Handbook of Philosophical Logic Second Edition 11

HANDBOOK OF PHILOSOPHICAL LOGIC 2ND EDITION VOLUME 11

HANDBOOK OF PHILOSOPHICAL LOGIC 2nd Edition Volume 11 edited by D.M. Gabbay andF. Guenthner

Volume 1 - ISBN 0-7923-7018-X Volume 2 - ISBN 0-7923-7126-7 Volume 3 - ISBN 0-7923-7160-7 Volume 4 - ISBN 1-4020-0139-8 Volume 5 - ISBN 1-4020-0235-1 Volume 6 - ISBN 1-4020-0583-0 Volume 7 - ISBN 1-4020-0599-7 Volume 8 - ISBN 1-4020-0665-9 Volume 9 - ISBN 1-4020-0699-3 Volume 10- ISBN 1-4020-1644-1

HANDBOOK OF PHILOSOPHICAL LOGIC 2nd EDITION

VOLUME 11 Edited by D.M.GABBAY King's College, London, U.K.

and

F. GUENTHNER Centrum fUr Informations- und Sprachverarbeitung, Ludwig-Maximilians-Universitiit Munchen, Germany

....

"

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-017-0466-3 (eBook) ISBN 978-90-481-6554-4 DOI 10.1007/978-94-017-0466-3

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All Rights Reserved

© 2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the hardcover 2nd edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

CONTENTS

Editorial Preface Dov M. Gabbay Modal Logic and Self-Reference Craig Smorynski Diagonalization in Logic and Mathematics Dale J acquette

VB

1

55

Semantics and the Liar Paradox Albert Visser

149

The Logic of Fiction John Woods and Peter Alward

241

Index

317

PREFACE TO THE SECOND EDITION It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good! The first edition was the second handbook published for the logic community. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisation on the one hand and to provide the theoretical basis for the computer program constructs on the other. The result was that the Handbook of Philosophical Logic, which covered most of the areas needed from logic for these active communities, became their bible. The increased demand for philosophical logic from computer science and artificial intelligence and computational linguistics accelerated the development of the subject directly and indirectly. It directly pushed research forward, stimulated by the needs of applications. New logic areas became established and old areas were enriched and expanded. At the same time, it socially provided employment for generations of logicians residing in computer science, linguistics and electrical engineering departments which of course helped keep the logic community thriving. In addition to that, it so happens (perhaps not by accident) that many of the Handbook contributors became active in these application areas and took their place as time passed on, among the most famous leading figures of applied philosophical logic of our times. Today we have a handbook with a most extraordinary collection of famous people as authors! The table below will give our readers an idea of the landscape of logic and its relation to computer science and formal language and artificial intelligence. It shows that the first edition is very close to the mark of what was needed. Two topics were not included in the first edition, even though D. GabbaI/ and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 11, vii-ix. © 2002, Kluwer Academic Publishers.

viii

they were extensively discussed by all authors in a 3-day Handbook meeting. These are: • a chapter on non-monotonic logic • a chapter on combinatory logic and A-calculus We felt at the time (1979) that non-monotonic logic was not ready for a chapter yet and that combinatory logic and A-calculus was too far removed. 1 Non-monotonic logic is now a very major area of philosophical logic, alongside default logics, labelled deductive systems, fibring logics, multi-dimensional, multimodal and substructural logics. Intensive reexaminations of fragments of classical logic have produced fresh insights, including at time decision procedures and equivalence with non-classical systems. Perhaps the most impressive achievement of philosophical logic as arising in the past decade has been the effective negotiation of research partnerships with fallacy theory, informal logic and argumentation theory, attested to by the Amsterdam Conference in Logic and Argumentation in 1995, and the two Bonn Conferences in Practical Reasoning in 1996 and 1997. These subjects are becoming more and more useful in agent theory and intelligent and reactive databases. Finally, fifteen years after the start of the Handbook project, I would like to take this opportunity to put forward my current views about logic in computer science, computational linguistics and artificial intelligence. In the early 1980s the perception of the role of logic in computer science was that of a specification and reasoning tool and that of a basis for possibly neat computer languages. The computer scientist was manipulating data structures and the use of logic was one of his options. My own view at the time was that there was an opportunity for logic to playa key role in computer science and to exchange benefits with this rich and important application area and thus enhance its own evolution. The relationship between logic and computer science was perceived as very much like the relationship of applied mathematics to physics and engineering. Applied mathematics evolves through its use as an essential tool, and so we hoped for logic. Today my view has changed. As computer science and artificial intelligence deal more and more with distributed and interactive systems, processes, concurrency, agents, causes, transitions, communication and control (to name a few), the researcher in this area is having more and more in common with the traditional philosopher who has been analysing 1 I am really sorry, in hindsight, about the omission of the non-monotonic logic chapter. I wonder how the subject would have developed, if the AI research community had had a theoretical model, in the form of a chapter, to look at. Perhaps the area would have developed in a more streamlined way!

PREFACE TO THE SECOND EDITION

ix

such questions for centuries (unrestricted by the capabilities of any hardware). The principles governing the interaction of several processes, for example, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based effective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. I believe the day is not far away in the future when the computer scientist will wake up one morning with the realisation that he is actually a kind of formal philosopher! The projected number of volumes for this Handbook is about 18. The subject has evolved and its areas have become interrelated to such an extent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupings of chapters. I would like to thank our authors are readers for their contributions and their commitment in making this Handbook a success. Thanks also to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers for their continuing support for the Handbook.

Dov Gabbay King's College London

x

Logic

II

IT Natural language processing

Program control specification, verification, concurrency

Artificial intelligence

Logic programming

Temporal logic

Expressive power of tense operators. Temporal indices. Separation of past from future

Expressive power for recurrent events. Specification of temporal control. Decision problems. Model checking.

Extension of Horn clause with time capability. Event calculus. Temporal logic programming.

Modal logic. Multi-modal logics

generalised quantifiers

Action logic

Planning. Time dependata. dent Event calculus. Persistence through timethe Frame Problem. Temporal query language. temporal transactions. Belief revision. Inferential databases

Algorithmic proof

Discourse representation. Direct computation on linguistic input Resolving ambiguities. Machine translation. Document classification. Relevance theory logical analysis of language

New logics. Generic theorem provers

General theory of reasoning. Non-monotonic systems

Procedural approach to logic

Loop checking. Non-monotonic decisions about loops. Faults in systems.

Intrinsic logical discipline for AI. Evolving and communicating databases

by Negation failure. Deductive databases

Real time systems

Semantics for logic programs

Constructive reasoning and proof theory about specification design

Expert systems. Machine learning Intuitionistic logic is a better logical basis than classical logic

Non-wellfounded sets

Hereditary finite predicates

Nonmonotonic reasoning

Probabilistic and fuzzy logic Intuitionistic logic

Set theory, higher-order logic, Acalculus, types

Quantifiers logic

Montague semantics. Situation semantics

in

Negation failure modality

by

and

Horn clause logic is really intuitionistic. of Extension logic programming languages A-calculus extension to logic programs

PREFACE TO THE SECOND EDITION

xi

Imperative vs. declarative languages

Database theory

Complexity theory

Agent theory

Special comments: A look to the future

Temporal logic as a declarative programming language. The changing past in databases. The imperative future

Temporal databases and temporal transactions

Complexity questions of decision proced ures of the logics involved

An essential component

Temporal systems are becoming more and more sophisticated and extensively applied

Dynamic logic

Database upand dates action logic

Ditto

Possible tions

Multimodal logics are on the rise. Quantification and context becoming very active

Types. Term rewrite systerns. Abstract interpretation

Abduction, relevance

Ditto

Agent's implementation rely on proof theory.

Inferential databases. Non-monotonic of coding databases

Ditto

Agent's reasoning is non-monotonic

A major area now. Important for formalising practical reasoning

Fuzzy and probabilistic data Database transactions. Inductive learning

Ditto

Connection with decision theory Agents constructive reasoning

Major now

Semantics for programming languages. Martin-Lof theories

Semantics for programming languages. Abstract interpretation. Domain recursion theory.

Ditto

Ditto

ac-

area

Still a major central alternative to classical logic

More central than ever!

xii

Clusical logic. Clusical Cragments

Basic ground guage

Labelled deductive systems

Extremely useful in modelling

A unifying framework. Context theory.

Resource and substructural logics Fibring and combining logics

Lambek calcuIus

Truth maintenance systems Logics of space and time

back-

lan-

Dynamic syntax

Program synthesis

Modules. Combining languages

A basic tool

Fallacy theory

Logical Dynamics Argumentation theory games

Widely applied here Game semantics

..

gammg

ground

Object level/ metalevel

Extensively used in AI

Mechanisms: Abduction, default relevance Connection with neural nets

ditto

Time-actionrevision models

ditto

Annotated logic programs

Combining features

PREFACE TO THE SECOND EDITION

Relational databases

Labelling allows for context control. and Linear logic

Linked databases. Reactive databases

Logical complexity classes

xiii

The workhorse of logic

The study of is fragments very active and promising.

Essential tool.

The new unifying framework for logics

Agents have limited resources Agents are built up of various fibred mechanisms

The notion of self-fibring allows for selfreference Fallacies are really valid modes of reasoning in the right context.

Potentially applicable

A dynamic view of logic On the rise in all areas of applied logic. Promises a great future

Important fea. ture of agents

Always central in all areas

Very important for agents

Becoming part of the notion of a logic Of great importance to the future. Just starting

A new theory of logical agent

A new kind of model

CRAIG SMORYNSKI

MODAL LOGIC AND SELF-REFERENCE

o

INTRODUCTION

Ever since Epimenides made his startling confession, philosophers and mathematicians have been fascinated by self-reference. Of course, mathematicians are not free to admit this. To the orderly mind of the mathematician the man who says 'I am lying' is witty, but not to be taken seriously, and the barber who shaves the heads of those in his village who do not shave their own heads simply does not exist-nor does anyone else in his lousy village and, besides, no selfrespecting mathematician would want to live there anyway. The Russell paradox is a different matter: R={x:x~x}

is not merely a linguistic trick, but something which, if admitted as an entity leads to real trouble: R E R iff R ~ R. The existence of R has a clear mathematical purpose-it shows Frege's set theory to be inconsistent. In short, whereas the philosopher takes self-reference, even the Liar, seriously, the mathematician associates it with inconsistency or inexpressibility. That is, the mathematician did so until 1930 when Kurt GOdel turned selfreference from a philosophically puzzling or mathematically suspect object into a respectable mathematical tool. GOdel's starting point was, oddly enough (or naturally enough), the Liar. The sentence, 1 am lying, when uttered can neither be true nor false, hence cannot be uttered. Well-it can be uttered (I just tried it); but it cannot be uttered coherently, i.e. not if it is to have a definite truth value (and we are to keep the usual laws of logic). But, observed G6del, if a language is expressible enough and a theory T in the language is both simple and powerful enough, it can express provability. If, moreover, a certain amount of self-reference is available, a watered-down Liar can assert I am unprovable, D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 11, 1-53. © 2002, Kluwer Academic Publishers.

