx + x = x, though it is true that x + x = 0 is equivalent tox = 0. I I do not write more it is not rom any unwillingness to discuss the subject with you, but simply because i we differ on this undamental point it is impossible that we should agree in other s.7
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Boole and Jevons never resolved their disagreement. Seen rom Boole’s perspective, this is understandable: or one thing, conceding Jevons’ point would have called or quite significant revisions o the overarching ramework o LO; or another, it would have amounted to just as significant a departure rom ordinary algebra. But this in turn would have meant departing rom the goal o the srcinal project, namely to show that ordinary algebra could model the laws o thought. Exercise 4
(a) ry to produce at least two counterexamples o your own which show the limitations o Boole’s method. (b) ry to explain in your own words why these are indeed counterexamples to the universality o Boole’s method.
4 Boole’s algebra, and Boolean algebra It is not uncommon to find Boole associated with – or even cited as the ather o – Boolean algebra, and it is natural to wonder whether and to what extent the latter owes its existence to the ormer. Tis final subsection very briefly addresses such questions. In mathematics, the term ‘Boolean algebra’ reers to the sub-discipline that studies the properties o two-valued logics with sentential connectives (and no quantifiers). Boolean algebra is also quite widely regarded as something o a theoretical precursor o computer science. Tus it has come to pass that Boole has sometimes been credited as no less than the grandather o computer science itsel. Now, there is no doubt that some conceptual link does exist between Boole’s account o logic and the algebra o logic developed in LO on the one hand, and the logic o Boolean algebras and computer science on the other. But such claims are historically inaccurate, at best. As several commentators have duly noted,8 there is next to no textual evidence to support the thesis that Boole had in mind any other, and more abstract, applications o his algebra o logic. What is in act known about Boole’s algebra o logic and about the development o Boolean algebra supports a rather weaker claim: namely, that while Boole’s algebra o logic is emphatically not identical with Boolean algebra, there is a considerable amount o overlap between the principles governing the two (about which more will be said shortly). And, o course, there is little doubt that the modern mathematical discipline owes its name to the author o LO:
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Te adjective ‘boolean’, sometimes even ‘Boolean’, is known to mathematicians or a powerul algebra, to electrical engineers and computing specialists who apply it widely in parts o their work, to philosophers as a component o logic, and to historically-minded people as one George Boole (1815–1864), who ormed an algebra o logic rom which we enjoy these modern ruits. Grattan-Guinness and Bornet, 1997: xiii
Against the backdrop o this crucial historical disambiguation, we now move to give a brie introduction to the notion o a Boolean algebra, which will occupy what remains o this section.9 In mathematics, a Boolean algebra (BA) is defined as a set A together with: ●
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a unary operation ¬ two binary operations (join) and (meet) two (distinct) elements 0 and 1
such that or all a, b, c ∈ A the ollowing axioms hold: ●
commutativity:
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associativity:
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distributivity:
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special laws I:
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special laws II :
ab=ba ab=ba a (b c) = (a b) c a (b c) = (a b) c a (b c) = (a b) (a c) a (b c) = (a b) (a c) a0=a a1=a a ¬a = 1 a ¬a = 0
It is not uncommon to find a Boolean algebra defined in terms othe closely related concept o a ring. In abstract algebra, a (Boolean) ring is defined asa set X with: ●
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a unary operation − two binary operations + and · two distinct elements 0 and 1.
Te characteristic axioms or a Boolean ring overlap, but do not coincide with, those or a Boolean algebra: ●
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commutativity o addition: associativity:
a+b=b+a a + (b + c) = (a + b) + c a · (b · c) = (a · b) · c
Boole ●
distributivity:
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special laws I:
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special laws II :
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a + (b · c) = (a + b) ·(a + c) a · (b + c) = (a · b) + (a · c) a+0=a a·1=a a + (−a) = 1 a ·(−a) = 0 2
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idempotence: a = a. Tus in talk o algebras the meet x y corresponds to the ring notationx + y and the join x y to xy.10 From the definitions just given we also immediately see that a Boolean algebra is none other than a commutative ring – i.e. a ring in which commutativity holds or both + (ring sum) and · (ring multiplication). Indeed all other defining conditions o a Boolean ring are satisfied by a Boolean algebra and vice versa. In particular idempotence is always a property o Boolean algebras: that is, or x an arbitrary element o a Boolean algebra we have thatx x = x. Suppose A is a Boolean algebra as defined above. Ten it is reasonably straightorward to prove the ollowing equivalence:
a=ba
iff a b = b.
Exercise 5
ry proving the above equivalence. Hint: the result ollows rom the basic properties o Boolean algebras stated earlier on in the section. o complete this brie overview o Boolean algebras we introduce one last notion, which serves to isolate a certain special class o algebras known as seldual algebras. Let A be a Boolean algebra as beore. We can then introduce a new relation ≤, defined the above Te relation ≤ defines ≤A b ,iff partial order onby theaset which in turnconditions entails thathold. the ollowing conditions areasatisfied or A: ●
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0 is the least element and 1 is the greatest element Te meet a b is the infimum o {a, b} Te join a b is the supremum o {a, b}
An algebra that meets all o the above conditions is termed seldual: or, i we exchange with or 0 with 1 within any axiom, the result is still an axiom o the algebra. Put differently,a sel-dual algebra is one that isinvariant under permutation.
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It must be emphasized that the above is a sketch o the most general (and thus most basic) type o Boolean algebra which can be defined. It should hopeully be straightorward enough to see that stronger systems can be obtained by adding urther conditions to the axioms listed above. Tere are many sources on Boolean algebras; or a conveniently compact and relatively accessible text, see or instance Halmos (1963).
Notes 1 Perhaps her most amous works are Philosophy and Fun o Algebra(London: C.W. Daniel, 1909) andTe Preparation o the Child or Science(Oxord: Clarendon Press, 1904). Incidentally, Mary’s uncle was Sir George Everest, afer whom Mount Everest is named. 2 On the other hand, there is some controversy in the literature over the nature o the philosophical views underwriting MAL and LO, as well as the question o whether there is a discontinuity between the ormer and the latter in this respect. See or instance Vassallo (1997) on this point. 3 One such notable exception, o course,is Bolzano, or which c. Chapter 4 o this volume. 4 For a comprehensive yet accessible introduction toalgebras in general, see or instance Pratt (2007). 5 Note that whereas both versions o the law are discussed in MAL, no mention o the index law is made in LO. 6 Analogous reasoning can be applied to show that the index law holds in its generalized version also, o course. 7 All o the quotations reproduced here are extracted rom Burris (2010) and Jourdain (1914). 8 Corcoran writes: ‘[Boole] did not write the first book on pure mathematics, he did not present a decision procedure, and he did not devise “Boolean algebra” ’ (2003: 282). 9 Note on terminology: in order to avoid ambiguities, Boole’s will here occasionally be reerred to as boolean algebra (lower case ‘b’), to distinguish it rom the specialist concept o Boolean algebra. 10 Although notation is not uniorm everywhere: in the context o ring theory, or instance, the join is sometimes defined asx y = x + y + xy. Te lack o uniormity in notation derives rom the ambiguity over the use o ‘+’, which should be interpreted as ring sum.
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Reerences Agazzi, E. (1974), ‘Te rise o the oundational research in mathematics’, Synthese 27: 7–26. Boole, G. (1847), Te Mathematical Analysis o Logic, Being an Essay owards a Calculus o Deductive Reasoning, Cambridge: Macmillan, Barclay & Macmillan. Boole, G. (1854), An Investigation o the Laws o Tought, on which are ounded the
Mathematical Teories o Logic and Probabilities, Cambridge and London: Walton & Maberly. Burris, S. (2010), ‘George Boole’, in Edward N. Zalta (ed.),Te Stanord Encyclopedia o philosophy, Stanord: University o Stanord Press. Corcoran, J. (2003), A ‘ ristotle’s Prior analytics and Boole’sLaws o thought,’ History and Philosophy o Logic24: 261–88. Frege, G. (1979), ‘Boole’s logical Calculus and the Concept-script’, in Hermes, H. et al. (eds), Posthumous Writings(P. Long and R. White,transl.), Oxord: Basil Blackwell. Grattan-Guinness, I., and Bornet, G. (eds) (1997), George Boole. Selected Manuscripts on Logic and its Philosophy, Basel: Birkhaüser Verlag. Halmos, P. 1( 963), Lectures on Boolean Algebras, Van Nostrand Mathematical Studies (Hal- mos and Gehring, eds), Princeton and London:Van Nostrand. Jevons, W.S. 1( 890), , ed. R. Adamson and H.A. LogicHill andPub. Other Works Jevons, New York:Pure Lennox & Minor Dist. Co. , repr. 1971. Jourdain, P.E.B. 1( 914), ‘Te development o the theories o mathematical logic and the principles o mathematics. William Stanley Jevons’,Quarterly Journal o Pure and Applied Mathematics, 44: 113–28. MacFarlane, A. (1916), en British Mathematicians o the Nineteenth Century, Mathematical monographs vol. 17 (Merriman and Woodward, eds). Pratt, V. 2( 007), A ‘ lgebra’, in Edward N. Zalta (ed.),Te Stanord Encyclopedia o Philosophy, Stanord: Stanord University Press. Putnam, H. (1982), ‘Peirce the logician’, Historia mathematica9: 290–301. Van Heijenoort, J. (1967), ‘Logic as calculus and logic as a language’, Synthese 17(3): 324–30. Vassallo, N. (1997), ‘Analysis versus Laws: Boole’s explanatory psychologism versus his explanatory anto-psychologism’, History and philosophy o logic18: 151–63.
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Mathematical Logic
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C.S. Peirce Peter Øhrstrøm
Charles Sanders Peirce (1839–1914) was a spokesman or an open and undogmatic understanding o logic. He gothis inspiration first and oremost rom the medieval juxtaposition o three o the seven ree arts into the so-called trivium. Te trivium consisted o the disciplines o Grammar, Dialectics (or Logic) and Rhetoric. Peirce’s work rom 1865 to 1903 shows a constant development o reflections on the content and application o this tripartition (see Fisch 1978). In the spring o 1865, he subdivided the general science o representations into‘General Grammar’, ‘Logic’ and ‘Universal Rhetorics’. In May the same year, he called this division ‘General Grammar’, ‘General Logic’ and ‘General Rhetorics’, and in 1867 it was presented as ‘Formal Grammar’, ‘Logic’ and ‘Formal Rhetorics’. wenty years later, in 1897, it had become ‘Pure Grammar’, ‘Logic Proper’ and ‘Pure Rhetorics’. In 1903 Peirce – within his own now more matured ramework – determined the tripartition as ‘Speculative Grammar’, ‘Critic’ and ‘Methodeutic’. Not only reflections on the trivium as such, but also several other elements o medieval logic had an impact on Peirce’s analyses. One example is the tripartition o the subjects o logic into terms, propositions and arguments – a division which can be ound in almost every medieval introduction to logic. It was clear to Peirce that this classification was relevant, not only within logic (in the narrow sense), but also within both grammar and rhetorics, a act which had also been recognized by the ancients and the medievals. It should be mentioned, however, that Peirce rejected the idea o completely non-assertoric terms. In his opinion, even terms are in general assertoric (see his Collected Papers – hereafer CP – section 2.341). Peirce’s extensive knowledge o classic and scholastic logic also meant that he would not accept logic as a completely untemporal or atemporal science. He could well imagine a new development o a logic, which would take time seriously. In act, Peirce’s philosophy contains eatures which could be interpreted 165
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as an emergent logic or events. For example, he defined the notion o a ‘oken’ as applying to ‘A Single event which happens once and whose identity is limited to that one happening or a Single object or thing which is in one single place at any one instant o time’ (CP 4.537). However, Peirce was hesitant about advancing a ormal logic o time himsel, but nevertheless it is relatively easy in his authorship to find clear ideas, which can be used in a presentation o a ormal logic time. Tis a great inspiration Prior, who part in the and 1960sosucceeded inwas re-establishing the logicor o A.N. time as a proper o 1950s logic. In Prior’s first great time logical work ime and Modality (1957), he gave a brie presentation o the history o the modern logic o time in an appendix; about a quarter o this exposition is devoted to the importance o Peirce with respect to the development o the new logic o time. Tis broader view o logic is also evident in the most remarkable logical ormalism suggested by Peirce, the existential graphs. His idea was that in symbolic logic ‘it will be convenient not to confine the symbols used to algebraic symbols, but to include some graphical symbols as well’ (CP 4.372). Peirce introduced three kinds o existential graphs: Alpha, Beta and Gamma Graphs. Te system o Alpha Graphs corresponds to what we now call propositional logic, the system o Beta Graphs corresponds to firstorder predicate calculus, whereas the systems o Gamma Graphs correspond to various kinds o modal (including temporal) logic. When Peirce died in 1914 his system’s Gamma Graphs were still very tentative and unfinished. In the ollowing, we shall investigate the main structures o the axiomatic systems which Peirce suggested with his Alpha and Beta Graphs, and also some o the ideas o the Gamma Graphs he was considering. It should, however, be kept in mind that the system o Beta Graphs is an extension (or generalization) o the system o Alpha Graphs, whereas any system o Gamma Graphs will be an extension (or generalization) o the system o Beta Graphs.
1 Te Alpha Graphs In Peirce’s graphs, the statements in question are written on the socalled ‘sheet o assertion’, SA. In act, the empty sheet o assertion is supposed to be an axiom, essentially a sort o a constant corresponding to ‘ruth’. Tis is the only axiom in the Peircean Alpha-system. Propositions on the SA may be enclosed using socalled ‘cuts’, which in act correspond to negations (we also speak o ‘negated contexts’). Tat is, the ollowing combination o graphs means thatP is the case, Q is not the case, and the conjunction o S and R is not the case.
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Here the box represents the SA, whereas the curved closures symbolize the cuts, i.e. the negated contexts. In terms o the established ormalism o propositional logic, the above graph is equivalent to
P ~Q ~(S R) It is obvious that such conjunctional orms are rather easy to represent in Peircean graphs. Disjunctions and implications are, however, slightly more complicated. Te implication,P ⊃ Q, is represented by the ollowing graph:
In standard ormalism, this graph can be expressed as ~(P ~Q), which is exactly the traditional definition o material implication in terms o negation and conjunction. Clearly, in this case, we have an example o nested cuts, since theQ is placed in a cut, which is inside another cut. In this way, we may in act have very complicated systems o several nested cuts. In such cases, we may speak o sub-graphs being oddly or evenly enclosed. In the present case, we may say that Q is evenly enclosed, whereas P is oddly enclosed. Peirce made several ormulations o his logic o his existential graphs (see CP Vol. 4 book 2). In a slightly modernized orm inspired by Roberts (1973: 138), the rules or the Alpha Graphs can be listed as ollows:
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R1. Te rule o erasure. Any evenly enclosed graph may be erased. R2. Te rule o insertion. Any graph may be scribed on any oddly enclosed area. R3. Te rule o iteration. I a graph P occurs in the SA or in a nest o cuts, it may be scribed on any area not part o P, which is contained by the place o P. R4. Te rule omay deiteration. iteration be erased.Any graph whose occurrence could be the result o R5. Te rule o the double cut. Te double cut may be inserted around or removed (where it occurs) rom any graph on any area. In addition to these rules, there is as mentioned above just one axiom: the empty graph (SA). It is a remarkable theoretical result that this system o Peirce’s Alpha Graphs corresponds exactly to what is now called standard propositional calculus (see Sowa 1992, 2010). Tis means that any theorem which is provable in ordinary propositional logic (i.e. any proposition which may be shown to be a tautology using ordinary truth-tables) may also be demonstrated in Peirce’s system. A ew examples may be given in order to illustrate how the Peircean system o existential graphs works:
o the lef a double cut is introduced (by R5). o the right the propositionQ is inserted (by R2).
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o the lef the propositionQ is iterated (by R3). o the right the proposition P is inserted (by R2). Tis constitutes a proo o the theorem:
(P Q) ⊃ Q o prove a theorem corresponding to a certain graph one must transorm the empty SA into thesignificantly graph in question. John Sowaa (1992, 2010) has argued that it is in many cases easier to prove theorem by using the graphs rather than the established logical procedures.
2 Te Beta Graphs In the Beta Graphs, Peirce introduced a predicate calculus with a quantification theory ormulated in terms o what he called ‘lines o identity’ (ligatures). Tese graphs are immediately designed or existential statements. Te statement which is now normally ormalized as x:q(x) is represented by the graph:
Universal statements have to be represented in a slightly more complicated way using two cuts (i.e. two negations) corresponding to the ormula ~(x:~q(x)):
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It is evident that the introduction o lines o identity (ligatures) gives rise to a number o complicated questions which Peirce had to deal with. It seems that or a longer period he was trying to find the optimal ormulation o the logical rules or the Beta Graphs which should obviously be seen as an extension o the system or Alpha Graphs. Roberts (1973: 138) has suggested a reasonable exposition o Peirce’s ideas and enumerates the rules or the Alpha and Beta Graphs as ollows: R1. Te rule o erasure. Any evenly enclosed graph and any evenly enclosed portion o a line o identity may be erased. R2. Te rule o insertion. Any graph may be scribed on any oddly enclosed area, and two lines o identity (or portions o lines) oddly enclosed on the same area may be joined. R3. Te rule o iteration. I a graph P occurs in the SA or in a nest o cuts, it may be scribed on any area not part o P, which is contained by the place o P. Consequently, (a) a branch with a loose end may be added to any line o identity, provided that no crossing o cuts results rom this addition; (b) any loose end o a ligature may be extended inwards through cuts; (c) any ligature thus extended may be joined to the corresponding ligature o an iterated instance o a graph; and (d) a cycle may be ormed, by joining by inward extensions, the two loose ends that are the innermost parts o a ligature. R4. Te rule o deiteration. Any graph whose occurrence could be the result o iteration may be erased. Consequently, (a) a branch with a loose end may be retracted into any line o identity, provided that no crossing o cuts occurs in the retraction; (b) any loose end o a ligature may be retracted outwards through cuts; and (c) any cyclical part o a ligature may be cut at its inmost part. R5. Te rule o the double cut. Te double cut may be inserted around or removed (where it occurs) rom any graph on any area. And these transormations will not be prevented by the presence o ligatures passing rom outside the outer cut to inside the inner cut. In addition to these rules there are two axioms: the empty graph (SA) and the unattached line o identity. From these two axioms it is possible to derive a number o theorems and rules using (R1–5). For instance, the graph corresponding to the ollowing implication
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x: q(x) ⊃ x: q(x)
turns out to be provable, as demonstrated by Roberts (1992). Te axiom o the unattached line o identity has a crucial role to play in this proo. Tis means that quantification cannot be empty in the logic o Beta Graphs (which is in act also the case with Prior’s quantification theory). With that restriction, it can be shown that theorems which can be proved in first-order logic can also be proved in terms o existential graphs – and vice versa. Utilizing some illustrative examples, we shall consider how some o the classical syllogisms can be handled in terms o Peirce’s existential graphs. Te so-called ‘Barbara’ is the most amous among the Aristotelian syllogisms: All M are P All S are M Tereore: All S are P In terms o Peirce’s existential graphs, it can be demonstrated that the conclusion ollows rom the two premises o the argument:
Te two premises in the argument can be represented on the sheet o assertion in terms o the first diagram.
In the second diagram the graph to the lef has been iterated into the interior o the graph to the right.
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A ligature is added to M, extended inwards and joined with the other occurrence o M.
Here M is deiterated, extended inwards and joined with the other occurrence o M.
Here M is erased rom an even area.
Here a double cut is removed.
Here the graph to the lef (the first premise) has been erased and the graph to the right (the conclusion) has been made a bit nicer.
Another syllogistic example would be the so-called ‘Ferio’, which may be stated in this way:
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No M are P Some S are M Tereore: Some S are not P Te conclusion o this syllogism may be proved rom the premises in the ollowing way using Peirce’s existential graphs:
Here a ligature is added joining the two occurrences o M. Ten deiteration o M and erasure o M lead to the conclusion (made a bit nicer in the last diagram). In this way the syllogism ‘Ferio’ in the first figure is proved.
3 Te need or more than Alpha and Beta Graphs Te logic o Beta Graphs is clearly useul in many cases. Tey can be shown to correspond to what has later been termed first-order predicate calculus. Peirce realized, the Alphathe and Beta Graphs are not satisactory in all cases. Forhowever, instance, that he considered ollowing two propositions (see CP 4.546):
1. Some married woman will commit suicide, i her husband ails in business. 2. Some married woman will commit suicide, i every married woman’s husband ails in business. Peirce argued that these two conditionals are equivalent i we analyse them in a merely classical and non-modal logic – i.e. in terms o Beta Graphs within his own logical system. For the sake o simplicity, we reormulate the problem using only predicates with one argument.
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According to Peirce’s rules or Beta Graphs and their lines o identity, the graphs corresponding to (1) and (2) can be proved to be equivalent, i.e.
– where ail(x) means ‘x is a woman married to a businessman who ails in business’, and suicide(x) means ‘x commits suicide’. Tis equivalence can be established by the rules o transormation or Beta Graphs. Te two graphs respectively correspond to the ollowing two expressions o standard predicate notation (where quantification is understood to be over the set o women married to businessmen): (1a) (x)( ail(x) ⊃ suicide(x)) (2a) (x)((y)ail(y) ⊃ suicide(x)) Both o these expressions are equivalent with
( x)~ail(x) (x) suicide(x) Te inerence rom (1a) to (2a) appears rather natural. I there is a wie who will commit suicide i her husband ails in business, then it is obvious that this woman will also commit suicide in the case that all husbands ail in business. However, the opposite inerence is clearly counterintuitive. Even i there is a woman who will commit suicide in the case that all husbands ail in business, it does not mean that this woman will commit suicide in the case that her own husband ails in business. Nevertheless, (1a) and (2a) turn out to be logically equivalent, as long as we are moving strictly within classical predicate logic, respectively the Beta Graphs. Tereore, as long as we are trying to represent our case within those systems, we are obliged to accept the counterintuitive inerence. However, it may be more natural to ormulate the problem in terms o three predicates, so let wie(x) stand or ‘x is the wie o a businessman’. When the
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statements (1) and (2) are represented with these three predicates, the crucial question will be the transition between the ollowing graphs:
Again, these graphs can be shown to be equivalent. Essentially, their equivalence is due to the act that the term wie is outside t he scope o the negations. Tereore, the rules o iteration and deiteration or Beta Graphs can be applied to the inner copies. Te proo using Beta Graphs could run as indicated below.
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In the step (S1) a ligature is added joining to two occurrences o wie-. In step (S2), the inner occurrence o wie- is deiterated. In step (S3), a double cut is removed leading to the result. In terms o standard ormalism (1) and (2) are represented by: (1b) ( x)(wie(x) (ail(x) ⊃ suicide(x))) (2b) ( x)(wie(x) ((y)(wie(y) ⊃ ail(y)) ⊃ suicide(x))) Using undamental equivalences rom first-order logic, (1b) and (2b) can be written as disjunctions (1b’) (x)(wie(x) ~ail(x)) (x) (wie(x) suicide(x))) (2b’) (x)(wie(x) ((y)(wie(y) ~ail(y)) suicide(x))) By the ‘omission’ o wie(x) in the first part o the disjunction in (2b’), it becomes evident that (1b’) ollows rom (2b’). I act, it is evident that the two propositions are equivalent. Peirce stated that the equivalence o these two propositions is ‘the absurd result o admitting no reality but existence’ (CP 4.546). As Stephen Read (1992) has pointed out, Peirce’s analysis is a strong argument against anyone inclined to assert that conditionals in natural language are always truth-unctional. But the Peircean analysis is also an argument or the need o a new tempo-modal logic. Peirce ormulated his own solution in the ollowing way: I, however, we suppose that to say that a woman will suicide i her husband ails, means that every possible course o events would either be one in which the husband would not ail or one in which the wie will commit suicide, then, to make that alse it will not be requisite or the husband actually to ail, but it will suffice that there are possible circumstances under which he would ail, while yet his wie would not commit suicide. CP 4.546
Tis means that we have to quantiy over ‘every possible course o events’. A.N. Prior’s tense-logical notation systems provide the means or doing just that. Te operator suited or the problem at hand is G, corresponding to ‘it is always going to be the case that’. Prior established a system designed to capture Peirce’s ideas on temporal logic – appropriately called ‘the Peircean solution’. In the Peircean system, G means ‘always going to be in every course o events’. Using the operator in this way, we can express (1) and (2) as respectively (1c) ( x)G(ail(x) ⊃ suicide(x)) (2c) ( x)G(( y)ail(y) ⊃ suicide(x))
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(It should be mentioned that a linguistically more appropriate representation perhaps should take the orm N(p ⊃ Fq), where N and F stands or ‘necessarily’ and ‘at some uture time’, respectively. However, (1c) and (2c) are sufficient or the conceptual considerations which are important here.) Expression (1c) clearly means that there is some married woman w or whom the proposition, ~ail(w) suicide(w), holds at any time in any possible uture course o events. means that there is a married woman or whom the suicide(w) proposition, ( y) (2c) ~ail(y) , holds at any time in anyw possible uture course o events. For this reason it is ormally clear that (1c) entails (2c), but not conversely. And this corresponds exactly to intuition with respect to the two statements (1) and (2): the inerence rom (1) to (2) is valid, but Peirce was justified in maintaining that the inerence rom (2) to (1) should be rejected. Generally speaking, some kind o tempo-modal logic is required or describing conditionals in natural language reasoning in a satisactory way. Peirce’s considerations on the example discussed above clearly demonstrate that he realized this. In act, his observations served as a strong case or the development o the Gamma Graphs.
4 Te Gamma Graphs Peirce himsel made some attempts at solving the problems o modality by introducing a new kind o graph. In what he called ‘Te Gamma Part o Existential Graphs’ (CP 4.510 ff.), he put orward some interesting suggestions regarding modal logic. Some o his considerations on this topic were linked to what is now called epistemic logic, i.e. the logic o knowledge. In the ollowing we shall describe his ideas. In epistemic logic, the idea is that relative to a given state o inormation a number o propositions are known to be true. In Peirce’s graph theory, propositions describing the inormation in question should be written on the ‘sheet o assertion’ SA, using just Alpha and Beta Graphs. Other propositions, however, are to be regarded as merely possible in the present state o inormation. Peirce represented such propositions using ‘broken cuts’, combined with the ‘unbroken cuts’, which we already know rom the Alpha and Beta Graphs. A broken cut should be interpreted as corresponding to ‘it is possible that not . . .’. Tis means that ‘it is possible that . . .’ must be represented as a combination o a broken and an unbroken cut:
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Now, consider a contingent proposition, i.e. a proposition, which is possible, but not necessary according to the present state o inormation. In this state, the sheet will include at least two propositions:
Now, suppose that p is contingent relative to some state o inormation, and that we then learn that p is in act true. Tis would mean that the SA should be changed according to the new state o inormation. Te graph corresponding to Mp (‘possibly p’) should be changed into the graph or ‘it is known with certainty that p’, i.e. ~M~p. Obviously, this means that the graph or M~p should be dropped, which results in a new (and simpler) graph representing the updated state o inormation. In this way, Peirce in effect pointed out that the passage o time does not only lead to new knowledge, but also to a loss o possibility. With respect to this epistemic logic, Don D. Roberts (1973: 85) has observed that the notions o necessity and possibility both may seem to collapse into the notion o truth. Roberts himsel gave an important part o the answer to this worry by emphasizing how ‘possibility and necessity are relative to the state o inormation’ (CP 4.517). In the context o existential graphs, Peirce in act established an equivalence between ‘p is known to be true’ and ‘p is necessary’. In consequence, ‘p is not known to be alse’ and ‘ p is possible’ should also be equivalent in a
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Peircean logic. Tereore, the kind o modal logic which Peirce was aiming at was in act an epistemic logic, which should be sensitive to the impact o time.
5 empo-modal predicate logic and existential graphs Peirce was concerned with the epistemic aspect o modality, but he also wanted to apply his logical graphs to modality in general – that is, to use them or representing any kind o modality. However, he was aware o the great complexity in which a ull-fledged logic involving temporal modifications would result. Tis is probably the reason why Peirce’s presentations o the Gamma Graphs remained tentative and unfinished. In the ollowing, we intend to explain some o the problems he was acing, and suggest some ideas regarding the possible continuation o his project. Our analysis o the problem rom CP 4.546 suggests that the two statements should in act be understood as ollows: (1’) Some married woman will (in every possible uture) commit suicide i her husband ails in business. (2’) Some married woman will (in every possible uture) commit suicide i every married woman’s husband ails in business. We intend to ormulate a graph-theoretical version o the tense-logical solution. So, we have to make sure that there are proper graphical representations o (1’) and (2’) such that the graphs are non-equivalent. In act, it is not difficult to create graphs corresponding to the modal expressions in (1’) and (2’). Obviously, a graph with a broken cut inside an unbroken cut with q would clearly correspond to the statement ‘in every possible uture q’. A representation o (1’) and (2’) could then be (omitting the SA):
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In order to treat problems like the one we have been discussing, we must be able to handle graphs involving the two kinds o cuts (broken and unbroken) as well as lines o identity. In consequence, we have to establish rules or modal conceptual graphs, specifically such that (1’) and (2’) would be non-equivalent. Although (2’) should entail (1’), the opposite implication should not hold. Te question regarding the relation between modaloperators and quantifiers is crucial or any modal predicate logic. Peirce was aware o this problem. He stated: Now, you will observe that there is a great difference between the two ollowing propositions: First, Tere is some one married woman who under all possible conditions would commit suicide or else her husband would not have ailed. Second, Under all possible circumstances there is some married woman or other who would commit suicide or else her husband would not have ailed. CP 4.546
It is very likely that what Peirce had in mind was the insight that we cannot with complete generality derive x: Ns(x) rom N(x: s(x)) – that is, not without making some restrictions. Tis is in act a very old wisdom which was also known to the medieval logicians. One cannot deduce ‘there is a man who will live orever’ rom ‘it will orever be the case that there is a man’. Te logic o Peirce’s existential graphs (in particular the Gamma Graphs) turns out to be very complicated. From a modern point o view, a lot has to be carried out in order to give the system the kind o mathematical rigour which would now be expected in ormal logic. Although an important contribution to this project has been made by Frithjo Dau (2006), there is still much to do in order to develop the ormal and mathematical aspects o the Peircean graphs. On the other hand, enough is known already about the graphs to conclude that they present genuine and interesting alternatives to propositional logic, firstorder predicate logic and various kinds o modal logic. In addition, John Sowa (1992, 2010) has demonstrated that they can be used or knowledge representation in a very illustrative and intuitively appealing manner.
Reerences Dau, Frithjo (2006), ‘Mathematical Logic with Diagrams. Based on the Existential Graphs o Peirce’, U Dresden,http://www.dr-dau.net/Papers/habil.pd (accessed 10 August 2016)
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Fisch, Max (1978), ‘Peirce’s Central Teory o Signs’, in Tomas A. Seboek (ed.),Sight, Sound and Sense, Advances in Semiotics, Bloomington: Indiana University Press, pp. 31–70. Peirce, Charles Sanders (1931–58), Collected Papers o Charles Sanders Peirce, Vols I–VIII , ed. C. Hartshorne, P. Weiss and A. Burke , Cambridge, MA: Harvard University Press [CP]. Prior, A.N. (1957), ime and Modality, Oxord: Clarendon Press. Read, Stephen (1992), ‘Conditionals are not truth-unctional: an argument rom Peirce’, Analysis, January 1992, pp. 5–12. Roberts, Don D. (1973), Te Existential Graphs o Charles S. Peirce, Te Hague: Mouton. Roberts, Don D. (1992), ‘Te existential graphs’, Computers & Mathematics with Applications, Vol. 23, Issue 6, pp. 639–63. Sowa, John F. (1992), ‘Logical Foundations or Representing Object-Oriented Systems ’, Swiss Society or Medical Inormatics. Sowa, John F. (2010), ‘Existential Graphs. MS 514 by Charles Sanders Peirce: A commentary’, http://www.jsowa.com/peirce/ms514.htm(accessed 10 August 2016).
