Vann McGee Truth, Vagueness, And Paradox an Essay on the Logic of Truth 1990

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1.ihrarj of Congres\ ('ataloging-in-Publicatio~~ Data hlc(icr. Vann. 1040I'ruth. vayuene\\. and paradox. an e\\;I) o n the logic o f truth' Vann McGee. Pcn1. Itlclude\ hibliogr;tph~c;lI references. ISBN 0 87110-087-6 (all\. paper) ISBN 0-87220-086-8 ipbk. 1 I . Truth 2. Krterence tPhilo\ophq) H1)171 . M i 7 1900 160-dcl0

0. I. 2. 3. 4.

5. 6. 7. 3 . Liar pat-adox

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Preface Our Project Formalized Versions o f the Semantic Antinomies Logical Necessity Tarcki's Solutions to the Liar Antinomy Kripke and 3-valued Logic Kripke's Construction and the Theory of Inductive Definitions Rule-of-revision Semantics Partially lnterpreted Languages Truth in Partially Interpreted Languages Definite Truth in Partially lnterpreted Languages Toward a Semantics of Natural Language Bibliography Index

vii 1 18 31 67 87 107 127 148 158 196 209 223 23 1

Preface To my parents, with love and gratitude

This book is an investigation into the logic of truth. The investigation is provoked by the liar paradox, which shows that our naive understanding of truth, which is characterized by the acceptance of Tarski's schema (T)

r$7

is true if and only if $

is inconsistcnt. The aim of the investigation is to develop a new understanding of truth that does not fall prey to contradictions. There are scarcely any philosophical problems of greater urgency than the liar paradox, for there are scarcely any concepts more central to our philosophical understanding than the concept of truth. The notions of truth and reference lie at the very center of all our attempts to understand how our language is linked to the world around us. These are the notions we need to use if we want to understand the astonishing fact that my utterance of the sentence 'The Yuan emperors ruled harshly' is son~ehowintimately connected with events that happened seven hundred years ago half a world away. The liar antinomy and the closely related antinomies involving reference show us, quite unmistakably, that our present way of thinking about truth and reference is inconsistent. Unless we can devise new ways of thinking about truth and reference which rise above the antinomies, we shall not have even the beginning of a satisfactory understanding of human language. We want to replace our naive conception of truth by a scientific conception that serves the same purposes without falling prey to inconsistencies. The relation between our old and new conceptions of truth will be the same as the relation between our old, prescientific understanding of space and time and the understanding of space and time that we get from modern science. Where do we begin'? Schema (T) is so deeply embedded in our ordinary thinking about truth that we might fear that, once we decide to give (T) up, we should become so badly disoriented that we would not be able to talk about truth at all. A starting point is provided by some advice of Wittgenstein. In trying to understand a philosophically troublesome concept, do not focus all your attention upon how the concept behaves when it is on philosophical holiday. Pay attention to the everyday, unproblematic, nonphilosophical work the concept does. When we look at the nonphilosophical work done by the concept of truth,

vii

what strikes us most proniincntly is that we can use the notion of truth in order to endorse or to deny a statement o r set of statements without being required actually to repeat the statements; it is enough that we be able to namc thc statements. Thus. if l say Every