2

CRAIG SMORYNSKI

or, rather, What I am saying is unprovable. This sentence will, in fact, be unprovable. As one might guess, GOdel's observation was a big hit. Everyone, the mathematician as well as the philosopher, was impressed by GOdel's argument and the conclusions he drew from it. If T is a sufficiently powerful formal theory and T is sufficiently sound, then T is incomplete; i.e. there are true sentences undecided by T.

GODEL'S FIRST INCOMPLETENESS THEOREM.

1fT is a sufficiently powerful formal theory, then T cannot prove its own consistency.

GODEL'S SECOND INCOMPLETENESS THEOREM.

The story of GOdel's Theorems and their effect on the Philosophy of Mathematics need not be repeated here. What should be emphasised is that a mathematical theory of self-reference was long in developing. Philosophers tried simulating a few paradoxes other than the Liar and mathematicians developed Recursion Theory (cf. the chapter by Van Dalen (this Handbook volume 1)) as a safe alternative. A theory of self-reference could have emerged in the 1950s when Leon Henkin asked the question: we know that GOdel's sentence asserting its own unprovability is unprovable; what about the sentence asserting its won provability? For example, if a sentence declares I am provable, is it telling the truth? At the International Congress of Mathematicians in 1954 Martin H. Lob proved that the answer was yes. Unfortunately, the referee for his paper, which appeared the following year, noticed that Lob's argument established something a bit more general-a bit more philosophically interesting-and the cute fact of the provability of the statement asserting its own provability was overlooked. Proof theorists emphasised the philosophical importance of Lob's Theorem and the mathematical dabblers backed off. In the early 1970s, however, the story changed. Suddenly, from several directions at once it was recognised that, modulo the background analysis by Lob of the representation of provability within a system, the proofs of GOdel's Theorems and Lob's Theorem were propositional in character, that is they used propositional logic with an additional operator and some familiar laws-Leo modal logic. In the sequel I shall exposit some of the modal analysis of self-reference. My plan is fairly simple: in the immediately following section I shall discuss the arithmetical background-self-reference in (say) arithmetic, Lob's

MODAL LOGIC AND SELF-REFERENCE

3

Derivability Conditions, GOdel's theorems, and Lob's Theorem. There follows in Section 2 a description of a system of modal logic called Provability Logic, or PrL. The analysis of self-reference in PrL is given in the next section. The key result is the De J ongh-Sambin Theorem: every appropriate formula has a unique, explicitly definable fixed point, i.e. self-referential sentences arising from modal contexts have genuine meanings determinable without resort to self-reference. Up to this point, modal logic will only have been used notationally. In Section 4, I discuss the model theory of PrL. This is not only of interest in its own right, but it also serves as a tool for a further analysis of arithmetic derivability. In Section 5, I discuss arithmetical interpretations of PrL. The main theorems here are Solovay's two Completeness Theorems. The First Completeness Theorem asserts that PrL is the logic of arithmetic provability, whence the modal analysis is complete in a sense. The Second Theorem characterises the schemata valid with respect to truth and is, in effect, the strongest single incompleteness theorem known. Having exhausted, to some extent, the study of pure provability by the end of Section 5, I next lead the reader into the applied theory of selfreference. This material is both a bit more advanced and more skimpily presented: first, in Section 6, I discuss Rosser's sentences and their relatively complete modal analysis. In Section 7, the goal is different-to use modal logic to unify many different self-referential formulae and explain all their known applications at once. For the instances of self-reference falling under the scope of the explanation, the explanation is completely satisfactory; for other similar self-referential instances, I can only refer to the literature for the beginnings. 1 THE INCOMPLETENESS THEOREMS Nowadays, mathematical logicians would prefer a discussion of the set theoretic encoding of syntax in a weak set theory-except, of course, for the proof theorists, who would find a theory of finite sequences most natural. Traditionally, however, one discusses the language of arithmetic-or, rather, a language of arithmetic. In view of the handy discussion of indexing in Van Dalen's chapter on recursion theory (this Handbook volume 1), I find it easiest to assume the reader is familiar with such encoding and simply discuss the end result rather than the process of such an encoding of syntax within the language of arithmetic. In the sequel, we will mostly need this only as an explanation of the modal systems and the types of questions we will ask about them. To begin with, we should specify the language of arithmetic and declare some axiomatisation for a formal theory of arithmetic. For the language, we

CRAIG SMORYNSKI

4

have, in addition to the logical apparatus (variables, equality, connectives and quantifiers), the individual constant 0, function constants 5 (successor) + (addition), . (multiplication), and / for each (primitive recursive definition of a) primitive recursive function f, and a binary relation ~ (order). Numerals 1,2,3, ... are abbreviations for 50,550,5550, ... , respectively. The axioms of formal number theory, called Peano Arithmetic, or PA, consist, in addition to the usual logical ones, of the following:

I.

0=1 5x 5x= 5y -+ x = y M(x}' ... , Xt) = Xi &';(X1, ... ,Xt) =n

II.

X ~

III.

y ++ 3z(x + z = y)

x+O=x x+5y=5(x+y) x·O=O x·5y = x·y+x /(0, Xl, . .. ,Xt) = g(x}, . .. , Xt) /(5x, Xl , ... ,Xt) = /,,(/(X, Xl , ... ,Xt),X1, ... ,Xt,x),

if f is defined from g, h by primitive recursion /(x}, . .. ,Xt) = g(/"l (Xl, ... ,Xt), . .. ,/"p(X1, . .. ,Xt)), if f is defined from g, hI, .. . , hp by composition.

IV.

cpO 1\ Vx(cpx -+ cp(5x)) -+ Vxcpx, all cpx with

X

free.

The first group of axioms merely concerns the initial functions from which the primitive recursive ones are generated; the second defines order; the third simply gives equations corresponding to the definitional principles for the primitive recursive functions generated from the initial ones; and the fourth schema is just induction. The axioms for + and . are redundant insofar as + and . are defined by recursion from 5. These functions are, however, special: the full system PA can be shown to be a definitional extension of that generated by using only 5, +,. and their axioms (including the subschema of IV in the restricted language). Now, using the primitive recursive indexing of primitive recursive functions discussed in Van Dalen's chapter (this Handbook volume 1), one can get a primitive recursive encoding of syntax: there is an assignment of numbers r t', r cp' to terms t, formulae cp, respectively, such that the usual syntactic operations are primitive recursively simulated. E.g. there are primitive recursive functions con, neg, sub such that con(r cp" r tP ,) = r cp 1\ tP' neg(r cp') = r -,cp' sub(r cpx', r x', n) = r cp7i'.

MODAL LOGIC AND SELF-REFERENCE

5

Moreover, formal derivations can be viewed as finite sequences of formulae and the relation 'x codes a derivation of the formula with code y' is primitive recursive, i.e. there is a formula, Prov(x,y): p(x,y) =

0,

satisfying: Prov(x, y) is true iff x = (r CPI I, . . . , r CPn ') and y = r CPn I and CPI, ... ,CPn is a formal proof. From this we get a formula, Pr(y) : 3x Prov(x,y), asserting the provability of the formula with code y. Suppose now that I(XI,"" Xk) is a primitive recursive function. If mt, ... ,mk are given and I(rri l , ... , mk) = n, we can actually calculate this value using the defining clauses for I-and, hereditarily, those for the functions entering into the definition of I. But, these defining clauses are, in fact, axioms of PA, whence the calculation exhibiting I(ml,' .. ,mk) = n is virtually a formal derivation of /(ml,' .. ,mk) = ii, i.e. we have (i)

I(ml,"" mk) = n

=}

PA I- /(mt, ... , mk) = ii.

If we look carefully at this argument, we see that it is inductive-an overall induction on the number of steps used to define I, and in the case I is defined by primitive recursion, an additional induction on the variable of the recursion. Since PA has the induction schema IV, it follows that this argument can be formalised:

(ii)

PA I- \lXI,' .. , xky[/(xt, . .. ,Xk) = y -t -t Pr(sub(r I(XI,' .. , Xk) = y I, r Xl I, ... ,ry I, Xl, ... ,y))].

Actually, the proof of (ii) presupposes decent behaviour of Prov(x, y). The crucial property, which is easily built into the definition of Prov(x, y) is the provable closure of Pr(x) under modus ponens: (iii) PA I- Pr(r cP ') " Pr(r cP -t

t/J ') -t Pr(r t/J ').

Properties (i)-(iii) are all we will need to know about Pr(x). They are, however, not in a very elegant form. Applying (i) and (ii) to the primitive recursive characteristic function p(x, y) for Prov(x, y) we can derive the following: THEOREM 1 (Lob's Derivability Conditions). For all sentences cP, t/J, Dl. PA I- cP

=}

PA I- Pr(rcp')

6

CRAIG SMORYNSKI

D2. PA I- Pr(r If' I) A Pr(r If'

~

D9. PA I- Pr(r If' I)

~

'IjJ I)

~

Pr( 'IjJI)

Pr(rpr(r If' I) I).

Proof. Dl. Note that PAl- If' implies that there is a derivation If'l, ... If'n-l, If' of If'. Thus p«r If'l I, ... , r If'n-l I, r 1f'1), r If' I) = O. By (i), we have PA I- proV«rlf'l', ... ,rlf'n_l',rlf"),rlf"), Le. PA I- 3xProv(x,rlf"), Le. PA I- Pr(rlf'l). D2. This is just point (ii) above. • D3. This is just the formalisation of Dl.

Conditions D1-D3 are the key properties of Pr(x). Essentially, they are all we will need to know about Pr(x) until Section 6, where we will replace it by a new predicate. For this purpose, we introduce here a little terminology and discuss the generalisation of D 1 and D3 that we will need. Both of these are quite simple. DEFINITION 2. A formula If' is a PR-formula if it is of the form /(tl, ... , tk-d = tk, where each term ti is either a variable or a numeral and / is a primitive recursive function constant. A formula If' is an RE-formula if it has the form 3x1jJx, where 'ljJx is aPR-formula. Thus, a PR-formula is a canonical definition of a primitive recursive relation and an RE-formula is such for an RE relation (as defined in Van Dalen's chapter (this Handbook volume 1». In Section 6 we will need the following generalisations ofDl and D3, both

of which follow from (i) and (ii) respectively, in the same manner in which D1 and D3 followed therefrom: THEOREM 3 (RE-Completeness). Let If'Xl, ... ,Xk be an RE- (or PR- )formula.

(i) For all ml, ... ,mk, cpffll, ... ,mk holds

=> PA I- cpffl!, ... , mk

(ii) PA I- I(JYl, ... , Yk ~ Pr(sub(r If'Xl, ... , Xk I, r X1 I, ••• , r Xk I,y!, ... ,Yk». Getting back to our immediate needs, we will require one additional powerful principl~the Diagonalisation Lemma: THEOREM 4 (Diagonalisation Lemma). Let 'ljJx have only x free. There is a sentence If' such that

MODAL LOGIC AND SELF-REFERENCE

7

Proof. We use sub in much the same way in which the s:!'-function is used to prove the Recursion Theorem. Fix the variable x and consider

Ox: Let m

= rox' and cp = Om.

'I/J (sub (x, rx',x».