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Frege Walter B. Pedriali
Introduction Here’s a neat story. Afer centuries o stagnation, logic was dramatically transormed, more or less at a stroke, by the appearance o Frege’sBegriffsschrif in 1879, the annus mirabilis in the history o logic, the watershed year when, the story goes, modern logic was born. 1 In that epoch-changing work Frege single-handedly modern logic, finally ree rom its restrictive Aristotelian invented mould. His achievement is all breaking the moreitextraordinary given that Frege’s groundbreaking work was developed in nearcomplete isolation and greeted with little or no peer recognition at all. Tis heroic picture o Frege’s solitary and unrewarded intellectual efforts is explained mostly by two circumstances: the awkward, cumbersome two-dimensional notation he employed (a notation that most readers have since ound hard to decipher), and the act that the ormal system Frege later developed in Grundgesetze (1893) turned out to be inconsistent, thus destroying both his hope o establishing a new era or logic as a scientific discipline properly so-called as well as whatever reputation he had gained since the Begriffsschrif days. Frege wasn’t just a heroic figure in the intellectual history o our subject, then, but a tragic one too. Te story, thus narrated, is very neat; indeed, it is a littletoo neat. It is too neat because it overlooks the importance o the work o De Morgan, Boole, Jevons, McColl, Schröder and Peirce (the modern turn in logic actually predates 1879). It is also too neat because it glosses over the act that modern logic differs in important respects rom that o Frege (logic became truly ‘modern’ a lot later than 1879). Tere is, however, a sense in which the story gets things essentially right. Begriffsschrif is without question the first work in which we find a airly 183
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systematic presentation o a ormal system that is recognizably similar to those developed today – the modern way o doing logic, that is, was undoubtedly born with that slim book. Moreover, no other single work in logic beore or since contains as many wide-ranging technical innovations packed within a ew pages (eighty-eight in all). Finally, it was only with Begriffsschrif that a unitary treatment o propositional andbetween quantificational logicand wasStoic at last provided, resolving the longstanding dispute Aristotelian logic by giving them a common home. In short, while it is certainly true that the ull story o the birth o modern logic is a rather complex affair, hardly reducible to a single event, it is also the case that i one nevertheless wants to single out a pivotal point or expository purposes, then the appearance o Begriffsschrif in 1879 is arguably the best place where to locate the watershed dividing ancient rom modern approaches to logic. My task in this chapter is to spell out the respects in which Begriffsschrif (and Frege’s contribution more generally) marks the onset o modern logic. o that aim, I will bring out the key innovative eatures o Frege’s pioneering work in logic, all the while indicating the ways in which that work differs rom our present conception o logic (and its implementation in the various systems o logic currently being employed).2 I will start by giving a very brie sketch o the birth o modern logic (somewhat qualiying the standard story sketched above). In section 2, I discuss Frege’s philosophical outlook, while in section 3 I lay out the system given in Begriffsschrif, highlighting its main novelties. In section 4 I close the chapter by briefly sketching the contradiction that arose in Frege’s late work,Grundgesetze.
1 Locating the watershed We start by asking: in what sense could Frege be said to have inaugurated modern logic? And what is modern logic anyway? Let’s tackle the second question first. At a first approximation, we can say that logic became modern when it became mathematical logic, the study o mathematical reasoning, done by mathematical means and or mathematical purposes. More specifically, logic became modern when a proper apparatus o quantification able to cope with the complexities o multiple generality in mathematical reasoning was developed (we’ll see in a moment what all o this means). Finally, and relatedly, we can say that logic became modern when it itsel
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became an object o mathematical study: that is, logic became modern when meta-logic (the study o the structural properties o logical systems) was born. And clearly, metalogic could only be developed once ways o making logical systems precise had been made available. o sum it all up in a slogan, then, logic became distinctively modern when it became the study o proo rather than o argument. With that roughassketch on board we can now askWell, the first question: is Frege properly regarded the ounder o modern logic? on the standard story the answer is a resounding yes, and this is so because Begriffsschrif displays – right rom its subtitle, ‘a ormal language o pure thought modelled afer that o arithmetic’ – all o the characteristic marks o modern logic that we’ve just reviewed, above all in its presentation o the first extended treatment o quantification.3 Moreover, Frege made it possible pretty much or the first time to treat logic as a ormal system. Tat is to say, he gave us logic as anobject o study, an object that we could test or consistency so as to veriy the good epistemic standing o our ‘logical convictions’.4 Te story, however, and as already anticipated, needs some qualification. o finesse the story ever so slightly, then, let’s first distinguish three main strands in the evolution o modern logic, namely, i) the algebraic tradition that went rom Boole to arski and beyond, spawning in its wake the model-theoretic tradition that was to be developed in the 1920s and 1930s (in this tradition one studies mathematical structures, the operations that can be defined over them, and the operation-preserving mappings between those structures); ii) the broadly Aristotelian tradition that studied argument-schemata (in this tradition, one tries to capture with some degree o ormality maximally general truthpreserving modes o reasoning irrespective o subject matter); and finally iii) the proo-theoretic tradition in which Frege and his ormalist rival Hilbert worked (in this tradition, one sets up deductive systems, studies their ormal properties, and engages in theorem proving).5 Now, as it happens, Frege had little inclination towards what was to become the model-theoretic approach to logic. In particular, he had no truck with the idea that a ormal language could be amenable to multiple interpretations. His conception o logic was universalist, by which it is meant that the quantifiers would range unrestrictedly over all objects, that the domain o quantification was not variable but fixed once and or all.6 And the task o the logically perect language that Frege strived to construct was to capture the unique inerential relations that hold together ‘the natural order o truths’, an order that was immutable and o the widest generality.7 In addition, Frege also thought that his
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system was a substantial improvement over those o Boole and Schröder precisely because it went against (or rather beyond) the algebraic approach to logic.8 In particular, Frege’s syntax comes with an interpretationalready packed into it.9 Better put: unlike Boole’s, it is a language o such absolute generality as not to stand in need o interpretation (since any interpretation would represent a restriction o that generality).10 now the claim Frege theextent ather that o modern logic begins o to look bit And problematic. Why?that Well, to isthe one’s conception logica is characterized by one’s conception o logical consequence, the contemporary view o consequence as truth-preservation in all models is one that is heavily shaped by the model-theoretic tradition and it is one that would have struck Frege as alien and indeed contrary to what he thought o as the proper view o logic – rom his universalist perspective, it is a grave mistake to try and explain the notion o logical truth in terms o truth in all models. In short, i conronted with it Frege might well have disowned modern logic as not counting as logic at all by his lights, given its mistaken conception o logical truth.11 Tis act alone seems to somewhat weaken the claim that Frege was the ounder o modern logic; perhaps, and to switch metaphors, we should rather see him as just one (highly striking) thread in a much wider tapestry than the one being woven in Begriffsschrif.12 And there is more. Frege’s logic is higherorder (it quantifies over unctions, not just objects) and it is meant to be that way (Frege’s logicist project, o which more later, required a logic with that kind o expressive power). But on the modern conception, higher-order logic involves existential assumptions that make it controversial to claim that itis logic.13 Indeed, the view that is still dominant today is that first-order logic is logic properly so called precisely because o its minimal existential assumptions – all it assumes is that its domains o quantification be non-empty. Higher-order logics, by contrast, include logical truths that make demands on the size o those domains, since they require infinite domains (not just non-empty ones) or their obtaining. It is, however, elt that the question o whether there are infinitely many objects is one over which logic should not adjudicate (nor is it one that logic should presuppose as settled one way or another). Furthermore, and, again, relatedly, higherorder logics lack certain key metatheoretical properties that arenow taken to be definitive o the logical.14 Accordingly, the claim that 1879 is the year modern logic was born also needs qualiying with respect to the kind o logic that was born then.15 Concerns about existential assumptions and desirable metatheoretical properties eventually led to a airly stable (but not entirely uncontroversial) consensus that logic proper
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(i.e. the genuinely logical core o our thinking) is first-order logic, and thus only a ragment o what counted as logic or Frege. Tat consensus, however, arguably became truly entrenched only in the 1960s when Lindström’s theorems gave it a solid ormal grounding.16 Modern logic is thus a airly recent affair and one whose concerns are ar removed rom those o Frege. And 1879’s privileged status looks no less problematic when we consider Frege’s hisdeep moreincomprehension immediate successors. workhad wasbothered met, almost withoutinfluence exception,onwith by the His ew who to read it.17 wo aspects contributed to this. First, Frege’s two-dimensional notation (o which more later) lef readers baffled. It was unanimously declared hard to read, and counterintuitive in the extreme. In contrast, the notation developed by Peano and Russell and Whitehead would immediately find avour with the mathematical community (in modern terms, we could say it had a riendlier interace). Secondly, as Frege himsel noted in the preace to Grundgesetze, the heady (and airly seamless) mixture o mathematics and philosophy in his work meant that mathematicians were put off by the philosophy and philosophers by the mathematics. Frege’s immediate impact on the history o logic was thus airly limited (and mostly due to Russell’s highly appreciative AppendixA to his Principles o Mathematics).18 In addition, Russell’s discovery that Frege’s system was in act inconsistent did little to improve Frege’s reputation but gave instead rather good copy to the story o Frege as a tragic figure who in trying to make logic ampliative, contra Kant, had broken its centuries-old sterility at the cost o making it engender a contradiction – to use Poincaré’s rather crude jibe.19 And indeed, i we look at the way logic and mathematics developed in the immediate postFrege, post-paradox era, there is a good sense in which it bears very little trace o Frege’s influence.20 All that changed in the 1950s, however, when Frege’s Grundlagen was translated into English by J.L. Austin, and Alonzo Church published his Introduction to Mathematical Philosophy. Frege’s importance as a philosopher o logic was finally recognized. His status as a classic would then be consolidated by Dummett’s 1973 opus magnum (and all o his subsequent writings), establishing Frege’s reputation as the ounder not just o modern logic but o analytic philosophy as a whole.21 And interestingly enough, the 1950s are also the time when the standard story gets established, mainly through Quine’s influence. Now, last paragraph aside, at this point you might legitimately be worried that the standard story just got things badly wrong. On closer examination, the
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commonly made claim that Frege is the ounder o modern logic has looked more and more problematic. rue, he beat everyone else to giving the first treatment o quantification.22 But he did so in a notation that no one else adopted and rom a position that was deeply hostile to ideas that were to become central to the model-theoretic tradition that inorms much o the contemporary conception o logic. Finally, it is also hard to deny that his work had little 23
immediate influence. So, in what sense, i any at all, can Frege be said to have ‘discovered’ the wonders o modern logic in his Begriffsschrif? As I said already, Frege’s genuine novelty lies in the inauguration o a new way o doing logic and to that extent he ully deserves the accolade standardly granted him. What I want to claim, that is, is that Frege’s importance is not (merely) historical: it isoperational, as it were – Frege showed us or the first time how we should operate when we want to do serious logical work; he gave us the first and most powerul logical toolbox. I’ll try to substantiate this claim in section 3. Beore that, though, we need to take a short detour through Frege’s philosophical stance.
2 Frege’s philosophy o logic As we have seen, Frege’s conception o logic is in several respects significantly different to the contemporary one. And yet it is precisely Frege’s idiosyncratic conception that moved him to develop the most strikingly innovative aspects o his work. So much so that the groundbreaking aspects o Frege’s logic are powered precisely by those aspects o his philosophical approach that are now considered to be archaic. My task in this section is to give a brisk summary o those aspects, and to explain their peculiarities and the way in which they triggered Frege’s technical innovations.24 Recall now the subtitle to Begriffsschrif: a ormal language or pure thought modelled afer that o arithmetic. But why should the language o pure thought be modelled on that o arithmetic, one might ask? Well, the answer, rom Frege’s perspective, is twoold. In the first instance, Frege’s proessional interests (he was a mathematician/geometer by training) put mathematical reasoning centre stage – Frege’s driving concern was to make that subject-specific reasoning as rigorous as possible. But he also thought that mathematics and logic were ‘reason’s nearest kin’.25 I you want to get to the roots o rationality, or Frege there’s no better place to start (and end) than reasoning about mathematical objects, that is. More specifically, Frege held a thesis, later called logicism, whereby mathematical
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truths were deemed to be cleverly disguised truths o logic. And another key thesis o Frege’s logicism is that mathematical objects (the objects those truths are about) are in act (reducible to) logical objects. Frege was thus a rationalist: the language o pure thought is the language o pure reason, the language in which the laws regulating any truth-oriented discourse can be stated and put to good use.26 And the target o those laws and o thought is, astho we’ve also already seen, what Leibniz hadFrege’ calleds highly the natural order o truths – and se truths are, o course, a priori truths. ambitious aim, then, was to show that those truths stretched a lot urther into our conceptual territory than Kant had thought. Frege’s project was thus a dual one: on the one hand, it was a oundational project in the philosophy o mathematics (a project that aimed to show how arithmetical truths could be analysed away in terms o logical truths and definitions); on the other, it was anepistemological project that aimed to show how we are justified in taking to be true just those mathematical statements that we so take, and that, as it turns out, underscoreall knowledge irrespective o subject matter.27 And or Frege justification was proo. But not just any old notion o ‘proo’ would do. Rather, proper proos had to begap-ree, in the sense that all assumptions underlying the proo should be made explicit and that any step taken therein could only be made in accordance with a specified rule.28 Tat is, all axioms, lemmas, definitions and rules o proo appealed to in the course o the proo must be made ully maniest in the ormalism itsel – and this is without question the way in which Frege can truly be said to have inaugurated the modern way o doing logic. A second sense in which proo had to be gap-ree concerned the notion o content. For Frege, content is a logical notion: a proposition, or a thought, in his later terminology, is that or which the question o truth and alsity can arise (and thus the natural order o truths is a natural order o true thoughts). What individuates a given thought is its reerence-determining properties, the way the thought determines its truth condition. In Frege’s later terminology that means a thought is a way o naming the rue, one o the two basic logical objects out o which all others are built. According to Frege’s conception at the time o Begriffsschrif, a thought is instead individuated in terms o its inerential potential, that is, in terms o what ollows rom it (what other thought you can reach rom it under the licence o logic) and what justifies it (how you can get to it under the licence o logic). But in act the two conceptions coincide: inerential properties or Frege are reerence-determining properties, since it is the latter properties that ensure that
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in correct inerence we never move rom names o the rue to names o the False. Furthermore, a thought is individuated in terms o its logical properties thus constituted (i.e. reerence and inerence-determining properties).29 So, in exactly the same way in which proo had to be gap-ree (in the sense that the justification or a given step cannot be lef to guesswork but must be precisely indicated by citing the law being applied in reaching that step and the premises on which that application rests) so content too (the inerential profile o the propositions involved in the proo) had to be gap-ree (in the sense that the thoughts concerned must be taken as eternal, ully specified propositional contents whose truth-value is specified in an absolute manner).30 Beore we can finally move to examining Begriffsschrif, two more eatures o Frege’s conception deserve mention, namely hisantipsychologism and his view that numbers are objects. 31 Regarding the first eature, it is important to note that when we speak o mathematical reasoning or o the laws o thought, on Frege’s view we must do so in complete disregard o the actual cognitive mechanisms that thinkers employ. For Frege, logic has to capture truth-preserving modes o inerence between thoughts without any concern or their specific cognitive implementation by a particular reasoner. Tat is to say, the laws o logic hold absolutely; they prescribe how one ought to judge in a way that applies whenever anyone makes a reasoning step, irrespective o the contingent psychological make-up o the reasoner.32 Te actual psychological means whereby those movements are implemented is o no concern whatsoever to Frege. Logic, or him, is in this sense a transcendental enterprise. It is constitutive o pure thought and indeed o rationality. Given that perspective, the language o pure thought and the system o proo developed by Frege is not intended to model the mental processes that mathematicians go through in their proos. Te purpose o the Begriffsschrif is rather to bring orth the justificatory structure o a given theorem: a proo must make clear the structure o the reasons on the basis o which the theorem can be asserted.33 Regarding the second point, Frege’s version o logicism was not limited to the claim that arithmetic is logic in disguise. It was also a thesis concerning the nature o number: numbers are objects, they are logical objects; that is, a particular kind o abstract objects that while not endowed with spatio-temporal properties are nevertheless ully objective, in his technical sense o objective, namely things about which we can issue judgements that are answerable to publicly assessable normative criteria.34 One very last point: Frege’s antipsychologism is best seen as an expression o his strong realism about mathematical objects viewed as strongly mindindependent
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entities. You might then be puzzled to learn that Frege gave great importance (and indeed explanatory priority) to the notion o judgement. Isn’t judgement something we do, you will immediately complain? And i so, how is that to be reconciled with Frege’s Platonism about number (his view that numbers exist independently o human reflection upon them)? Well, the tension is only apparent. While it is true that Frege gives great importance to the o judgement, thethat roleis,played by that notion epistemological, notnotion ontological. Judgement, is our entry point into isthe order o truths. o unravel that order, we start rom judgement. We first entertain a thought (a putative truth) and we then come to a judgement regarding its structure (its inerential potential) and its truth-value. Te act o judgement is simply the act o placing that thought in the order o truths (i it is alse, we’ll place its negation there, o course). But why is judgement central to Frege’s conception o logic? Because it is the act o judgement that generates concepts, thus making maniest a thought’s inerential relations.35 As I have just said, a judgement is not just recognition o truth-value (judgement as to whether a given thought is true or alse): it is also (and perhaps above all) recognition o its logical structure, it is an act o understanding why the thought is true (or alse), an acknowledgement o the inerential relations that hold that particular thought at a particular location in the order o truths. Now, the question that you will want to ask at this point is: what kind o structure do we recognize in an act o judgement, then? Tis question is in act at the heart o Frege’s contribution to the history o logic. And to see how Frege answered it, we can now, at last, dip intoBegriffsschrif, the work where logic turned truly modern.36
3 Begriffsschrif It should by now be clear that Frege’s rationalism is deeply revisionary .37 o achieve his aim o showing that the class o truths o reason includes that o mathematical truths, he had to provide a conceptual analysis o key mathematical notions by means o exclusively logical means. In order to do that, Frege had to uncover what he took to be the real conceptual structure underlying the largely misleading surace structure both in mathematical as wellas in ‘natural’reasoning. Both natural and mathematical language had to be reormed, regimented, made maximally unambiguous. Let’s examine how Frege set about doing so.
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I said that it is through acts ojudgement that we make apparent the inerential structure holding together the natural order o truths. Tat structure was Frege’s target. What he needed was a set o tools that would allow himto capture within a that were used in arithmetic. None o single, unified language all orms o inerence the languages that had been tried beore were able to achieve that – indeed, there is a very good sense in which prior to Frege there wereno ormal languages 38
properly so called. he set o himsel wasspecified to defineand a Begriffsschrif ormal system whereTe thetask meaning every with term the would be precisely kept constant in all its occurrences and whereevery orm o inerence required or arithmetical proos would be codified. In short, the system had to be unified and complete, accounting or all valid orms o inerence within a single language. Let’s now look very briefly at the gaps lef by pre-Fregean logics that Frege sets himsel to close off. Here’s a classic inerence in Aristotelian logic:
1. 2.
All As are Bs All Bs are Cs All As are Cs Stoic logic would instead concerns itsel with the codification o inerences o this orm:
1. I A then B 2. A B Te contrast, then, is between a logic o terms (the Aristotelian) and a logic o propositions (the logic o the Stoics).39 Very roughly, in Aristotelian logic one’s ocus is on relations between concepts, in particular relations o class-inclusion o the sort displayed in the syllogism above. In contrast, Stoic logic is designed or capturing relations between propositions.40 Now, while it is true that Boole had provided a ramework where both orms o inerence could be expressed, there was a severe limitation to the expressive power o his language. One had to switch interpretation o the language in order to capture those inerences (the symbols would be interpreted either as expressing relations between propositions or between concepts, but they could not do both at the same time, within a single interpretation). What was wanted, then, was a language that could capture both kinds o inerence within the same interpretation o its symbols. Te additional problem aced by Boole’s system was that there were intuitively valid ‘mixed’ inerences crossing the boundary between the two logics and that
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could not be represented in any language so ar provided precisely because o the need to switch interpretation halway through the inerence. Boole himsel gave one such case. Here’s the example:
1) Either all As are F, or all As are G 2) All As are either F or G Te move rom (1) to (2) is valid. Te move rom (2) to (1) isn’t. Exercise 3.1
Give a counterexample to show that the move rom (2) to (1) is not a valid inerence. Boole’s system was unable to give a ormal account o the validity o the one move and the invalidity o the other. A related problem was that due to its rigid treatment o the subject-predicate structure o a sentence, Aristotelian logic could not deal with inerences containing premises involving quantifiers in positions other than the subject place, that is, premises such as ‘Every student read some book’. Sentences o this orm display the phenomenon o multiple generality, a phenomenon o crucial importance to mathematical discourse (e.g. ‘For every number, there is some number greater than it’), and a phenomenon that also gives rise to ambiguities that one’s ormal language should disentangle. Frege’s logic was designed to solve all o these problems.41 How, then, did Frege solve these issues? Clearly, the undamental limitation in the work o his predecessors was the lack o a proper analysis o the genuine constituents o propositions. In modern terms, Stoic logic had concentrated on the compositional analysis o sentences, on how sentences are built up by means o the sentential connectives that determine a range o validities in virtue o their meaning. Aristotelian logic, instead, had concentrated on the kind o logical analysis that tries to extract relations between concepts. Frege’s main groundbreaking innovation was the provision o a logic that reflected a duallevel analysis o sentential structure. On one level, the analysis reflects how sentences are assembled compositionally rom their constituents (giving an internal analysis o their structure). On another level, the analysis disclosed the inerential relations that connect the sentence to other sentences in the language in virtue o the concepts figuring in that sentence (thus giving a kind o external analysis o the structure o the sentence in relation to other sentences).42
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o make that new kind o analysis possible, Frege’s first move was to abjure the subject–predicate distinction that had been at the heart o traditional logic.43 In its place, Frege proposed a unction/argument analysis.44 A given thought (a given candidate or judgement, a piece o judgeable content) admits o (and orces one into acknowledging) multiple analyses whereby its location in the order o truths is disclosed.45 Te keyinto ideaitsbehind the unction/argument is that down a thought constituent parts (in making aanalysis judgement as in to breaking which concept is being predicated o which individual(s)) we split it into a predicate part (corresponding to the concept being predicated) and an argument part (corresponding to the entity o which the concept is being predicated). Te concept part is represented by a unction; the argument part is the placeholder where individuals (or rather signs denoting them) may be slotted in.46 Incidentally, this is where another crucial advance over Boole took place, or in Boole’s notation acts about individuals were expressed as acts about concepts (to say that there is an F in Boolean notation is to say that the extension o a certain concept is non-empty, not to say that an individual satisfies the corresponding property).47 In analysing a thought, then, we uncover its inerential relations to other thoughts. Identity o content between thoughts is settled by identity o the set o their consequences: two propositions are equivalent in conceptual contenti they entail the same set o consequences.48 Accordingly, conceptual content or Frege is inerential potential and the concept-script to be developed is really an inerential-script, a script representing the logical orm o the inerential structure o thoughts.49 So, what structures both the order o truths and the logical content o a given thought is unctional structure, as well as compositional structure. And indeed here we have what is arguably Frege’s key insight: the linear structure o language hides away the constructional historyo s entences.50 So, on Frege’s view you do not grasp a thought (what a sentence expresses) until you’ve grasped its compositional structure (the way the sentence is put together rom its constituent parts)and its unctional structure (how many concepts can be generated by decomposing the thought expressed by it). Afer you’ve done that, you’ve finally properly placed that thought in the order o truths. You know how to iner to and rom it (according to the insight oBegriffsschrif§9) and you know what truth-conditions the sentence determines (according to the insight oGrundgesetze §32). Tis was all a bit heady, so let’s now look at a very simple example to see how it all works. Consider the thought that Cato killed Cato. Frege’s unctional
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analysis o its structure proceeds by abstraction.51 o uncover (and indeed generate) the underlying conceptual structure, we remove singular terms rom the expression and replace them with variable terms so as to obtain all possible unctions that could be discerned in the thought.52 So we might first get ‘Cato killed ’, and then ‘ killed Cato’ and finally ‘ killed ’, where marks argument the place(s), the gap(s) lef in the thought afer we have abstracted away the specific individual(s) the thought wassrcinal (partly)thought about. In so, we the inerential properties o the (ordoing instance, webring have out made apparent how we can move rom ‘Cato killed Cato’ to ‘Cato killed somebody’ or to ‘Somebody killed Cato’, or to ‘Somebody killed somebody’) while simultaneously showing how we can generate concepts rom a given thought.53 Note that the thought itsel remains unchanged. Te possibility o multiple unctional analyses o the same thought merely shows the multiplicity o inerential relations a given thought stands in. Te thought itsel retains a unique structure. Te other groundbreaking aspect o Frege’s unction/argument analysis is that the decomposition process makes available building blocks that can then be recombined to orm new thoughts, new concept/object combinations. And the combinatorial process can be iterated indefinitely. o stick with our example, once we have moved rom ‘Cato killed Cato’ to ‘Cato killed ’, it is then easy to move on and replace with another singular term and obtain, say, ‘Cato killed Brutus’, and so on or the other patterns. Tis is already powerul enough, in explanatory and expressive terms, but the ull power o Frege’s language will become apparent when we will finally introduce the quantifiers below.
3.1 Frege’s notation It’s now time to say something about Frege’s (in)amous notation. It is supposedly a highly counterintuitive way o representing thoughts (and actual reasoning patterns). I expended much time in spelling out Frege’s overarchinglogicoepistemological aims in the hope that it would be clear that, contrary to the standard story, the notation we are about to examine is a highly appropriate and indeed perspicuous way o mirroring on paper the deep structure o thought and inerence. Te lack o linearity o Frege’s notation, that is, is not a bug but a valued eature, since it is intended to signal the need or a departure rom the linear surace structure o standardly written linguistic strings (be they in natural language or in that o arithmetic). Tat is, Frege’s two-dimensional notation, not unlike the generative grammarian’s tree structures, makes maniest logical orm in a way that purposeully defies linearity. It reflectshow concepts are ormedand
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how the logical orm o a sentence is constructed, both eminently non-linear processes. Let’s look at the details now. Given Frege’s overall ramework, the first sign he needed was one that would mark judgeable content, that is, the content o possible assertions. Here it is: ——
which prefixed to the symbol or a given thought, say ‘A’, gives us: ——A Tis could be rendered as ‘the thought thatA’ or ‘the proposition thatA’.54 It represents the stage where a thought is merely entertained, or grasped, as Frege ofen put it. Tis horizontal stroke is called thecontent stroke. From merely entertaining a thought, we may then move to a judgement regarding its truthvalue; we may, that is, move toasserting the thought, making the judgement that, or example,A is true. o signal that the thought is being asserted, Frege added a short vertical stroke to the content stroke, giving what he called the judgement stroke:55 A Note two things here. First, the letter symbol represents that part o the content o a thought which has no logical significance in the sense that it is logically simple – no urther decomposition in the sense above is possible. Te judgement stroke instead ulfils a dual role. Its presence is essential to making an expression into a sentence o the ormal language: as Frege notes, it is a predicate common to all judgements.56 But it also makes what we could call a metalinguistic gesture: it signals that the thought that A is being asserted. Let me stress once again that assertion here carries no psychological connotations. It just says: there is a proo o A. And in act, given Frege’s axiomatic system, it says:A is a theorem.57 Frege’s calculus was thus intended to be a calculus o asserted, that is, proven truths – he had no truck with the idea that we could do proos rom alsetatement s s.58 Note also that or Frege it was crucial that the representation o the logical structure o a thought be kept separate rom those thought-components that are logically inert, as it were. In short, the task o the two-dimensional notation that he adopted is to vividly represent the logical orm o the thought, i.e. its logical properties. Tat is: the letters indicated on the right-hand side represent the individual contents; the network o lines on the lef-hand side represent the inerential relations holding between them. In other words, Frege’s twodimensional notation analyses the asserted thought into its constituent thoughts and the logical relations holding between them.59 o see how this all works, let’s now look at the way Frege represents conditional statements and generality. Here’s how Frege renders the thought that iA then B:
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(1)
Frege called the (thought expressed on the) upper line in a conditional the supercomponent (what we now call the consequent) and the lower line the subcomponent (i.e. the antecedent).60 Note that we have a content stroke governing the entire conditional, and then two context strokes preceding the two letter symbols. Te condition stroke is the vertical stroke joining up the two content strokes preceding the letter symbols. It indicates that the subcomponent is a sufficient condition or the obtaining o the supercomponent; it also signals that assertion o A logically commits you to assertion o B, that it logically leads you up there, as it were.61 Note that the judgement stroke governs the entire conditional (the thought-components are not asserted). Te notation may well strike those o us trained in the Peano–Russell notation (say ‘A ⊃ B’ or ‘A → B’) as counterintuitive or cumbersome, but Frege argued that his notation was in act ar more conducive to making the logical structure o thought maniest.62 One way o unpacking Frege’s claim is to say that his notation makes clear that, or example, allA-cases, all circumstances in which A obtains, are B-cases, cases in which B obtains too. Te spatial representation, withA below B, is thus meant to convey the notion o case-containment, as it were.63 Now, how does Frege define the conditional?64 He considers the our possible cases one could have here, interestingly enough not (yet) in terms o truth and alsity, but rather o affirmation and denial.65 And he takes his conditional to exclude only the case in which A is affirmed and B denied.66 Frege then immediately notes that this won’t capture in ull generality what we would render in natural language with i . . ., thenbecause his conditional won’t presuppose or convey any causal connection between A and B.67 Causality, he notes, is something that involves generality, something he has not defined yet. A second advantage that Frege claims or his two-dimensional notation is that it affords a clearer demarcation o ambiguity o scope. Consider a nested conditional such as: (2)
It is immediately clear what the antecedent is (namely,Γ). In Peano notation, we would have to use brackets (or dots):
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(3)
Γ .⊃ .A⊃B
(4)
or
and Frege thought this introduced unneeded complexity, especially with more complex conditionals, while also masking the logical orm o the thought.68 Let’s look at negation next. Frege expresses negation by adding a short vertical stroke under the content stroke:
A
(5)
A
(6)
Te asserted version is as expected:
Te gloss given is also as expected: the negation sign expresses the circumstances that the content does not obtain.69 And now it is time or some exercises.70 Exercise 3.2
Define conjunction using Frege’s notation (i.e. write outA B in two-dimensional notation). Exercise 3.3
Define disjunction using Frege’s notation (i.e. write out A B in two-dimensional notation). Exercise 3.4
Define exclusive disjunction using Frege’s notation. Exercise 3.5 Write out ¬(A B) in two-dimensional notation. Exercise 3.6
State the principle o contraposition, (¬ψ → ¬φ) → (φ → ψ), in Frege’s notation. Exercise 3.7
Te set o logical connectives used by Frege, {¬, →} in modern notation, is unctionally complete, that is, it suffices to express all (two-valued) truth-
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unctions (unctions taking truth-values as arguments and returning truthvalues as values). Show that the set {¬, ∧} is truth-unctionally complete. (Hint: show that any ormula expressible using the first set is expressible using the second set as well. Check Bostock (1997: §2.7) i you get stuck. You don’t have to do this in Frege’s notation but you get bonus points i you do!).