el- c,crtlzcdr.cr pronounccmcnt

of the Pope is true

1 have endorsed all of the r.1- (,crthcdt.rrpronouncement.; of the Pope. I have. in a sense. asserted the conjunction of all the Pope's o.r c~rthrrli-rrpronouncements. Without employing the notion of truth. I could not do this. for 1 surely cannot repeat 1111 of the Pope's pronouncements. Using thc notion of truth in the ordinary way. we are able. in effect. to produce the conjunction or thc disjunction of an arbitrary named set of sentences. Enabling us to d o this is essential to the nonphilosophical usefi~lnessof the ordinar) notion of truth. and if our scientifiu l that cally reconstructed notion of truth is to continue to perform the ~ ~ s e f work our ordinary notion performs. then it. too. must enable us, in effect. to for111 conjunctions and disjunctions of named sets of sentences. Although this requirement by no means uniquely determines our new theory of truth. it tells us a great deal about what the new theory ought to look like. tiere I would like to develop a specilic proposal for a way of thinking about truth which will. I hope, preserve those logical features which make our present notion of truth so singularly useful as a practical means for conveying information. yet avoid the contradictions that niake our present notion of truth s o singularly unsuitable as a vehicle for theoretical understanding. The specific proposal is to treat 'true' as a vague term. I d o not suppose that. in ordinary usage, 'true' is simply a vague term like other vague terms. 'True,' in ordinary usage, displays many of the characteristics typical of vague terms. but it displays other characteristics all its own, notably the propensity to paradox. The proposal here is that we replace our ordinary usage of 'true' by a scientifically respectable usage that treats 'tl-ue' simply as a vague predicate like other vague predicates. This reformed usage of 'true' will, I shall claim, bc satisfactory both as a basis for a theoretical understanding of the connection between language and the world and as a means for accon~plishingthe practical. nonphilosophical work now ably performed by our naive usage. We shall develop rules of inference governing the reformed usage of 'true' and show that these rules enable us to employ the reformed usage in just the ways we ernployed the naive usage to simulate con.junction and disjunction of named sets of sentences. The paradoxes arise. it will be argued, from the misapplication of these rules of inference in natural but fallacious ways. The ultimate aini of this endeavor is to develop a theory of truth for English, but I d o not attenlpt anything so ambitious here. Here I work entirely with formal languages, doing work that is preliminary to the development of a theory of truth for English. The plan is to devisc techniques that enable us to develop a theory

of truth for a formal language : j within 'Y itself. then to see if these sarrie techniques will not enable us to develop a theory of truth for E n g l ~ s hwithin English itself. We are employing Wittgenstein's method of language games. practicing our philosophical moves in a simplified setting before trying them out on English. l y promote is the unity The big philosophical cause this book aims ~ ~ l t i m a t e to of science. The dominant opinion has it that the liar antinomy proves that it is never possible to develop a successful theory of truth for a language within the language itself: instead. one must develop the theory of' truth for a language Y' within a metalanguage that is richer than Y in expressive power. This implies that, since we have no metalanguage richer than English. we cannot develop a theory of truth for English, or for any natural language. We can develop theories al for exariiple we can develop of truth for various fragments of a n a t ~ ~ r language; a theory of truth for thc fragment of the language that we use when we talk about chemistry. But we cannot extend the theory to encompass the language we use when we talk about language. We can develop a unified zoology that takes account of all the animals, and a unified astronomy that takes account of all the the heavenly bodies. But we cannot. according to the dominant view, develop a unified linguistics that takes account of a11 natural languages: we cannot even develop a linguistic theory that takes account of the entirety of any particular natural language. Unlike natural phenomena. human languages lie mysteriously beyond the reach of scientific inquiry. By providing an alternative to the dominant view, this work aims to encourage the prospects for a unified science that treats nature and language as parts of a united whole. I hopc to promote the outlook that human language is a product of human culture and human culture is part of the natural order, not inherently either more mysterious or less intelligible than the planetary orbits. This book started out, several revisions ago. as my doctoral dissertation for the Logic and Methodology of Science program at the University of California at Berkeley. Berkeley is not only a fun place to visit, it is an excellent place to g o to graduate school. and I owe a great deal to the faculty there and to my fellow students. My dissertation adviser was Charles Chihara. who spent a great dcal of time and effort helping mc with this project. His insights have proven invaluable; without his help, this book could not have been written. Jack Silver has given me a trernendous amount of help. He was very generous with his time and ideas. and his extraordinary combination of mathernatical and philosophical abilities have made his assistance invaluable. Let me express niy thanks to three other members of the faculty, Ernest Adams, George Myro, and Bruce Vermazen. and to two of my fellow students, Shaughan 1,avine and Steven Yablo. Since leaving Berkeley. I have been at the University of Arizona. I used a