Notice:

• I have stated the Diagonalisation Lemma in a weak form. In full strength it is as general as the Recursion Theorem-indeed, the two are basically the same: the differences are (1) the choices of languages to which they apply, and (2) the fact that one deals with relations and one with functions. For a little about the expositional history of the Diagonalisation Lemma, cf. [Smorynski,1981]. Once we have the Diagonalisation Lemma and Lob's Derivability Conditions, GOdel's Incompleteness Theorems are easy exercises: THEOREM 5 (GOdel's Incompleteness Theorems). Let

Then:

(i) PA¥ cp (ii) PA ¥ ,cp (iii) PA ¥ Con(pA), where Con(pA) is the sentence ,Pr(rO = I'). Proof. (i) Observe PA I- cp =>

PA I- Pr(rcp'), by Dl

=> PA I- 'cp, by the definition of cp.

But this contradicts the consistency of PA, whence PA ¥ cp. (ii) Again, PA I- ,cp =>

PA I- Pr(rcp'), by choice of cp

=> PA I- cp,

CRAIG SMORYNSKI

8

since PA proves only true theorems. But again, we have an inconsistency unless PA ¥ 'cp. (iii) Since PA ¥ cp, it suffices to show PA r Con(PA) ~ cpo We prove the contrapositive: PA r 'cp ~ Pr(rO = I'). A few applications of D1 and D2 to PA r ,cp ++ Pr(r cp ') yield: PA r Pr(r ,cp,) ++ Pr(rpr(r cp,) '). But whence But

PA

r cp ~ (,cp ~ 0 = I)

and a few additional applications of D1, D2 yield PA r ,cp ~ Pr(ro = 1').

It is probably worth noting at this point the following:

•

COROLLARY 6 (Kreisel's Fixed Point Calculation). Let PA r cp ++ ,Pr(rcp'). Then: PA r cp ++ Con(PA).

Proof. We have already shown PA r Con(PA) apply D1, D2: PArO=I~cp

=} =} =}

~

cpo For the converse,

PArPr(rO=I~cp')

PA r Pr(rO = I') ~ Pr(rcp') PA r ,Con(PA) ~ 'cp.

•

There are a few quick remarks that should be made. First, with respect to the formulation of the Incompleteness Theorems, it is customary to incorporate the safety assumptions into the statements. Thus, e.g. instead of saying CPA ¥ cp', one says 'H PA is consistent, then PA ¥ cp'. Frankly, I object to this latter version because it misleads the reader into believing the consistency of PA to be in question. However, there is a good reason to discuss the safety assumptions: by the first Incompleteness Theorem (Le. 5(1», PA cannot directly formalise 5(i): PA ¥ ,Pr(r cpl).

MODAL LOGIC AND SELF-REFERENCE

9

What can be proven (indeed, what was the proof of 5(3)) is the implication from the safety assumption: PA I- Con(PA)

~

-,Pr(cp').

similarly, 5(3) is formalised as PA I- Con(PA)

~

-,Pr(Con(PA)').

Our second remark incorporates another approach to the problem just cited. Recall that, in the Introduction, we stated the Incompleteness Theorem in terms of 'sufficiently strong formal theories T'. The fact is, the proof of Theorem 1.5 required only that (1) PA be strong enough to carry out some encoding of syntax, and (2) PA have a decently encodable syntax. Now, the former is true of any theory T containing PA (Containment via interpretability suffices.) and the latter can be met by T's having a recursively enumerable set of axioms: If we call a theory T satisfying this recursive enumerability condition an HE theory, we obtain a rigorously stated general form of the Incompleteness Theorems: THEOREM 5'

Let T be an RE theory containing PA and let

where PrT(x) is the proof predicate for T. Then: 1. If T is consistent, T

jot

cp,

2. If T is sufficiently sound, T 3. If T is consistent, T

jot

jot

-'cp,

Con(T), where Con(T) is -,PrT

nj = 1')

In the sequel, we will primarily restrict our attention to T = PA. There is yet a third remark I want to make about the Theorem, or, rather, about the Corollary. The equivalence of any sentence asserting its own unprovability with the assertion of consistency allows one, first of all, to assert the uniqueness up to provable equivalence of such sentences and thus to refer to the sentence asserting its own unprovability. But even more important, the equivalence shows the sentence to be explicitly definable. Does this remove some of the mystery of the self-reference? The reader will recall Henkin's question about sentences asserting their own provability. Lob's Theorem and its formalisation answer this question readily: THEOREM 7 (Lob's Theorem). Let t/J be any sentence.

CRAIG SMORYNSKI

10

COROLLARY 8 (Henkin's Problem). Let PA I- cp PA I- cpo

t+

Pr(rcp").

Then:

The Corollary follows immediately from the Theorem. Proof.[of Theorem 7] We could content ourselves with a proof of 7(1) and the remark that 7(2) is just its formalisation. And we will see the inter-deducibility of these two assertions in the next section. Nonetheless, I present proofs of both results here. 1. (Lob's proof). Assume PA I- Pr(r'IjJ") -+ 'IjJ and choose cp by Diagonalisation so that

Now Dl, D2 yield PA I- Pr(r cp")

t+

Pr(rpr(r cp") -+ 'IjJ.,) -+ Pr(rpr(r cp")") -+ Pr(r'IjJ")

and D3 eliminates the redundant part to yield

From the assumption, we conclude

i.e.

PA I- cpo

Dl yields PA I- Pr(rcp"), whence (*) and modus ponens yield PA I- 'IjJ. 2. (Kreisel's proof; cf. [Kreisel and Takeuti, 1974]). The formalisation is easier with the fixed point:

(Using Dl, D2, and the equivalence of cp with a statement of the form Pr(·)) D3 yields PA I- cp -+ Pr(r cp"), which, by choice of cp and D2 yields PA I- cp -+ Pr(r'IjJ"). However, D2 and the tautology 'IjJ -+ (cp -+ 'IjJ) yield PA I- Pr(r'IjJ.,) -+ Pr(r cp -+ 'IjJ"),

MODAL LOGIC AND SELF-REFERENCE

i.e.

11

PA I- Pr(r 1jJ ') -+ cpo

Hence, cp is equivalent to Pr(r 1jJ ,) and substitution into the defining equivalence for cp (legitimate by 01, 02) yields

PA I- Pr(r1jJ') ++ Pr(rpr(r1jJ') -+ 1jJ'), which is slightly more than required.

•

I have already remarked that Lob's Theorem settles Henkin's question. Georg Kreisel has often remarked that Lob's Theorem is a generalisation of the Second Incompleteness Theorem: choosing 0 = I for 1jJ, it reads

PA I- Pr(ro i.e.

= I') -+ 0 = I *

PA I- 0 = I,

PA ¥ -,Pr(ro = I').

Lob's Theorem is, in fact, 'merely' the contraposition to GOdel's Second Incompleteness Theorem for all finite extensions of PA:

PA + -,1jJ consistent

* *

PA + -,1jJ ¥ -,Pr(r1jJ ,) PA ¥ -,1jJ -+ -,Pr(r1jJI).

Because of this proof, it has become fashionable to call Lob's Theorem the Second Incompleteness Theorem and credit it to GOdel. This is not quite fair. Where GOdel's Theorem gives the important information of the underivability of consistency, Lob's Theorem goes further and actually characterises the provable instances of soundness (Le. the truth of theorems). Although the reduction of Lob's Theorem to the validity of the Second Incompleteness Theorem in a class of theories is easy, it is by no means obvious: it is true that this proof was independently hit upon by several people (including the author in 1974), but the earliest I've been able to trace it is 1967, when Saul Kripke showed it to various people at the UCLA set theory meeting-a full 12 years after Lob's Theorem had been published. The reader-particularly the one who has filled in the missing steps wherever I wrote 'by 01, 02, ... ,'-should have noticed that, in proving all these results, we only used propositional logic, 01-03, and the existence of fixed points. In [Macintyre and Simmons, 1973], Angus Macintyre and Harry Simmons attempted to replace this last tool by some powerful principle like 01-03. The principle they hit upon was Lob's Theorem, 7(1). They showed, among other things, the equivalence of 7(1), 7(2), the existence of the fixed point CPI ++ .Pr(r CPI ,) -+ 1jJ, the existence of the fixed point CP2 ++ Pr(r CP2 -+ 1jJ '), and the respective explicit calculations CPI ++ .Pr(r1jJ,) -+ 1jJ and CP2 ++ Pr(r 1jJ I)-they showed all these equivalences using only propositional logic and 01-03.

CRAIG SMORYNSKI

12

2 THE SYSTEM PRL OF PROVABILITY LOGIC The language of modal logic consists of Propositional variables: p, q, r, . .. Troth values: T, 1. Propositional connectives: -', ", V, ~ Modal operator: D.

Modal formulae will be denoted by capital letters A, B, C, .... The system PrL of provability logic is a simulation of the proof theory outlined in the preceding section. As indicated by the results of Macintyre and Simmons, there are several possibilities for the simulation of the 'advanced' part of the theory. To enable us to discuss this easily, I shall fist introduce a neutral system of Basic Modal Logic, BML, simulating Lob's Derivability Conditions: DEFINITION 9. The modal system BML is the system of logic whose axioms and rules of inference are the following schemata: Axioms

(AI) All (Boolean) tautologies (A2) DA" D(A

(A3)

~

B)

~

DB

DA~DDA

Rules

(Rl) A,A -+ BIB (R2) AIDA.

The system BML is a known system of modal logic and is almost certainly discussed in Bull and Segerberg's chapter (of this Handbook volume 3), where it appears under a modally more familiar name (K4?). For our purposes, it is more convenient not to place it in the context of a multitude of disparate systems; to us, BML is merely a convenient background for PrL.

Before we discuss PrL, let us acquaint ourselves slightly with BML. First, a short list of useful modal tautologies: LEMMA 10. 1. BML I- D(A" B) 2.

++ DA" DB

BML I- DA V DB -+ D(A V B)

MODAL LOGIC AND SELF-REFERENCE

9. BML I- D(A

~

4. BML I- D(A # 5. BML I- 0..1

~

B)

~

.DA ~ DB

B)

~

.DA # DB

13

DA

6. BML I- -,0..1 # .DA

~

-,D-,A.

The derivations of these are routine exercises and I omit them. The converse implications of 2.2(2)-2.2(5) are not derivable in BML or PrLnor are they generally true under arithmetical interpretation. Following a list of simple tautologies is usually a proof of the Deduction Theorem. This theorem fails for BML, however, because of R2: although A I- DA by R2, we cannot generally derive A ~ DA. A good arithmetical counterexample is given by interpreting A as Con(PA): since PA I- Con(PA)

~

-,Pr(rCon(PA)'),

we cannot have PA I- Con(PA)

~

Pr(rCon(PA)'),

as this would entail PA's proving its own inconsistency. However, R2 is the only obstruction to the Deduction Theorem.

THEOREM 11 (Modified Deduction Theorem). If r is a set of sentences and there is a derivation of B from r + A over BML which does not use R2, then there is a derivation of A ~ B from rover BML which also does not use R2. The proof of Theorem 11 is a routine induction and I omit it. [Incidentally, another solution to the problem of the Deduction Theorem is to drop R2 and augment the axioms of BML by adding DA as a new axiom for every instance A of an axiom. R2 is then a derived rule of inference, but no longer an obstacle to the validity of the Deduction Theorem.] We are almost past the routine stuff. First, a useful derived rule: LEMMA 12. Let ML be any system of modal logic containing BML and closed under R2. Then: ML I- DA

~

B

=> ML I- DA ~ DB.