3.2 Generality Te most celebrated technical innovation introduced inBegriffsschrif is Frege’s treatment o generality. Here we ace yet another slightly paradoxical situation, or on the one hand Frege is supposed to have inaugurated the modern treatment o quantification, but onthe other hand his own treatment is again highly idiosync ratic and widely divergent rom the standard textbook account. Frege, that is, treated quantification as a orm o predication. According to Frege, in making a quantificational statement such as ‘every philosopher is smart’ what we are really doing is saying that a certain property, in this case that o being smart i one is a philosopher, is satisfied by everything. In quantiying, we are thus predicating a property o a property (we say that a property has the property o being either satisfied by everything, orby something, or bynothing).71 Actually, the real technical and philosophical advance made by Frege was in his treatment o the variable. Previously, variables were treated as being symbols that reer to indeterminate individuals in the sense o individuals that were taken to have indeterminate properties. Frege insisted that names and variables behave differently and that variable symbols indicate (andeuten) arbitrary individuals rather than reer (bedeuten) to them.72 Let’s now see how Frege rendered generality in his notation. First, let’s quickly recall the unction/argument analysis we have already discussed. Let Φ denote an indeterminate unction (i.e. any o a possible range o unctions). We mark the argument place in a way still amiliar to us by enclosing the argument in parenthesis ollowing the unction sign: Φ(A).73 Tis statement lacks ull generality, or ‘A’ indicates a specific, i arbitrary, argument. o indicate generality, Frege used two devices. Let’s first look at the simplest one:
X (a)
(7)
In modern terms, this is a ormula with one ree variable, the small Roman letter ‘a’. But Frege is here taking the universal closure o this ormula (he’s tacitly ronting the ormula with a universal quantifier binding whatever ree variables there are in the ormula).74
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Tis approach, while efficient, is very blunt, since the scope o the generality claim coincides with the entire ormula. Frege’s srcinal contribution comes with the other way o expressing generality that he proposed. What we do this time is slot a German letter in the argument place, X(a), and bind that letter by placing another token o the very same letter-type in what Frege called the concavity sign interposed in the content stroke. We thus obtain: (8) Tis is read as the judgement that or any argument whatever the unction Φ is a act, as Frege rather clumsily put it in Begriffsschrif. In his later terminology, we would say: it names the rue or all arguments. In contemporary terminology: it is true under all assignment unctions, i.e. no matter what individual is assigned to the variable.75 Now, the role o the concavity is to delimit ‘the scope that the generality indicated by the letter covers’. And with this, Frege secured exactly what he needed to model the multiple generality mode o reasoning so requent in mathematics. Here’s an example: (9)
Exercise 3.8
Write out ormula 9 in modern notation. Here we have two quantifiers, expressing a multiple generality claim. Note how Frege’s notation makes the act that the scope o a is wider than that o e immediately apparent. More interestingly still, it also clarifies right away that the scope o e is more restricted than the entire ormula, a flexibility that had eluded Boole’s attempts. Indeed, the introduction o the concavity sign is where the watershed we talked about earlier really takes place.76 Te crucial innovation made by Frege, then, is that his system, unlike Boole’s, can express within the same language valid arguments in which propositional and quantificational inerence steps occur. Adding the concavity to the content stroke simply expands the language, leaving the previous ragment o the language still in place.77 Having presented his way o dealing with generality, Frege goes on to demonstrate the interaction between negation and generality, now made possible by the concavity device, noting the contrast between:
(10)
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(11) Exercise 3.9
Write out ormulae 10 and 11 in modern notation. Frege next shows how to make existential generalizations in his system using only generality and negation: Exercise 3.10
Express the existential statement ‘Tere is one F’ in Frege’s notation. Exercise 3.11
Express in Frege’s notation the classic multiple generality statement that or every natural number there is a greater one. In rounding off Part I o Begriffsschrif, Frege does two more things. He first shows how causal connections may be expressed in his system:78 (12)
Note here another didactic advantage o Frege’s notation: it clearly represents set-subsumption (that the Xs are contained in the Ps) by having the contained set (the subordinate concept, and subcomponent o the conditional) below the container set (the superordinate set, and supercomponent o the conditional).79 Te second thing he does is to quietly point out how the traditional Aristotelian square o opposition can be expressed in his system.80 In effect, having already shown how he can deal with multiple generality, he’s closed the first part o his groundbreaking work by showing that his ormal system is a proper extension o Aristotelian logic. It should be clear rom the examples above that there is another crucial respect in which the modern turn in logic truly occurred only with Begriffsschrif: generality, negation and the conditional can all be iterated and combined without limit, thus allowing the language to keep pace with the inexhaustible nature o mathematical structures. A crucial instrument here is Frege’s unstated rule o substitution, whereby ormulae can be substituted or other sentential variables in any o the theorems,
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without restrictions other than uniormity o replacement (the substituend must be replaced across all occurrences o the substituens).81 Here’s a quick example. In (13)
we can substitute the ormula itsel or its subcomponent obtaining: (14)
Te process is clearly iterable indefinitely. No system beore Frege had such expressive combinatorial power. For the first time in the history o logic, mathematical (and to some extent, natural) reasoning could be captured with precision and flexibility. Moreover, no system beore Frege had succeeded in writing out the logical structure o expressions and the iterability o the rules o ormation into the very language used to capture logical and mathematical thoughts.
3.3 Te ormal system Now that we have taken a look at some o the details o Frege’s notation and its philosophical underpinnings, we can say a bit more about the system o logic presented in Begriffsschrif. What we find there is the first axiomatic system or a higher-order predicate calculus with identity.82 Frege’s presentation is more inormal than is customary today, but it was ar more precise than anything that had been done beore and remained unsurpassed in rigour or quite some time.83 In essence, Frege’s language had only two connectives, the conditional and negation, and one quantifier, the universal quantifier. His system had nine axioms and one (explicit) rule o inerence, or rather, o proo, namely, modus ponens (MP).84 Frege had two reasons or such parsimony: first, to keep the metatheory simple; second, to make the epistemological status o the system as a whole as clear as possible (the ewer primitive notions, the clearer the assumptions on which the system rests). Unusually to modern eyes, Frege does not introduce the other standard connectives and quantifiers by means o definitions based on the primitive notions. He limits himsel to noting how he can express those other connectives and quantifiers by means o his chosen ones.
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All in all, the system given in Begriffsschrif is arguably the first system o logic that can properly be called ormal. Lines o proo are numbered, axioms are numbered, ormulae are numbered and cross-reerenced, inerence steps and conclusions o derivations are clearly marked, the axioms and rules o inerence used are properly flagged.85 Tere is no question at all that properly disciplined deduction starts in 1879. No step is taken without proper justification, and all moving the is system are up inspection, withinsisting nothingthat lef to guesswork, as Frege parts put it.oTis the sense in or which I have been Begriffsschrif inaugurated the modern way o doing logic.86 Let’s now look at the axioms and the rule(s) o proo used by Frege. In part II o Begriffsschrif, Frege states his nine axioms, interspersed with proos to exempliy how his system works in practice and to build up theorems that are then used in subsequent proos as abbreviations o their proos. Axiomatic presentations o predicate logic are seldom used in introductory courses, chiefly because o the difficulty in finding proos within such systems.87 On the other hand, they make it a lot neater to prove the classic metatheoretical results as well as making very clear the meaning o the connectives and quantifiers involved in the axiomatization by means o the axioms – you look at the axioms and you know exactly what system you’re dealing with and what the connectives ‘mean’. All such presentations find their kernel in Begriffsschrif. Indeed, it is now customary to call them Frege–Hilbert systems.88 In §6 Frege introduced his main rule o inerence, namely modus ponens. Te rule was introduced and justified in terms o the definition o the condition stroke already given in §5.89 Let me stress once again that the system in Begriffsschrif is a proo system, a system where proos always start rom axioms.90 Frege thus needed to list the axioms that he required or his purposes. He does so, albeit rather perunctorily, in §14. Let’s look at the first three o Frege’s axioms:91 Axiom 1
Tis is pretty much a direct consequence o the definition or the condition stroke. Frege gives a direct justification in terms that we might call semantic: given the meaning o the connective, the only possibility or an instance o axiom 1 to be alse is the case where we affirm the antecedent (a given thought represented here by a) and deny the consequent, that is, the nested conditional. But in turn that conditional will only be alse when its antecedent is affirmed and
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its consequent (the very same thought represented by a) is denied. And this, Frege notes, amounts to both affirming and denying the thought expressed by a, which, he confidently concludes, is impossible.
Axiom 2.
Axiom 2 is a kind o ‘distributive’ axiom that is very useul indeed in proos, as we shall see in a moment. Frege’s justification is similar in structure to the one given or axiom 1.
Axiom 3.
Axiom 3 states a principle that would become a rule o inerence in Grundgesetze: the interchangeability o subcomponents. In other words, their order doesn’t matter.92
Exercise 3.12 ranscribe axioms 1 to 3 into modern notation. And here are three more axioms that should strike you as very amiliar indeed:93
Axiom 4.
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Axiom 5.
Axiom 6.
Exercise 3.13
ranscribe axioms 4 to 6 into modern notation and state their current names. Most o part II is devoted to proos in propositional logic. From §20 onwards, identity and quantification enter the scene, announced by the final three axioms.94 Here are the two axioms defining identity: Axiom 7.
Axiom 8.
Frege’s treatment o need identity in Begriffsschrif has been object much 95 discussion. We don’t to enter into all its subtleties here .the I will just o note that the treatment o identity is one o the aspects that Frege will substantially revise in his later system o logic in Grundgesetze. At any rate, the two axioms in Begriffsschrif have an obvious use in proo (or instance, axiom 7 allows that two names with the same content be interchangeable in any proo context). Finally, Frege introduces his axiom or the quantifier: Axiom 9.
Tis axiom should be sel-explanatory. Exercise 3.14
ranscribe axiom 9 into modern notation. Exercise 3.15
Note that Frege uses German and Roman letters in the axiom. Why? One thing to note in connection with axiom 9 is that as it stands it is a typeneutral axiom,96 in that Frege took it not to be restricted to first-order variables
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only. Te German letter could also be taken to range over unctions too, that is (with the predicate letter then ranging over unctions o higher order). In Grundgesetze, by contrast, Frege will ormulate two distinct axioms, one firstorder and another higher order.97 For the remainder o part II, Frege shows how his system can model three amiliar Aristotelian syllogisms. He then leaves it as an exercise or the reader to show how thiswrap can be to all other valid orms. Let’s now upextended our discussion o Frege’s ormal system inBegriffsschrif by talking a bit more about his (unstated) rule o substitution. As I said, Frege–Hilbert systems make finding proos a notoriously hard task. Te trick consists in ‘spotting’ how to substitute ormulae into axioms so as to enable the derivation o the needed theorems by appropriate applications o modus ponens. Tere’s no better way o grasping what the modern turn in logic was all about than by doing proos. So let’s look very briefly at a proo to see how it all works in practice. We go or an easy proo. We want to prove: (15) (15) is a way o stating the law o identity (sometimes called reflexivity in its metatheoretical version), so Frege’s system had better be able to prove it.Modus ponens (MP) is the main rule o proo in our system. So the last step in the proo is going to be an application o MP that would yield (15) as the last ormula in the proo. Te ormula on the preceding line o proo will then have to have (15) as supercomponent to be detached in the concluding line. What could the subcomponent look like? It’d better be an axiom (so that we can detach our target theorem). Well, a plausible candidate is: (16)
Is (16) an axiom? Yes it is. We have just replaceda or b in axiom 1
to obtain (16).98 Now, i we plug (16) as subcomponent o a ormula that has
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(15) as supercomponent, the resulting ormula doesn’t look like any o our axioms:
(17)
So, (17) will itsel be the result o the application o another MP step. Again, by the same reasoning what we now need is a ormula with (17) as its supercomponent and a subcomponent which is an axiom. You will not be surprised to hear that, once more, our riend here is the substitution rule. Te ollowing ormula is an instance o axiom 1:
(18)
or we have simply substituted
or b.99 And now we can put
together (18) as subcomponent and (17) as supercomponent and end up with:
(19)
Te question is: have we finally reached an instance o an axiom so that we can get our proo started? Well, let’s remind ourselves o what axiom 2 looks like:
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Is (19) then an instance o axiom 2? Yes it is, although it may not be easy immediately to see it is so. Let’s unpack the two moves that get us there. First, we can replace c with a in axiom 2 to obtain
(20)
We then substitute
or b to get us (19) as desired:
And we’re done at last! We finally got to an axiom and that will allow us to start our proo (a good rule o thumb to find proos in this kind o system is to start
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rom the conclusion and reverse-engineer a proo until we reach an axiom, as we have just done). Here then is the entire proo:
A ew things to note: the ‘2’ next to the first ormula flags out which axiom it is an instance o. Te horizontal line below the first ormula signals that an inerence has been drawn. Te colon next to ‘(1)’ signals that the inerence was an MP step and that the major premise has been omitted rom the statement o the proo. ‘(1)’ signals that the omitted ormula in the MP step, that is, the subcomponent o the initial ormula, was an instance o axiom 1. I we had instead omitted the minor premise, we would have marked that with ‘::’. Te same procedure is repeated or the second step in the proo. Finally, at each step, we speciy to the lef o the derived ormula which subormula we have replaced to derive the substitution instance o the axioms involved.
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Exercise 3.16
Te proo given above is the standard one given in the streamlined Frege–Hilbert system or propositional logic. Te Begriffsschrif system, however, had more axioms, allowing or a different proo o , one that uses axioms 1 and 3 (the one excised by Łukasiewicz), substitution and two applications o MP. Can you work out this proo? (i you can’t, check the end o §16 o Begriffsschrif and see how Frege did it. It’s a beautiul little proo, more ingenious than the one above and it will reveal much o the innovative proo-theoretic power o Frege’s system. Beore checking that out, do try to find the proo yoursel! And remember, the key is to find the crucial substitution moves that allow you to appeal to one o the axioms.)100 Tis, then, is Frege’s system in action, painstakingly precise in the conduct o proo, leaving nothing to guesswork, and establishing, when needed, the good epistemic standing or even the most commonly accepted propositions, as exemplified in our little proo.
4 Te rise and all o Frege’s project Frege’s treatment o quantification was groundbreaking, as we have repeatedly noted, but it was also unusual by modern standards in that it included, without much anare, quantification overproperties, not just over objects. Tis was o a piece with Frege’s commitment to logicism. As we have seen already, Frege took mathematics to be part o logic in the sense that its statements were to be analysed by means o purely logical notions. And right at the opening o Begriffsschrif he had announced his first goal towards that aim, namely, to show that ‘the concept o ordering in a sequence’, the undamental notion in arithmetic, the device that generates all numbers, could be reduced to and explained in terms o that o logical ordering.101 o do so, Frege had to include properties within the scope o his quantifiers, or the notion o ordering is, in effect, the principle o mathematical induction, the principle that states the constraints under which certain properties can be inherited by all members o a particular sequence.
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Frege’s definition o the notion o ordering was as ollows:
Tis is proposition (76) o Begriffsschrif, and it defines what is now called the strong ancestral. Frege glosses this definition roughly like this: i we can iner rom the two propositions that property F is hereditary in the -sequence and that any element that ollows x in the -sequence has property F that, or any F whatsoever, y has the property F, then we can say that y ollows x in the -sequence or that x precedes y in that sequence. Exercise 4.1
Why is there no assertion stroke in ront o the conceptual notation or (76)? Exercise 4.2
Write out Frege’s definition in modern notation. Tere are many remarkable things about this definition and the use to which it is put in the proos that ollow.102 For our purposes let’s isolate two acts. First, in proposition (76) Frege introduces, without any comment at all, quantification over properties, not just over objects – the letter F placed above the concavity that governs the entire ormula in (76) quantifies over any property whatsoever, with being just arbitrary instance. Secondly, in the Frege helpsFhimsel to aone substitution rule in which letters thatproos rangethat overollow, properties and relations are replaced by letters ranging over ormulae. Tis practice amounts in effect to a comprehension schema (a schema pairing membership conditions to a specific set), and rom there to paradox it is a very short road indeed. Te next substantive step Frege took in the pursuit o his logicist project was the introduction, in Grundgesetze, o a new kind o objects, namely extensions o concepts, or what he called value-ranges – in modern terms, something like the
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graph o a unction, the set o ordered pairs o argument and values that characterize the unction extensionally. With concepts and extensions in his ontology and an unrestricted substitution rule (i.e. an unrestricted comprehension schema), Frege’s expanded and more precisely regimented logical system in Grundgesetzewould be proved inconsistent by Bertrand Russell’s amous letter o 16 June 1902.103 Aswith Frege presents the paradox in the to Grundgesetze start thehimsel plausible assumption that there areAferword classes that do not belong, we to themselves (the class o human beings, or instance). We then consider the case o the concept class that does not belong to itsel. Under that concept all all and only those classes that do not belong to themselves. Its extension will then be the class o all classes that do not belong to themselves. Call that class K. We now ask the question o whether K belongs to itsel, and we end up with a contradiction whichever answer we give. Te paradox arises because we are treating properties (classes) as entities over which the variables can range; we treat them as potential arguments or higher-order predicates, that is. Without some kind o restriction on the range o those variables in place, paradox will inevitably arise. In Begriffsschrif Frege had not yet introduced extensions o concepts and thereore he did not have classes in his system. But rom Grundlagen onwards, he wanted to define the concept o number in terms o extensions o concepts, and he thereore introduced a theory o classes into his system with catastrophic results.104 His lie’s work had been all but destroyed.105 Te irony is that Frege’s mature system ailed him just where he had been at his most brilliant in his early path-opening work. Tat is, the strikingly srcinal proo o the property o connectedness o the ancestral, the proo that had shown the young Frege at his best, was to unleash, ourteen years later, the contradiction that scuppered his logicist ambitions. Frege had wanted the history o logic to be the history o arithmetic. Tat proved to be an impossible goal to satisy and at the end o his lie he admitted as much.106 Afer him, logic and the theory o classes, what we now call set theory, parted company. What is now called logic is the consistent part o Frege’s system, namely the first-order ragment. Set theory, in turn, is still grappling with questions o consistency to this day. What, then, is Frege’s place in the history o logic? Gödel (1944: 119) gave perhaps the most penetrating characterization o Frege’s peculiar role in that history. On the one hand, Frege’s main interest was in ‘the analysis o thought’. And we could add: Boole may well have titled his treatise Te Laws o Tought but it was Frege who first devoted at least as much effort to
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clariying what thought was, rom a logical point o view, as he’d spent clariying what its laws were.107 Indeed, that’s why I gave so much space in this chapter to Frege’s philosophy o thought – there is no understanding his philosophy o logic without understanding his philosophy o thought. On the other hand, Gödel also noted that Frege’s progress had been unduly slowed down by his ‘painstaking analysis o proo’. And we could add: it was Frege’s obsessive insistence on precise definitions or looming the purposes o proo in that had made him miss the crucial warnings about the contradiction Cantor’s writings and alienated the much-needed sympathy o his colleagues.108 In the opening section, I said that whatever qualifications one might want to add to the standard story, Frege could indeed be taken to be the ounder o modern logic because he had shown us how to do logic. But arguably Frege did more than that. Logic is something we do. It is, again as Frege (1897a: 128) himsel insisted, a normative science in its own right, directing us in our thinking and, more generally, in our lives. And logic’s main concern is with truth. In assessing Frege’s unique contribution to the history o logic, then, his unremitting dedication to the cause o truth is perhaps even more remarkable and praiseworthy than his technical achievements. Tere is thus no better way o closing our account o Frege’s position in the history o logic than by quoting Russell’s glowing tribute: As I think about acts o integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire lie’s work was on the verge o completion, much o his work had been ignored to the benefit o men infinitely less capable, his second volume [o Grundgesetze] was about to be published, and upon finding that his undamental assumption was in error, he responded with intellectual pleasure clearly submerging any eelings o personal disappointment. It was almost superhuman and a telling indication o that o which men are capable i their dedication is to create work and knowledge instead o cruder efforts to dominate and be known. 109
Frege, then, didn’t just show us how todo logic. He also showed us how to be a logician, how to put truth first, at all times, no matter what the costs may be. And that, surely, is the hardest lesson o all to learn.
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Notes 1 Te story is common currency in writings about Frege (e.g. rom the seminal Dummett (1973: xxxiii–xxxvi) all the way to the more recent Burge (2005: 69), Potter (2010: 3) and Zalta (2013)), as well as in textbooks (e.g. rom the introduction to the first two editions o Quine (1950) and his (1960: 163) to Soames (2010: 1–2)). It probably srcinates with a remark in Łukasiewicz ( 1935: 112). Putnam (1982) and Boolos (1994) strongly (perhaps a tadtoo strongly) oppose the standard account. A more balanced story is in Hodges (1983/2001: 1–4) and Sullivan (2004: §2.1). 2 A good discussion o the current conception is in e.g. Beall and Restall (2006: §2.5). On Frege’s conception, see Goldarb (2010) and Blanchette (2012). 3 Tere is a lively debate concerning the extent towhich Frege’s work may properly be said to be concerned with metatheoretical considerations. See e.g. Stanley (1996), appenden (1997) and Heck (2010) or the view that he was, Ricketts (1986, 1996) or the contrary view, and Sullivan (2005) or a sophisticated assessment o the controversy.Read (1997) is also useul. 4 Frege (1893/2013: xxvi). Te irony is that in modern terms by this move Frege inaugurated the now common distinction between logic and logics, a distinction that he, however, would have ound repugnant or the reasons I discuss in the next page and in note 6 below. 5 Very roughly,deductive systemscan be divided into systemso proo properly so called (axiomatic systems that only allow axioms – orpreviously derived theorems – to figure as premises in proo) or deductive systems more generally, that is, systems o deduction, where assumptions other than axioms (or theorems) can figure as premises in a derivation. See Sundholm1983/2001) ( and Socher-Ambrosius and Johann (1997). 6 Van Heijenoort (1967b). As Dummett (1991a: 219) observes, Frege seemed to have in mind a ‘single intended interpretation’ or his language, defined over absolutely all objects (or at least, all objects or which the language had names). See Blanchette (2012: ch. 3) or a recent discussion o this issue. 7 See Grundlagen §17. 8 See Frege (1880/1881; 1882; 1897b: 242). 9 Again, the issue is not as simple as that. See Heck (2012: 40–1) or useul qualifications. 10 Boolos (1975: 47) labels a view o this kind as ‘archaic’. So much or Frege’s being the ounder o modern logic then! See Blanchette (2012: ch. 6) or urther discussion. 11 It is surely not by chance that a model theorist like Hodges1983/2001: ( 1) instead situates the watershed in 1847, the year Boole published hisMathematical Analysis o Logic. See Dummett (1959) or some powerul reasons to disagree with Hodges. 12 Just as Hilbert and Ackermann (1928: 1–2), and Putnam (1982: 260), would have it.
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13 Te classic criticism is in Quine (1970: 66). See Boolos (1975: 44–5) and Shapiro (1991) or critical discussion. Very roughly, the problem is that in quantiying over properties one appears to be naming them, and hence reiying attributes into entities, thus inflating our ontology to ‘staggering’ proportions. 14 Tat is, first-order logic is the strongest logic that can satisy compactness, completeness and the downward Löwenheim–Skolem theorem, a result known as Lindström Teorem(s). See Ebbinghaus et al. (1994: ch. 13) or the technical details. See also Read (1997) or a good account o the early history o logic in this connection. 15 Sure, it is true that Frege inaugurated a new way odoing logic, that he established new methods o logic, but the object (what counted as logic) and targets (what it was or) o those methods were significantly different to those that are common currency today. 16 For those theorems, see note 14. Depending on where we stand on this issue, we can say that modern logic was born in 1879 because Frege developed its first-order ragment then or because higher-order logic is logic anyway. Te latter is without question a minority view. See Shapiro 1( 991) or the most sustained argument in its support. Interestingly, first-order logic’s privileged status was consolidated when it showed itsel to be the perect tool o the model theorist, and model theory became in turn settled in the 1950s and 1960s. 17 Potter (2010: 23) quotes a remark rom one o Frege’s colleagues that even at his own university ‘hardly anyone took Frege seriously’. Frege did not are much better in the wider world. See Grattan-Guinness (2000: §4.5.1–2). 18 Another early exposition o Frege’s system was Jourdain1912). ( 19 Frege’s system wasn’t the only victim. Hilbert and Zermelo had been aware o the paradoxes a ew years ahead o Russell (see e.g. Hilbert’s last letter to Frege in Frege 1980: 51). Hilbert took the paradoxes to show that ‘traditional logic is inadequate’. So, there you have it: by 1903 modern logic was already traditional! 20 Opinion is divided here. Putnam 1( 982) takes the view that Frege’s work had a very limited impact. Deenders o the standard story (e.g. Reck and Awodey 2004 : 40–1) have instead argued that Frege’sindirect influence was nevertheless a determinant actor in the evolution o logic in that the logicians he taught, met or corresponded with (i.e. Carnap, Wittgenstein, Russell and Peano) were the lead players in the field, and in particular that Gödel’s undamental results o 1930–31 were heavily indebted to Carnap’s 1928 lecture course on metalogic that Gödel had attended. For their part, Whitehead and Russell wrote in the preace toPrincipia that ‘in the matter o notation’ they ollowed Peano’s lead. In ‘all questions o logical analysis’, however, their ‘chie debt’ was to Frege (p. viii). Basically, Frege’s axiomatic system was rewritten into Principia in Peano’s notation. Tat’s how Frege’s work entered the bloodstream o modern logic.
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21 Dummett (1973, 1981, 1991a, 1991b, 1994). 22 It was a close shave though. See Putnam 1982: ( 255–59) and Goldarb (1979) or details. 23 It’s even more serious than that. In a very good sense, had Frege’s peculiar approach to logic become dominant, it would have actuallyhindered the ull development o modern logic. Here’s just one example. Frege’s notation or the quantifier, o which more later, made it completely unsuitable to the crucial treatment given by Skolem, where quantified ormulae are divided into matrix (quantifierree component) and prefix (the quantifier component), allowing the ormulation (and proo) o various laws about quantifier elimination that are essential to proving undamental metalogical results. I you’re curious about this, Kelly1997: ( §9.2.3) gives a very gentle introduction to the delights o Skolemization. 24 Indeed, Frege 1( 897b: 234, 237, 248) himsel insists that we can make sense o a system o deduction only i we are ully apprised o the aims and purposes o the writer. 25 Grundlagen, §105. Te view was o course by no means unique to Frege. Boole (1854/2009: 422) or one held that view too. 26 On Frege’s srcinal take on rationalism, see e.g. Burge (1992: 299–302; 1998), Gabriel (1996), Jeshion (2001: 969), Ruffino (2002) and Reck (2007). 27 For Frege (1879: 7) the language o pure thought captured by the Begriffsschrif constituted the core on which any scientific (i.e. truth-oriented) discourse would be based. 28 Frege argued convincingly that mathematicians’ habits in this regard were unacceptably sloppy. 29 Note that this is why Frege’s language comes with an interpretation already packed into it. It makes no sense to speak o alternative interpretation o the Begriffsschrif ormulae because their logical properties cannot be altered without changing the thought, since those properties are determined by the position that the thoughts expressed by those ormulae occupy in the order o truths. 30 Beaney (1996: ch. 6) has a penetrating discussion o the problems that Frege’s conception o absolute thoughts give rise to. 31 Psychologism was, very roughly, the view that mathematical entities are essentially defined in terms o our mental acts. For instance, according to early Husserl, a set consists in a ‘uniying act o consciousness’. Frege’s strongest attack on psychologism is in Frege (1894). See also the preace toGrundlagen. 32 Frege (1893/2013: xvii). 33 See Frege (1879: §13; 1897b: 235). By contrast, Gentzen’s systems o natural deduction were intended to model more closely the actual way in which mathematicians prove results. But it is no objection against Frege to say that his axiomatic system bears no resemblance to how mathematicians actually reason. Tis charge simply betrays a deep misunderstanding o Frege’s project.
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34 Objective, then, in the sense o mind and judgement-independent, as against sensations and intuitions (in Kant’s sense) that are inevitably subjectrelative. 35 See Frege (1879/1891: 2), (1880/1881: 16). 36 Lest this be conusing, to recap: my suggestion is that while the modern turn began beore Frege, the new direction in logic only became firmly established in his work so that the modern way o doing logic was born with him. Modern logic (in our sense o logic) was finally codified only in the 1960s. 37 On the revisionary/descriptive contrast, see Strawson (1959: 9). 38 From Frege’s perspective, the ailings o the ‘language’ developed by Boole had been threeold: i) its expressions were given multiple interpretations within the same language; ii) the language as a whole admitted o (and required) multiple interpretations (roughly, a propositional and a calculus o classes interpretation); iii) the language lacked the expressive power to account or all orms o inerence (specifically, it was unable to account or multiple generality inerences). See Frege (1880/1881). 39 I use the notion o proposition loosely here and interchangeably with that o sentence. Te sense at stake is: whatever it is that the operators o socalled propositional logic operate on. 40 An excellent treatment o Aristotelian logic geared towards explaining the Fregean revolution is in Beaney (1996: ch. 1). 41 Not entirely without residue though. See Beaney 1996: ( §2.3–2.4) or the loss o some subtleties o syllogistic logic in Frege’s system. 42 Te contrast is between what Dummett (1981: 271ff.) has called, respectively, analysis proper and decomposition. 43 Frege (1879: §3; 1895: 120; 1897b: 239). Te rejection o the logical significance o that distinction goes hand in hand with Frege’s introduction o the quantifiers: e.g. Frege deliberately chose to express existential generalizations such as ‘some numbers’ as the negation o generality (‘not all numbers are such that not’) so as to eliminate the possibility o treating ‘some numbers’ as alogical subject. It is probably better to say that Frege did notquite reject the claim that all propositions are o subject– predicate orm. Rather, his point was that the subject–predicate distinction is neither fixed nor properly tracked by standard grammar. Logical analysis is required to show what is predicated o what. 44 Very roughly, a unction is an operation mapping objects (calledarguments) rom one set to objects (called values) in another (not necessarily distinct) set. 45 At the time o Begriffsschrif, Frege had not yet drawn his sense/reerence distinction. Te notion o judgeable content would later be split into grasp o thought (sense) and truth-value appraisal (judgement concerning reerence, or according to Frege the reerence o a sentence is its truth-value). See Frege (1893/2013: 9, n. 2).
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46 In Begriffsschrif Frege is a bit sloppy regarding the use/mention distinction, so he switches between talk o objects and signs airly reely. See Stevenson (1973) or a good discussion o this issue. Te point to bear in mind is that or the later Frege unctional expressions reer to unctions and variables to objects. 47 As Frege (1880/1881: 18) points out, this makes it impossible to distinguish between individuals and concepts under which they uniquely all. Peirce made the same move as Frege, only a ew years later. See Haaparanta 2009: ( §5). 48 I keep switching between thought, content and proposition. Frege changed his terminology (and his conception o these notions) over his career. o keep things simple, I slide back and orth between these notions as appropriate. No harm should ensue, as long as you keep in mind that the details o the story vary over time. 49 I mentioned the subtitle o Begriffsschrif. Tis is probably the place to point out that the title itsel means concept-script, a conceptual calculus, a calculus o concepts, or even better: a calculus generating concepts. See Frege (1880/1881: 15m n. *). 50 As argued convincingly in Dummett (1973: 2, 9, 11). 51 Note that abstraction is here a purely technical notion, not to be conused with the mental process o abstraction (i.e. introspective removal o properties) that Frege criticized in authors such as J.S. Mill and Husserl. 52 Frege’s notion o unction is radically general (much wider than the one standard in mathematics): the result o removing a singular term rom any (well-ormed) sentence (expressing a thought) will be a unction(-name). See Grundgesetze I §2. 53 For Frege concepts are a subspecies o unctions (see Frege 1891). Tey map individuals to truth-values: e.g. ‘Cato killed ξ’ is a unction that maps all individuals such that they were killed by Cato to the rue and everything else to the False (where we replace the name o an individual or and evaluate the resulting thought or truth and alsity). 54 Tis is slightly anachronistic. At the time oBegriffsschrif, Frege hadn’t yet drawn the sense/reerence distinction. Te way he put things then was thatA‘’ stands or a state o affairs, a worldly entity, rather than something in the realm o sense like his later notion o Gedanke. I choose to speak o thought instead to stress the continuity in Frege’s approach to logic (and in any case Frege himsel also speaks interchangeably o thought and content in e.g. §8). ‘Te content thatA’ could be a more neutral way o putting matters. 55 Whitehead and Russell (1910/1997: 8) shortened the horizontal stroke and called it the assertion sign. It then became the standard sign or the derivability relation in a proo system. Martin-Lö (1972) will resurrect Frege’s conception o a calculus o asserted proposition. 56 Frege (1879: §3). Te requirement will be made more precise inGrundgesetze I §5. 57 Teorems are ormulae provable in the system rom the axioms. Te axioms are treated as theorems by courtesy.