Proof. ML I- DA ~ DB

=> ML I- D(DA -+ B), by R2 => ML I- DDA ~ DB, by 2.2(3)) => ML I- DA ~ DB, by A3.

•

14

CRAIG SMORYNSKI

With Lemma 12, we have completed our first group of preliminaries. Our next goal is to handle substitutions. This is motivated not merely by the customary metaphysical question of substitution into modal contexts, but also by mathematical necessity: in Section 1, the steps I avoided giving in the proofs of the Incompleteness Theorems and Lob's Theorem were precisely those corresponding to such a substitution. Slicker proofs are obtained when we know how to perform substitutions. There are essentially two types of substitutions to be made-inside a modal context and outside such. The latter substitution can be handled by the usual result from propositional logic: if all occurrences of pin A(P) lie outside the scopes of boxes, then BML I- (B

t+

C) -+ .A(B)

t+

A(C).

Substitution inside a modal context will clearly require more than mere equivalence; it will require at least D(B t+ C). By axiom A3, this will be enough and substitution in general contexts will require (B t+ C) /\ D(B t+ C). Before proving this, it is convenient to introduce an abbreviating operator and list some of its properties. DEFINITION 13. The strong box [s], is defined by: [s]A

= A /\ DA.

LEMMA 14. BML(D) I- BML([s]), i.e. 1. BML I- [s]A /\ [s](A -+ B) -+ [s]B

2. BML I- [s]A -+ [s][a]A 3. BML I- A ~ BML I- [alA.

By Lemma 14, [a] is as good a modal operator as D. In particular, Lemma 10 holds with 0 replaced by [s]. Moreover, 14(3) holds for any modal logic ML containing BML closed under R2. Thus, Lemma 12 also holds with 0 replaced by [a]. The following Lemma lists a few additional properties of [a]. LEMMA 15. 1. BML I- [alA -+ A

2. BML I- [alA

t+

[a][a]A

3. BML I- D[s]A t+ DA t+ [a]DA.

The proof of this Lemma makes yet another exercise in axiom pushing for the reader. Such things are tedious, but necessary. And they do payoff; we can now prove some lemmas of substance-the Substitution Lemmas:

MODAL LOGIC AND SELF-REFERENCE

15

LEMMA 16 (First Substitution Lemma; FSL). For all A(P), B, C,

BML f- [s](B +-* C) -+ .A(B) +-* A(C). LEMMA 17 (Second Substitution Lemma: SSL). For all A(P),B,C,

BML f- D(B +-* C) -+ D[A(B) +-* A(C)]. These Lemmas are equivalent, as we shall see later. For the moment, it suffices to note that the First readily implies the Second, and then to prove the First. Proof that FSL implies SSL: Write D for B +-* C,E for A(B) +-* A(C), and notice that

BML

ffff-

[s]D -+ E, D([s]D -+ E), D[s]D -+ DE, DD -+ DE,

by by by by

FSL R2 2.2(3) 2.7(3).

Proof of FSL: By induction on the complexity of A(P).

(i)

A(P) is p: BML f- [s](B +-* C) -+ .B +-* C by 2.7(1).

(i')

A(P) is q: BML f- [s](B +-* C) -+ .q +-* q by AI.

(ii)-(iii)

A(P) is T or ..l: the proof is as in case (i').

•

(iv)-(vii) A(P) is obtained from simpler Al and A2 by means of propositional connectives: Apply the induction hypothesis and the substitution lemma for propositional calculus. (viii)

BML

A(P) is DD(P): This is the interesting case. Note f- [s](B +-* C) -+ .D(B) +-* D(C), by induction hypothesis, f- D[s](B +-* C) -+ D[D(B) +-* D(C)], by 2.2(3» f- [s](B +-* C) -+ D[D(B) +-* D(C)], (*)

by the definition of [s] and 15(3) (the use of A3 mentioned before). From (*), one additional application of 10(3) yields the desired equivalence. • We now have all the syntactic preliminaries for which we needed BML and can now consider the problem of axiomat ising the 'advanced' properties of the proof predicate. The most elegant solution uses the Formalised Lob Theorem: DEFINITION 18. The modal system PrL is the extension of BML by the addition of the axiom schema

CRAIG SMORYNSKI

16

(A4) D(DA -t A) -t DA. As proven by Macintyre and Simmons [1973], one can also use the unformalised Lob Theorem: LEMMA 19. PrL is equivalent to the system obtained by adding to BML the rule of inference: (LR) DA -t A/A. Proof. It is easy to see that Pr L is closed under LR: PrL I- DA -t A

'* PrL I- D(DA -t A), by R2 '* PrL I- DA, by A4 '* PrL I- A, by assumption DA -t A.

Conversely, let T denote the extension of BML by the addition of the rule LR. By A3, BML I- D(DA -t A) -t DD(DA -t A)

and by A2, BML I- D[D(DA -t A) -t DA] " DD(DA -t A) -t DDA.

Combining these yields BML I- D[D(DA -t A) -t DA]/\ D(DA -t A) -t DDA.

But again A2 yields BML I- D(DA -t A) /\ DDA -t DA,

whence BML I- D[D(DA -t A) -t DA] -t .D(DA -t A) -t DA.

A single application of LR yields T I- D(DA -t A) -t DA.

•

By this Lemma, the choice of Formalised or Unformalised Lob Theorem to axiomatise the more advanced results of informal provability logic (Le. the stuff of Section 1) is immaterial and we can make the choice on aesthetic grounds. We chose the Formalised version because an axiom schema is generally easier to handle model theoretically than a rule of inference.

MODAL LOGIC AND SELF-REFERENCE

17

What I have not directly addressed is the justification of basing PrL on Lob's Theorem rather than on the more obvious Diagonalisation Lemma. Systems based on Diagonalisation can be given and, proof- theoretically, they are not totally inelegant. But, it happens that they are no stronger than PrL-a fact that will require the rest of this and all of the next section to prove. That part of the proof occupying the rest of this section consists of the slicker modal derivation of the Incompleteness and Lob's Theorems accessible once the Substitution Lemmas have been established. First, the range of diagonalisation must be isolated: DEFINITION 20. The variable p is boxed in A(P) if every occurrence of p in A(P) lies within the scope of a D. (I am tempted to say 'p is boxed in in A(P).') The point of this definition is that, in arithmetic interpretations (cf. Section 5 below), the property of p's being boxed in A(P) corresponds to that of a sentence cp's occurring only in contexts of the form Pr(r ... cp .. .') in another sentence 'Ij;. In this case, we can write 'Ij; as 'Ij;(r cp') and apply the Diagonalisation Lemma to obtain a sentence cp such that PA I- cp

H

'Ij;(cp').

In other words, if p is boxed in A(P), the equivalence pH A(P)

will always be solvable in arithmetical interpretations. Hence, a modal simulation of diagonalisation must allow for solutions to pH A(P) whenever p is boxed in A(P). How do we modally simulate diagonalisation? An ugly, but workable method is to add, for each p and A(P) with p boxed in A(P), a new constant CA and axiom CA H A(CA)' A more elegant approach that is proof theoretically, if not obviously model theoretically equivalent is to treat the CA'S S eliminable, i.e. to add a Diagonalisation Rule to BML. DEFINITION 21. The modal system DiL of Diagonalisability Logic is the extension of BML by the addition of the rule of inference: (DR) [s][P

H

A(P)]-t BjB,

where p is boxed in A(P) and has no occurrence in B. The form of eliminability of self-reference, i.e. the assumption of a strongly boxed equivalence rather than a mere equivalence or boxed equivalence is explained by the FSL: in actual practice, as we shall see, we need to substitute p and A(P) for each other in general contexts.

CRAIG SMORYNSKI

18

The first major result about PrL was Dick de Jongh's proof that PrL is closed under DR, i.e. that PrL coincides with DiL. This proof was modeltheoretic and has been superseded by later developments. After reading the introduction in Section 4 to the model theory of PrL, the interested reader can consult [Smorynski, 1978] for De Jongh's original proof. By way of proving the coincidence of PrL with DiL, let me slowly show DiL to contain PrL. LEMMA 22 (Incompleteness Theorems).

(i) DiL I- [s][P ++ ,Op] A ,0.1 -+ ,Op (ii) DiL I- ,0.1 -+ ,0,0.1 (iii) DiL I- [s][P ++ ,Op] -+ .p ++ ,0.1.

Proof. (i) Assume· [s][P ++ ,Op]. Now, [Op -+ OOp] -+ [Op -+ O,p]

by the FSL. On the other hand, Op -+ Op, whence Op -+ OpAO,p -+ O(PA ,p) -+ 0.1.

Hence BML I- [s][P ++ ,Op] -+ .Op -+ 0.1, and contraposition yields

BML I- [s][P ++ ,Op] A ,Ol. -+ ,Op. (ii) Let us skip this for a moment. (iii) Obviously, BML I- Ol. -+ Op I- ,Op -+ ,0.1.

With (i), this yields

BML I- [s][P ++ ,Op]

-+ .,Op ++ ,0.1 -+ .p ++ ,0.1.

(ii) By (iii), we have

BML I- [s][P ++ ,Op]-+ .p ++ ,0.1, whence

BML I- [s][P ++ ,Op] -+ [s][P ++ ,0.1]

MODAL LOGIC AND SELF-REFERENCE

19

and we can substitute ,OJ.. for p. Do so in (i): BML I- [s][P ++ ,Op]

~

.,OJ..

~

,Op

to conclude BML I- [s][P ++ ,Op]

~

.,OJ..

~

,O,OJ...

~

,O,O ..L

A final application of DR yields DiL I- ,OJ..

• Of course, what we really want to prove in DiL is A4. LEMMA 23 (Lob's Theorem). DiL I- O(OA

~

A)

~

.p ~ Op

~

OA.

Proof. Assume [s][P ++ O(p ~ A)]. Again [O(P ~ A)

~

00(p ~ A)]

p

~

Op 1\ O(p ~ A)

by the FSL. Thus whence p~OA.

Conversely, BML I- OA Thus:

O(p

~

A), and the assumption on p yields

I- [s][P ++ O(p ~ A)] I- [s][P ++ O(p ~ A)] I- [s][P ++ O(p ~ A)]

~

.p ++ OA

~ ~

.OA

~

OA~p.

BML

.[s][P ++ OA], by 2.4 ([s])

++ O(OA ~ A), by FSL,

whence DR yields DiL I- OA ++ O(OA ~ A).

• REMARK The proofs of 22 and 23 are not really different from those of the Incompleteness Theorems and Lob's Theorem in Section 1; they are merely more explicit in their use of FSL.

20

CRAIG SMORYNSKI

3 SELF-REFERENCE IN PRL Ostensibly, the goal of the present section is the proof that PrL is closed under the Diagonalisation Rule. We will actually encounter something a bit stronger, namely the existence of explicitly definable fixed points to any legitimate self-referential equivalence p ++ A(P). That is, for p and A(P) with p boxed in A(P), we will find a sentence D such that

PrL I- D ++ A(D). This will immediately yield the closure of PrL under DR. For, applying R2 yields PrL I- [s][D ++ A(d)].

If, on the other hand, we have a proof of [s]fp ++ A(P)] -+ B with p not occurring in B, we can replace every instance of p in the proof by D and see that PrL I- [s][D ++ A(D)] -+ B. From this and (*), modus ponens yields PrL I- B. Before I show how the fixed points are constructed, let me first point out that they are unique. THEOREM 24 (Uniqueness Theorem). Let p be boxed in A(P). Then:

PrL I- [s]fp ++ A(P)] " [s][q ++ A(q)] -+ .p ++ q. Proof. Obviously, we have to apply Lob's Theorem. Thus, our goal is to derive p ++ q from D(P ++ q). Write A(P) = B[DC} (P), . .. ,DCn(P)], with p not occurring in B[q}, ... , qnl. Observe BML I- D(P ++ q) I- D(P ++ q) I- D(P ++ q) I- D(P ++ q)

-+ -+ -+ -+

D[Ci(P) ++ Ci(q)], by SSL, .DCi(p) ++ DCi(q), [S][DCi(P) ++ DCi(q)], by 2.4, .A(p) ++ A(q), by FSL.

We now drag in the fixed point assumptions to conclude

BML I- [s]fp ++ A(P)] " [s][q ++ A(q)] -+ [D(P ++ q) -+ (P ++ q)].

By 12 we can add a box to the right hand side to get

BML I- [s]fp ++ A(P)] " [s][q ++ A(q)] -+ D[D(P ++ q) -+ (P ++ q)].

MODAL LOGIC AND SELF-REFERENCE

21

Thus, A4 yields

PrL I- [s]fp

t+

A(P)] A [s][q

t+

A(q)] -+ D(P

t+

q),

which, with (*), yields the conclusion:

PrL I- [s]fp

t+

A(P)] A [s][q

t+

A(q)] -+ (p t+ q).

• The Uniqueness Theorem is due independently to Dick de Jongh, Claudio Bernardi, and Giovanni Sambin. The above proof is Bernardi's. De Jongh's model theoretic proof can be found in [Smorynski, 1978]; Sambin's rather more difficult syntactic proof appears in [Sambin, 1976]. The existence proof for fixed points, i.e. the construction of explicitly definable fixed points in PrL, is also not too difficult. It requires one clever use of Lob's Theorem to generalise Lob's Theorem, and then the rest is a simple algebraic computation. LEMMA 25. PrL I- DC(T)

t+

DC[DC(T)].

Proof. The left-to-right implication is fairly simple: BML I- DC(T) -+ .T t+ C(T), I- DC(T) -+ [s][T t+ DC(T)], (*) I- DC(T) -+ .DC(T) t+ DC[DC(T)], by FSL whence

BML I- DC(T) -+ DC(DC(T)]. For the converse implication, start with (*):

BML I- DC(T) -+ [s][T t+ DC(T)], I- DC(T) -+ .CDC(T) t+ C(T), by FSL, I- DC(T) -+ .CDC(T -+ C(T), I- CDC(T) -+ .DC(T) -+ C(T), I- DC[DC(T)] -+ D[DC(T) -+ C(T)], by 2.2(3), whence A4 yields

PrL I- DC[DC(T)] -+ DC(T).

COROLLARY 26. Let A(P) = B[DC(P)]. Then

PrL I- AB(T)

t+

A[AB{T)].

•

22

CRAIG SMORYNSKI

Proof. By the Lemma, PrL I- DCB(T) ++ DCB[DCB(T)]. Applying R2 and FSL we get PrL I- BDCB(T) ++ BDCB[DCB(T)], i.e.

PrL I- AB(T) ++ A[AB(T)].

•

With Corollary 26, we already have enough to determine the fixed points for the historically most important modally expressible instances of selfreference:

Godel's sentence. A(P) = ,Dp. Here, B(q) = ,q and the fixed point is D

= AB(T) = ,0,T = ,D.L

Henkin's sentence. A(P) = Dp. here, B(q) = q and the fixed point is D

= AB(T) = DT = T.

Lob's sentence. A(p,q) = Dp is D = AB(T)

~

q. Here, B(r)

= r ~ q and the fixed point

= OCT ~ q) ~ q = Oq ~ q.

Kreisel's variant. A(p,q) = O(p ~ q). Here, B(r) = r and the fixed point is D = AB(T) = O(T ~ q) = Oq. We can now proceed to the general case: THEOREM 27 (De Jongh-Sambin Theorem). Let A(P,q,l , ... ,qn) have only the propositional variables p, ql, ... ,qn and let p be boxed in A. There is a modal sentence D(ql,' ., ,qn) containing only the propositional variables ql, ... ,qn such that 1. PrL I- [s][P ++ A(P)] ~ .p ++ D

2. PrL I- D ++ A(D). Proof. By 24, we need only prove (2). Suppressing ql, ... , qn, A(p, ql, ... , qn) can be written in the form B[DC1 (P), .. . , OC1;(P)] , where the DCi(P)'s do not overlap and every occurrence of p lies in some occurrence of a DCi (P).

MODAL LOGIC AND SELF-REFERENCE [REMARK.

23

The decomposition is not unique. For example,

A(P) = D(D--,p V Dp) -+ Dp can be written in the form B[DCl (P),DC2(p)) with

or with

We prove the Theorem by induction on k. For k = 0, there is nothing to prove. For k = 1, we can simply refer to Corollary 26. Suppose k > 1. Let A*(p,qI, ... ,qn,qn+d = B[DCl(P), ... ,DCk-l(p), DCk(qn+d]. A* has only k - 1 components DCi (P) , and, by induction hypothesis, has a fixed point D* = D*[ql, ... , qn+l], i.e. PrL I- D*[ql' ... ,qn+d

f-t

B[DCl (D*), ... ,

Ck-l(D*),DCk(qn+d)·

(*)

Let D = D[ql, ... , qn) be a fixed point of D*[ql' .. . ,qn, qn+l] in the variable qn+l, i.e. PrL I- D f-t D*[ql, ... , qn,D]. Letting D'

= D* [ql , ... , qn, D) and replacing qn+ 1 by D in (*) yields: PrL I- D'

f-t

B[DCl (D'), ... , DCk - l (D'), DCk(D)).

Using FSL to replace D' by D in this yields PrL I- D

f-t

B[DCl (D), ... , DCk - l (D), DCk (D)],

•

i.e. a fixed point for A(P). The whole procedure behind the proof is best clarified by an example: EXAMPLE 28. Let A(P,ql,q2) A = B[DCl(P), DC2 (P)), where

= D(P -+ qd V D(P -+

(2). Then we have

By the above procedure, we replace the second occurrence of p by one of a new variable q3 and consider

24

CRAIG SMORYNSKI

We can find the fixed point to this by appeal to 3.3: A*(P) = B*[DC*(P)], where B*(r) = r V D(q3 -+ Q2), C*(P) = p -+ ql· The fixed point is

which simplifies to

We now replace the newly introduced q3 by p : A' (p, ql, q2) = Dql VD (p -+ q2) and find its fixed point. Corollary 26 readily yields D = Dql V DQ2. The De Jongh-Sambin Theorem was independently proven by Dick de Jongh and Giovanni Sambin. De Jongh's original proof was model theoretic and more difficult;· Sambin's was syntactic and complicated. The present simple version is essentially due to De Jongh. Claudio Bernardi and C. Smorynski independently proved a special case somewhat earlier and their proofs are still interesting. Cf. [Bernardi, 1975] and [Smorynski, 1979] for these. There are now a number of proofs of this theorem-cr. [Boolos, 1979; Sambin, 1976] and [Smoryriski, 1978]. Most of these proofs are model theoretic. We turn now to this model theory. 4

MODEL THEORY FOR PRL

For a full discussion of the Kripke model theory of modal logic, I refer the reader to the chapter by Bull and Segerberg (this Handbook volume 3). Here I will only describe as much model theory as necessary for arithmetical discussion. This means that (i) I will not define Kripke models in full generality, and (ii) I will not prove the basic theorems-these proofs can be gleaned from Segerberg and Bull's discussion. DEFINITION 29. A frame is a triple (K, R, ao), where K is a non-empty set of nodes a, (3, ,,(, ... (in fact ao E K), R ~ K x K is transitive (Le. for a,(3,"( E K,aR(3 and (3R"( imply aR,,(), and ao is a minimum element with respect to R (Le. for (3 E K other than ao, aoR(3). DEFINITION 30. A Kripke model is a quadruple K = (K,R,ao 11-), where K, R, ao) is a frame and II- is a satisfaction relation between nodes a and modal sentences. The assertion 'a II- A' is read either 'a forces A' or 'A is true at a' and is assumed to satisfy the following conditions: (i)

Nothing special is assumed for atomic formulae

MODAL LOGIC AND SELF-REFERENCE

25

(ii)-(iii) a II- T; a If ..L (iv)

a II- -,A iff a If A

(v)-(vii) a II- A 0 B iff (a II- A) (viii)

a II- DA iff V{3[aR{3

0

(a II- B) for

0

E {I\, V, -t}

=> (3 II- AJ.

The next Remarks collect some trivial lemmas that follow immediately from the definition. REMARK 31. (i) A forcing relation II- on a frame K, R, ao) is completely and freely determined by its decisions on the atoms. That is, any decision on the truth or falsity of atomic formulae at nodes (Le. the decision for each a and p whether or not a II- p) extends uniquely to a forcing relation II- making those same decisions. In particular, in describing a model (K, R,ao, II-) we need only specify the choices a II- p or a If p. (ii) The relation a II- A depends only on a and those {3 such that aR{3. Thus, given K = (K,R,ao,m 11-) and a E K, one can define Ka = (Ka, Ra,a, II-a) by (a) Ka = {a} U {{3 E K: aR{3}

(b) Ra = R

r Ka

x Ka

(c) II-a: For {3 E Ka,{3ll-a p iff {311- p.

Ka is a Kripke model and, for all {3 E Ka and all sentences A, {3 II-a A iff {3 II- A. (iii) 0 is persistent with respect to R: If a II- DA and aR{3, then (3 II- DA. For, let aR{3 and note that a II- DA

=> V'Y(aR'Y => 'Y II- A), => V'Y({3R'Y => 'Y II- A), => {311- DA,

by definition by transitivity by definition.

A model theory must have notions of truth and semantic consequence: DEFINITION 32. (i) Let K = (K, R, ao, II-) be a Kripke model. A sentence A is true in K, written K 1= A, iff A is forced at ao : K 1= A iff ao II- A. A set r of sentences is true in K, written K 1= r iff every sentence B E r is true in K. (ii) r semantically entails A, written r 1= A, iff, for all models K, if K 1= r then K 1= A.

26

CRAIG SMORYNSKI

The customary thing to do with formal systems and model theories is to prove completeness: THEOREM 33 (Strong Completeness Theorem). For all BML iff r 1= A.

r,A,r I-

A over

Proof. A proof of this can be gleaned from Bull and Segerberg's chapter .