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58 Frege (1918–19: 375; 1923: 261). 59 In Dummett’s terminology (see n. 42), his notation representsboth the decomposition and the analysis o the thought. 60 Tere is the temptation to suggest that Frege intended this notation to represent the conceptual (or perhaps epistemological) dependency o the supercomponent on the subcomponent (see e.g. Boolos 1998: 144). Te temptation should, however, be resisted. Frege’s condition stroke is equivalent to the material conditional and no dependency o any kind is thereby asserted. Te gloss in Landini (2012: 43) in terms o a metalinguistic conditional is equally misplaced. Frege is very aware his condition stroke does not characterize the natural language conditional in ull generality. All that the stroke expresses is that the subcomponent is a sufficient condition or the obtaining o the supercomponent. See e.g. Frege1879: ( 37). 61 For Frege’s two senses o assertion Aussage ( ) see Ebert and Rossberg (1893/2013: xvii). Sometimes Frege seems to take the dependency expressed by the condition stroke as a stronger relation yet, as in §16 where he glosses the stroke as expressing the thought that the supercomponent is aconsequence o the subcomponent. 62 See Frege (1897b: 236). 63 Tis applies even more clearly to the way Frege renders the classic conceptcontainment statement A ‘ ll As are Bs’. 64 I distinguish between the conditional (the ormula as a whole) and the condition stroke (the vertical stroke joining up two content strokes). 65 In Grundgesetze he will switch to talk o naming the rue or the False, because in his later philosophy he took sentences to be names o logical objects, that is, the two truth-values. 66 In effect, this is the truth-conditional definition o the conditional, first deended by Philo o Megara and currently called the material conditional reading. 67 Frege is also dismissing the so-called paradoxes o material implication as irrelevant precisely because he is not modelling all uses o i . . ., then. 68 Frege (1897b: 247). As we shall see, Frege used only two connectives, conditional and negation, which meant that ormulae in his system ofen consisted o deeply nested conditionals. Most commentators have accepted the verdict o history (or o common practice) and sided with Peano. For a vigorously dissident opinion, see Cook (2013). See also Landini (2012). 69 Note that i you remove the vertical component o the judgement stroke and the content stroke preceding A ‘ ’, you’ll be lef with the modern way o representing negation, ‘¬’. 70 Te answers to most exercises are to be ound in Begriffsschrif itsel, but o course you should attempt thembeore reading Frege’s work! 71 Quantifiers are second-level concepts, in Frege’s later terminology: they take properties, rather than objects, as their arguments. Tere is a urther twist to the
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story. Afer work by Mostowski in the 1950s, so-called generalized quantifiers have become standard, especially in natural language semantics. And the inspiration or that second revolution came rom Frege! See e.g. Barwise and Cooper (1981) and Peters and Westerståhl (2006: ch. 2). 72 Te issue o the variable is one that looms large in early analytic philosophy (see e.g. Russell (1903/1996) and Wittgenstein’sractatus). In many ways, the modern treatment o the variable was born only with arski 1935). ( Frege’s treatment represents an alternative to arski’s notion o assignment. See Heck2012: ( ch. 3) or discussion. 73 It may seem a little conusing that Frege is here using the same sort o letter to denote propositions (as in the conditional ormulae discussed above) and arguments o unctions. But one should bear in mind that or Frege unctions could take as argument just about anything, including unctions themselves. Note, however, that, in his ormal language, when unctions figure as arguments o other unctions, they are really ‘eeding’ the values they return to the higherorder unctions (see Grundgesetze I, §21). Moreover, inGrundgesetze unction names, once ‘saturated’ with an argument, are names o truth-values, and so what a higher-level unction does is really operate on the truth-values names provided by the lower-level unction which is their argument. 74 One reason why Frege adopts this variant notation is that it validates the transitivity o chains o generalized statements (i.e. the old Barbara syllogism) not otherwise available in his system (the concavity sign – to be explained below – would get in the way, as it were). See Frege (1893/2013: I, §17). Note that this is one more way in which Frege strived to provide a language that would give a unified treatment o all valid orms o inerence. Frege also noted the need or restrictions to be in place, namely, thata should not occur in X other than in its argument places. He noted similar and now amiliar restrictions in the case o inerences o the orm
,
a must also not occur in A. It is in remarks like these that we recognize that the modern turn has now ully taken place: this is, notation apart, just like whatwe do in logic today. 75 In modern notation, this would be the amiliar ∀x F(x), or, in slightly more exotic textbooks, ∀x φ(x). Russell (1908: 65–6) gave a similar gloss. o make Frege’s awkward ormulation more precise, one would say that the thought expressed by the proposition obtained by applying the unctionΦ to any argument is true. Tere is some uncertainty as to whether Frege had a substitutional conception o quantification (replacement o the variable by linguistic expressions rom a given class) or an objectual conception (replacement o the variable by a constant denoting an object in the domain). In this context, Frege glosses the notion as ‘whatever we may put in place o a’. Te ‘whatever’ is not quite so unrestricted though. See Stevenson (1973: 209–10) or discussion.
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76 Equally important is §9 where the unction/argument analysis is introduced. 77 Tink o Boole’s language as a multi-use tool: you can change the head o the tool to carry out different jobs, but you can never carry out more than one job at a time. Frege’s language abolishes the need or a change o head, as it were. 78 Frege had little time or modalities, but given his view o absolutely unrestricted generality (all substitution instances were available in his system), (12) comes pretty close to expressing a naïve view o causal connection. 79 See also Frege (1893/2013: I §12). 80 In response, early critics in the algebraic tradition such as Jevons had been crudely dismissive, saying that Frege had clothed Aristotle in mathematical dress without making any real advance over Boole. Frege 1882) ( gives the lie to that claim. 81 Frege puts a (vague) restriction in place, namely, that when the substituend is a unction sign, this be taken into account. InGrundgesetze, I §48 rule 9, Frege will state his substitution principles much more rigorously but ar too liberally. Indeed, those rules will be the source o the contradiction that scuppered his system. Tere is a good sense in which Begriffsschrif is consistent because it lacks the precision o Grundgesetze with regard to the rules o substitution. See Kamareddine and Laan (2004: §1b1). 82 Te calculus happens to be complete and sound. In §13 Frege gestured towards an awareness o the need to ensure that a system satisy those two conditions (roughly, soundness means that its axioms be true and its rule(s) o inerence truthpreserving; completeness, that the system be strong enough to prove all intuitively true propositions in the subject matter at hand). 83 As Gödel (1944: 120) amously noted, Russell’s and Whitehead’s Principia represented a retrograde step in expository rigour, particularly with respect to the treatment o syntax. 84 It is a rule o proo because it takes as premises only axioms or previously established theorems. Here I assume amiliarity with MP.As I have mentioned already, Frege also made recourse to a rule o substitution, not clearly stated as such. 85 Section 6 is where Frege states most o these now amiliar bookkeeping conventions. 86 Frege also seems to be the first to introduce the distinctions between object and metalanguage (Frege 1923: 260) and variable and meta-variable (Grundgesetze I, 6, n. 1). 87 Mendelson (1964–1997), Hamilton (1978) and Hodel (1995) are among the exceptions. 88 Most current axiomatic presentations are derived rom the amendments given in Łukasiewicz (1935), where the completeness o a streamlined version o Frege’s system was established. One o the things that Łukasiewicz showed was that Frege’s axioms could be reduced in number (or instance, axiom 1 and 2 entail axiom 3, and
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so the latter is redundant). In case you’re tempted to check out that paper, be warned that the axioms are given using Polish notation! 89 As hinted already, Frege also used an unstated rule o uniorm substitution, widely used in his proos (a ormula may be substituted or another uniormly throughout the containing ormula). He also used the equivalence between (7) and (8) as another rule o inerence (given in two variants). See §11 o Begriffsschrif. 90 And so strictly speaking modus ponens is here a rule o proo: it can be applied only to theorems, not to assumptions (as one would do in a Gentzen-like natural deduction system). Frege also allows that proos be commenced rom previously established theorems (see or instance the last proo in §15). 91 Strictly speaking, we should speak o axiom schemata: all instances o these axioms are true. Frege, however, is here using small German letters with the understanding that they are universally closed in the sense o ormula (7). On this second reading, the axioms are not schematic at all. A schematic reading, however, preserves the contrast between axiom 9 that explicitly involves (and indeed defines one direction o the notion o ) generality and the other eight that instead do not. 92 Te interchangeabilityo subcomponentsallows Frege to treat conditionals ashaving the conjunction o all their interchangeable subcomponents as a single subcomponent. Following Cook’s (2013: A-9) elegant suggestion, we can thus read the condition stroke as ann–ary unction
where the
s are all the
ormula’s subcomponents. Section 17, ormula 28, §18 ormula 31 and §19 ormula 41 respectively. Section 20 ormula 52, §21, ormula 54 and §22, ormula 58 respectively. I you’re interested in the topic, take a look at Mendelsohn (2005: ch. 4). In Sullivan’s (2004: 672) useul terminology. Basic Law IIa, §20, and Basic Law IIb, §25, respectively. Again, depending on whether we read Frege’s axioms as schemata or as universally closed ormulae, we may decide to say that itis axiom 1. Incidentally, i you are unamiliar with substitution techniques, you may be surprised to see that we can instantiate axioms that bear different variables by using the same variable. Nothing wrong with that at all. And Frege exploited this technique over and over again in his proos. 99 I’m omitting quotation marks around subormulae here in keeping with Frege’s cavalier attitude to the use/mention distinction! 100 Interestingly enough, in Grundgesetze, I, §18, Frege treats (15) as a special case o axiom 1 by appeal to his orm o the contraction rule, what he called the union o subcomponents (two occurrences o the same subcomponent can be absorbed/ contracted into one). 93 94 95 96 97 98
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101 Frege’s target here was again Kant,who had argued that mathematical induction can only be explained in mind-dependent terms. A purely logical explanation o the kind Frege gave was meant to alsiy Kant’s claim. 102 For a detailed assessment o Frege’s technical achievements in Part III o Begriffsschrif see Boolos (1985). 103 o be precise, the ormin which Russell stated the contradiction did not apply to Frege’s system. Frege pointed this out in his reply, but he also immediately restated
104 105
106 107 108
109
the contradiction in terms that would indeed be syntactically appropriate, and atal, to his system. See Frege’s own diagnosis in Frege 1980: 191, n. 69. For good introductory accounts o Frege’s lie and work, see Kenny (1995), Weiner (1990) and Noonan (2001). For an account o Frege’s logic inGrundgesetze see Cook (2013). Frege (1924, 1924/1925). Rather, what the laws o the laws o thought were,to adaptGrundlagen. On the first point see Frege (1892) and ait (1997: 246). On the second, see the increasing irritation in Hilbert’s letters F( rege 1980) and Peano’s doubts, expressed in his 1895 review o Grundgesetze, regarding Frege’s ‘unnecessary subtleties’. Van Heijenoort (1967a: 127).
Reerences Barwise, John and Cooper, Robin (1981), ‘Generalized Quantifiers and Natural Language’. Linguistics and Philosophy, 4: 159–219. Beall, J.C. and Restall, Greg (2006), Logical Pluralism. Oxord: Clarendon Press. Beaney, Michael (1996), Frege: Making Sense. London: Duckworth. Beaney, Michael and Reck, Erich H. (eds) (2005), Gottlob Frege: Critical Assessment. London: Routledge. Blanchette, Patricia A. (2012), Frege’s Conception o Logic . Oxord: Oxord University Press. Boole, George (1854/2009), An Investigation o the Laws o Tought. Cambridge: Cambridge University Press. Boolos, George (1975), ‘On Second-Order Logic’. In Boolos (1998), pp. 37–53. Boolos, George (1985), ‘Reading the Begriffsschrif’. In Boolos (1998), pp. 155–70. Boolos, George (1994), 1879?’ In Boolos (1998), pp. 237–54. Boolos, George (1998), Logic, Logic, and Logic. Cambridge, MA: Harvard University Press. Bostock, David (1997), Intermediate Logic. Oxord: Clarendon Press. Burge, yler (1992), ‘Frege on Knowing the Tird Realm’. In Burge (2005), pp. 299–316. Burge, yler (1998), ‘Frege on Knowing the Foundation’. In Burge (2005), pp. 317–55.
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Burge, yler (2005), ruth, Tought, Reason. Essayson Frege. Oxord: Clarendon Press. Cook, Roy (2013), ‘How o ReadGrundgesetze’. In Frege (1893/2013), pp. A1–A42. Dummett, Michael (1959), ‘George Boole’. In ruth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 66–73. Dummett, Michael (1973), Frege. Philosophy o Language. 2nd edn. London: Duckworth. Dummett, Michael (1981), Te Interpretation o Frege’s Philosophy . London: Duckworth. Dummett, Michael (1991a), Frege and Other Philosophers. Oxord: Clarendon Press. Dummett, Michael (1991b), Frege. Philosophy o Mathematics. London: Duckworth. Dummett, Michael (1994), Origins o Analytical Philosophy. Cambridge, MA: Harvard University Press. Ebbinghaus, H.-D., Flum, J. and Tomas, W. 1( 994), Mathematical Logic. 2nd edn. New York: Springer. Ebert, Philip A. and Rossberg, Marcus (1893/2013), ‘ranslators’ Introduction’. In Frege (1893/2013), pp. xiii–xxvi. Frege, Gottlob (1879), ‘Begriffsschrif’. In van Heijenoort 1( 967a), pp. 5–82. Frege, Gottlob (1879/1891), ‘Logic’. In Frege 1( 979), pp. 1–8. Frege, Gottlob (1880/1881), ‘Boole’s logical Calculus and the Concept-Script’. In Frege (1979), pp. 9–52. Frege, Gottlob (1882), ‘Boole’s logical Formula-language and my Concept-script’. In Frege (1979), pp. 47–52. Frege, Gottlob (1891), ‘Function and Concept’. In Frege 1( 984), pp. 137–56. Frege, Gottlob (1892), ‘On Sense and Reerence’. In Beaney, Michael (ed.)Te Frege Reader. Oxord: Blackwell, pp. 151–71. Frege, Gottlob (1893/2013), Basic Laws o Arithmetic. Oxord: Oxord University Press. Frege, Gottlob (1894), ‘Review o E.G. Husserl,Philosophie der Arithmetik I’. In Frege (1984), pp. 195–209. Frege, Gottlob (1895), ‘Comments on Sinn und Bedeutung’. In Frege 1( 979), pp. 118–25. Frege, Gottlob (1897a), ‘Logic’. In Frege 1( 979), pp. 118–51. Frege, Gottlob (1897b), ‘On Mr Peano’s Conceptual Notation and My Own’. In Frege (1984), pp. 234–48. Frege, Gottlob (1918–19), ‘Negation’. In Frege (1984), pp. 373–89. Frege, Gottlob (1923), ‘Logical Generality’. In Frege (1979), pp. 258–62. Frege, Gottlob (1924), A ‘ New Attempt at a Foundation or Arithmetic’. In Frege (1979), pp. 278–81. Frege, Gottlob (1924/1925), ‘Numbers and Arithmetic’. In Frege 1( 979), pp. 275–7. Frege, Gottlob (1979), Posthumous Writings. Oxord: Blackwell. Frege, Gottlob (1980), Philosophical and Mathematical Correspondence . Oxord: Blackwell. Frege, Gottlob (1984), Collected Papers on Mathematics, Logic, and Philosophy . Oxord: Blackwell. Gabbay, Dov and Guenthner, Franz (eds) (1983/2001), Handbook o Philosophical Logic, vol. 1. 2nd edn. Dordrecht/Boston/Lancaster: D. Reidel Publishing Company.
Frege
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Gabriel, Gottried (1996), ‘Frege’s “Epistemology in Disguise”*’. In Schirn, Matthias (ed.) Frege: Importance and Legacy. Berlin: Walter de Gruyter, pp. 330–46. Gödel, Kurt (1944), ‘Russell’s Mathematical Logic’. In Feerman, Solomon (ed.)Collected Works, vol. II. Publications 1938–1974. Oxord: Oxord University Press, pp. 119–41. Goldarb, Warren (1979), ‘Logic in the wenties: Te Nature o the Quantifier’.Journal o Symbolic Logic, 44, 3: 351–68. Goldarb, Warren (2010), ‘Frege’s Conception o Logic’. In Potter and Ricketts 2010), ( pp. 63–85. Grattan-Guinness, Ivor (2000), Te Search or Mathematical Roots, 1870–1940. Logic, Set Teories and the Foundations o Mathematics rom Cantor through Russell to Gödel . Princeton and Oxord: Princeton University Press. Haaparanta, Leila (2009), ‘Te Relation between Logic and Philosophy, 1874–1931’. In Haaparanta, Leila (ed.)Te Development o Modern Logic. Oxord: Oxord University Press, pp. 222–62. Hamilton, A.G. (1978), Logic or Mathematicians. Cambridge: Cambridge University Press. Heck, Richard (2010), ‘Frege and Semantics’. In Potter and Ricketts 2010), ( pp. 342–78. Heck, Richard (2012), Reading Frege’sGrundgesetze. Oxord: Clarendon Press. Hilbert, David and Ackermann, Wilhelm (1928), Principles o Mathematical Logic. Providence, Island: AMS Chelsea Hodel, Richard Rhode E. (1995), An Introduction to Publishing. Mathematical Logic. Boston: PWS Publishing Company. Hodges, Wilrid (1983/2001), ‘Elementary Predicate Logic’. In Gabbay and Guenthner (1983/2001), pp. 1–129. Jeshion, Robin (2001), ‘Frege’s Notion o Sel-Evidence’. Mind, 110, 440: 937–76. Jourdain, Philip E.B. (1912), ‘Gottlob Frege’. Quarterly Journal o Pure and Applied Mathematics, 43: 237–69. Kamareddine, Fairouz and Laan, wan (2004), A Modern Perspective on ype Teory From its Origins until oday. Dordrecht: Kluwer Academic Publishers. Kelly, John (1997), Te Essence o Logic. Harlow: Prentice Hall. Kenny, Anthony (1995),Frege. Harmondsworth: Penguin Books. Landini, Gregory (2012), Frege’s Notations. What they are and how they mean . Basingstoke: Palgrave Macmillan. Łukasiewicz, Jan (1935), ‘Zur Geschichte der Aussagenlogik’. Erkenntnis, 5, 111–31. Martin-Lö, Per (1972), A ‘ n Intuitionistic Teory o ypes’. In Sambin, Giovanni and Smith, Jan M. (eds),wenty-five Years o Constructive ype Teory. Oxord: Clarendon Press, pp. 127–72. Mendelsohn, Richard L. (2005), Te Philosophy o Gottlob Frege. Cambridge: Cambridge University Press. Mendelson, Eliott (1964–1997),Introduction to Mathematical Logic. 4th edn. Boca Raton: Chapman & Hall. Noonan, Harold W. 2( 001), Frege: A Critical Introduction. Cambridge: Polity Press.
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Peters, Stanley and Westerståhl, Dag (2006), Quantifiers in Language and Logic. Oxord: Clarendon Press. Potter, Michael (2010), ‘Introduction’. In Potter and Ricketts 2010), ( pp. 1–31. Potter, Michael and Ricketts, om (eds) (2010), Te Cambridge Companion to Frege. Cambridge: Cambridge University Press. Putnam, Hilary (1982), ‘Peirce the Logician’. In Realism with a Human Face. Cambridge, MA: Harvard University Press, pp. 252–60. Quine, W.V. 1950), ( Methods o Logic. Cambridge, MA: Harvard University Press. Quine, Willard V. 1( 960), Word and Object. Cambridge, MA: Te MI Press. Quine, Willard V. 1( 970), Philosophy o Logic. 2nd (1986) edn. Cambridge, MA: Harvard University Press. Read, Stephen (1997), ‘Completeness and Categoricity: Frege, Gödel and Model Teory ’. History and Philosophy o Logic, 18: 79–93. Reck, Erich H. (2007), ‘Frege on ruth, Judgment, and Objectivity’. In Greimann, Dirk (ed.) Essays on Frege’s Conception o ruth . Grazer Philosophische Studien, Amsterdam and New York: Rodopi, pp. 149–73. Reck, Erich H. and Awodey, Steve (eds) (2004), Frege’s Lectures on Logic. Carnap’s Student Notes 1910–1914. Chicago and La Salle, Illinois: Open Court. Ricketts, Tomas (1986), ‘Objectivity and Objecthood: Frege’s Metaphysics o Judgement’ In Haaparanta, L. and Hintikka, J. (eds)Frege Synthesized. Reidel: Dordrecht,.pp. 65–95. Ricketts, Tomas (1996), ‘Logic and ruth in Frege’. Proceedings o the Aristotelian Society, Supplementary Volumes , 70: 121–40. Ruffi no,Marco (2002), ‘Logical Objects in Frege’s Grundgesetze, Section 10’. In Reck, Erich H. (ed.) From Frege to Wittgenstein. Oxord: Oxord University Press,pp. 125–48. Russell, Bertrand (1903/1996), Te Principles o Mathematics. London: Routledge. Russell, Bertrand (1908), ‘Mathematical Logic as Based on the Teory o ypes’. In Marsh, Robert C. (ed.) Logic and Knowledge. London: Routledge, pp. 59–102. Shapiro, Stewart (1991), Foundations without Foundationalism. A Case or Secondorder Logic. Oxord: Clarendon Press. Soames, Scott (2010), Philosophy o Language. Princeton, New Jersey: Princeton University Press. Socher-Ambrosius, Rol and Johann, Patricia (1997), Deduction Systems. New York: Springer. Stanley, Jason (1996), ‘ruth and Metatheory in Frege’. In Beaney and Reck (2005), pp. 109–35. Stevenson, Leslie (1973), ‘Frege’s wo Definitions o Quantification’. Te Philosophical Quarterly, 23, 92: 207–223. Individuals. London and New York:Routledge. Strawson, P.F. 1959), ( Sullivan, Peter (2004), ‘Frege’s Logic’. In Gabbay, Dov M. and Woods, John (eds.) Handbook o the History o Logic, vol. 3. Te Rise o Modern Logic: From Leibniz to Frege. Amsterdam: Elsevier, pp. 671–762.
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Sullivan, Peter (2005), ‘Metaperspectives and Internalism in Frege’. In Beaney and Reck (2005), pp. 85–105. Sundholm, Göran (1983/2001), ‘Systems o Deduction’. In Gabbay and Guenthner (1983/2001), pp. 133–88. ait, William (1997), ‘Frege versus Cantor and Dedekind: On the Concept o Number’. In Te Provenance o Pure Reason. Essays in the Philosophy o Mathematics and Its History. Oxord: Oxord University Press, pp. 212–51. appenden, Jamie (1997), ‘Metatheory and Mathematical Practice in Frege’.Philosophical opics, 25, 2: 213–64. arski, Alred (1935), ‘Te Concept o ruth in Formalized Languages’. In Logic, Semantics, Metamathematics, 2nd (1983) edn. Indianapolis: Hackett, pp. 152–278. van Heijenoort, Jean (ed.) (1967a), From Frege to Gödel. Cambridge, MA : Harvard University Press. van Heijenoort, Jean (1967b), ‘Logic as Calculus and Logic as Language’. Synthèse, 17, 3: 324–30. Weiner, Joan (1990), Frege in Perspective. Ithaca, NY: Cornell University Press. Whitehead, Alred North and Russell, Bertrand (1910/1997), Principia Mathematica to *56. Cambridge: Cambridge University Press. Zalta, Edward (2013), ‘Frege’s Teorem and Foundations or Arithmetic’. In Zalta, Edward (ed.) Te Stanord Encyclopedia o Philosophy, all 2013 edn,http://plato. stanord.edu/archives/all2013/entries/rege-logic/.
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Introduction Gottlob Frege is customarily awarded the honorific title ‘the ather o modern logic’. Jean van Heijenoort (1992: 242) tells us that ‘Modern logic began in 1879, the year in which Gottlob Frege (1848–1925) published his Begriffsschrif’, while Michael Dummett (1993: xiii) regards Frege as the ‘initiator o the modern period in the study o logic’. Yet, the praise so requently accorded to the srcinality and quality o Frege’s logic and philosophy has obscured the significance and influence o his contemporary, Giuseppe Peano.Te contribution o the latter has been underestimated; indeed in terms o causal consequence, Peano’s work had a greater impact on the development o logic at a crucial period surrounding the turn o the nineteenth century. Peano’s influence was elt principally through his impact on Bertrand Russell (indeed, much the same can be said o Frege). In this chapter we shall examine this line o influence, looking first at Peano’s logic and then at Russell’s.
1 Giuseppe Peano Giuseppe Peano was born in Piedmont, in 1858, ten years afer Frege and ourteen years beore Russell. He was educated in the Piedmontese capital, urin, where he became proessor o mathematics, teaching and researching in analysis and logic. As with several other mathematicians, his interest in logic was in part motivated by the need to promote rigour in mathematics. Te emphasis he places on rigour can be appreciated through a debate between Peano and his colleague Corrado Segre. Segre had made important contributions to algebraic geometry, and with Peano made urin a notable centre or geometry. Nonetheless, 229
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they had quite different attitudes towards rigour, proo and intuition. Segre elt that in order to make progress in a new area o mathematics, it would ofen be necessary to resort to less rigorous, incomplete procedures (Borga and Palladino 1992: 23). Peano (1891: 66) responded that ‘a result cannot be considered as obtained i it is not rigorously proved, even i exceptions are not known.’ Te difference between the intuitive approach and the rigorous approach is apparent in the mathematical understanding o continuity. notion intuitive o a continuous curve is one that can be divided into parts,Te each o which is continuously differentiable. Curves o the latter sort cannot fill a space – however small a space we choose, the curve cannot go through every point in the space. Intuitively a one-dimensional thing, a continuous curve, cannot fill twodimensional space. However, Peano shows that some continuous curves can go through every point in a finite space, indeed in infinite space. In a paper published in 1890, Peano gave a ormal description o what became known as the ‘Peano curve’, a curve that passes through every point in the unit square. Te paper gave no illustration o the curve, precisely because, Peano held, it is thinking in visual terms about geometry and analysis that leads to the use o intuition in mathematics and the breakdown o rigour. Te other influence on the Peano curve was Cantor. Peano was intrigued by Cantor’s proo that the set o points in the unit interval (the real numbers between 0 and 1) has the same cardinality as (i.e. can be put into 1–1 correlation with) the number o points in the unit square. Since the cardinality o the unit interval is equal to that o the whole real line, this raises the possibility that there is a curve that proves Cantor’s conclusion by going though every point in the unit square. Peano’s curve does exactly that.
1.1 Peano’s logical symbolism In the year ollowing publication o his space-filling curve, Peano instituted his ‘Formulario’ project, the aim o which was to articulate all the theorems o mathematics using a new notation that Peano had devised. A little earlier Peano (1888) has published a book on the geometrical calculus, ollowing the approach o Hermann Grassmann’s Ausdehnungslehre. According to Grassmann, geometry does not have physical space as its subject matter. It is rather a purely abstract calculus that may have application to physical space. Conceiving it this way allows or a ormal approach to geometrical proo divorced rom intuition and thereby allows or flaws in intuition-based reasoning (or example in Euclid) to be avoided. Moritz Pasch, in hisVorlesungen über neuere Geometrie(1882) tools
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certain terms as primitive and undefined (Kernbegriffe as Pasch later called them) while certain theorems (Kernsätze) are taken as unproved, i.e. are the axioms. Derived theorems are to be proved by pure logic alone, independently o any intuition. Pasch’s approach influenced Peano’s I principii di geometria logicamente esposti (1889). For this programme to be realized ully, Peano held, the geometrical calculus needed the ramework o a logical calculus. Inwork 1889 (in Peano published his his Arithmetices principia, nova exposita Tis Latin) contained amous five postulates ormethodo arithmetic – the. Arithmetices principia. Te phrase ‘nova methodo exposita’ reers to the new symbolism that Peano introduced or logical relations: the now amiliar ‘ ’ or set/class membership and ‘Ɔ’ (a rotated ‘C’ or ‘consequentia’). In the second volume o the Formulaire Peano replaced ‘Ɔ’ with the now-amiliar horseshoe ‘⊃’). Peano had already invented several other symbols central to the symbolism o logic and set theory, such as the symbols or union and intersection, ‘ ∪’ and ‘∩’, and existential quantification, ‘∃’, and the tilde ‘∼’ or negation. Peano signified universal generalization using subscripts and his ‘Ɔ’. Te amiliarity o these symbols tends to mask the importance o their introduction. But the idea is also amiliar that having the appropriate terminology is crucial not just to expressing thoughts clearly but ofen also to having the relevant thoughts at all. Te development o a symbolism or logic was not just a lubricant to the accelerating vehicle that was mathematical logic, but was an essential part o its motor, as Bertrand Russell was soon to acknowledge.
2 From Peano to Russell In the final summer o the nineteenth century, the first International Congress o Philosophy took place in Paris (it was to be ollowed immediately by the second International Congress o Mathematics). Peano was on the committee or the Congress and one o its leading lights. Also attending was twentyeight-year-old Bertrand Russell. Born into the aristocracy and grandson o a Prime Minister, Russell camerom quite a different backgroundto Peano.Nonetheless, their interests converged on the intersection o mathematics, logic and philosophy. Russell went up to Cambridge in 1890, where he completed both the Mathematical and Moral Sciences (Philosophy) riposes. Like Peano,Russell was much taken by thework o Georg Cantor. He had begun to study Cantor’s work in 1896 (Grattan-Guinness 1978: 129), and had been thinking a great deal when he met Peano in August 1900. In his autobiography, Russell records the significance o his meeting thus:
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Te Congress was the turning point o my intellectual lie, because there I met Peano. I already knew him by name and had seen some o his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better o any argument on which he embarked. As the days went by, I decided that this must be owing to his mathematical logic . . . It became clear to me that his notation afforded an instrument o logical analysis such as I had been seeking or years. Russell 1967: 144
And thus: I went to [Peano] and said, ‘I wish to read all your works. Have you got copies with you?’ He had, and I immediately read them all. It was they that gave the impetus to my own views on the principles o mathematics. Russell 1959: 65
2.1 Burali-Forti’s paradox and Cantor’s theorem Russell, who had been intending to stay or both congresses, lef Paris in order to return home to study Peano’s work in greater detail. He was, however, already aware o Burali-Forti’s paradox and o Cantor’s paradox. Cesare Burali-Forti was Peano’s assistant and lectured on Peano’s mathematical logic and used Peano’s symbolism in his own work. In 1897 Burali-Forti had published his amous paradox concerning ordinals. In a simple terms the class o all ordinals, Ω, is an ordinal or corresponds to an ordinal. But then we can construct a greater ordinal, which is both a member o Ω (because it is an ordinal) and not a member o Ω (because it is greater than Ω). Georg Cantor had earlier (1891) published his diagonal proo that the power set o a set is always larger than the set itsel, as ollows. Let us assume that there is a 1–1 correspondence between S and its power set (so : S → (S) is a bijection); or every member o x o S, (x) (S), and or every member y o (S), there is a unique x S such that (x) = y. (S), which we will call
able 9.1 is a 1–1 correlation between S and (S)
S
( S)
x
(x)
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Te various (x) are subsets o S. And so some members o S may be correlated by with sets o which they are themselves members, i.e. x (x). Let us use ‘Σ’ to denote the set o such members o S, i.e. Σ = {x : x S x (x)}. Now consider the remaining members o S; call this set ‘ Σ*’. Σ* contains those members o S which are not members o the sets with which they correlate under ; i.e. Σ* = {x : x S x (x)}. Every member member othey one are or other o Σ and * but notmeans, o both. S is a o Both Σ and Σ* are o subsets S, hence members o Σ (S). Tis by the definition o , there must be members o S which correlate with them. Let ‘x*’ denote the member that correlates with Σ* (i.e. (x*) = Σ*). able 9.2 Some members o S correlate with sets they are members o; these are members o Σ. Others correlate with sets they are not members o; these are members o Σ*.