•

Our interest is not in BML, however; it is in PrL. By proving a Strong Completeness Theorem for BML, one can conclude strong completeness of PrL with respect to models of PrL; in particular, one can prove weak completeness: PrL I- A iff PrL 1= A. Unfortunately, such a result is not very useful. A good model theory for a formal theory provides recognisable models. We can get something like this for PrL at the cost of the strength of the completeness result. The fact is that we can recognise the frames which always yield models of PrL, but these are not good enough for strong completeness. DEFINITION 34. A sentence A is valid in a frame (K, R, ao) if a II- A for all a E K and all models K = (K, R, ao, II-) on the frame. A set of sentences r is valid in a given frame if every sentence in r is valid in the frame. To determine the frames PrL is valid in, we simply write down what it means for O(Op -+ p) -+ Op to be valid in (K,R,ao). For notational convenience in doing this, we let R denote the converse relation to R and X ~ K any set of nodes we intend to be those at which p is to be forced. In terms of X,a II- O(Op -+ p) -+ Op iff

V.8Ra[V'YR.8h E X) :::} .8 E X] :::} V.8Ra(.8 EX). In words: A4 is valid iff transfinite induction on

R holds.

DEFINITION 35. A frame (K, R, ao) is reverse well-founded if it has no ascending sequence oflength w, i.e. ifthere is no infinite sequence aORal R .... THEOREM 36 (Characterisation Theorem). The frames in which PrL is valid are precisely the reverse well-founded frames. I have already indicated why this is true. There will probably be some disagreement as to whether or not this indication constitutes a proof; but I shall let it go at that. As already remarked, what is really needed is a completeness theorem. A filtration argument (cf. Bull and Segerberg's chapter) yields the following: THEOREM 37 (Completeness Theorem). For any sentence A, the following are equivalent:

MODAL LOGIC AND SELF-REFERENCE

27

1. PrL f- A

2. A is valid in all (finite) reverse well-founded frames 9. ao II- A for all models K = (K, R, ao, 11-), with (K, R, ao) a (finite) reverse well-founded frame

4. ao

II- A for all models K = (K, with root ao.

Qo, from ao II- DA it follows that {3 II- A. Thus, Q-I II- DA. Now, if PrL I- DA ~ DB, it would follow that Q-I II- DB, and hence that Qo II- B, which is false. Thus, from the underivability of [s]A ~ B we conclude that ofDA~ DB. •

MODAL LOGIC AND SELF-REFERENCE

29

5 ARITHMETICAL INTERPRETATIONS The reader will recall the raison d'etre of PrL, namely the desire to analyse the predicate Pr(x). To see how good an analysis PrL offers, we introduce the notion of an arithmetical interpretation of PrL and cite some completeness theorems therefore. DEFINITION 41. An arithmetical interpretation * of the modal language is an assignment of arithmetical sentences A * to modal formulae satisfying the following: for all A, B,p, (i)

if p is atomic p* is a sentence of the language of arithmetic

(ii)-(iii)

T* is 0 = OJ ..L * is 0 = 1

(iv)-(vii)

* respects propositional connectives

(viii)

(DA)* = Pr(r A*").

As was the case with Kripke semantics, this is an inductive definition and an interpretation * is uniquely determined by the assignment p t-t cp of sentences to atoms. As our choice of axioms and rules for PrL was based on metamathematical knowledge of PA, it should surprise no one that the interpretation is sound: THEOREM 42 (Soundness Lemma). For all modal sentences A, PrL f- A

~

V*(PA f- A*).

The proof by induction on the length of a derivation in PrL is omitted. As I said, PrL was defined in such a way as to guarantee soundness. The hope, however was to analyse Pr(x) in the sense of actually axiomatising the schemata valid under all interpretations. In [Solovay, 1976], Robert M. Solovay showed that PrL does this: THEOREM 43 (Solovay's First Completeness Theorem). For all modal sentences A, V*(PA f- A* ~ PrL f- A. Solovay's Completeness Theorem is of tremendous importance in that it tells us that PrL axiomatises the schemata provable about Pr(x) in PA. It is, however ,not of much use. Of course, it does yield some information: we can, for example, get a quick new proof that PrL is closed under the Diagonalisation Rule DR of 21 (Exercise). But this is less than the actual calculation. More useful is Solovay's Second Completeness Theorem.

CRAIG SMORYNSKI

30

DEFINITION 44. By PrL + we mean the system of modal logic whose axioms consist of all theorems of Pr L and the schema of Reflexion, (Rfn) DA -+ A,

and whose sole rule of inference is Rl. DEFINITION 45. Let A be a modal sentence. We define SeA)

= {C : DC is a subformula of A}.

THEOREM 46 (Solovay's Second Completeness Theorem). The following are equivalent: 1. A· is true for all interpretations

*

3. PrL I- ACES(A)(DC -+ C) -+ A.

What does this say? By Lob's Theorem, the only provable instances of soundness, are those trivially so:

PA I-

k > i, since otherwise the principle would be violated by the same sequence of inferences in the unauthorized iteration: TRUEmrTRUEmrpROVABLEir sub(n)'''grPROVABLEir sub(n)" = n" The conclusion in (6) is not logically inconsistent, but states instead that while it is true that the GOdel sentence is true in one system, it is not true in another. This parallels the solution to arithmetized liar sentences in a Tarskian semantic hierarchy. The distinction between implicit and explicit semantic predication iterations works in analogous ways for the arithmetized liar. The implicit truth iteration of the GOdelized liar has this form in the first horn of the liar dilemma: ====~i

i

TRUEmrTRUE r sub(n)'" grTRUE r sub(n)" = n' The explicit iteration is reformulated from this sentence when the substitution function sub(n) is appropriately engaged:

Unpacking sub(n) in further explicit iterations of the liar sentence continues indefinitely in principle for suitable semantic indices i, k, m, ... , n. Similarly, in the second horn of the liar paradox for semantic predication TRUEmrL '. The GOdel sentence in the inconsistency half of this version of the incompleteness-inconsistency dilemma requires the GOdel sentence if false to be true on the assumption that it is provable. The inference holds, but by generalized Tarskian semantic stratification, the formal system in which the Godel sentence is assumed to be false is not the same formal system in which it can be shown to be true when its falsehood implies its provability and its provability implies its truth.

DIAGONALIZATION IN LOGIC AND MATHEMATICS

B.9

125

Rosser's Conditional AntiprooJ Metatheorem

The second version of the dilemma follows from Rosser's method of setting upper bounds on the lengths of conditional antiproofs in GOdel-style unprovability diagonalizations. It is easy to adapt the integrated Tarskian truth and provability stratification to neutralize Rosser's metatheoretical dilemma. To begin, notice that there is only a difference in complexity between the Rosser and the original GOdel sentence, but that both apply ordered truth predicates to constructions containing ordered provability predicates. This is all that is needed to adjust the previous solution to Rosser's theorem. The resulting formulas are somewhat more complicated in internal content, but the essential limitation of applying truth or provability predicates of a certain order only to constructions containing semantic predicates of lower order remains the same. The application to the Rosser formula can now be exhibited, and the metatheoretical dilemma neutralized similarly as above. On the first horn of the Rosser dilemma, informally, if PROVABLEr R", then PROVABLEr R". In the generalized Tarskian semantic hierarchy, this amounts to: [PROVABLEmrPROVABLEir sub(n)"-t[3xfgrpROVABLEi r sub(n)" = x A sub(x) A x ~ n] A grpROVABLEir sub(n)" = n"]-t [PROVABLEmr -,PROVABLEir sub(n)"-t[3x[grpROVABLEi r sub(n)" = xA sub(x) A x ~ nlJ A grpROVABLEir sub(n)" = n"] The conditional, required to prove Rosser's theorem in a Tarskian semantic hierarchy, is by no means logically true. This is best seen by asking within the generalized Tarskian semantic framework whether it follows from the construction of sentence R and the assumption that R is m-provable that the negation of R is also m-provable. Of course, the inference does not follow. The reason is that what R now says in the Tarskian hierarchy is that if R is i-provable, then R is at least as economically i-provable. The m-provability of Il by hypothesis, on the other hand, cannot be deduced from the i-provability of Il, where m > i. From the m-provability of the i-provability of R, the i-provability but not the m-provability of R only can be derived by the diagonal construction of R. The result is that there is no valid channel from Rosser's sentence R to either PROVABLEmr R" A PROVABLE mr ll" or to PROVABLEir R" A PROVABLEir ll", but at most only to PROVABLEmr R" A PROVABLEi

rll".

It might be objected that appeal to m-provability is unnecessary in the first horn of Rosser's metatheoretical dilemma, that it obtains directly on the assumption that PROVABLEir R". This move merely leads us back to the implicit-explicit iteration of semantic properties ambiguity that hounds

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arithmetized diagonalizations. Since R itself contains ordered provability and unprovability predications, the attachment of the 'PROVABLEi ' predicate to 'R' is permitted in the generalized Tarskian framework only if i is higher-order than the provability and unprovability predications that R contains. Similar considerations defeat the second horn. As one exit from the dilemma is sufficient, the remaining part is left as an exercise.

8.10 Godel's w-Consistency Dilemma The w-consistency version of GOdel's inconsistency-incompleteness dilemma is equally amenable to the neutralization strategy. The first horn is immediately blocked in the same way as the Rosser theorem, since only the second horn requires the w-consistency assumption. Here again, the needed inference from PROVABLErG' to PROVABLE r G' is invalidated by the stipulation that every truth or provability predication be hierarchically stratified, semantic predicates of a given order attaching only to conStructions containing semantic predicates of lower order. Since 'G' contains the unprovability predicate 'PROVABLE', the assumption that G is provable must be formulated this way:

From this, however, it does not follow that:

The inference is blocked as previously shown even on the soundness or intrasystemic provability => truth assumption for the third version of the metatheoretical dilemma, and cannot be deduced without it. The second horn of the dilemma in Godel's original derivation is more interesting, since it requires the w-consistency assumption. The informal exposition of this part of Godel's theorem states that if PROVABLEr G', then there is no GOdel number of any proof of G; that is, PROVABLEr grG" =IoA PROVABLEr grG" =I- 1 A PROVABLErgrG" =I- 2 A ... . By wconsistency, it follows that PROVABLEr 3xf.qrG' = xl'. This contradicts the assumption, because, where G asserts its own unprovability, PROVABLE rG' by GOdel-coding entails PROVABLErpROVABLErG'AgrG' = PROVABLEr existsx[grG' = xl', which by conjunction elimination and existential generalization implies PROVABLE r 3x[grG' = xl'. Now the solution is at once apparent, since the contradiction is avoided when the sentence PROVABLErQl is reformulated under generalized Tarskian constraints as:

DIAGONALIZATION IN LOGIC AND MATHEMATICS

127

This in turn implies:

However, this sentence does not contradict the said implication of the w-consistency assumption in the second hom of GOdel's dilemma, which in this ordering states:

It follows that neither half of GOdel's metatheoretical dilemma holds if the generalized Tarskian hierarchy of truth and provability predicates is enforced.