S
(S)
x
(x) where x (x) – such sets are members o Σ
z
(z) where z (z) – such sets are members o Σ*
able 9.3 x* correlates with Σ* = (x*)
S
( S)
x*
Σ*
Now we ask: Is x* a member o Σ or Σ*? I x* Σ* i.e. x* Σ, then by definition o Σ, x* (x*) (see able 9.2). But by definition o x* (see able 9.3), (x*) = Σ*, so x* Σ*. So we have shown that x* Σ* → x* Σ*. I on the other hand x* Σ*, then by definition o Σ* (see able 9.2), we have x* (x*). Since (x*) = Σ* (see able 9.3), it ollows that x* Σ*. Tat is, we have shown that x* Σ* → x* Σ*. So together we have: x* Σ* ↔ x* Σ*, which is a contradiction. Tus the assumption that there is a 1–1 correlation between S and (S) is alse. But Russell had not initially seen the ull significance o this result. He wrote: ‘[I]in this one point the master [Cantor] has been guilty o a very subtle allacy, which I hope to explain in some uture work’ (Russell 1963: 69). Nonetheless, in due course, most probably as a result o thinking about Burali-Forti’s paradox and being assisted by Peano’s symbolism, he came to see the implications o Cantor’s proo and o Burali-Forti’s paradox. Cantor himsel, had, around the
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same time as Burali-Forti, drawn the conclusion rom his theorem that there is no largest cardinal. Russell later remarked, ‘I attempted to discover some flaw in Cantor’s proo that there is no largest cardinal . . . Applying this proo to the supposed class o all imaginable objects, I was led to a new and simpler contradiction’. Te contradiction to which he was led was that o the set o sets which do not contain themselves.
3 Bertrand Russell In 1903 Russell published his Principles o Mathematics, in which he set out the logicist programme he was to pursue with Alred North Whitehead. Although Russell had already been at work on a book with this title, he was impelled to rewrite the work completely as a result o meeting Peano, which he did in the last three months o 1900 (Russell 1959: 72–3). At this time Russell was not aware o Frege’s work. Indeed Russell came across Frege by reading the works o Peano, which Peano had presented to him in Paris. On seeing Frege’sGrundgesetze der Arithmetik (Basic Laws o Arithmetic) and Grundlagen der Arithmetik (Foundations o Arithmetic), Russell was struck by the philosophical significance o the latter and by the atal flaw in the ormer. Because Russell engaged with Frege’s work only afer he had completed writing thePrinciples o Mathematics, Russell chose, in the light o its philosophical significance, to add an extended appendix in which Frege’s work is discussed. Russell says: Te work o Frege, which appears to be ar less known than it deserves, contains many o the doctrines set orth in Parts I and II o the present work, and where it differs rom the views which I have advocated, the differences demand discussion. Frege’s work abounds in subtle distinctions, and avoids all the usual allacies which beset writers on Logic. His symbolism, though unortunately so cumbrous as to be very difficult to employ in practice, is based upon an analysis o logical notions much more proound than Peano’s, and is philosophically very superior to its more convenient rival. In what ollows, I shall try briefly to expound Frege’s theories on the most important points, and to explain my grounds or differing where I do differ. But the points o disagreement are very ew and slight compared to those o agreement.
Te conclusion we should draw, I believe, is that despite the quality o Frege’s work, it was not especially influential in the early development o modern logic. Russell is the crucial figure (perhaps with Hilbert alongside him). But although Frege’s work may have sharpened Russell’s subsequent philosophical thinking,
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there is no reason to suppose that Russell’s logic would have been much different without Frege. Even i less worthy, Peano’s logic was, as we have seen, seminal or Russell. As or the atal flaw, Russell immediately saw that Frege’s axioms in his Basic Laws led to a contradiction. As Russell wrote to Frege, just as the completion o the latter was in print: Let w be the predicate: to be a predicate that cannot be predicated o itsel. Can w be predicated o itsel? From each answer its opposite ollows. Tereore we must conclude that w is not a predicate. Likewise there is no class (as a totality) o those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection does not orm a totality. 1 16 June 1902, letter to Frege, translation in van Heijenoort 1967: 125
Frege’s fifh axiom (V) claims that every predicate defines a set, its extension. Russell’s w seems to be precisely a predicate that does not have an extension. Let Fx be x x (de F). (Recall de : x y ↔ G(y =ˆG ∧Gx).) By (V) we have an extension or F,viz. ˆF. We now consider whether or not F(ˆF). (1) (2) (3) (4) (5) (6) (7) (8)
¬F(ˆF) ˆFˆF G(ˆF=ˆGG(ˆF)) ˆF=ˆG → (F↔G) G(F↔G G(ˆF)) ∧ F(ˆF) ˆFˆF ¬F(ˆF)
assumption (1, de F, double negation) (2, de ) (Vb) (3, 4 MP ) (5, simple logic) (rom de ) (de F)
And so considering lines 1, 6 and 8, we have: ¬F(ˆF) ↔ F(ˆF), which is Russell’s paradox. Another o the paradoxes considered by Russell was that discovered by the French mathematician Jules Richard. Russell gives it as ollows: consider all decimals that can be defined by means o a finite number o words; let E be the class o all such decimals. Since the possible finite expressions o a natural language are countable, the number o members o E is countable. Hence we can order (not necessarily in order o size) the members o E as the first, second, third etc. Let N be a number defined as ollows: i the nth figure in the nth decimal is p, let the nth figure in N be p+1, or 0 i p=9. Ten N is different rom all the members o E, since whatever finite value n may have, the nth figure in
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N is different rom the nth figure in the nth o the decimals in the ordering o E (i.e. N is constructed by Cantor’s diagonal method). Nevertheless we have defined N in a finite number o words, and thereore N ought to be a member o E. Tus N both is and is not a member o E. (A simpler version o the paradox is this: let M = the least integer not definable in ewer than twenty syllables. M has just been defined in nineteen syllables, so M both is and is not definable in ewer than twenty syllables.) Te paradox is dissolved, according to Richard, i we hold that the definition o E, as the totality consisting o all decimals finitely definable, cannot rightly be regarded as ranging over decimals which are defined by reerence to that totality. Tus given that N is defined that way, it is not a sensible possibility that N is in E.
3.1 Te vicious circle principle Richard’s dissolution exemplifies the vicious circle principle ormulated by Richard and by Henri Poincaré to exclude viciously circular definition. Russell states the principle thus: ‘ “Whatever involves all o a collection must not be one o the collection”, or, conversely: “I, provided a certain collection had a total, it would have members only definable in terms o that total, then the said collection has no total”.’ And in a ootnote he explains this last: ‘When I say that a collection has no total, I mean that statements about all its members are nonsense’ (Russell 1908: 30). Although the application o the principle is, as it stands, not entirely clear, it is probably unair to claim, as Kurt Gödel did, that Russell gives us not one but three principles. Tere is but one principle: i a definition is to be meaningul then i the definiens reers to all o some totality the definiendum may not be part o that totality. Te aim o the vicious circle principle is to exclude impredicative definitions. Te idea o impredicativity, never made entirely clear, was roughly this: a definition is not impredicative i the definiendum is defined in terms o predicates to which it is not a possible argument: i.e. the definiendum is defined in terms o totalities (the argument domains o the predicates) o which it is not a part. In impredicative definitions, the definiendum is a possible argument o the predicate and is included in the totality according to which it itsel is defined. Let us see how this works or contradictions resulting rom unrestricted comprehension (essentially the same as the contradiction emanating rom Frege’s axiom (V)). Unrestricted comprehension says: or any predicateφ there is a class y o objects satisying φ, i.e. or anyφ, yx(x y ↔ φx). Te contradiction arises rom taking φx to be x x, i.e. there is a y such that x(x y ↔ x x).
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Substituting y or x (since x ranges over all classes) we have y y ↔ y y. Tis definition o y is impredicative since y is being defined in terms o a predicate, x x, or which y is itsel a possible argument. Russell takes the vicious circle principle to exclude such definitions. Te vicious circle principle is negative, pointing to one source o contradiction which must be excluded. As such it is a criterion o meaningulness. We must construct our mathematical logicmeaningless. and its language in such a way that sentences which violate the principle are On which positive principles the logic should be constructed the principle does not directly suggest. However, different interpretations o why violation leads to contradiction will encourage different approaches to a positive logic. Poincaré, on the one hand, saw the illegitimacy in talking o all o the class o finitely definable decimals as deriving rom the act that that class is infinitely large. Hence logic must be constructed in such a way as to allow quantification only over finite classes.
3.2 Russell’s theory o types Russell, on the other hand, saw the issue as turning on logical homogeneity. Te things we talk about may not be all o the same logical kind or type. According to Russell, it should be improper to quantiy over things o a different logical type. I one assumes that things which involve all o a collection are different sorts o thing rom the members o the collection, then the requirement o logical homogeneity would orbid quantiying over members o a collection and the collection itsel at the same time. Te contradictions arise because the heterogeneity, the difference in kind, is hidden by the description given o the object. Te hidden heterogeneity o the collections appears to license including in the collection the very entity which is defined in terms o the collection. Consequently there would be no violations o the vicious circle principle in a logic which precludes talking o heterogeneous collections, but allows quantification only over entities o the same logical type. Tis is precisely what Russell and Whitehead’s theory o types aims to do. How it does this is again most easily seen in connection with the abstraction axiom (naïve/unrestricted comprehension): or all φ, yx(x y ↔ φx). We have seen how this axiom leads to contradiction. Te theory o types, by submitting to the vicious circle principle, avoids the contradiction in two ways. Firstly, classes are not allowed to be members o themselves or o classes o the same type as themselves; ormulae such as x x and x x are simply ill-ormed, they are not ormulae o the language and hence the question o their truth or alsity does not
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arise. Tus they do not represent predicates that can take the place o φ in the above abstraction axiom. More generally our stricture prevents the universal and existential quantifiers in the axiom, the ‘x’ and ‘y’ in yx(x y ↔ φx), rom overlapping. I their ranges did overlap, as they are allowed to do in the derivation o the contradiction, we would be orced to contemplate possibilities such as y y, which I have said are excluded as being nonsense. Te quantifiers are restricted each totypes one homogeneous type.at the very bottom o which comes the Tese logical orm a hierarchy, type o individuals, which are not really classes at all. Te next level up, the first type o classes, contains only those classes which contain individuals as members. Te next higher type is the type o classes whose members are the classes o the first type, i.e. the type o classes o classes o individuals. Te hierarchy continues in such a way that each class is assigned a particular type, and i this type is the n+1th, its elements are o the nth type, and it itsel may only be a member o those classes o the type n+2. Correspondingly we allow it to make sense to assert that one set is a member o another only i the type o the latter is one higher than that o the ormer. Hence quantification must be only within a single type; to be exact, in ormulas such as ‘xFx’ we should speciy which type o classes the apparent variable x ranges over. I this is specified as n, then the gloss or ‘xFx’ starts ‘or all classes o the nth type . . .’ which we might emphasize by writing: ‘ xnFx’. Our axiom o comprehension then looks like this: yn+1 xn(xn yn+1 ↔ φxn). It should be noted that our hierarchy means that it makes no sense to talk o ‘class’ (nor o ‘cardinal’ or ‘ordinal’ etc.) simpliciter, or the term is ambiguous, not reerring to one sort o object, but many sorts. Tus use o the term ‘class’ must be governed by a systematic ambiguity, that is, taken as reerring to classes o some one, albeit unspecified, type. Tis hierarchy o classes does a nice job o excluding the set theoretic or class paradoxes: Russell’s, Cantor’s, Burali-Forti’s. It seemed as i classes were all o one sort and that we could quantiy over all o them. In Principia Mathematica it turns out that classes may be rather different things and that we cannot quantiy over all with one variable.
3.3 Dissolving the semantic paradoxes On the other hand, the paradox rom which we first gleaned our guiding vicious circle principle, the Richard paradox, and those like it appear to be somewhat
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different. In the class case we applied the principle that a class which is defined in terms o some totality o classes is a different sort o class rom those in that totality, hence it could not be part o that totality. But in the case o the Richard paradox we tried to define one decimal, N, in terms o a particular class, E, o decimals. In this case it could not possibly be said that N is a different sort o object rom the members o E. On the contrary, N is a decimal just like the members E, and paradox isNdissolved in such aoway exist, thereomay wellassuming be classesthe containing and the members E. that N does It would appear that the semantic paradoxes are not to be dissolved by pointing to hierarchies o entities which differ in type. Let us return to the vicious circle principle. It warned us not to reer to illegitimate totalities. Russell’s approach was then to seek in each paradox the illegitimate totality reerred to. In his own paradox it was ‘all classes’; in the Burali-Forti contradiction it was ‘all ordinals’ etc. In the case o the Richard paradox our problem is not the totality ‘all decimals’ but ‘all definitions o decimals’, or the crucial question was whether the definition o N belongs to this totality. Similarly the paradox o the least indefinable ordinal reers to all definitions. Berry’s paradox reers to all names. What we must then do is to seek a hierarchy o definitions or names, which will amount to the same thing since the definitions and names are both given by description, that is, in terms o properties. And indeed, this part o Russell and Whitehead’s theory, the theory o orders, proceeds precisely analogously to the theory o classtypes. I a property (strictly, a propositional unction) is defined in terms o some totality o properties, then the defined property is o a different sort, i.e. o a higher order. Te term ‘property’ is itsel ambiguous, requiring specification, at least in principle, o some specific order.Variables range only over properties o one order.
3.4 Te no-class theory Russell thought that i classes really did exist as objects independent o unctions or o constructions then there would be no logical reason why a class might not be a member o itsel. More significantly, Russell held, like Frege, that there could be no restriction on variables preventing them rom ranging over all objects in the world, which would not be arbitrary. Tus i classes were real, there could be no objection to our talking o the totality o classes which do not contain themselves and other such totalities which lead to the contradictions. Tus the theory o types requires taking a non-platonist, non-realist view o classes By making classes derivative o unctions, the hierarchy o classes could be justified on the ground o the distinction between unctions. Te justification is
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most natural i unctions are seen as linguistic entities constructed in terms o other unctions already defined. Clearly such a view obeys the vicious circle principle as it was ormulated above; contravention would be a case o circular definition, obviously giving no meaning to the sign in question. Furthermore, it coheres with Russell’s logical atomism. Nonetheless there is nothing in this to prevent a unction taking arguments o more than one type so long as all such types are lower than thesettype o theInunction itsel, as in a cumulative hierarchyis like Zermelo–Frankel theory. Principia Mathematica this possibility excluded. In act Russell and Whitehead come later, in the second edition o Principia Mathematica (1925–7), to justiy certain contentious axioms (the axioms o reducibility and infinity) abductively; that is, by virtue o their ability to generate ‘true’ theorems. Tis marks a departure rom their srcinal logicist programme o ounding mathematics on evident logical axioms.
3.5 Problems with the theory o types Te theory o types does not lead to the sort o contradictions aced by Frege’s logic. However, it does depend on two axioms which do not seem to be true logical principles. One is the axiom o reducibility and the other is the axiom o infinity. Te ormer need not concern us here. Te need or the axiom o infinity arises because we are not able to quantiy over entities belonging to different types. Imagine that there are only a finite number o individuals in the world. Tus only a finite number o entities o type 0. Te entities o type 1 will be all the possible classes o entities o type 0. Te number o such entities will also be finite (2n to be precise). Tus there will only ever be a finite number o entities o any given type. Tis means there will be no type with an infinite number o entities. So we will not be able to get enough classes to represent the natural numbers (which Russell does in Frege’s manner: 3 = the class o three membered classes (o a certain type).) Tus we need to have as an axiom, the assertion that there are infinitely many individuals in the universe. Tis axiom seems to be one which cannot be a truth o logic.
Conclusion Frege was undoubtedly an important orce in the development o modern logic. But his significance can be overstated under the influence o the retrospective
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appreciation o the quality o both his logical and philosophical work. Frege’s work was not widely read in his lietime. But two thinkers who did read Frege were Peano and, thanks to Peano, Russell. Te influence o Peano and Frege on Russell and Whitehead may be summed up in the Preace to Principia Mathematica: In the matter o notation, we have asar as possible ollowed Peano,supplementing his notation, when necessary, by that o Frege or by that o Schröder . . . In all questions o logical analysis, our chie debt is Frege.
Note 1 It is worth remarking that in this letter Russell expresses the contradiction explicitly using Peano’s notation, thus: w = cls x з(x x).: w w. =. w w. Russell says that he had written to Peano about this.
Further reading Gödel, Kurt (1951), ‘Russell’s Mathematical Logic’, in Schlipp, Paul Arthur (ed.), Te Philosophy o Bertrand Russell, 3rd ed., New York: udor, 123–153. Reprinted in Benacerra, Paul and Putnam, Hilary (eds) (1983),Philosophy o Mathematics: Selected Readings, 2nd ed., Cambridge: Cambridge University Press. Grattan-Guinness, Ivor (2000), Te Search or Mathematical Roots, 1870–1940:Logics, Set Teories and the Foundations o Mathematics rom Cantor through Russell to Gödel (2000), Princeton: Princeton University Press. Kennedy, Hubert (1980), Peano: Lie and works o Giuseppe Peano. Dordrecht: D. Reidel. Russell, Bertrand (1908), ‘Mathematical Logic as Based on the Teory o ypes’. Reprinted in J. van Heijenoort, ed. (1967), From Frege to Gödel:A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press. Russell, Bertrand (1903), Te Principles o Mathematics, Cambridge: Cambridge University Press. Russell, Bertrand (1919), Introduction to Mathematical Philosophy, London: George Allen and Unwin. Selection reprinted in Benacerra, Paul and Putnam, Hilary (eds) (1983), Philosophy o Mathematics: Selected Readings, 2nd ed., Cambridge: Cambridge University Press Sainsbury, Mark (1974), Russell, London: Routledge and Kegan Paul. Shapiro, Stewart (2000), Tinking About Mathematics, Oxord: Oxord University Press. Wang, Hao (1974), From Mathematics to Philosophy, London: Routledge and Kegan Paul.
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Whitehead, Alred North and Russell, Bertrand (1910), Introduction toPrincipia Mathematica [ull reerence below].
Reerences Borga, M. and D. Palladino 1( 992), ‘Logic and oundations o mathematics in Peano’s school’, Modern Logic 3: 18–44. Dummett, M. (1993), Frege: Philosophy o Language. London: Duckworth. Grattan-Guinness, I. (1978), ‘How Bertrand Russell discovered his paradox’, Historia Mathematica 5: 127–37. Peano, G. (1888), Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann, preceduto dalle operazaioni della logica deduttiva. urin: Bocca. Peano, G. (1891), ‘Osservazioni del direttore sull’articolo precedente’, Rivista di Matematica 1: 66–9. Russell, B. (1908), ‘Mathematical Logic as Based on the Teory o ypes’, American Journal o Mathematics, 30(3): 222–62. Russell, B. (1959), My Philosophical Development. London: George Allen & Unwin. Russell, B. (1963), Mysticism and Logic. London: Unwin Books. Russell, B. (1967), Te Autobiography o Bertand Russell. London: George Allen & Unwin. van Heijenoort, J. (ed.) (1967), From Frege to Gödel. A Sourcebook in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press. van Heijenoort, J. (1992), ‘Historical development o modern logic’, Modern Logic 2: 242–55. Whitehead, A. N., and Russell, B. (1910, 1912, 1913), Principia Mathematica, 3 vols, Cambridge: Cambridge University Press; second edn, 1925 (Vol. 1), 1927 (Vols 2, 3).
10
Hilbert Curtis Franks
1 A mathematician’s cast o mind Charles Sanders Peirce amously declared that ‘no two things could be more directly opposite than the cast o mind o the logician and that o the mathematician’ (Peirce 1976: 595), and one who would take his word or it could only ascribe to David Hilbert that mindset opposed to the thought o his contemporaries, Frege, Gentzen, Gödel, Heyting, Łukasiewicz and Skolem.Tey were the logicians par excellence o a generation that saw Hilbert seated at the helm o German mathematical research. O Hilbert’s numerous scientific achievements, not one properly belongs to the domain o logic. In act several o the great logical discoveries o the twentieth century revealed deep errors in Hilbert’s intuitions – exempliying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that ‘[i]t is almost inconceivable that a man should be great in both ways’ (ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the undamental ideas and goals (indeed, even the name) o proo theory, the first systematic development and application o the methods (even i the field would be named only hal a century later) o model theory, and the statement o the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various subdisciplines o modern logic, Hilbert brought them each under the umbrella o mainstream mathematical activity, so that or the first time in history teams o researchers shared a common sense o logic’s open problems, key concepts and central techniques. It is not possible to reduce Hilbert’s contributions to logical theory to questions o authorship and discovery, or the work o numerous colleagues was made possible precisely by Hilbert’s influence as Europe’s pre-eminent mathematician together with his insistence that various
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logical conundra easily relegated to the margins o scientific activity belonged at the centre o the attention o the mathematical community. In the ollowing examination o how model theory, proo theory and the modern concept o logical completeness each emerged rom Hilbert’s thought, one theme recurs as a uniying moti: Hilbert everywhere wished to supplant philosophical musings with definite mathematical problems and in doing so made not evidently‘sonecessitated by the questions themselves, about how tochoices, rame investigations that’, as he emphasized in 1922, ‘an unambiguous answer must result’. Tis moti and the wild success it enjoyed are what make Hilbert the chie architect o mathematical logic as well as what continue to inspire misgivings rom several philosophical camps about logic’s modern guise.
2 Model theory Hilbert’s early mathematical work stands out in its push or ever more general solutions and its articulation o new, increasingly abstract rameworks within which problems could be posed and solved in the sought generality. Because o the deviation rom traditional methods brought about by these ambitions, Hilbert’s work engendered praise or its scope and srcinality as well as criticism or its apparent lack o concreteness and ailure to exhibit the signs o what was customarily thought o as proper mathematics. His undamental solution o Gordan’s problem about the bases o binary orms exemplifies the trend: Hilbert extended the problem to include a much wider range o algebraic structures, and his proo o the existence o finite bases or each such structure was ‘nonconstructive’ in that it did not provide a recipe or speciying the generating elements in any particular case. When Hilbert submitted the paper or publication in Mathematische Annalen Gordan himsel rejected it, adding the remark ‘this is not mathematics; this is theology’ (Reid 1996: 34). Te work was recognized as significant by other eminent mathematicians, however, and it was eventually published in an expanded orm. Felix Klein wrote to Hilbert about the paper, ‘Without doubt this is the most important work on general algebra that the Annalen has ever published’ (Rowe 1989: 195). Te split reception o Hilbert’s early work oreshadows an inamous ideological debate that ensued with constructive-minded mathematicians L.E.J. Brouwer and Herman Weyl in the 1920s – a debate which resuraces in the third section o this chapter. For now it will do to ocus on the pattern o abstraction and generality itsel, as it arose in Hilbert’s logical study o the oundations o
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geometry. Te question o the relationship o Euclid’s ‘parallel postulate’PP) ( to the other principles in his Elements had mobilized scholars since antiquity until in the nineteenth century Gauss, Lobachevski, Riemann, Beltrami and others offered examples o well-defined mathematical spaces in which Euclid’s principles each are true except or PP. Whereas the classical ambition had been to derive PP rom the other principles, showing it to be redundant, such spaces made o PP rom theprinciples others: Because one o these independence spaces evident could bethe read as an interpretation o the in the Elements, the truth o the other postulates does not guarantee the truth o PP; neither does the truth o the other postulates guarantee that PP is alse, or the long amiliar Euclidean plane can be viewed as an interpretation in which PP, alongside the other principles, is true. One burden shouldered by advocates o independence proos o this sort was to demonstrate that the several interpretations described are in act coherent mathematical structures – indeed, a avourite tactic among medieval and Renaissance thinkers was to argue indirectly or the derivability o PP by showing that any such interpretation would be sel-contradictory. Tis burden proved to be a heavy one because o a prevailing view that a ‘space’ should be responsible to human intuition or visual experience so that the content o one’s spatial perception was cited to undermine ‘deviant’ interpretations o the geometric postulates. Nineteenth-century advances in algebraic geometry helped quarantine independence proos rom this sort o objection, but some conusion persisted about whether the results were refinements o the concept o space or purely logical observations about the relationships among mathematical principles. Tus von Hemholtz said both that Riemann’s work ‘has the peculiar advantage that all its operations consist in pure calculation o quantities, which quite obviates the danger o habitual perceptions being taken or necessities o thought’, and that the resulting geometries should be considered as ‘orms o intuition transcendentally given . . . into which any empirical content whatever will fit’ (Helmholtz 1976: 673). In his Foundations o Geometryo 1899 , Hilbert provided a general setting or proving the independence o geometric principles and greatly sharpened the discussion o mathematical axioms. Along the way he offered new independence proos o PP and o the Archimedean principle and constructed interpretations o assemblages o axioms never beore thought o together. But the impact o the work lay not in any one o the independence proos it contained nor in the sheer number o them all, so much as in the articulation o an abstract setting in which the compatibility and dependence o geometric principles could be investigated methodically and in ull generality. ‘It immediately became apparent’, explained
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his close collaborator Paul Bernays, ‘that this mode o consideration had nothing to do with the question o the epistemic character o the axioms, which had, afer all, ormerly been considered as the only significant eature o the axiomatic method. Accordingly, the necessity o a clear separation between the mathematical and the epistemological problems o axiomatics ensued’ (Bernays 1922: 191). Hilbert managed this separation by dispensing with the stricture that the models used toway. interpret o geometric be o viewed as spaces in any traditional In thesets abstract setting oprinciples Foundations Geometry , Euclid’s principles were recast as collections o ormal axioms (as theories, 1, 2, . . .). Tough they contained words like ‘point’ and ‘line’, these axioms no longer had any meaning associated with them. Further, Hilbert exploded the distinction between the mathematical principles being studied and the structures used to interpret collections o them. In Foundations o Geometry, there are only theories, and an interpretation o a collection o geometric axioms is carried out in an algebraic theory (typically a field over the integers or complex numbers) S, so that each axiom in can be translated back into a theorem o S. From this point o view, Hilbert was able to articulate precisely the sense in which his demonstrations established the consistency o collections o axioms or the independence o one axiom rom others. In each case the question o the consistency o is reduced to that o the simpler or more perspicuous theory S used in the interpretation, demonstrating the ‘relative consistency’ o with respect to S. For i were inconsistent, in the sense that its axioms logically imply a contradiction, then because logic is blind to what meaning we ascribe to words like ‘point’ and ‘line’ this same implication holds also under the reinterpretation, so that a collection o theorems o S itsel implies a contradiction. Because theorems o S are implied by the axioms o S, in this case S is inconsistent. So the interpretation shows that is inconsistent only i S is. Similarly, the independence o an axiom like PP rom a collection o axioms C can be demonstrated relative to the consistency o (typically two different) theories S 1 and S2: one constructs theories 1 = C ∪ , 2 = C ∪ ¬ and demonstrates the consistency o 1 relative to that o S1 and o 2 relative to that o S2. O course the generality o Hilbert’s methods is suited or investigations unrelated to geometry, to metatheoretical questions about axiom systems in general. And while some contemporaries, notably Gottlob Frege, demurred rom the purely logical conception o consistency on offer, Hilbert’s techniques won the day. ‘Te important thing’, Bernays remarked, ‘about Hilbert’s Foundations o Geometry was that here, rom the beginning and or the first time, in the laying down o the axiom system, the separation o the mathematical and logical rom
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the spatial-intuitive, and with it rom the epistemological oundation o geometry, was completely carried out and expressed with complete rigour’ (Bernays 1922: 192). Indeed the abstract point o view Hilbert introduced, together with the idea o using interpretations o this sort to study the logical relationships among the axioms o amiliar mathematical theories, are the basic setting and tool o contemporary model theory. Amid the revolutionary turn o thought displayed in Hilbert’s work on geometry are two notorious doctrines worth attention because o their influence in logic. Te first is Hilbert’s insistence that mathematical existence amounts to nothing more than the consistency o a system o axioms. ‘I the arbitrarily given axioms do not contradict one another’, he wrote to Frege, ‘then they are true, and the things defined by the axioms exist’ (Frege 1980: 40). As a doctrine o mathematical existence, this idea is doubly dubious: it would later be clear rom discoveries o Skolem and Gödel that a consistent theory is typically not ‘categorical’ – its several models are not isomorphic – so the sense in which it is supposed to implicitly (partially?) define the terms that appear in its axioms is not clear. Further, the inerence rom the compatibility o a collection o axioms to the existence o a structure that models them is an inerence. As Gödel would emphasize, it is carelessto define existence in this way, because the validity o that inerence depends on the completeness o the underlying logic. Among the reasons that a contradiction might be underivable rom a set o axioms is the possibility that the logic used is too meagre to ully capture the semantic entailment relation. In the case o first-order theories, consistency does indeed imply the existence o a model, but the incompleteness o higher-order logic with respect to the standard semantics leaves open the possibility o consistent theories that are not satisfied by any structure at all. A second doctrine is Hilbert’s idea that his ‘axiomatic method’ would do more than allow a general setting or consistency and independence results but in act provide significant advances in the ordinary mathematical theories that were subjected to axiomatization. At times, Hilbert even expressed an ambition that axiomatics would open the door to the solution o all mathematical problems, perhaps even by rendering unsolved mathematical conjectures to systematic resolution in the abstract setting o ormal, uninterpreted sentences subject to combinatorial tests o derivability. Just posing this idea generated significant interest in the decision problem: the question whether the truth or alsity o any given sentence in the language o a ormal theory can be effectively determined. By the work o Church and uring it became known that even first-order quantification theory is undecidable, or although there is an algorithm or
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discovering o any valid ormula o quantification theory that it is valid, there is no corresponding procedure or discovering o a ormula that it is not valid (or, equivalently, that its negation has a model) in the event that it is not. Still worse, according to Gödel’s first incompleteness theorem, no axiomatic theory whatsoever in the signature o even basic arithmetic could ever be ‘syntactically complete’ in the sense that or any sentenceφ in that signature, either φ or
¬ φ. One cannot, as Hilbert had hoped to do, provide a ull axiomatization o number theory. However overreaching Hilbert’s ambitions may have been, his more modest prediction that, through axiomatics, symbolic logic would acilitate advances in ordinary mathematics was confirmed. In the first decade o the twentieth century, Hilbert himsel applied his model-theoretical techniques to problems in algebra, geometry and mathematical physics with considerable success: already in Foundations o Geometry one finds the description o non-Archimedean geometries, a topological definition o the plane, and new results about continuous unctions. With the urther maturation o model theory, Mal’tsev, arksi, Robinson and others successully proved results in group theory and the theories o algebraic classes defined via interpretations o the sort ound in Foundations o Geometry by applying the metatheorems o classical logic (such as the compactness theorem) to these domains. Robinson’s statement o the significance o this breakthrough can be read as an acknowledgement that Hilbert’s vision is being realized even i the dream that logic could answer all mathematical questions has been reuted: [Te] concrete examples produced in the present paper will have shown that contemporary symbolic logic can produce useul tools – though by no means omnipotent ones – or the development o actual mathematics, more particularly or the development o algebra and, it would appear, o algebraic geometry. Tis is the realization o an ambition which was expressed by Leibniz in a letter to Huygens as long ago as 1679. Robinson 1952: 694
3 Proo theory Whereas the consistency o a geometrical theory in which the axiom o parallels ails was a pressing open question to mathematicians in the nineteenth century, the consistency o basic arithmetic and even o mathematical analysis could be doubted only by the severest sceptics. Tough Kronecker had earned scorn or
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his repudiation o higher mathematics, Hilbert noted that ‘it was ar rom his practice to think urther’ about what he did accept, ‘about the integer itsel’. Poincaré, too,‘was rom the start convinced o the impossibility o a proo o the axioms o arithmetic’ because o his belie that mathematical ‘induction is a property o the mind’. Tis conviction, like Cantor’s insistence that ‘a “proo” o their “consistency” cannot be given’ because ‘the act o the “consistency” o finite multiplicities a simple, truth’ ( act ilbert 1922: struck Hilbert as short-sighted.isWhy shouldunprovable one iner, rom theH that the 199) consistency o a set o principles is not legitimately in doubt, the belie that the question o their consistency cannot be meaningully posed? Opposed to this way o thinking, Hilbert proposed that a definite mathematical problem can be ormulated about the consistency o any axiomatic system, and at the dawn o the wentieth Century he set out to show just this. A new tactic would be needed or the task, or the relative consistency proos o Hilbert’s earlier work appear to be unavailable in the case o arithmetic. Te consistency o various geometric theories had been proved relative to that o arithmetical ones, but relative to what could the consistency o arithmetic be meaningully established? ‘Recourse to another undamental discipline’, Hilbert remarked, ‘does not seem to be allowed when the oundations o arithmetic are at issue’ (1904: 130). Te consistency o arithmetical theories must be established in some sense ‘directly’. His wish to design direct consistency proos saw Hilbert return once more to the undamental insight o the metatheoretical perspective: the act that the axioms o a ormalized theory could be viewed as meaningless inscriptions. Rather than, as beore, using this act to construct reinterpretations o the axioms by changing the meanings o the terms they contain, Hilbert now proposed that the theory be lef uninterpreted, so that each axiom, and indeed each ormal derivation rom the axioms, could be treated as an object or the mathematician to reason about. Mathematical proo is an activity inused with meaning, but a proo in a ormal axiomatic theory is ‘a concrete object surveyable in all its parts’ (Bernays 1922: 195). Tis groundbreaking idea, which gives shape to the whole enterprise o proo theory, first appeared in Hilbert’s 1917 talk at Zürich: [W]e must – this is my conviction – take the concept o the specifically mathematical proo as an object o investigation, just as the astronomer has to consider the movement o his position, the physicist must study the theory o his apparatus, and the philosopher criticizes reason itsel. Hilbert 1918: 1115
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Te question o the consistency o a system o axioms can be posed ormally as the question whether there are proos in the system o two contradictory results. It is easy to see that this ormulation is equivalent to the question whether there is a proo in the system o some one predesignated evident alsehood. In the case o an arithmetical theory , a direct consistency proo would thus amount to an (inormal mathematical) argument that it is not possible to derive the expression 1 ≠Beore 1 romdescribing axioms o by means the rules o inerence designated or.about how such anoargument might unold, a ew words the philosophical debate surrounding the whole programme are due. Hilbert emphasized in many o his early papers that the inormal mathematical arguments comprising his consistency proos do not involve reasoning more ‘complex’ or ‘dubious’ than the principles o reasoning encoded in the axioms o the theory about which one is reasoning. It is not hard to see why: i one draws rom complex principles in order to show that a relatively simple theory is consistent, then it is not clear that one has demonstrated anything, or in even asking about the consistency o a theory (even i one does not literally harbour any doubts), one has presumably assumed a position rom which that theory’s own principles are not all available. o use them, and especially to use principles stronger than them, would seemingly be to drop the question one meant to be asking. Indeed, the proos issued by Hilbert and his colleagues Ackermann and von Neumann were criticized on precisely these grounds, their insistence that they had avoided any such circularity notwithstanding. Interesting and heated debates ensued, uelled in large part by Brouwer’s and Weyl’s allegiance to constructivist principles violated by the systems Hilbert aimed to prove consistent. By Gödel’s second incompleteness theorem, it is known that the consistency o any consistent arithmetical theory o the sort Hilbert studied cannot be ormalized in that very theory, that the metatheory in which the consistency proo is carried out must be in at least some ways stronger. Te debate about the sense in which Hilbert’s methods are circular as well as the debate about whether the epistemological gains o a consistency proo were actually o central importance to Hilbert continue to this day. o the latter issue, Kreisel’s report that ‘Hilbert was asked (beore his stroke) i his claims or the ideal o consistency should be taken literally’ and that Hilbert ‘laughed and quipped that the claims served only to attract the attention o mathematicians to the potential o proo theory’ is noteworthy (Kreisel 2001: 43). In any case, the mathematical and logical interest o Hilbert’s style o consistency proo is completely unscathed by any deect in the epistemological motivations o Hilbert or anyone else.
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In the vicinity o these issues, however, one finds Hilbert describing the logic o his proo theory as ‘finitist’. As others had objected in principle to the use o non-constructive existence proos and uses o the law o excluded middle on infinite totalities like those eatured in Hilbert’s solution o Gordan’s problem, now Hilbert himsel banned these techniques. But Hilbert’s stance was certainly not based on a principled opposition to classical logic. On the contrary, his aim was to demonstrate that and mathematical theories ladenowith non-constructive principles are consistent conservative extensions the finitary ones used to reason about their proos. Again, a convincing interpretation o Hilbert’s several remarks about the significance o finitism has proved to be elusive: was he hoping only to show that infinitary mathematics and non-constructive techniques are sae and efficient, though in act meaningless, tools or discovering acts about the ‘real’, finitary realm? Or was he rather ully in deence o the meaningulness o ordinary mathematics, adopting finitist restrictions in his proo theory in an attempt to avoid circularity and thereby arrive at meaningul consistency arguments? A third reading, closer to the attitude o contemporary proo theorists, is that the stipulation o restrictions in one’s logic was simply a response to the constructive nature o proo transormations. Each interpretation has its textual support, but whatever Hilbert’s motives were, it is undeniable that to him the logic o proo theory and the logic o ordinary mathematics are importantly different. For Hilbert there is no single ‘true logic’; rather, the logic appropriate or a particular investigation is derived a posteriori rom the details o that investigation – a position that oreshadows the contemporary notion o ‘logical pluralism’. In roughest outline, the reasoning in a direct consistency proo is the ollowing reductio ad absurdum: assume that a proo in the theory o 1 ≠ 1 has been constructed (so that is evidently inconsistent). Tis object will be a finite list o ormulas, the last o which is 1 ≠ 1, each o which is an axiom o or the result o applying one o the rules o inerence o to ormulas that appear earlier in the list. Following a general algorithm, first transorm this object into another proo in o 1 ≠ 1 containing only closed ormulas, typically in some perspicuous normal orm. Ten transorm the resulting proo into a third object (again this will be a finite list o ormulas, but not necessarily a proo) consisting entirely o ormulas containing only numerals, propositional connectives, and the equality sign. A recursive argument can then be used to veriy that every ormula (beginning with those that emerged rom axioms, continuing to those that emerged rom ormulas that were arrived at in one inerence, etc.) in this list is ‘correct’ according to a purely syntactic criterion (or this, Hilbert stipulated that
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numerals are finite strings o the symbol ‘1’ concatenated.) Tis is the birth o a common proo-theoretical technique: in contemporary parlance, one has shown that the axioms each have this property and that the inerence rules ‘preserve’ the property. Afer a finite number o steps, one will have verified that the ormula 1 ≠ 1 is ‘correct’ (although the way Hilbert defined this notion it, o course, is not). From this contradiction, one concludes that no proo o 1 ≠ 1 can be constructed is consistent. in A, special so that case to consider is the treatment o quantifiers , in this algorithm.
o acilitate the ‘elimination’ o quantifiers along the way to construct a list o purely numerical ormulas o the sort just described, Hilbert rendered his arithmetical theories in the ε-calculus. Tis is an extension o quantifier-ree number theory with a unction symbol ε that operates on ormulas A(a) to orm terms εaA(a), with the intuitive meaning ‘that which satisfies the predicate A i anything does’. (I this construction seems peculiar, bear in mind that x(yA(y) A(x)) and x(A(x) yA(y)) are first-order logical truths, the first corresponding to the ε-term, and the second corresponding to its dual τ term (that which satisfies A only i everything does.) Hilbert’s arithmetical theories included the transfinite axioms:
1. A(a) A(εaA(a)) and 2. εaA(a) ≠ 0 ¬A(δεaA(a)) (δ is the predecessor unction) It is not hard to see that these axioms allow one to derive the usual axioms or the quantifiers and induction. o accommodate the ε-terms that appear in a proo, the algorithm stipulates how to substitute numerical terms or each appearance o the ε-terms: In the case that only one ε-term appears in the proo, one simply replaces each o its occurrences with the numeral 0 and each occurrence o εaA(a) with the least n or which A(n) is true. When more than one ε-term appears in the proo and especially when there are nested ε-terms, the substitution becomes ar more complicated, and the algorithm loops in response to these complications. Now, the strength o the metatheory needed to conduct such reasoning is determined by the problem o veriying that the algorithm or proo transormation eventually comes to a halt. Te transormed proos are typically much larger than the objects with which one begins, and in order to rule out the possibility that any one o the proos in ends in 1 ≠ 1, one effectively considers the transormation o every one o the proos o . For this, because o the spiralling nature o the algorithm, one must use multiply embedded inductive arguments. Tis comes to light especially in the consideration given to the
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treatment o ε-terms. When several terms are present, some alling within the scope o others, these must be indexed, and an ordering defined on the indices, in order to keep track o how the proo transormation proceeds. One then observes that veriying that the algorithm eventually halts involves ‘transfinite induction’ through the ordinal number used to order the indices o terms. ω Already in 1924, Ackermann was explicit that induction to ωω was used in the proo o the consistency o a theory he considered. He wrote:
Te disassembling o unctionals by reduction does not occur in the sense that a finite ordinal is decreased each time an outermost unction symbol is eliminated [as in an ordinary inductive proo ]. Rather, to each unctional corresponds as it were a transfinite ordinal number as its rank, and the theorem that a constant unctional is reduced to a numeral afer carrying out finitely many operations corresponds to [a previously established act]. Ackermann 1924: 13.
Tis is the sense in which today one speaks o ordinals associated with mathematical theories: Te ‘consistency strength’ o a theory is measured by the ordinal used to track the induction used to prove its consistency. In the continuation o the passage just quoted, Ackermann claimed that the use o transfinite induction did not violate Hilbert’s ‘finitist standpoint’. Tis posture was later emulated by Gentzen, who presented his masterul consistency proo o first-order Peano Arithmetic (PA) together with a statement that although one ‘might be inclined to doubt the finitist character o the ‘transfinite’ induction’ through
used in his proo ‘even i only because o its suspect name’ it is important to consider that the reliability o the transfinite numbers required or the consistency proo compares with that o the first initial segments . . . in the same way as the reliability o a numerical calculation extending over a hundred pages with that o a calculation o a ew lines: it is merely a considerably vaster undertaking to convince onesel o this certainty. Gentzen 1938: 286
Te debate about the relationship between Hilbert’s wish to provide a finitist consistency proo o arithmetic, Gödel’s theorem to the effect that any consistency proo would have to extend the principles encoded in the theory one is proving to be consistent, and Gentzen’s proo (which is carried out in the relatively weak
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theory PRA (primitive recursive arithmetic) together with the relatively strong principle o transfinite induction up to ε0) is not likely to be resolved any time soon. From the point o view o logic, however, this debate is a distraction rom what seems to have been Hilbert’s main purpose: to show that the question o the consistency even o elementary mathematical theories could be ormulated as a mathematical problem requiring new perspectives and techniques or its solution and ushering insights along the way. An example o suchinanrewarding insight can be extracted rom Gentzen’s achievement. By showing that transfinite induction to the ordinal ε0 can be used to prove the consistency o PA, Gentzen demonstrated that this principle is unprovable in PA. (Tis ollows immediately rom Gödel’s ‘second incompleteness theorem’, mentioned above). But he also showed that transfinite induction to any ordinal below ε0 (a stack o ωs o any finite height) is provable in PA. Tis is the sense in which ε0 is sometimes described as the ordinal o PA: no smaller ordinal will suffice. But Gentzen did more. PA has as an axiom a principle o mathematical induction over all ormulas in its signature. One can also consider ragments o PA defined by restricting quantifiers to ormulas with a maximum quantifier complexity (call these the theory’s class o inductive ormulas). Gentzen showed that the size o the least ordinal sufficient or a proo o the consistency o such a ragment corresponds with the quantifier complexity o that theory’s class o inductive ormulas. In effect he established a correspondence between the number o quantifiers o ormulas in the inductive class and the number o exponentials needed to express the ordinal that measures the theory’s consistency strength. Tis correspondence ‘one quantifier equals one exponential’ has been called the central act o the logic o number theory and is rightly seen as the maturation o Hilbert’s technique o quantifier elimination, the realization o Hilbert’s idea that consistency proos could be used to analyse quantifiers. Tough observing, in 1922, that ‘the importance o our question about the consistency o the axioms is well-recognized by the philosophers’, (Hilbert 1922: 201), Hilbert strove to distinguish his interests rom theirs. ‘But in this literature’, he continued, ‘I do not find anywhere a clear demand or the solution o the problem in the mathematical sense’. Tat, we have seen, is what Hilbert demanded, and i Gentzen can be credited with providing the solution, Hilbert must be credited or ormulating the question in purely mathematical terms and or articulating a setting or its investigation. As or the philosophical significance o the result, one must appreciate that the ‘mathematical sense’ o the consistency question, unprecedented and unpopular
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at the time that Hilbert first put it orward, is the one that commands the interest o mathematicians today as well as the one that brought logical investigations once again, in a yet different way, into the mainstream o mathematical activity. Anticipating this revolution, Bernays remarked that ‘Mathematics here creates a court o arbitration or itsel, beore which all undamental questions can be settled in a specifically mathematical way, without having to rack one’s brain about subtleor logical whether judgmentsproo o a certain have a meaning not’ dilemmas (Bernays such 1923:as222). In Hilbert’s theory,orm he wrote elsewhere, ‘mathematics takes over the role o the discipline ormerly called mathematical natural philosophy’ (Bernays 1930: 236). Hilbert’s influence in proo theory does not stop with these well-known accomplishments, though. We will see in the next section that in his lectures o 1917–18, Hilbert pioneered the metalogical investigation o the semantic completeness o quantification theory, asking or the first time whether all universally valid ormulas o this theory could be syntactically derived. But remarkably, in his lectures rom 1920, he had so modified his ramework that ‘questions o completeness’ and ‘o the relationship between the ormalism and its semantics . . . receded into the background’. In their place, seeking ‘a more direct representation o mathematical thought’, Hilbert redesigned his logical calculus or quantification theory so that the rules governing the calculus, rather than a semantic theory external to that calculus, could be thought o as directly ‘defining’ or ‘giving the meaning’ o the propositional connectives and quantifiers (Ewald and Sieg 2013: 298). Tus one finds in Hilbert’s lectures rom early 1920 what is likely the earliest articulation o the concept o ‘proo-theoretical semantics’ that later characterized the approach to the study o logic o Gentzen, Prawitz and others: ‘Diese Regel kann als die Definition des Seinszeichens augeassst werden’ (ibid.: 323). Tis is remarkable not only because o how dramatically it extends the reach o Hilbert’s influence. rue to Hilbert’s pragmatic attitude, here he once more articulates a ruitul concept deeply at odds with entire rameworks o logical investigation that he had earlier pursued.
4 Logical completeness In the description o a ‘symbolic calculus’ with which he began his treatise on rigonometry and Double Algebra, Augustus De Morgan listed three ways in which a ormal system, even one whose ‘given rules o operation be necessary
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consequences o the given meanings as applied to the given symbols’, could nevertheless be ‘imperect’. Te last sort o imperection he considered is that the system ‘may be incomplete in its rules o operation’. He explained: Tis incompleteness may amount either to an absolute privation o results, or only to the imposition o more trouble than, with completeness, would be required. Every rule the want o which would be a privation o results, may be called primary: all which might be dispensed with, except or the trouble that the want o them would give, may be treated merely as consequences o the primary rules, and called secondary. De Morgan 1849: 351
Evidently, De Morgan would ault a system, not only or our inability in principle to prove with it everything we would like to know (all the truths or valid laws in some domain), i.e. or its lacking certain primary rules, but also or being cumbersome. But in distinguishing these two weaknesses, he clearly isolated a property o logical systems converse to the first one he mentioned. o say that all a system’s rules are necessary consequences o the given meanings is to say that the system is sound. o say that it has enough rules to derive each such necessary consequence is to say that it is complete. Some years later, in a paper called ‘On the algebra o logic’, Peirce boldly asserted, ‘I purpose to develop an algebra adequate to the treatment o all problems o deductive logic’, but issued this caveat: ‘I shall not be able to perect the algebra sufficiently to give acile methods o reaching logical conclusions; I can only give a method by which any legitimate conclusion may be reached and any allacious one avoided’ (Peirce 1885: 612). Te concern about efficiency had been dropped. Peirce sought only to present a sound (avoiding any allacious conclusion) and complete (reaching each legitimate one) logic. It is entirely mysterious why Peirce elt entitled to claim that his logical system is complete. No argument o any sort to this effect appears in his paper. De Morgan made no such boast. But the two logicians shared an appreciation o what a good logical system would be like. One wonders what the source o this commonality might have been, because outside the writing o these two, one scarcely finds a hint that the question o completeness even occurred. One exception is Bolzano, who a hal century earlier developed logical theory on two ronts. He designed one theory o ‘ground and consequence’ that was supposed to track the dependencies o truths and another o ‘derivability’ that was supposed to allow us to reason rom hypotheses to their necessary conclusions. Tose dependencies o truths were the things we are supposed to
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care about, and derivability was merely an analysis o well-regulated reason. Peculiar, though, is the act that his theory o ground and consequence closely resembles the logical calculi o the modern era, whereas his definition o derivability is a precursor to today’s notion o logical consequence. So when Bolzano asked i every ground–consequence relation is in act derivable, he seems to have our concept o completeness, which we inherited rom De Morgan and Peirce, wrong way around. therecent completeness seems todaythe so perectly natural andEvidently, central, is o vintage. question, which Bolzano despaired at not finding a way to ormulate his version o the completeness question so that he would know how to answer it, which is noble compared to De Morgan and Peirce’s apparent lack even o an attempt at such a ormulation. Te whole enterprise was rather ill-ated, and by the turn o the twentieth century, as logic began to settle into its modern guise, the question o logical completeness simply did not arise. Logicians began either to think o logical systems as primitive encodings o the principles o right reasoning, with no eye towards matters o logical truth, or not to think o them at all, ocusing entirely on matters o truth, satisaction o ormulas, and such semantic notions. Tose like Gentzen in the first camp could still ask i anything was missing rom their systems, but the question was a matter o psychological introspection or possibly an empirical study o the types o inerences that appear in mathematical journals. Skolem and others in the second camp had no systems to ask afer and pursued instead questions o decidability o classes o ormulas. In this setting, Hilbert stood alone. He was discontent equally with the idea o empirically validating a logical system and o ignoring them altogether. In an address at the Bologna International Congress o Mathematicians, Hilbert remarked: ‘[]he question o the completeness o the system o logical rules [o the predicate calculus], put in general orm, constitutes a problem o theoretical logic. Up till now we have come to the view that these rules suffice only through experiment’ (Hilbert 1929: 140). Te sentiment first appeared in Hilbert’s lectures rom the academic year 1917–18, in which he remarked that ‘whether [the predicate calculus] is complete in the sense that rom it all logical ormulas that are correct or each domain o individuals can be derived is still an unsolved question’, because our knowledge about this at the time was entirely ‘empirical’ (Ewald and Sieg 2013). As beore, it is not clear that Hilbert harboured any doubts about the completeness o quantification theory. Quite possibly, the empirical evidence that amiliar systems could not be improved upon with the addition o new principles was convincing to him. But even i the evidence was conclusive, what
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the world did not have and what the prevailing attitudes about logic precluded was a mathematical proo o a theorem about these matters. Once again, Hilbert seemed to be motivated to transorm a question rom philosophy or natural science into a mathematical problem and to see what sorts o mathematical ideas would be generated in the process. Te stock o undamental insights is vast. But Hilbert’s own intellectual trajectory to posing the question is subtle, and it isInrewarding to trace this completeness history beorein tallying the spoils.different ways. (Te 1905 Hilbert defined two apparently exposition beginning here ollows closely Mancosu, Zach, and Badesa 2009, which should be consulted or urther details.) He asked first (Hilbert 1905: 13) whether or not the axioms o a ormal theory suffice to prove all the ‘acts’ o the theory in question. On page 17 he ormulated a ‘completeness axiom’ that he claimed ‘is o a general kind and has to be added to every axiom system whatsoever in any orm’. Such an ‘axiom’ had first appeared inFoundations o Geometry. Hilbert explained: [I]n this case . . . the system o numbers has to be so that whenever new elements are added contradictions arise, regardless o the stipulations made about them. I there are things that can be adjoined to the system without contradiction, then in truth they already belong to the system.
Te first thing to notice is that Hilbert is both times speaking about axiomatic mathematical theories and is not yet asking about the completeness o a logical calculus. But the dissimilarities are also important: the completeness axiom is an axiom in a ormal theory; the first notion o completeness is a property o such a theory. One wonders why the same word would be used in these two different ways. A partial answer can be ound in 1917–18 where Hilbert elided these notions together as he turned his attention to logical calculi. In this passage, he is discussing propositional logic: Let us now turn to the question o completeness. We want to call the system o axioms under consideration complete i we always obtain an inconsistent system o axioms by adding a ormula which is so ar not derivable to the system o basic ormulas. Ewald and Sieg 2013: 152
Here, or the first time, the question o the completeness o a logical calculus has been posed as a precise mathematical problem. But although the question is being asked about a system o logic rather than being ormulated as a principle
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within the system, the question bears more structural resemblance to the axiom o completeness than to the primitive question about all ‘acts’ (or tautologies) o propositional logic being proved: the completeness axiom says that the addition o any new element in, or example, an algebraic structure will result in a contradiction; the completeness o the propositional calculus is the conjecture that the addition o any ormula to a set o theorems will result in a contradiction. Tese lecture notes proos both ointhe o a calculus or propositional logic andcontain o the completeness, theconsistency sense just described, o that same system. o prove consistency, Hilbert used the ollowing interpretation strategy (the system under consideration contains connectives only or disjunction and negation): let the propositional variables range over the numbers 1 and 0, interpret the disjunction symbol as multiplication and the negation symbol as the unction 1 – x. In this interpretation, every ormula in the classical propositional calculus (CPC ) is a unction on 0 and 1 composed o multiplication and 1 – x. Hilbert observes that the axioms are each interpreted as unctions that return the value 0 on any input, and that the rules o inerence each preserve this property (so that every derivable ormula is constant 0). Furthermore, the negation o any derivable ormula is constant 1 and thereore underivable. Tus, no ormula is derivable i its negation also is, and so the system is consistent. Te same interpretation figures in the completeness argument: It is known how to associate with every ormula φ another φcn in ‘conjunctive normal orm’, so that φ and φcn are each derivable rom the other in CPC. (A cn ormula is a conjunction o clauses, each o which is a disjunction o propositional variables and negations o propositional variables.) By the previous argument, a ormula is provable only i it is constant 0 under the interpretation, which, in the case o a cn ormula, occurs precisely when each o its clauses contains both a positive (unnegated) and negative (negated) occurrence o some propositional variable. Now let φ be any unprovable ormula. Ten its associated ormula φcn must also be underivable and thereore must contain a clause with no propositional variable appearing both positively and negatively. o show that CP C+φ (CPC augmented with φ as an additional axiom) is inconsistent, let ψ be any ormula whatsoever, and label χ the result o substituting, into φcn, ψ or every variable that occurs positively in c and ¬ψ or every variable that occurs negatively in c. It is easy to show that CP C +φ φ, CP C φ φcn, CP C φcn χ, and CPC χ ψ. Tus CPC +φ ψ or any ormula ψ. Suppose, now, that some ormulaφ were constant 0 under this interpretation but unprovable in CPC . Ten the same consistency argument as beore would carry through or CPC +φ, contradicting the completeness result just proved. It
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ollows that every ormula interpreted as a constant 0 unction is a theorem o CPC . Tis reasoning, reproduced rom Hilbert’s lectures, establishes that CPC is complete with respect to the unctional interpretation and oreshadows the concept o ‘semantic completeness’ amiliar today. It might be surprising initially that these two notions o completeness, the first purely syntactic and the second establishing a bridge between a ormal system and its interpretive scheme, mesh so nicely criterion or propositional It might be Hilbert’s more surprising, the syntactic held thelogic. primary role in thought still, – histhat school customarily reerred to it, not only as the ‘stronger’ sense o completeness (which it is), but also as the ‘stricter’ sense o the word. Te influence o the concept o the completeness o mathematical theories is palpable. Te coincidence o these two notions o completeness served the Hilbert school well or their investigations o propositional logic, or even i the primitive notion o logical completeness is the one about the tractability, with one’s logical system, o all truths, they always held talk about truth and content at arm’s distance or being insufficiently ‘ormal’. Hilbert preerred to use interpretations as tools or discovering purely mathematical acts and wished to avoid debates about which interpretations are correct. So i the primitive question o logical completeness could be sharpened into one entirely about ormal provability, Hilbert viewed this as progress. As it happens, however, such was not the ate or the concept o completeness as it arises or quantification theory. By the publication o Hilbert and Ackermann’s Grundzüge der theoretischen Logik in 1928, Ackermann had discovered that unprovable ormulas can be consistently added to the classical predicate calculus. o see this, one again begins by veriying an axiom system’s consistency with an interpretation: to interpret a ormula o first-order quantification theory, first erase all quantifiers. Interpret propositional variables and propositional connectives as beore (variables range over {0, 1}, disjunction is the multiplicative product, etc.) Further, ignore how the argument places o the predicate letters are filled, and interpret these also as variables ranging over {0, 1}. As beore, each axiom gets interpreted as a constant 0 unction, and each rule o inerence preserves this property. But the negation o any theorem is a constant 1 unction and thereore underivable, so the system is consistent. (Unlike the case o propositional logic, however, this interpretation in no way oreshadows a semantic theory. It is only a tool or metatheoretical investigations, not an intended interpretation.) Te incompleteness, in the strong sense, o the predicate calculus is witnessed by the ormula xF (s) xF(x), which gets interpreted as a constant 0 unction
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in the scheme just described. o see that this ormula is underivable, one need only observe that in any domain with more than one object, the sentence that results rom this ormula i we interpret F as any predicate true o only one thing is alse, and appeal to the soundness o the system under consideration. Hilbert’s scruples are evident when he describes this reasoning as merely making the underivability o the ormula ‘plausible’ – he proceeds to present a ‘strictly ormal proo’ its underivability with no appeal to the system’s soundness or matters o truthoand alsity. In any case this result certainly does not establish that there are logically valid ormulas o quantification theory that are unprovable in the predicate calculus. Instead, it drives a wedge between the two senses o completeness that coincide in the case o propositional logic. Te demonstration that ‘any legitimate conclusion may be reached and any allacious one avoided’ so that we are lef with no ‘absolute privation o results’ cannot be ‘sharpened’ in this case so as to eliminate all talk o truth. It is well known that Gödel proved the semantic completeness o the predicate calculus in his 1929 thesis. Less well known is that the question he answered in the process was not, despite his remarks, one that‘arises immediately’ to everyone who thinks about logic. It could not arise less than a century earlier, when the distinction between logical systems and the truths they are meant to track was completely reversed in the work o Bolzano, and it could barely figure in the thought o most logicians as a respectable problem in the ensuing years, because it had not been shown that such a question could be ramed in precise mathematical terms. Tings changed rapidly afer Hilbert turned his attention to logic. I his attempts to treat the question o quantificational completeness did not materialize, the conceptual clarification ushered in by those attempts paved the way or Gödel’s success. Tere is little doubt, however,that Hilbert’s distrust o semantic notions hindered his research on quantificational completeness as much as it helped him in the case o propositional logic. Tere is no ‘strictly ormal proo’ o the completeness o quantification theory precisely because the necessary semantics are ineliminably infinitary and thereore not accessible by the methods in which Hilbert aimed to cast all o his metatheoretical investigations. Tis is witnessed doubly by the existence, on the one hand, o quantificational ormulas that can be true but only in an infinite domain and, on the other hand, by the prominent role o the law o excluded middle applied to an infinite domain in Gödel’s completeness proo. Again, it remains a matter o some debate exactly what Hilbert’s reasons were or requiring all metatheoretical investigations to be finitary precisely as his
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most celebrated work in ordinary mathematics is not. But the seeds o this conviction were sown at the very beginning o Hilbert’s logical investigations. Te ‘implicit definition’ doctrine that mathematical existence amounts to no more than the consistency o a set o axioms, Gödel pointed out, is hard to reconcile with Hilbert’s insistence that the completeness o the predicate calculus should not be believed based on evidence but requires mathematical proo. For the completeness o the predicate calculus is equivalent claim thatinevery consistent set o first-order ormulas has a model. o see to thethe implication one direction, assume that i a collection o ormulas is consistent, then they are all true in some model. By contraposition, i a ormula is alse in every model, then a contradiction can be derived rom it. Let φ be a first-order validity (a ormula true in every model). Ten ¬φ is alse in every model. Tereore, a contradiction can be derived rom ¬ φ. So ¬¬φ is derivable, as is φ by double negation elimination. Te hackneyed story about how Hilbert’s logical programme was undermined by Gödel’s results must be toned down. o begin with, we have seen that Hilbert’s investigations ar extend the search or finitistic consistency proos o mathematical theories and that the maturation o proo theory, in the work o Gentzen, and o the connections between proo theory and model theory, in the work o Gödel, are actually realizations o Hilbert’s broader aims. As or Gödel’s arithmetical incompleteness theorems, which would seem to reute at least some o Hilbert’s more popular statements, their appropriation by the Hilbert school is instructive. In short time, Hilbert and Bernays applied those same techniques to the problem o logical completeness and showed that i φ is a first-order quantificational ormula that is not reutable in the predicate calculus, then in any first-order arithmetical theory there is a true interpretation o φ. By Gödel’s own completeness theorem, it ollows that a first-order ormula φ is valid i, and only i, every sentence o that can be obtained by substituting predicates o or predicate letters in φ is true. ‘Te evident philosophical advantage o resting with this substitutional definition, and not broaching model theory’, wrote Quine, ‘is that we save on ontology. Sentences suffice, sentences even o the object language, instead o a universe o sets specifiable and unspecifiable’ (Quine 1970: 55). Whether a distrust o modern semantics such as Quine’s motivated Hilbert can probably not be known. But it is a truly Hilbertian final word on the matter when one observes not only that what Hilbert would have deemed an ‘inormal notion’ is here reduced to a matter o arithmetical truth, but also that an exact relationship to the ‘stricter sense’ o
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completeness is hereby salvaged: as a corollary to this arithmetical completeness theorem Hilbert and Bernays showed that the addition o any unprovable ormula o quantification theory as a new axiom causes the ormal arithmetical theory PA based on the predicate calculus to become ω-inconsistent.
5 Conclusion In remarks published alongside papers that he, Heyting and von Neumann wrote to articulate the philosophical positions known as logicism, intuitionism and ormalism, Rudol Carnap distinguished the outlook o a typical logician, or whom ‘every sign o the language . . . must have a definite, specifiable meaning’, with that o the mathematician. Te attitude o the latter, he thought, was exemplified by Hilbert when he said, ‘We eel no obligation to be held accountable or the meaning o mathematical signs; we demand the right to operate axiomatically in reedom, i.e. to set up axioms and operational specifications or a mathematical field and then to find the consequences ormalistically’ (Hahn et al. 1931: 141). History has shown that the latter cast o mind, though surely not destined to supplant the ormer in all investigations, has its place in the advancement o logical theory.