8.11

Toward an Integrated Hierarchical Semantics for Standard First-Order Logic with Arithmetic

The consequences of forestalling GOdel's limiting metatheorems in firstorder logic are likely to seem liberating or disorienting, depending on one's philosophical and mathematical temperament. To be complete and decidable, it should be emphasized, does not mean to be completed and decided. Thus, unproven and undisproven propositions like Goldbach's conjecture and the status of the Continuum Hypothesis and Generalized Continuum Hypothesis need not threaten the decidability of standard first-order logic with arithmetic. 58 It might be observed that a generalized Tarskian semantic hierarchy, integrating ordered truth and provability predications, merely accomplishes what GOdel himself postulates in the second main metatheorem, where he proves that no logically consistent formal system of requisite complexity can 58Godel remarks in 'On Formally Undecidable Propositions', 614, n. 61: 'Theorem X implies, for example, that Fermat's problem and Goldbach's problem could be solved if the decision problem for the r.f.c. [restricted functional calculus] were solved.' The implication might be understood to be that the undecidability of first order logic with arithmetic established by Gooel's arithmetized diagonalizations somehow explains the facts that Fermat's problem and Goldbach's conjecture are so far neither proven nor disproven. There is nevertheless no direct connection between unsettled hypotheses of ordinary mathematics and Gooel's proof, just as there is no immediate connection between the Heisenberg uncertainty principle in quantum mechanics (with which Godel's limiting metatheorems are often compared) and unsettled philosophical problems about free will.

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DALE JACQUETTE

prove its own consistency. The evident implication is that this can only be accomplished in a higher-order metatheory. In an addendum to his proofs, "On Completeness and Consistency", GOdel writes: ... the undecidable propositions constructed for the proof of Theorem 1 become decidable by the adjunction of higher types and the corresponding axioms; however, in the higher systems we can construct other undecidable propositions by the same procedure, and so forth. To be sure, all the propositions thus constructed are expressible in Z (hence are number-theoretic propositions); they are, however, not decidable in Z, but only in higher systems, for example, in that of analysis. 59 The projection of a hierarchy of metaianguages, each of which can prove undecidable sentences and hence the syntactical consistency of subordinate systems, is structurally similar to the generalized Tarskian semantic hierarchy. There are nevertheless important differences. The Tarskian ordering of formal systems is not motivated by, but ultimately prevents, intrasystem undecidability. The generalized Tarskian semantic hierarchy integrates ordered truth and provability predications in a language and metalanguage stratification that avoids both incompleteness and inconsistency horns of GOdel's metatheoretical dilemma. It is not that appeal must be made to higher-order systems in the hierarchy to decide undecidable sentences in or demonstrate the consistency of subsystems, but rather that by the generalized Tarskian stratification arithmetized GOdel diagonalizations are not undecidable, and cannot sustain any of the three versions of the metatheoretical dilemma. GOdel's arithmetization avoids Russell's syntactical hierarchy, but not Tarski's semantic hierarchy. What, finally, if anything, might be said to justify the generalized Tarskian semantic hierarchy? The answer is the same as for the Tarskian semantic hierarchy of truth predications. Tarski's truth hierarchy is postulated for no other reason than as a preventive measure to avoid semantic paradox. The generalized integrated Tarskian semantic hierarchy of truth and provability predication types similarly has as its philosophical rationale the desire to avoid not formal inconsistency alone, but the dilemma of incompleteness or syntactical inconsistency. GOdel's proof demonstrates the previously unappreciated need for a generalized truth and provability hierarchy, just as 59GOdel, "On Completeness and Consistency", 617. See Godel, 'On Formally Undecidable Propositions', 599: 'From the remark that [R(q)j qj says about itself that it is not provable it follows at once that [R(q)j qj is true, for [R(q)j qj is indeed unprovable (being undecidable). Thus, the proposition that it is undecidable in the sI/stem PM still was decided by metamathematical considerations. The precise analysis of this curious situation leads to surprising results concerning consistency proofs for formal systems, results that will be discussed in more detail in Section 4 (Theorem XI).'

D1AGONALIZATION IN LOGIC AND MATHEMATICS

129

the liar paradox demonstrates the need for a Tarskian stratification of truth predications. The principle that provability in a formal system implies truth in the same system also supports the generalized Tarskian truth and provability hierarchy. Consider the statement that it is provable that proposition p is true, PROVABLEkrTRUEirp II. If provability in formal system i implies truth in i, then there would be a direct violation of Tarski's original truth predication hierarchy unless k -I i, since it follows by the intrasystemic provability => truth principle that TRUEkrTRUEirp". The generalized integration of typed or ordered truth and provability predications is therefore not only consistent with Tarski's semantic stratification, but necessary in order to preserve it. The effect of an attempt to undermine GOdel's incompleteness results is mitigated by the consideration that the proposed neutralization in no way diminishes GOdel's contribution in achieving important new perspectives in mathematics. The arithmetization of syntax is not sacrificed, but remains a powerful metamathematical tool. The proposal more importantly does not overturn GOdel's historical success for his specific target of defeating the logicist theory of arithmetic in Principia M athematica and related systems, though it may somewhat narrow the range of formal languages to which the incompleteness results apply. It is usually supposed that the metatheorems prove the incompleteness of any and all standard first-order arithmetics. If the above neutralization proposal is essentially correct, however, then the limitation holds only for systems of logic that do not avail themselves of a generalized Tarskian hierarchy of truth and provability predicates. Gadel poses a challenging dilemma for standard first-order logic. Here we have considered another kind of dilemma for GOdel's proof. If GOdel's limiting metatheorems are not constructed within a generalized Tarskian semantic hierarchy, then his conclusions are vastly understated, and Godel arithmetization can be wielded not only to force a choice between incompleteness and inconsistency, but to establish by the arithmetized liar the outright formal inconsistency of any sufficiently powerful logic. If, on the other hand, GOdel's proofs are attempted within a generalized Tarskian semantic hierarchy, then the theorems simply fail, and all three versions of the incompleteness-inconsistency dilemma are neutralized. In this application we see exactly how diagonalization in the self-non-application sense of Godel's, Rosser's, and Church's theorems function syntactically, and how, in view of the self-non-application/infinite regress polarity, they can be dissolved. We thereby learn something valuable about the power and limitations of diagonalized metatheorems in mathematical logic and their interrelation with particular forms of infinite regress.

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130

9

WITTGENSTEIN'S CRITIQUE OF DIAGONALIZATION

It is appropriate to continue the philosophical appraisal of diagonalization with Wittgenstein's critique. There have been few logically and mathematically trained writers as consistently critical of diagonalization methods as Wittgenstein, both in his early and later work. By considering Wittgenstein's arguments against diagonalization, we gain a better appreciation for its unique contributions to formal reasoning. We have seen above that Russell regarded Wittgenstein's mathematical logic in the Tractatus as incomplete by virtue of failing to accommodate transfinite numbers. Russell also remarks that he sees no insuperable obstacle to Wittgenstein's adding the necessary axioms to make his system compatible with transfinite 'classical' mathematics as Russell conceives of it. Wittgenstein, it may be needless to say, would certainly not have allowed this kind of 'improvement' to his theory to extend its principles to higher orders of infinity. If it is true that there is a common underlying diagonalization structure in the construction of Cantor's proof for the existence of transfinite numbers and for self-non-application paradoxes like the liar and Russell paradox and their cousins, then the fact that Wittgenstein in Tractatus 3.333 dismisses self-applicational and self-non-application constructions alike as violating the picture theory of meaning and thereby disposes of the need for a Russellian theory of types should equally disqualify the possibility of Cantor's diagonalization. The picture theory prohibits self-applications and self-non-applications alike. It also thereby precludes literal self-intra-substitution of the sort Smullyan discusses. Wittgenstein's requirements for a correct logical notation in its nonperceptible symbolic aspect make it incoherent for a sign to contain itself, let alone to say anything about or predicate or deny any property of itself. The passage in its entirety makes this point evident when Wittgenstein declares: 3.332 A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself. 3.333 If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition "F(F(fx))", and in this the outer function F and the inner function F must have different meanings; for the inner has the form (fx), the outer the form 'IjJ((fx)). Common to both functions is only the letter "F", which by itself signifies nothing. This is at once clear, if instead of "F(F(u))" we write "(3 is both true and false. For example we may only admit judgement of things that are uniquely true. Another possibility is that we employ an extension of van Fraassen Supervaluations. For further information see Priest [1969; 1984]. Dowden [1979; 1984], Woodruff [1984]. Visser [19841 Failure of -,E does not only occur in paraconsistent treatmenL M, then, for all tPl (x, X), ... , tPk (x, X) there are tlh (X), ... , tPk (X) such that for i = 1, ... , k:

tPi(X) ~s tPi(X,{Xl I tPd,···,{Xl I tPk}).

6.3

Examples

Examples of fixed points for stipulations are given in figure 1. The following fact provides an example for of a fixed point satsfaction.

196

ALBERT VISSER

FACT 38. Let M be as usual. Let P be any partial valued subset of M, i.e. P : M -+ {T, F, JL}. Then there is a fixed point 8 of ~M and a fonnula 'I/J(xd having just Xl free, such that'I/J represents P in M (8), i.e., for every n in M and any g, P(n) = ['I/J(n)](M(X),g).

Proof. Consider the fonnula (Xl E X) of C(X). Let'I/J be the corresponding diagonal sentence constructed in Lemma 36. We have: (Xl E {Xl I Sat(X2, (XI,X2»)}) Sat(X2, (Xt,X2», r X", Sat(m, (Xl, m». Define:

8( r).-{ P(n)ifq=m,r={n,m) p q, .JL otherwise.

We have: ~M(8p)(m, (n,m»

= [X](M(8p ), h(n,m» = 8p(m, (n,m».

Hence: 8p ~ ~M(8p). Applying the function O;M to 8 p, we obtain a fixed point 8 of ~M which is minimal among the fixed point'> extending 8p. We find:

P(n)

=

=

8p(m,{n,m» ~ 8(m,{n,m» [Sat(m, (n,m»](M(8),g) ['I/J(n)](M(8),g).

To see that identity holds, consider:

P(n) if P(n) is T or F P + with P + (n):= { Totherwise

and

P- with P-(n) := { P(n) if ~(n) is T or F F otherwIse

Define 8 p+, 8 p- similar to 8 p above. Just as for 8 p, there are minimal fixed point'> 8+ and 8- extending 8 p+ and S p- respectively. We have: 8 ~ 8+, 8 ~ S-. So, if P(n) is undefined, we have:

s(m, (n,m» ~ 8+(m, (n,m» = T 8(m,{n,m» ~ 8-(m,{n,m» = F. Hence, 8(m, (n,m»

= JL. We may conclude that 'I/J(xd represents P in M(8) .

•

Fact 38 shows that '['M can be embedded in Fix(M). As a consequence 1'(M) and lR = (R,~) can be embedded in Fix(M).