Problems 1. Show that the ollowing are equivalent ways o ormulating the consistency o CPC : (a) For no φ is φ and ¬φ (b) For some φ, φ (c) p&¬p 2. Show that xA(x) A(εx(A(x))). 3. Determine o each ormula whether it is unsatisfiable, satisfiable in a finite domain, or unsatisfiable in every finite domain but satisfiable: (a) xzy(R(x, y)&¬R(x, x)&(R(y, z) R(x, z))) (b) xyz(R(x, y)&¬R(x, x)&(R(y, z) R(x, z))) (c) yxz(R(x, y)&¬R(x, x)&(R(y, z) R(x, z))) 4. Veriy the our claims used in the final step o the proo o strong completeness or CP C.
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5. Explain the other direction o the equivalence between ‘Every first-order validity is provable’ and ‘Every consistent collection o first-order ormulas has a model’; i.e. show that i every validity is provable, then every ormula rom which no contradiction can be derived has a model.
Reerences Ackermann, W. 1( 924), ‘Begründung des “tertium non datur” mittels der Hilbertschen Teorie der Widerspruchsreiheit’. Mathematische Annalen 93: 1–36. Bernays, P. 1( 922), ‘Hilbert’s significance or the philosophy o mathematics ’. reprinted in Mancosu 1998, pp. 189–97. Bernays, P. (1923), ‘On Hilbert’s thoughts concerning the grounding o arithmetic ’, reprinted in Mancosu 1998, pp. 215–22. Bernays, P. 1( 930), ‘Te philosophy o mathematics and Hilbert’s proo theory ’, reprinted in Mancosu 1998, pp. 234–65. De Morgan, A. (1849), ‘rigonometry and Double Algebra’, reprinted in Ewald 1996a, pp. 349–61. Ewald, W. (1996a), From Kant to Hilbert: a sourcebook in the oundations o mathematics (Vol. 1), New York: Oxord University Press. Ewald, W. 1( 996b), From Kant to Hilbert: a sourcebook in the oundations o mathematics (Vol. 2), New York: Oxord University Press. David Hilbert’s Lectures on the Foundations o Ewald, W. and W. Sieg (eds)2013), ( Arithmetic and Logic 1917–1933, Berlin and New York: Springer. Feerman, S.J., W. Dawson, Jr., S.C. Kleene, G.H. Moore, R.M. Solovay and J. van Heijenoort (eds) (1986), Kurt Gödel:Collected Works (Vol. 1), New York: Oxord University Press. Frege, G. (1980), Philosophical and Mathematical Correspondence, ed. G. Gabriel et al., Oxord: Blackwell. Gentzen, G. (1938), ‘Neue Fassung des Widerspruchereiheitsbeweises ür die reine Zahlentheorie’, Forschungen zur logik und zur Grundlegung der exackten Wissenschafen, New Series 4, 19–44. ranslated as ‘New version o the consistency proo or elementary number theory’ in Szabo 1969, pp. 252–86. Hahn, H. et al. (1931), ‘Diskussion zur Grundlegung der Mathematik’, Erkinntness 2, 135–49 (translated by J. Dawson, Jr. 1984). Helmholtz, H. (1976), ‘Te srcin and meaning o the geometric axioms ’, reprinted in Ewald 1996b, pp. 663–88. Hilbert, D. (1899), ‘Grundlagen der Geometrie’, inFestschrif zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, pp. 1–92. Leipzig: eubner. Hilbert, D. (1904), ‘On the oundations o logic and arithmetic’, reprinted in J. van Heijenoort (ed.) (1967), From Frege to Gödel: a sourcebook in mathematical logic, 1879–1931, Cambridge, MA: Harvard University Press.
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Hilbert, D. (1905), ‘Logische Principien des mathematischen Denkens. Vorlesung, Sommer-Semester 1905’. Lecture notes by Ernst Hellinger. Unpublished manuscript, 277 pp. Bibliothek, Mathematisches Institut, Universität Göttingen. Hilbert, D. (1918), A ‘ xiomatic Tought’ reprinted in Ewald 1996b, pp. 1105–15. Hilbert, D. (1922), ‘Neubergründung der Mathematik. Erste Mitteilung’, translated by William Ewald as ‘Te new grounding o mathematics: first report’ in Mancosu 1998 , pp. 198–214. Hilbert, D. (1929), ‘Probleme der Grundlegung der Mathematik’, Mathematische Annalen 102, pp. 1–9, as translated in B. Dreben and J. van Heijenoort,‘Introductory note to Godel 1929, Godel 1930, and Godel 1930a’, in Feerman et al. 1986 , pp. 44–59. Hilbert, D. and W. Ackermann (1928), Grundzüge der theoretischen Logik. Berlin: Springer. Kreisel, G. (2011), ‘Logical hygiene, oundations, and abstractions: diversity among aspects and options’, in Baaz, M., C.H. Papadimitrou, H.W. Putnam, D.S. Scott and C.L. Harper (eds), Horizons o ruth, New York: Cambridge University Press, pp. 27–56. Mancosu, P. 1( 998), From Brouwer to Hilbert: the debate on the oundations o mathematics in the 1920s, New York: Oxord University Press. Mancosu, P., R. Zach and C. Badesa (2009), ‘Te development o mathematical logic rom Russell arski, , in Haaparanta, Modern Logicto , New York:1900–1935’ Oxord University Press. L. (ed.),Te Development o Peirce, C.S. (1885), ‘On the Algebra o Logic’ reprinted in Ewald 1996a, pp. 608–32. Peirce, C.S. (1976), ‘Notes on Benjamin Perice’s linear associative algebra ’, reprinted in Ewald 1996a, pp. 594–5. Philosophy o Logic, 2nd edn, Cambridge, MA: Harvard Quine, W.V.O. 1970), ( University Press. Reid, C. (1996), Hilbert, Berlin: Springer. Robinson, A. (1952), ‘On the application o symbolic logic to algebra’,Proceedings o the International Congress o Mathematicians, Providence, RI : American Mathematical Society, pp. 686–94. Rowe, D.E. (1989), ‘Klein, Hilbert, and the Gottingen Mathematical radition ’, Osiris 5, 186–213. Szabo, M.E. (ed.) (1969), Te Collected Papers o Gerhard Gentzen, London: North Holland.
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wentieth-Century Logic
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Gödel P.D. Welch
Gödel’s achievement in modern logic is singular and monumental . . . a landmark that will remain visible ar in space and time . . . Te subject o logic will never again be the same. von Neumann
1 Introduction Kurt Gödel is claimed by some to be the greatest logician since Aristotle.Te ideas that Gödel is associated with in logic are the Completeness Teorem which appeared in his 1929 PhD thesis, but more particularly theIncompleteness Teorem (actually a pair o theorems), and they have both been crucial in almost all theoretical areas o twentieth century logic since their inception. Whilst Frege’s keen insights into the nature o quantifier ensured a great leap orward rom the Syllogism, and his attempts to ormulate a conception o arithmetic purely as a logical construct were groundbreaking and influenced the course o the philosophy o mathematics in the late nineteenth, and through Russell, in the early twentieth century, and determined much o the discourse o that period, ultimately it is Gödel’s work that encapsulated the nature o the relationship betweendeductive processes acting on symbolic systems, and the nature otruth (or satisaction), or more widely interpretedmeaning or semantics in the Completeness Teorem. It is ofen said that the much deeper Incompleteness Teorems that came a year later illustrated a limitation o the axiomatic method, and in particular brought to a halt David Hilbert’s programme o putting mathematics on a secure,finitistically provable, ground o consistency. It may even be true that more ink and paper has been expended on the Incompleteness Teorem and its consequences (imagined or otherwise) than any other theorem in mathematics. 269
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We shall give an introductory account o these two theorems but one should be aware that Gödel made significant contributions to other areas o logic (notably giving an interpretation o intuitionistic logic in the usual predicate logic) which we do not cover here. Gödel’s contributions to set theory (his hierarchy o constructible sets with which he showed the consistency o the Axiom o Choice and the Continuum Hypothesis, one might consider a orm o ‘ramified’ or ‘iterated logic’) have had almost as much oundational impact in set theory as the logical theorems reerred to above have been in all areas o current logic. We do not cover these here either.
2 Te Completeness Teorem We shall describe the Completeness Teorem presently, but to set it in some context we must see what people meant by ‘Logic’. Russell and Whitehead in the Principia Mathematica (PM ) had set up a deductive system whereby, as in the axiomatic system o Euclid, the concepts o the subject under discussion, here o logic, and then it was hoped, also o arithmetic, could be codified and reduced to a small number o selevidently true postulates, and rules o inerring rom those postulates. What Russell meant by ‘logic’ was perhaps not entirely the same as what has come down to us as a ‘logic’ in the twenty-first century. A line in a deductive proo o the logical calculus o PM would be an interpreted ormula, about something, but in a modern deductive proo need not be so interpreted. However, Russell was trying to ollow Frege and ormulate such a system, ultimately rom which it was hoped the laws o arithmetic could be derived. Tis latter aim is not our concern here; merely the idea o a deductive system is what is important. Tus, one might have a collection o axioms, or postulates, P say (which might be PM ), and one may derive by applying one or more ‘rules o deduction’ rom a collection R o rules, a particular proposition A, say. Now what exactly a ‘proposition’ is here, we can ignore (although Russell could not) because all that mattered later was that there was some symbolic, or ormal language in which the postulates, the proposition A, and the intermediate propositions B1, B2, . . .,Bn could be written. Here, the point was thatBn was the final proposition A and any Bi was either (i) a postulate rom the collection P or ollowed rom one or more earlier propositions on the list, by an application o a deductive rule rom R. In modern notation this state o affairs is rendered ‘P A.’
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What is allowed as a ‘deductive rule’? Tis would have had a different answer to different people in different eras. Bolzano took a ‘deductive rule’ as one that appealed to notions o truth, or something like meaning. For Frege and Russell deductive rules should at least be meaningul and preserve the truth o the consequent A rom the truth o the axioms P. By the the time Gödel emerged as a graduate student in Vienna in the later 1920s, Hilbert and Ackermann (1929)emerged had codified system o deduction, ‘restricted unctional calculus’ which romatheir deliberations on the the deductive system o PM . Tis could be applied to sets o postulates in a fixed type o language which we now call a ‘first order language’ and consisted o constant symbols, and symbols o both unctional and relational type that might be used to express possible unctions and relations amongst variables v0, v1,. . . . However it is important to note that what those variables will be interpreted as and what actual unctions and relations those unctional and relational symbols denote play absolutely no part in this specification whatsoever. Te language is to be thought o as merely symbol strings, and the ‘correct’ or ‘well- ormed’ strings will constitute the relevant ‘ormulae’ which we shall work with. Te rules determining the well-ormed ormulae are thus purely syntactic. Te set o derivation or deduction rules or determining which sequences o ormulae constituted permissible proos had emerged rom PM as we have mentioned, but the question o the correct set, or adequate set o rules had not been settled. Hilbert had remarked that empirical experience with the rules then in use seemed to indicate they were indeed adequate. Te question was stated explicitly in Hilbert and Ackerman (1929). Hilbert had, over the previous decades, embarked on an ambitious scheme o proving the consistency o all o mathematics. Te motivation or this came rom the disturbances caused by the discovery by Russell o a ‘paradox’ in Frege’s system – really a undamental error or inconsistency in Frege’s basic conception – and the similar ‘paradoxes’ in set theory o Cantor and Burali-Forti. We shall not discuss these here, but only remark that Hilbert had envisaged a thorough-going rethinking o mathematical axioms, and a programme or showing the consistency o those axioms: namely that one could not prove both A and its negation, ¬A, using rules rom the given set R. Te danger had suraced that set theory might be inconsistent and whilst all o mathematics could be seen to be developed rom set theory how could we be sure that mathematics was sae rom contradiction? Hilbert’s thinking on this evolved over the first two decades o the twentieth century and set out to reassure mathematicians o the logical saety o their field. Hilbert had thus, as part o the programme, axiomatized
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geometry in a clear modern ashion, and had shown that the problem o establishing the consistency o geometry could be reduced to the problem o the consistency o analysis. Hilbert had thus ounded the area which later evolved into proo theory, being the mathematical study o proos themselves, which, as indicated above, were to be regarded as finite strings o marks on paper. It was in this arena that the basic question o completeness o a set o rules had arisen. What would i arules deductive system was not to be useul was some(in reassurance that be a) needed deductive themselves could introduce ‘alsity’ short: only true statements could be deduced using the rules rom postulates considered to be true; this is the ‘soundness’ o the system), and more pertinently b) that any ‘universally valid’ ormula could in act deduced. We have reerred already to a) above, and orthe system derived romPM in Hilbert and Ackermann (1929) this was not hard to show. But what about b) and what does it mean? For a well-ormed ormula B to be universally validwould mean that whatever domain o objects the variables were thought to range over, and whatever interpretation was given to the relation and unction symbols o the language as actual relations and unctions on that domain, then the ormula B would be seen to be true with that given meaning. In modern parlance again, we should say that B is universally valid, or more simply just ‘true’ in every relevant domain o interpretation. In modern notation, the idea that B is universally valid is written B. Given a set o sentences Γ o the language one may also define the notion o a ormulae B being a logical consequence o Γ: in any domain o interpretation in which all o Γ was satisfied or deemed to to be true, then so must also B be true. Denoting this as Γ B the Completeness Teorem can be stated thus: Teorem 1 (Gödel: Te Completeness Teorem (1929)) (Gödel 1930)
For any first order language and any sentence B o that language then: B
B.
More generally or any urther set o sentences Γ rom : ΓB
Γ B.
Te () direction o both statements represents thesoundness theorem o the deductive system’s set o rules: iB is provable in the system then B would be universally valid; that is, true in every interpretation. Similarly in the second part, i B were provable rom the assumptionsΓ, then in any interpretation o in
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which all the sentences o Γ were satisfied, (that is interpreted as true statements), B would also have to be made true in that interpretation. A paraphrase is to regard the rules as being truth-preserving. As an example let us take Γ as the set o axioms or group theory in mathematics, with B some assertion in the appropriate language or this theory. Te Soundness Teorem states that iB is derivable rom the axioms using the rules o inerence, then in any structure in which the axioms Γ are true, i.e. in any group, B will necessarily be true. Tis direction o the theorem was essentially known in some orm to Hilbert, Ackermannet al.
Te harder, and novel, part is the implication ( ) which sometimes alone is called the completeness (or adequacy) theorem. aking the example o Γ the axioms o group theory, i Γ B, which asserts that i in any group B is satisfied, then the conclusion Γ B is read as saying that rom the group axioms Γ, the statement B can be deduced. Tis is the sense o ‘completeness’: the rules o deduction are complete or adequate, they are up to the task o deriving the validities o the system. I one looks at the proo one can see how it goes beyond a very strictly finitary, syntactically based argument. We shall see that the argument intrinsically involves infinite sets. We give a modern version o the argument due to Henkin, and or the sake o exposition assume first that the set o assumptionsΓ is empty and concentrate on the first version. Suppose when trying to prove the () direction, we assume that B is not deducible: B. We wish to show thatB is not universally valid and hence we seek an interpretation in which ¬B is satisfied. Tis suffices, as no interpretation can satisy bothB and ¬B. We thus need a structure. In this argument one enlarges the language to a language by adding a countable set o new constants c0, c1, . . .cn . . . not in. One enumerates the ormulae o the language as φ0, φ1, . . .φk, . . . taking ¬B as φ0. One then builds up in an infinite series o steps a collection o sentences Δ = ∪n Δn. At the k’th stage, when defining Δk one considers whetherΔk−1 ¬φk. I this is the case, thenφk is added to Δ to obtain Δ , which by the case assumption, is not inconsistent. I this is not k−1 k the case then ¬φk is so added, and consistency is still maintained. Additionally, i φk is to be added and it is o the ormvmψ(vm) then a new constant, not yet used so ar in the construction,cr say, is chosen and ψ(cr) is added as well. Tese are the two essential eatures. For our discussion the first is notable: what one does is to make an infinite sequence o choices whether to takeφk or ¬φk when building Δ. We are thus picking an infinite branch through an implicitly defined infinite binary branching tree: binary since we are making yes/no choices as to whether to add φk or ¬φk. o argue that such a branch must exist one appeals toKönig’s ree Lemma: any finitely branching infinite tree must contain an infinite branch
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or path through . Te second eature, the choice o constants cr allows us rom the branch, that is the sequence o ormulae that make upΔ, to construct a structure whose domain will be built rom sets o constants. Because the set o constants is countably infinite, the structure so built will also be countable. Tis will be important or consideration below. Tese details and how the languageʹ is interpreted in the resulting structure will be suppressed here.
Γ B and In the starts roma structure a set o sentences wishes to case showthat Γ one B, one needs in whichΓalland o Γassumes {¬B} is satisfied. Te process is the same as beore but starting with Δ0 as Γ {¬B}.
Te crucial use o the ree Lemma argument marks the step Gödel took beyond earlier work o Herbrand and Skolem. He saw that use could be made o this infinitary principle to construct a semantic structure. wo immediate corollaries can be drawn rom Gödel’s argument: Teorem 2 (Te Löwenheim-Skolem Teorem) I Γ is a set o sentences in that is satisfiable in some structure then it is
satisfiable in a countable structure. Te argument here is just the proo o the Completeness Teorem itsel: first i Γ is satisfiable in some structure then it is consistent. Now build a structure using the proo o the Completeness Teorem starting rom Δ0 = Γ. In the resulting countable structure all o Γ will be true. Tis theorem had been proven much earlier by the logicians afer whom it is named. Skolem’s 1923 argument had come quite close to proving a version o the Completeness Teorem. Later Gödel said: Te Completeness Teorem, mathematically, is indeed an almost trivial consequence o Skolem 1923. However, the act is that, at that time, nobody (including Skolem himsel) drew this conclusion neither rom Skolem 1923 nor, as I did, rom similar considerations o his own. Tis blindness (or prejudice, or whatever you may call it) o logicians is indeed surprising. But I think the explanation is not hard to find. It lies in the widespread lack, at that time, o the required epistemological attitude toward metamathematics and toward nonfinitary reasoning. Gödel 2003
Te second Corollary is also a simple observation but apparently was unnoticed by Gödel and others or some years beore it was explicitly stated.
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Teorem 3 (Te Compactness Teorem) Suppose a set o sentences Γ o a first order language is not satisfiable in any structure. Ten there is a finite subsetΓ0 ⊆ Γ that is inconsistent.
Te argument here is that i Γ is not satisfiable, then no structure makes Γ true; this implies ‘Γ σ ¬ σ’ (interpretation: any structure that makes Γ true also makes a contradiction theo latter happen). by Completeness Γ σ ¬ σtrue . But aand proo σ ¬ σcan romnever assumptions in ΓHence is a finite list o ormulae, and so can only use a finite set Γ0 o assumptions rom Γ. Tus Γ0 is a finite inconsistent subset. What we see here is that the inconsistency o the set Γ can always be localized to a finite subset o sentences responsible or an inconsistency (o course there could be infinitely many non-overlapping inconsistent finite subsets depending on what Γ is). Returning to the example o general theory above: then many such theories Γ require an infinite set o axioms. I Γ B, then we know that in any structure M1 in which all o Γ is true, then B will be true. Te Compactness argument shows that we do not have to give one reason why in another structure M B is true, and 1 another reason or being true in structure M2: because Γ B, a finite subset Γ0 suffices as assumptions to prove B, and the proo Γ0 B gives us one single reason as to why in any structure in which Γ is satisfied, B will be true.
2.1 In conclusion o summarize: we now use the idea o a logic more generally as a system comprising three components: a syntactic component o a language ʹ in some orm, a deductive component o rules o inerence that acts on the ormulae o that language, and a third very different semantic component: the notion o a structure or interpretation o the ormulae o the language ʹ. It has become second nature or logicians when studying the plethora o different logics to define such by reerence to these components. However, it was the Completeness Teorem that showed, certainly in the case o standard first order logic, the necessary interconnectedness o these concepts or demonstrating the adequacy o the rules o inerence. Given a logic, one o the first questions one asks, is ‘Is it complete? Is there a Completeness Teorem or it?’ For a host o logics, modal logics, logics o use to computer science, completeness theorems are provable. For others they are not and the reasons why not are themselves interesting. For socalled second-order
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logic where we allow the language to have quantifiers that range not over just individuals but sets o individuals and relations between individuals, the logic is incomplete: there is a deductive calculus or the logic but the lack o a Completeness Teorem renders the logic intractable (apart rom special cases or areas such as in finite model theory) or proving useul theorems. It is partly or such reasons that set theorists ollowed Skolem into ormulating set theory and thinking reasoning in model first order logic,tools because a Completeness Teorem and and all the useul building (suchthere as theisCorollaries o the Löwenheim–Skolem Teorem and the Compactness Teorem) that come with it. Whilst the Teorem is nowadays not seen as at all difficult, Gödel’s 1929 result can be seen as distilling out exactly the relationship between the different threads or components o first order logic; by answering Hilbert and Ackermann’s question he demonstrated that semantical concepts, concepts o structure, satisfiability and o truth in such structures, had to be brought in to answer the drier questions o whether the rules o the deductive calculus which told only how strings o marks on paper could be manipulated, were sufficient or producing all validities in structures satisying a set o assumptions. It is not insignificant that the ree Lemma brought out what would otherwise have remained implicit and hidden: the nature o the argument also required infinite sets. Indeed rom our present vantage point we know that the Completeness Teorem must use the ree Lemma: i one assumes the Completeness theorem then we can prove the ree Lemma rom it: they are thus equivalent. But these discoveries were to come much later.
3 Incompleteness o understand the Incompleteness Teorems we need to discuss urther Hilbert’s programmatic attempt to put mathematics on a consistent ooting by returning to the position in the late 1920s. Hilbert had the belie that mathematics could be made secure rom possible paradoxes that dogged the early years o set theory by means o a series ofinitary consistency proos. A consistency proo or a theory stated in a language would be some orm o proo that one could not have φ ¬φ or some (or any) ormula φ o . But what should count as a legitimate proo or argument that was consistent? o be o any value the argument had to use indubitably secure means, that themselves were not open to question. Hence the notion o ‘finitary’. What Hilbert meant by this term was never given an absolutely explicit definition
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by Hilbert or his ollowers, even though the term was discussed by Hilbert on many occasions. Hilbert divided mathematical thinking into the ‘real’ and the ‘ideal’. Te ormer was essentially the mathematics o number theory, and the subject matter was ‘finitist objects’ the paradigm being stroke sequences ‘||||, . . ., |’ representing numbers, together with simple repetitive operations perormed on them. Hilbert ormulated various epistemological constraints on such objects, such surveyability etc.again Operabe tions them kind. (suchTe as concatenation corresponding toasaddition) should o aon simple truth or otherwise o such finitarily expressed statements was open to inspection as it were. At a later stage Hilbert and Bernays used the idea o primitive recursion (a scheme o building up number unctions by simple recursion schemes) and orms o induction that could be expressed in the language o arithmetic A by quantifier-ree ormulae, stating that such should count as finitary. As a deeper discussion o what constituted finitary methods would take us too ar afield, we shall let the idea rest with this version. Ideal mathematics according to Hilbert could prove or us, or example, quantified statements in number theory. But i we really were in possession o finitary means or proving the consistency o that piece o idealized mathematics, we could have confidence in the truth o that quantified number theoretical statement, in a way that we could never have otherwise, because o the complexity involved in surveying all the natural numbers required by the quantifiers. Hilbert is sometimes paraphrased as saying that ‘consistency o a theory yields the existence o the mathematical objects about which it speaks.’ But his intentions here were more nuanced, and more restricted. Hilbert had given a thorough-going axiomatization o geometry in Foundations o Geometry (1899). Te consistency o the axioms o higher dimensional geometry could be reduced to that o plane geometry alone, and in turn the latter could be seen to be consistent by interpreting it in analysis, and thus as reducing the problem to that o the consistency o analysis. Given the emergence o the set theoretical paradoxes at the time, Hilbert wanted a proo o the consistency o analysis that was direct and did not involve a reduction using, say Dedekind cuts o sets o reals (as this might be in danger o importing unsae methods). In his 1900 problems list, at second place he gave this consistency problem or analysis. Hilbert’s thinking was that the logical system o Principia Mathematica was inadequate or their purposes and so developed a new calculus or logical expressions (the ε-calculus). In 1924 Ackermann developed an argument or the consistency o analysis, but von Neumann, who took a deep interest in the
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oundations o mathematics at that time (ormulating an axiomatic set theory, and the notion o ‘von Neumann ordinal’ number), saw an error. By the late 1920s Ackermann had developed a new ε-substitution method and there was optimism that with this, a new consistency proo or analysis given in this calculus could be given. Tis was even announced by Hilbert at the 1929 International Congress o Mathematics. But it was to be short-lived.
3.1 Gödel’s First Incompleteness Teorem Gödel worked in Vienna, and quite independently o Hilbert’s Göttingen school. He expressed surprise concerning attempts at proving the consistency o analysis. It is mysterious why Hilbert wanted to prove directly the consistency o analysis by finitary methods. I saw two distinguishable problems: to prove the consistency o number theory by finitary number theory and to prove the consistency o analysis by number theory . . . Since the domain o finitary number theory was not well-defined, I began by tackling the second hal . . . I represented real numbers by predicates in number theory . . . and ound that I had to use the concept o truth (or number theory) to veriy the axioms o analysis. By an enumeration o symbols, sentences and proos within the given system, I quickly discovered that the concept o arithmetic truth cannot be defined in arithmetic. I it were possible to define truth in the system itsel, we would have something like the liar paradox, showing the system to be inconsistent . . . Note that this argument can be ormalized to show the existence o undecidable propositions without giving any individual instances. (I there were no undecidable propositions, all (and only) true propositions would be provable within the system. But then we would have a contradiction.) . . . In contrast to truth, provability in a given ormal system is an explicit combinatorial property o certain sentences o the system, which is ormally specifiable by suitable elementary means. quoted in Wang 1996
In September o 1930 there was to be a joint meeting o several academic societies in Königsberg. Various members o the Wienerkreis, Carnap, Feigl and Waismann would speak at the Conerence on Epistemology o Exact Sciences. At the meeting Carnap, Heyting and von Neumann would give hourlong addresses on logicism, intuitionism and ormalism respectively. Gödel was to give a short contributed talk on results relating to his thesis. A ew days beore, Gödel met Carnap in a cae to discuss the trip and then, out o the blue, related to Carnap his theorem concerning incompleteness o systems similar to PM . It
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seems that although Carnap noted in a memorandum ‘Gödel’s discovery: incompleteness o the system o PM, difficulty o the consistency proo’, he could not have understood exactly what Gödel had achieved. He met Gödel three days later or a urther discussion, but a week later, in Königsberg, he would still in the ensuing discussions there emphasize the role o completeness o a system being an overriding criterion or a ormal theory. ormal system sufficient syntax to talk about numerals P has ‘0’, Let ‘1’, us . . suppose . k‘ ’, . . . a(as names or the corresponding actual numbers) as well as symbols or some basic arithmetical operations such as the successor operation o adding one, x + 1, addition and multiplication in general. Te system embodied in Principia Mathematicais such a system. Also the Dedekind-Peano axioms or number theory (‘PA ’) are expressed in such a language and allow the use o such operations with their normal properties, together with the notion o mathematical induction. A ormal theory P is called ω-consistent, i whenever we have that P proves all o φ(‘0’), φ(‘1’), . . ., φ(‘k’), . . . individually, then it is not the case thatP proves v¬ φ(v). Because o the infinite hypothesis here, this is a stronger requirement on a theory P than simple consistency alone. Teorem 4 (Gödel : Te First Incompleteness Teorem (1930)(Gödel 1931)
Let P be a ormal theory such as that o Principia Mathematica expressed in a suitable language L. I the theory P isω-consistent, then there is a sentenceγP o such that: P γP and P ¬γP Te existence o such a sentence γP (a ‘Gödel sentence’ or the systemP) shows that P cannot derive any sentence or its negation whatsoever. Te system is ‘incomplete’ or deciding between γP and ¬γP. I the sentence γP is purely a number-theoretic statement then the conclusion is that there will be sentences which are presumably true or alse o the natural number structure (as the case may be) but the system P is necessarily incapable o deriving either. I some version o ‘ideal’ mathematics could decide between the two, then that ideal mathematical argument could not be given within the ormal system P. Gödel said nothing about the Incompleteness results during his own talk in Königsberg, and only mentioned them in a rather casual manner during a roundtable discussion on the main talks that took place on the last day o the conerence. Hilbert was attending the conerence and would give his arewell address as President o the German Mathematical Union, but did not attend the session at
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which Gödel made his remarks. It may well have been the case that, with the exception o von Neumann, no one in the room would have understood Gödel’s ideas. However, von Neumann realized immediately the import o Gödel’s result. We discuss here the proo o the theorem, which proceeded via a method o encoding numerals, then ormulae, then finite sequences o ormulae (which might proo in system P) all bywhen numbers. Tis coding knownconstitute as ‘Gödel acoding (orthe numbering)’ and a numeral ‘k’ or has the become number k is inserted into a ormula φ(v0), resulting in φ(‘k’), then the latter can be said to be assigning the property expressed by φ to the number k (the which may encode a certain proo, or example). Tus indirectly ormulae within the language can talk about properties o other ormulae, and properties o sequences o ormulae, and so orth, via this coding. How the coding is arranged is rather unimportant as long as certain simplicity criteria are met. It is common to use prime numbers or this. Suppose the language L is made up rom a symbol list: (,), 0, S, +, ×, =, ¬, →, , v0, v1, . . .,vk, . . . (where S denotes the successor unction, and there is an infinite list o variables v etc.) o members on the list we respectively assign code numbers: k 1, 2, . . ., 9, 10, 11, 12+k (or k ). (Hence the variable v2 receives code number 12 + 2 = 14.) For the symbol s let c(s) be its code in the above assignment. A string such as: v1(0 = S(v1)) expresses (the alse) statement that some number’s successor is 0. In general a string o symbols s1 s2 . . .sk rom the above list can be coded by a single number:
c(s1s2 … sk ) = 2c(s ).3c(s ). … .pkc(sk) 1
2
where pm is the m’th prime number. Given a number, by computing its prime actors we can ascertain whether it is a) a code number o a proo, or b) o a ormula, or c) o just a single symbol, and moreover, which symbol, ormula, or proo, in a completely algorithmic ashion. Te number 2 has the numeral term or name: S(S(0)) rom and we abbreviate the latter name as ‘2’. We shall write ‘φ’ or the numeral o the code number o the ormula φ. ‘φ’ unctions thus as a name or φ. Ten the ormula 0 =S(0) has as code: c(0 = S(0)) = 23.37.54.71.113.132. Suppose the latter value is k say, then ‘0 = S(0)’ is ‘k’. Te efficiency o the coding system is entirely irrelevant: we only require the simple algorithmicity o the coding processes that maps the syntax o the language in a (1–1) ashion into in a recoverable ashion. Tose codes are then named by the appropriate numerals which are terms in .
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A urther minimal requirement on the ormal system P is that it be sufficiently strong to be able to represent predicates or properties o natural numbers which are definable over the standard structure. By this is meant the ollowing: let φ(v0, . . .,vk) be any ormula. P represents φ i or all natural numbers n0, . . .,nk both o the ollowing hold: φ[n0,
. . .,nk] ⇒ P φ(‘n0’, . . ., n‘ k’);
φ[n0,
. . .,nk] ⇒ P φ(‘n0’, . . ., n‘ k’).