SEMANTICS AND THE LIAR PARADOX

6.4

197

Comparing Strong Kleene and Van Fraassen

We just consider stipulations, Sat being analogous. How are we to compare Strong Kleene to Van Fraassen fixed point'i? One idea is to look at natural ways to go and VF := (-)FVF. Clearly, from one kind to the other. Consider SK := (-)FSK 8 8 F~K ~ FXF. Let f be a strong Kleene and 9 be a Van Fraa'isen fixed point. By Theorem 28(1), f ~ FXF(f) and 9 ~ F~K(g). Hence, by Remark 17: VF(f) is the minimal Van Fraassen fixed point extending f, and SK (g) is the maximal Strong Kleene fixed point below g. By Theorem 27(2), SK and VF are monotonic respectively from FixVF(S) to FixsK(S) and from FixSK(S) to FixVF(S). Moreover, as is easily seen, if f is a Strong Kleene fixed point: f ~ SKoVF(f). So, SK, VF, FixsK(S) and FixvF(S) satisfy the conditions of Theorem 27. Call a Strong Kleene fixed point f stable if f = SK 0 VF(f). Call a Van Fraassen fixed point 9 stable if 9 = VF 0 SK(G) The structures of these stable fixed point'i are: FixSK(S) :=Fix(FixSK(S),SK 0 VF) Fix VF (S) := Fix(FixvF (S), VF

0

SK)

By Theorem 27(2), FixSK (S) and Fix VF (S) are isomorphic. By Theorem 27(3), Int(FixSK(S)) = Int(Fnc8 K(S)). In other words, the intrinsic Strong Kleene fixed points are precisely the intrinsic stable fixed points. By Theorem 27(4), the minimal Van Fraassen fixed point is stable. EXAMPLE 39. Let S be:

f: of a : a V (f V -,f)

b : c t\ (f V -,f) c: b V -'c. We have: FixSK(S)

FixvF(S)

r r'c

FixSK(S) FixVF (S) oa

So, the maximal intrinsic VF fixed point is not stable, nor is the minimal SK fixed point stable.

6.5

Fixed Point Valued Semantics

We develop Fixed Point Valued Semantics here for the case of Stipulations. The ca'ie of Sat is fully analogous (except for the problem ofaxiomatisation). We treat

ALBERT VISSER

198

just the SK case. From the technical point of view, there is no obstacle to do the VF case, philosophically, however, it is very awkward: doubling an idea as it were. As will be illustrated, Fixed Point Valued Semantics does not evade the problem of choice betwen fixed points: one can always consider interesting substructures of fixed points to build the meaning values. := ({T, F, JL},!;;;) where!;;; is given by: Let

r

r

Clearly is a complete lattice. Let A be a set and let S be a stipulation list on A. Consider the following possible set'i of meaningvalues:

{w I wmonotonic from FixSK(S) to 1!'} {w I wmonotonic from Int(FixSK(S)) to 1!'} {w I wfrom {m I m is maximal in FixSK(S)} to {T,F,JL}}.

VF := V1-:= VM:=

(Here: 'F' stands for 'Full', '1' for 'Intrinsic', 'M' for 'Maximal', 'V' for'VaIues'.) Let r be in {F, I, M}, let!;;; be the order on Vr induced by!;;; ofT+ , i.e. W !;;; w' iff, for all f in the appropriate domain, W(f) !;;; W' (f). Define r-val(S) := (Vr. !;;;). As is ea'iily seen, r-val(S) is a complete lattice. We write '[4>]r' for the element ofVr. such that, for all f of the appropriate domain, [4>]r(f) = [4>]SK f. EXAMPLE 40 (Coloured samesayers). Samesayers receive different meaningvalues by having different 'colours'. E.g. let S be: a : a, b : b. Then, [a]r i- [b]r. This illustrates that meaningvalues are sensitive to aparently meaningless details of syntax. Let r, ~

~

C(A), define (using the r-va1 ordering):

Call the domain of tP E Vr : Dr. We have:

r

Fr

~

iff, for all f in Dr. nHtP]rf I tP E r}(r) !;;; UHt/J]rf I t/J E ~}(1!'+) iff, for all f in Dr if, ( VtP E r [tP]SK f = T ) =} ( 3t/J E ~ [t/J]SK f = T ) { and if, ( Vt/J E ~ [t/J]SK f = F ) =} ( 3tP E r, [tP]SK f = F ).

SEMANTICS AND THE LIAR PARADOX

As is ea'iily seen:

199

r FF ~:::} r FI ~ :::} r FM ~.

EXAMPLE 41. Let S be: a: ,b

b: ,a c: d /\ (c V ,c) d: d. Then, d F I C, but d I1F c. Moreover, F M ,(a /\ b), but 111 ,(a /\ b). Thus, F M captures our tendency to say of a and b, that they can't both be true.

Axiomatisation

Given S, can we axiomatise 1= F? Consider the relation I- F generated by: (i) the rules of propositional partial valued logic, (ii) a -II- F cf>, for a E A, where cf> = S(a). (Here, X -II-F e, stands for: X I-F e and e I-F X). A partial valuation f is in Fix( {T, F, JL}A, F~K) precisely if, for all a E A and cf> = S(a), we have [a]1K !; [cf>l1 K and [7r]1 K !; [a]SK f, in other words, precisely if [a]SK f = [cf>]SK f. Hence, a partial valuation f is in Fix( {T, F, JL}A, F~K) precisely if it satisfies a -II-F cf>, for all a E A and cf> = S(a). Applying the completeness theorem for propositional partial logic, we find I- F = 1= F. PROBLEM 42. Axiomatise 1= I and 1= M. To get a similar result for Sat we must adapt our approach by also considering 'non-standard acceptable structures'. The result will then be that the proper axiomatisation (for the F-case) of Sat extending a given theory having only nonstandard acceptable structures as models, is of the form:

7

THE REVISION PICTURE OF TRUTH

Gupta and Herzberger independently discovered the idea of treating the paradoxes using iterations of classical structures. Suppose that Socrates says Aristotle speaks falsely and suppose that Aristotle says Socrates speaks truly. Suppose further that these are the only things they say. Call the above sentences respectively S and A. Clearly if S is true, A is true and then S must be false. S's falsity will make A false. So S becomes true. Etc. Of course, we can formalise the above reasoning in predicate logic and arrive at a contradiction. But let us view what is going on more semantically: a'i the unwinding of our seman tical intuitions. The rea'ioning can be represented a'i follows:

200

ALBERT VISSER

S

A

This representation is not yet the kind of process Gupta and Herzberger consider. Their idea is that a stage is like a classical world, a world in which the truth predicate is already fully evaluated-either by stipulation or by inheritance from earlier stages. So we should start e.g. by taking for example S true and A false. We get:

S

A

Note that the re-evaluations occur with the ticks of an inexorable clock for all sentences at the same time. Let Lo be Lo is false and let Ll be Ll is false. Gupta/Herzberger unwindings are e.g.:

Stage 0

Lo T

1

F

2

T

3

F

.J. .J. .J.

Stage 0

Lo T

1

F

2

T

3

F

Ll

F

.J.

T

.J. and

F

.J.

T

.J. .J.

.J.

Ll

T

.J.

F

.J.

T

.J.

F

SEMANTICS AND THE LIAR PARADOX

201

But not the following 'Einsteinean' process Stage 0

Lo T

Ll

.l.

1

F

2

T

3

F

Stage 0'

.l.

l'

.l. 2'

A consequence is that in Gupta/Herzberger process the sentence C := "Lo is false

f+

Ll is false"

has a stable value: it is either 'alwll.Ys' true or 'always' false. A first pressing question is: what to take as initial stages? Herzberger seems to have some preference to start with taking everything false -or at least this choice seems most coherent with his limit rule. See below. Such a choice could be defended by claiming some fundamental asymmetry between truth and falsity, e.g. because falsity is just a lack of truth, i.e. falsity is really the gap. Gupta takes a different track: all choices have equal status. We should consider all possible initial stages and, subsequently, 'quantify out' the arbitrariness of choice. One could view the initial stages as fully classical worlds that one puts 'experimentally' for one's mind's eye. To get anything like a realistic picture we should admit transfinite stages. Consider for example the sequence: '1'0 := "Snow is white" tpn+l := r"CPn" is true" tpw := "For all n, tpn is true".

A Gupta/Herzberger process for the tpi looks as follows 14: tpw

Stage 0 1

2 3 4

F F

F F F

14We give the extension of the truth predicate in each stage: the declared truths and falsities, not what is true or false at that stage considered as a model. This is'the reason that !Po can be false in stage 0: we start with nothing in the extension of the truth predicate.

202

ALBERT VISSER

At all finite stages , we are not saying that it is true in some fictional text that he believes that cI>. What is intended by the I -belief that cI> is the belief that 1(cI». It is a kind of short-hand. Similarly, the I-truth that cI> is the truth that I( cI». The same hold for I-facts.

THE LOGIC OF HCTION

275

One of the objectives of the Sperber-Wilson restrictions is to keep inference a finite accomplishment. A like reason clearly applies here. It is counterintuitive to suppose that, in telling a story of any kind in any way that stories can be told, the author is trying to get his readers to believe infinitely much, still less that the story in question is infinite. The restrictions are also aimed at discouraging irrelevancies from creeping into a rea~oner's inference. At the deductive level, introduction rules give obvious offence. Maximum account~ call for the same kinds of restrictions. It is undesirable to have irrelevances verified by an author's efforts, but precisely this will happen if we give introduction rules frec rein. A case in point is the following fictional adaptation of the classical theorem ex [also quodlibet, which asserts that any inconsistency entails any proposition whatever. Let {A, r -,A ,} be the inconsistency concerning Keith's election to the American presidency in the Bradbury story mentioned in Heintz [1979]. 1. A sayso 2. -,A sayso 3. A V B 1, V-introduction 4. B 3,4, disjunctive syllogism If V-introduction is a~sumed for the maximum account of Bradbury's story, then it will contain every proposition whatever, which very clearly was not his intent. 44 In applying their constraint", Sperber and Wilson seek to model the human cognitive agent a~ a maker of inferences that are both finite and relevant. Concerning this last point, it is also clear that the constraint that helps discourage the inference of irrelevancies also - when adapted to the problem of constructing maximum account~ - discourage the admittance of true propositions made true in ways other than their membership in the given maximum account. Of course, putting the point in just these words is circular. But the implied recursion is not hard to make explicit if necessary. More pressing in the question of how aggressive we should be in excluding such extraneous propositions. Bearing on this in a central way is the question of the incompleteness of fictional entities. Those, such a~ Parsons, who see the fictional as intrinsically incomplete are influenced by the following pair of fact~. One is that for all sort~ of propositions CP, neither it nor it~ negation is in the author's text or it~ deductive closure. Such propositions include those that say that Holmes had (or hadn't) a mole on his shoulder, that Dr. Watson habitually shaved (or didn't) in his pyjama~, or that, among the people mentioned in Tolstoy's fictive sentence, "When the train came into the station, Anna got out into the crowd of passengers, ...", one of them wa~ (or was not) named "Pyotr". If it were correct simply to identify the maximum account of a story with it~ own deductive closure, this view of incompleteness would also be entirely correct. It is ea~y to see, however, that even where the deductive rules are held to restrictions in the manner of Sperber and Wilson, this represent~ a considerable shortfall from what a maximum account more plausibly 44 For doubts about whether these measures suffice for finite output, see Levinson [2O