One may show that any primitive recursive (p.r.) predicate, as mentioned above, can be represented in a system such as PM . Gödel then developed a series o lemmas that showed that operations on syntax could be mimicked by p.r. operations on their code numbers. For example, there is a p.r. predicateR(v0, v1, v2) so that i χ is φ ∧ ψ, then in P we may prove R(‘φ’, ‘ψ’, ‘χ’). Tis is merely a reflection o the act that we can calculate a code number or χ once those or φ and ψ are given. Ultimately though, the act thatR(‘φ’, ‘ψ’, ‘χ’) may hold is because o certain arithmetical relationships between the code numbers, not because o any ‘meaning’ associated to those numbers (which we attribute to them because they code particular ormulae). Similarly i we have three ormulae ϕ, ψ, χ and ψ happens to be the ormula ϕ → χ, then we could view this triple i it occurred in a list o ormulae which may or may not constitute a derivation in P, that they stand in or a correct application o Modus Ponens on the first two ormulae yielding the third. Gödel shows that the relation PMP (v0, v1, v2) which holds o a triple o numbers as above i indeed that third ollows by application to the first two in the appropriate order, is primitive recursive: then we have: PMP
[c(φ), c(φ → χ), c(χ)], and hence P PMP (‘φ’, ‘φ → χ’, ‘χ’).
In short, syntactic operations, the checking o ormulae or correct ormation, and substitution o terms or variables, etc., up to the concept o the checking o a number that codes a list o ormulae whether that list constitutes a correct proo o the last ormula o the list, these are all p.r. relations o the code numbers concerned. For the last then Gödel constructs a p.r. predicate Pr(v0, v1) which is intended to represent in P (in the above sense) ‘v0 is the code number o a proo o the last ormula which has code number v1.’
I then, k is the code number o a correct proo in P o the last ormula σ o the proo, then the number relation Pr[k, c(σ)] holds, and indeed is itsel a relation
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between numbers; moreover it is provable in P because P can represent any p.r. predicate: P Pr(‘k’, ‘σ’). Again, to repeat, the act that Pr(x, y) holds just says something about a particular numerical relationship between x and y, which either holds (or does not) irrespective o the interpretation we may put on it in terms o ormulae, correct proos, etc., etc. Having done this, v0Pr(v0, v1) is naturally interpreted as ‘Tere is a proo o 1 the ormula number .’ Tis existential statement Prov( v1). Tiswith final code relation, due tov the existential quantifier turns we out abbreviate not to be p.r., but this does not matter.
Lemma 1 (Te diagonal lemma)
Given any ormula φ(v0) o the number-theoretic language we may find a sentence θ so that P θ ↔ φ(‘θ’). Proo: We let se be the string with code e, thus c(se) = e. Define the unction r : × → by: (1) r (e, n) = c(∀v1(v1 = ‘e’ → sn)). Ten r (e, n) is p.r. being a composition o simple p.r. unctions, indeed just multiplications involving some primes, the number n, and the codes or the symbols s, ∀, =, → etc. occurring in the string ∀v1 (v1 = ‘e’ → sn). Tis entails, inter alia, that r is representable in P. We now define adiagonal unction d : → by d(e) = r (e, e). Ten d is p.r. and so representable too: there is a ormula D(v1, v0) that represents the graph o d as above. Moreover it can be shown: (2) P ∀v0(D(‘n’, v0) ↔ v0 = ‘d (n)’). Given our φ let ψ(v1) be v0(D(v1, v0) φ (v0)). Let h = c(ψ(v1)). Ten, to spell it out: sh is ψ(v1). Let θ be ∀v1 (v1 = ‘h’ → sh). Ten by our definition o sh:
P θ ↔ ∀v1(v1 = ‘h’ → ψ(v1)) by logic:
↔ ψ(‘h’)
Using (2), De o ψ: ↔ φ(‘d (h)’) But ‘θ’ is ‘d(h)’, so we are done.
Q.E.D.
Te lemma, despite its construction, is less mysterious than it seems: it is just a fixed point construction. Indeed there is nothing terribly particular about the
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choice o θ: one may show or each choice o φ that there are in act infinitely many different ormulae θʹ satisying the lemma: the argument has just provided one o them. One should note, however, that the proo is completely constructive (and can be run in intuitionistic logic): a θ is given, together with a proo that it satisfies the biconditional. We can immediately derive a corollary which is ofen reerred to as that ‘truth is What that there is no ormula o the language ( 1) orundefinable’. which we have thatthis ormeans anyk: is τ(‘k’) i and only i σ where c(σ)τ=vk. For, assume the axioms oP are true in . Suppose τ were such a ormula, thence we should have by applying the Diagonal Lemma to the ormula ¬τ(v1), and then Soundness Teorem, that there is some sentence θ: θ ↔ ¬ τ(‘θ’).
(**)
However, this θ is like a liar sentence: or i it is true in , then so is ¬τ(‘θ’); but by the assumption on τ then we also have τ(‘θ’). Hence ¬θ is true in ; but this immediately leads to the same orm o contradiction. Hence there is no such ormula τ(v1). We have shown that the set o arithmetical truths is not arithmetically definable: Corollary 1 (arski: Te Undefinability of ruth)
Tere is no ormula (v1) o or which we have that or any k: P τ(‘k’) i and only i σ where c(σ) = k. Tis theorem, usually attributed to arski, is easier to establish than the Incompleteness Teorem to come, and seems to have been also known to Gödel (see the quotation at the beginning o this section). Gödel seems to have come to the realization that an Incompleteness Teorem would be provable precisely because provability within a ormal system such as P was, unlike truth, representable within P. Tat was the key. In the proo o Teorem 4 we shall apply it with ¬ v0Pr(v0, v1) as φ. Tis yields P γ ↔ ¬v0Pr(v0, ‘γ’)
( *)
We emphasize once more that the diagonal lemma says nothing about truth or meaning or satisaction in the structure IN: it says something only about provability o certain ormulae in the ormal system P, ormulae which express certain equivalences between sentences and ormulae containing certain numeral terms. And that holds or the expression ( *) too.
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Proof of Teorem 4
Suppose γ is as at ( *) above. Suppose or a contradiction that P γ. Let n be the code number o such a proo. Combining that proo with (*) yields that P ¬ v0 Pr(v0, ‘γ’). However n is afer all a code o a proo o γ and as P represents Pr, then P Pr(‘n’,‘γ’). Te conclusions o the last two sentences imply that P is inconsistent. Tis is a contradiction. Hence P γ. Now rom the last statement we conclude that no natural number n is a code o a proo o γ in P. Hence, as P represents Pr, we have or alln: P ¬Pr(‘n’,‘γ’). Te assumption o ω-consistency now requires that P ¬γ Q.E.D. Note that the assumption o ω-consistency o P is only deployed in the second part o the argument to show P ¬γ. Rosser later showed how to reduce this assumption to that o ordinary consistency by the clever trick o applying the Diagonal Lemma to the ormula ‘v0(¬Pr(v0, v1) v2(v2 is the code o a shorter proo thanv0 o the ormula ¬v1)).’ Another remark: it is ofen asserted that the Incompleteness Teorem states that ‘there are true sentences (in arithmetic, or in a ormal theory, or in . . .) that are not provable’. Tis is not a strictly accurate account o the theorem: the theorem itsel mentions only deduction in ormal theories and says nothing about truth. However, the Gödel sentenceγ is indeed true i we assume the consistency o P: by assuming the consistency o P we concluded that P γ, that is ¬ v0 Pr(v0, ‘γ’) is true, which o course is γ itsel. We thus have ‘Con(P) γ’. But note this is not (yet) an argument within the deductive system P. Yet another remark: the use o the sel-reerential Gödel sentence γ that asserts its own unprovability sometimes leads to the impression that all undecidable statements unprovable in such a theory as PM must o necessity have some degree o sel-reerence. However, this is alse. We comment on this again below. Similarly we do not reer tothe Gödel sentence γ or PM , since one can show there are infinitely many such.
3.2 Te Second Incompleteness Teorem Von Neumann lef the room in Königsberg realizing the import o what Gödel had achieved. He may have been the only person to do so: Hans Hahn, Gödel’s thesis supervisor, who was also present, made no mention o the Incompleteness results. Neither the transcript o the session, nor the subsequent summary prepared by Reichenbach or publication made any mention even o Gödel’s participation. Although Gödel attended Hilbert’s lecture, the two never met (or
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corresponded later), and the Viennese party then returned home. I von Neumann approached Hilbert whilst at the meeting to appraise him o the results, then it was not recorded. Von Neumann shortly realized that more could be obtained by these methods. We can express the consistency o the ormal system P by the assertion that rom P we cannot prove a contradiction, 0 = 1 say. We thus letCon(P) be the sentence
¬ v0Pr(v0, ‘0 = 1’). Teorem 5 (Gödel Second Incompleteness Teorem)
Let P be a ormal system as above. Ten P Con(P). Proo: Te essence o the argument is that we may ormalize the argument o ‘Con(P) ⇒ γ’ at the end o the last section in number theory, and so in a system such as P. We should thus have shown P Con(P) → γ.
(**)
We know rom the First Incompleteness Teorem thatP γ. Hence P Con(P). Q.E.D. Von Neumann realized that the something akin to the Second Incompleteness Teorem would ollow by the same methodsGödel had used or the First, and, in the November afer Königsburg, wrote to Gödel. However, Gödel had himsel already realized thisand submitted the Second Teorem or publication in October. O course the above is extremely sketchy: the devil then is in the detail o how to ormalize within the theoryP, the inerence above romCon(P) to ¬ v0 Pr(v0, ‘γ’), we thus need to show ‘P Con(P) implies P v0 Pr(v0, ‘γ’)’ within P itsel. In other words we must establish ** ( ) above. Gödel did not publish these rather lengthy details himsel, they were first worked out by Hilbertand Bernays in 1939.
4 Te sequel Tere are many points o interest and possibilities or elaboration in these theorems, and hence the extensive academic literature on them. Gödel lef deliberately vague what he meant by ‘ormal system’. He said at the time that it was not clear what a ormal system was or how it could be delineated. He stated his theorems as being true in the system o PM and or ‘related systems’. It was clear that a similar system that had sufficient strength to prove the arithmetical acts needed in the coding and deduction processes would do. Hence the theorems were more general than had they been restricted to just PM . It was lef
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to Alan uring five years later (1936) to give a mathematical definition o ‘computable’ that could be used to demarcate what a ormal system was: a set o axioms and rules that could be ‘recognized’ by a uring machine, and so that a programmed machine could decide whether a derivation in the system was correct. In the intervening period Gödel had speculated regarding on what an ‘effectively given’ ormal system could subsist, and rejected proposals rom Church definitive:or such. However, he recognized that what uring proposed was When I first published my paper about undecidable propositions the result could not be pronounced in this generality, because or the notions o mechanical procedure and o ormal system no mathematically satisactory definition had been given at that time . . . Te essential point is to define what a procedure is. Tat this really is the correct definition o mechanical computability was established beyond any doubt by uring. Gödel 1995a
We give now a more modern statement o the First Incompleteness Teorem. Teorem 6 (Gödel, First Incompleteness Teorem)
Let P be a computable set o axioms or number theory that contain the axioms PA. Ten i P is consistent, it must be incomplete: there is a sentenceγP so that P γP and P ¬γP
4.1 Consequences or Hilbert’s programme Te most dramatic consequences o the theorems were or Hilbert’s programme o establishing the consistency o mathematics, and in particular ocussing on arithmetic, by ‘finitary means’. As we have discussed above finitary methods were to be o a restricted kind: the writing o, and operations on, finite strings o marks on paper, and using intuitive reasoning that ‘includes recursion and intuitive induction or finite existing totalities’ (Hilbert in a 1922 lecture). However finitary reasoning was also lef somewhat vague, but clearly the usual arithmetical operations on numbers (and this or him included exponentiation) counted as finitary. Hilbert and Bernays (1929) seem to have settled on primitive recursive arithmetic, PRA, which allows the definition o unctions by primitive recursion schemes, and induction on quantifier-ree ormulae. I this constituted the ‘finitary means’ o Hilbert, then indeed the Incompleteness Teorems dealt a death blow to this programme. Von Neumann thought so, and Weyl, in his 1943
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obituary o Hilbert, described it as a ‘catastrophe’. Gödel was initially more circumspect: he did not consider at that time that it had been argued that all methods o a ‘finitary’ nature could be ormalized in, say, PM . At a meeting o the Vienna Circle in January 1931 he said that he thought that von Neumann’s assertion that all finitary means could be effected in one ormal system (and thus the Incompleteness Teorems should have a devastating effect on Hilbert’s programme) was the weakthat point in von Neumann’s Again his 1931 paper Gödel wrote there might be finitaryargumentation. proos that could notinbe written in a ormal system such as PM . It was hard to bring orward convincing arguments in either direction at this point: there was no clear notion o ‘ormal system’; this had to wait until 1936 or uring and there was also something o a conusion about intuitionistic logic: both von Neumann and Gödel thought that intuitionistic logic could count as finitary reasoning. However, this turned not to be the case: in 1933 Gödel showed that classical arithmetic could be interpreted in the intuitionistic version (known as Heyting Arithmetic, HA, and which only uses intuitionistic axioms o logic and rules o inerence), thus ruling out the idea o using intuitionistic logic to help codiy finitary reasoning, since the consistency o HA alone would now give the consistency o PA . However, by 1933 (Gödel 1995b) he had changed his mind and acknowledged that all finitary reasoning could indeed be ormalized in the axiom system o Peano Arithmetic (‘PA ’), which in particular allowed mathematical induction or ormulae with quantifiers. He later remarked in several places (as in the quotation above) that uring’s precise definition o a ormal system convinced him that the Incompleteness Teorems reuted Hilbert’s programme.
4.2 Salvaging Hilbert’s programme It was not recorded when precisely Hilbert learnt o the Incompleteness results. Bernays, when he had previously suggested to Hilbert that afer all a completeness proo might not be possible, reported that Hilbert reacted with anger, as he did eventually to the results themselves. Nevertheless, attempts were made to recover as much o the programme as was consistent with the Incompleteness Teorems. Bernays (who in correspondence with Gödel indicated that he was also not convinced that all finitary reasoning could be captured by a single system) in particular sought to discover modes o reasoning that could count as finitary but avoid being captured by the ormal systems o the kind Gödel discussed. Hilbert and Bernays soon aferwards reacted positively by trying to see what could be done. By 1931 Hilbert was suggesting that an ‘ ω-rule’ might be deployed where
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rom an infinite set o deductions proving P(0), P(1), . . . , P(n), . . . one would be allowed to iner kP(k) might be permissible as a orm o reasoning. It is unknown whether Hilbert was reacting directly to the hypothesis o ωconsistency in the first version o the First Incompleteness Teorem, since indeed the displayed Gödel sentence was a pure sentence which would be ‘proved’ i the ω-rule were allowed. But the rule itsel was not perceived as being finitary. However, the major and mostostriking advance here came G. Gentzen, who showed that the consistency Peano Arithmetic could be rom established afer all, i one allowed inductions along well orderings up to the first ‘epsilon number’ (ε0 being the first fixed point o the ordinal exponentiation unction α → ωα). Clearly these are not finitary operations in any strict sense, but nevertheless Gentzen’s work opened a whole area o logical investigation o ormal systems involving such transfinite inductions thus opening the area o proo theory and ‘ordinal notation systems’. (Gentzen also had the result exactly right: inductions bounded below ε0 would not have sufficed.)
4.3 Afer Incompleteness Had the Hilbert programme succeeded, it would have shown that idealmathematics could be reduced to finitary ‘real’ mathematics: the consistency o a piece o ideal mathematics could be shown using just real, finitistic mathematical methods. A relativized Hilbert programmeseeks to reduce an area o classical mathematics to some theory, necessarily stronger than finitary mathematics. Feerman has argued that most o mathematics needed or physics, or example, can be reduced to predicative systems which can be proo-theoretically characterized using ordinal notation systems albeit longer thanε0, but still o a small or manageable length. As the Second Incompleteness Teorem had shown, given a ormal theory 0 (such as PA) we have that0 Con(0). But we may addCon(0) as a new axiom itsel to thereby obtaining a somewhat stronger deductive theory. (It is not that 0 the Teorem casts any doubt on the theory0 or its consistency, it is only that it demonstrates the impossibility o ormalizing a proo o that consistency within0.) Tus setting:
1 : 0 + Con(0) Te thinking is that since we accept PA and believe that its axioms are true o the natural number structure, we should also accept that PA is consistent. (Whilst a 0, i consistent, neither provesCon(0) nor its negation, it would be presumably perverse to claim that ¬ Con(0) is the correct choice o the two to make here.)
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Continuing, we may define:
Having collected all these theories together as ω, we might continue:
ω+1 = ω + Con(ω) etc. We thus obtain a transfinite hierarchy o theories. What can one in general prove rom a theory in this sequence? uring called these theories ‘Ordinal Logics’ and was the first to investigate the question as to what extent such a sequence could be considered complete:
Question: Can it be that or any problem, or arithmetical statement A there might be an ordinal α so that α proves A or ¬A? And i so can this lead to new knowledge o arithmetical acts? Such a question is necessarily somewhat vaguely put, but anyone who has considered the Second Incompleteness Teorem comes around to asking similar or related questions as these. Te difficulty with answering this, is that much has been swept under the carpet by talking rather loosely o ‘ α’ or α ≥ ω. Tis is a subtle matter, but there seems no really meaningul way to arrive at urther arithmetical truths. Such iterated consistency theories have been much studied by Feerman and his school (see Franzén 2004). Lastly we consider the question o Gödel sentences themselves. Much has been studied and written on this theme alone. However, the use o the diagonal lemma leading to the sel-reerential nature o such a Gödel sentence gives a contrived eeling to the sentence. (One should also beware the act that not any fixed point o a ormula φ(v1) is necessarily stating that it says o itsel that it satisfies φ.) Could there be propositions that were more genuinely mathematical statements, and were not decided by PA? Gödel’s methods produced only sentences o the diagonal kind, and the problem was remarkably difficult. Some orty years were to pass beore the first example was ound by Paris and Harrington (concerning so-called Ramsey-like partition principles). Since then many more examples o mathematically interesting sentences independent oPA have been discovered.
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Further reading Dawson, J. (1997), Logical Dilemmas: Te Lie and Work o Kurt Gödel, Wellesley, MA: A.K. Peters. Franzén, . (2004), Inexhaustibility: A non-exhaustive treatment, vol. 16 o Lecture Notes in Logic, Wellesley, MA: ASL/A.K. Peters. Franzén, . (2005), Gödel’s Teorem:An Incomplete Guide to Its Use and Abuse, Wellesley, MA: A.K. Peters. Kennedy, J. (2011), ‘Kurt Gödel’, Stanord Encyclopedia o Philosophy, . Smullyan, R. (1993), Gödel’s Incompleteness Teorems , Oxord and New York: Oxord University Press. Zach, R. (2003), ‘Hilbert’s program’, Stanord Encyclopedia o Philosophy, .
Reerences Franzén, . (2004), Inexhaustibility: A non-exhaustive treatment, volume 16 o Lecture
Notes in Logic., Wellesley MA: ASL/A.K.Peters. Gödel, K. (1930), ‘Die Vollständigkeit der Axiome des logischen Functionenkalküls?, Monatshefe ür Mathematik und Physik’, Monatshef ür Mathematik und Physik, 37 (trans. in Gödel (1986)), pp. 349–600. Gödel, K. (1931), ‘Über ormal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshef ür Mathematik und Physik, 38 (transl. in Gödel (1986)), pp. 173–98. Gödel, K. (1986), Collected Works: Publications 1929–36, ed. S. Feerman, J.W. Dawson Jr., S.C. Kleene, G.H. Moore, R.M. Solovay and J. van Heijenoort, pp. 304–23, Oxord: Oxord University Press. Gödel, K. (1995a), Collected Works Vol III:Unpublished Essays and Lectures, ed. S. Feerman, J.W. Dawson Jr., S.C. Kleene , G.H. Moore, R.M. Solovay and J. van Heijenoort, pp. 166–8, Oxord: Oxord University Press. Gödel, K. (1995b), ‘Te present situation in the oundation o mathematics ’, in S. Feerman, J.W. Dawson Jr., S.C. Kleene , G.H. Moore, R.M. Solovay and J. van Heijenoort (eds), Collected Works Vol III:Unpublished Essays and Lectures, Oxord: Oxord University Press. Gödel, K. (2003), Collected Works Vol V:Correspondence H–Z, ed. S. Feerman, J.W. Dawson Jr., S.C. Kleene, G.H. Moore, R.M. Solovay and J. van Heijenoort, pp. 304–23, Oxord: Oxord University Press. Hilbert, D. and W. Ackermann1929), ( Grundzüge der theoretischen Logik, Berlin: Springer-Verlag.
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Grundlagen der Mathematik, vol. I, Berlin: SpringerHilbert, D. and P. Bernays1934), ( Verlag, Berlin. Grundlagen der Mathematik, vol. II, Berlin: SpringerHilbert, D. and P. Bernays1939), ( Verlag. uring, A.M. (1936), ‘On Computable Numbers with an application to the Entscheidungsproblem’, Proceedings o the London Mathematical Society, 42(2). Wang, H. (1996), A Logical Journey: From Gödel to Philosophy, Boston: MI Press. Zach, R. (2003), ‘Hilbert’s program’, Stanord Encyclopedia o Philosophy, http://plato. stanord.edu/entries/hilbert-program, accessed 12 August 2016.
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12
arski Benedict Eastaugh
1 Introduction It is hard to overstate Alred arski’s impact on logic.* Such were the importance and breadth o his results and so influential was the school o logicians he trained that the entire landscape o the field would be radically different without him. In the ollowing chapter we shall ocus on three topics: arski’s work on ormal theories o semantic concepts, particularly his definition o truth; set theory and the Banach–arski paradox; and finally the study o decidable and undecidable theories, determining which classes o mathematical problems can be solved by a computer and which cannot. arski was born Alred ajtelbaum in Warsaw in 1901, to a Jewish couple, Ignacy ajtelbaum and Rosa Prussak. During his university education, rom 1918 to 1924, logic in Poland was flourishing, and arski took courses with many amous members o the Lvov–Warsaw school, such as adeusz Kotarbiński, Stanisław Leśniewski and Jan Łukasiewicz. Prejudice against Jews was widespread in interwar Poland, and earing that he would not get a aculty position, the young Alred ajtelbaum changed his name to arski. An invented name with no history behind it, Alred hoped it would sound suitably Polish. Te papers confirming the change came through just beore he completed his doctorate (he was the youngest ever to be awarded one by the University o Warsaw), and he was thereore awarded the degree under his new name o Alred arski. Struggling to obtain a position in line with his obvious brilliance, arski took a series o poorly paid teaching and research jobs at his alma mater, supporting himsel by teaching high-school mathematics. It was there that he met the woman who would become his wie, ellow teacher Maria Witkowska. Tey married in 1929, and had two children: their son Jan was born in 1934, and their 293
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daughter Ina ollowed in 1938. Passed over or a proessorship at the University o Lvov in 1930, and another in Poznan in 1937, arski was unable to secure the stable employment he craved in Poland. Despite these proessional setbacks, arski produced a brilliant series o publications throughout the 1920s and 1930s. His work on the theory o truth laid the ground not only or model theory and a proper understanding o the classical but alsodecision or research on the o truth thatlogical is stillconsequence bearing ruitrelation, today. arski’s procedure orconcept elementary algebra and geometry, which he regarded as one o his two most important contributions, was also developed in this period. In 1939 he embarked or the United States or a lecture tour, with a thought o finding employment there. Seemingly oblivious to the impending conflagration, arski nevertheless contrived to escape mere weeks beore war with Germany broke out, but leaving his wie and children behind. Working as an itinerant lecturer at Harvard, the City College o New York, Princeton and Berkeley, arski spent the war years separated rom his amily. Back in Poland, Maria, Jan and Ina were taken into hiding by riends. Despite intermittent reports that they were still alive, arski spent long periods without news, and his attempts to extricate them rom Poland were all in vain. It was not until the conclusion o the war that he learned that while his wie and children had survived, most o the rest o his amily had not. His parents perished in Auschwitz, while his brother Wacław was killed in the Warsaw Uprising o 1944. About thirty o arski’s close relatives were amongst the more than three million Polish Jews murdered in the Holocaust, along with many o his colleagues and students, including the logician Adol Lindenbaum and his wie, the philosopher o science Janina Hosiasson-Lindenbaumowa. In 1945, arski gained the permanent position he craved at the University o Caliornia, Berkeley, where Maria and the children joined him in 1946. Made proessor in 1948, arski remained in Caliornia until his death in 1983. Tere he built a school in logic and the philosophy o science and mathematics that endures to this day: a testament to his brillianceas a scholar, his inspirational qualities asa teacher, and his sheer orce o personality. Te most universally known and acclaimed part o arski’s career consists o his work on the theory o truth, so it is natural that we begin our journey there. Section 2 starts rom the liar paradox, and then turns to arski’s celebrated definition o truth or ormalized languages. Tis leads us to the undefinability theorem: that no sufficiently expressive ormal system can define its own truth predicate. Much o arski’s early research was in set theory. Although he remained interested in the area or the rest o his working lie, his best-known contribution
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to the field remains the paradoxical decomposition o the sphere which he developed in collaboration with Stean Banach, colloquially known as the Banach–arski paradox. Tis striking demonstration o the consequences o the Axiom o Choice is explored in section 3. As a logician, only Kurt Gödel outshines arski in the twentieth century. His incompleteness theorems are the singular achievement around which the story o section to 4 pivots. Gödel, problems. logicians still held a general algorithm decide Beore mathematical Many o out this hope area’sor successes in the 1920s are due to arski and his Warsaw students, such as the discovery that when ormulated in a language without the multiplication symbol, the theory o arithmetic is decidable. In 1936, five years afer Gödel’s discovery o incompleteness, Alonzo Church and Alan uring showed that the general decision problem or first-order logic was unsolvable. Te ocus then turned rom complete, decidable systems to incomplete, undecidable ones, and once again arski and his school were at the oreront. Peano Arithmetic was incomplete and undecidable; how much could it be weakened and retain these properties? What were the lower bounds or undecidability? Tis is only intended as a brie introduction to arski’s lie and work, and as such there are many ascinating results, connections and even whole areas o study which must go unaddressed. Fortunately the history o logic has benefited in recent years rom some wonderul scholarship.Te encyclopaedic Handbook o the History o Logic is one such endeavour, and Keith Simmons’s chapter on arski (Simmons 2009) contains over a hundred pages. arski is also the subject o an engrossing biography by Anita Burdman Feerman, together with her husband and arski’s ormer student, Solomon FeermanFeerman ( and Feerman 2004). Entitled Alred arski: Lie and Logic, it mixes a traditional biography o arski’s colourul lie with technical interludes explaining some o the highlights o arski’s work. Finally, in addition to being a logician o the first rank, arski was an admirably clear communicator. His books and papers, ar rom being o merely historical interest, remain stimulating reading or logicians and philosophers. Many o them, including early papers srcinally published in Polish or German, are collected in the volumeLogic, Semantics, Metamathematics(arski 1983).
2 Te theory o truth Semantic concepts are those which concern the meanings o linguistic expressions, or parts thereo. Amongst the most important o these concepts are
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truth, logical consequence and definability. All o these concepts were known in arski’s day to lead to paradoxes. Te most amous o these is theliar paradox. Consider the sentence ‘Snow is white’. Is it true, or alse? Snow is white: so the sentence ‘Snow is white’ is true. I snow were not white then it would be alse. Now consider the sentence ‘Tis sentence is alse’. Is it true, or alse? I it’s true, then the sentence is alse. But i it’s alse, then the sentence is true. So we have a contradiction whichever value we assign to the sentence. Tis sentence is known as the liar sentencetruth . In everyday speech and writing, we appear to use truth in a widespread and coherent way.ruth is a oundational semantic concept, and thereore one which we might naively expect to obtain a satisactory philosophical understanding o. Te liar paradox casts doubt on this possibility: it does not seem to require complex or ar-etched assumptions about language in order to maniest itsel, but instead arises rom commonplace linguistic devices and usage such as our ability to both use and mention parts o speech, the property o bivalence, and the typical properties we ascribe to the truth predicate such as disquotation.
2.1 arski’s definition o truth Both their apparent ambiguity and paradoxes like the liar made mathematicians wary o semantic concepts. arski’s analyses o truth, logical consequence and definability or ormal languages were thus major contributions to both logic and philosophy. Tis paved the way or model theory and much o modern mathematical logic on the one hand; and renewed philosophical interest in these semantic notions – which continues to this day – on the other. In his seminal 1933 paper on ‘Te Concept o ruth in Formalized Languages’ (1933), arski offered an analysis o the liar paradox. o understand arski’s analysis, we need to make a ew conceptual points. Te first turns on the distinction between use and mention. I we were to say that arski was a logician, we would be using the name ‘arski’ – but i we said that ‘arski’ was the name that logician chose or himsel, we would be mentioning it. In the written orms o natural language we ofen distinguish between using a term and mentioning it by quotation marks. When we say that ‘Snow is white’ is true i, and only i, snow is white, we both use and mention the sentence ‘Snow is white’. Te liar paradox seems to rely on our ability not merely to use sentences – that is, to assert or deny them – but on our ability to reer to them. Te locution ‘Tis sentence’ in the liar sentence reers to (that is, mentions) the sentence itsel, although it does not use quotation marks to do so. Consider the ollowing
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variation on the liar paradox, with two sentences named A and B. Sentence A reads ‘Sentence B is alse’, while sentence B reads ‘Sentence A is true’. We reason by cases: either sentence A is true, or it is alse. I A is true, then B is alse, so it is alse that A is true – hence A is alse, contradicting our assumption. So A must be alse. But i A is alse, then it is alse that B is alse, and so B is in act true. B says that A is true, contradicting our assumption that it is true. So we have a contradiction either way. In his analysis o the liar paradox, arski singles out two key properties which a language must satisy in order or the paradox to occur in that language. Te first consists o three conditions: the language must contain names or its own sentences; it must contain a semantic predicate ‘ x is true’; and all the sentences that determine the adequate usage o the truth predicate must be able to be stated in the language. Tese conditions are jointly known as semantic universality. Te second property is that the ordinary laws o classical logic apply: every classically valid inerence must be valid in that language. arski elt that rejecting the ordinary laws o logic would have consequences too drastic to even consider this option, although many philosophers since have entertained the possibility o logical revision; see section 4.1 o Beall and Glanzberg (2014) or an introductory survey. Since a satisactory analysis o truth cannot be carried out or a language in which the liar paradox occurs – as it is inconsistent – arski concluded that we should seek a definition o truth or languages that are not semantically universal. Tere are different ways or a language to ail to be semantically universal. Firstly, it could ail to have the expressive resources necessary to make assertions about its own syntax: it could have no names or its own expressions. Secondly, it could ail to contain a truth predicate. Finally, the language might have syntactic restrictions which restrict its ability to express some sentences determining the adequate usage o the truth predicate. Tis seems to exclude the possibility o giving a definition o truth or natural languages. Not only are they semantically universal – quotation marks, or instance, allow us to name every sentence o English within the language – but they actually aim or universality. I a natural language ails to be semantically universal then it will be expanded with new semantic resources until it regains universality. arski goes so ar as to say that ‘it would not be within the spirit o [a natural language] i in some other language a word occurred which could not be translated into it’(arski 1983: 164). When English ails to have an appropriate term to translate a oreign one, in cases like the German schadenreude or the
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French aux pas, the oreign term is simply borrowed and becomes a loanword in English. arski thereore offered his definition o truth only or ormal languages. Tese tend to be simpler than natural languages, and thus they are more amenable to metalinguistic investigation. Te particular example that arski used was the calculus o classes, but essentially the same approach can be used to define or o anytruth, ormal As is standard the current literature on ormaltruth theories welanguage. shall use the language oin arithmetic. A ormal language is typically constructed by stipulating two main components. Te first is the alphabet: the collection o symbols rom which all expressions in the language are drawn. In the case o a first-order language like that o arithmetic, the alphabet includes (countably infinitely many) variables v0, v1, . . .; logical constants , , ¬ , , →, ↔, =; punctuation (,). Tis is then enriched by the addition o nonlogical constants, unction symbols and relational predicates. In the case o the first-order language o arithmetic this includes the constant symbols 0 and 1; the two binary unction symbols + and ×; and the binary relation symbol
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