version of the book in a seminar in which Marian David. Charles Latting. Steven Laurence, and Scott Sturgeon went carefully through the text, making valuable suggestions. Let me also thank Keith Lehrer for his help. Portions of the paper have been read to the philosophy colloquia at the University of California at Irvine, at the University of Arizona, and at Rutgers University, to the mathematics colloquium at the University of Colorado at Boulder, to a conference on paradoxes and type-free theories at the University of Texas at Austin, and to a symposium at the Pacific Division meetings of the American Philosophical Association. I have received some extremely valuable comments. Let me list a few of the other people who have helped me: Nicholas Asher, Nuel Belnap, George Boolos, Anil Gupta, Brian McLaughlin, William Keinhardt, Brian Skyrms, Albert Vissar, and Peter Woodruff. A summer stipend from the Social and Behavioral Sciences Research Institute here at the University of Arizona gave me a valuable opportunity to work on this. The book was lucky enough to win the Johnsonian prize in philosophy. for which I am very grateful. As prizewinner, the book was published through a joint effort on the part of the Journal of Philosophy and Hackett Publishing Company, both of whom have been very helpful to me. I owe a special debt to Shaughan Lavine, who, in his capacity as one of the editors of the Journal, went painstakingly through the mathematical portions of the text, working diligently to remove obscurities and confusions. Whatever glimmers of clarity you may find in the text are most likely due to Shaughan. Michael Kelly at the Journal ofPhilosophy edited the text, putting a great deal of thoughtful effort into it. At Hackett Publishing Company. Frances Hackett, James Hullett, and Dan Kirklin have been extremely helpful. Kirklin's thoughtful and tirelessly diligent efforts have been especially valuable. Finally, I would like to thank my wife, Roberta Hayes-Bautista, whose patience this project stretched to (sometimes a little past) the breaking point.

Our Project It is the aim of science to find out what is true. This is an enormously difficult aim to accomplish, so we have made the task easier by dividing up the workload, parceling out the task among the various specialized disciplines. Thus, it is the aim of astronomy to find out the truth about the heavens and the aim of zoology to find out the truth about animals. Each of thcsc specialized disciplines aims to find out a portion of the truth, but it remains to philosophy to try to understand truth as such. Each of the sciences aims to find out the truth about its subject matters. One of the subject matters of philosophy is truth. So one of the aims of philosophy is to find out the truth about truth. When we attempt to find out the truth about truth, an unusual difficulty confronts us. Most of the time, when we try to understand something complicated, our problem is that we do not know what to say, or perhaps we know a few things to say, but what we know to say is altogether too little to constitute a satisfactory account. When we try to understand truth, we encounter precisely the opposite difficulty. When asked to give a theory of truth, we know exactly what to say, and what we know to say is altogether too much to constitute a satisfactory account. What we find ourselves almost irresistably inclined to say is this: the statement that a sentence is true expresses exactly the same thought that the sentence itself does.' If that is so, we must have

5 is true iff 4 whenever is a quotation name of 4. The two sentences '5 is true' and '4' express exactly the same thought, so that the biconditional conjoining them must be not only true but analytic. However attractive this account may be. it cannot be right. According to it, I

More precisely. the statement that an English sentence is true in English exprehses the same thought that the English sentence expresses. When an English bpeaker says, without qualification, that what looks like an English sentence is true, we presume that she means that the sentence is true in (her dialect of) English, just as, when an English speaker says, without qualification, "The wcather is unpleasantly hot." wc presume she Illcans "The weather is unpleasantly hot here now." "True" always means true in some particular language. but we do not usually need to bc told explic~tlywhat language is intended. Cf. remark 0.1 below.

\4,e would havc to havc 'The starrcd scntcncc is not truc' is true iff the stan-ed sentence is not true. Yet. as we can see from the exhibit below:

*

The starred sentence is not true

'The starrcd scntcncc is not truc' is identical to the starrcd sentence, so that. silbstituting equals for equals. we derive. absurdly. The starred sentence is true iff the starred sentence is not true Consideration of the starrcd sentence shows us that there is something drastically defective about our ordinary understanding of what it is for a sentence to be true. W e may think of the biconditionals is truc iff ct, for € a cli~otationnarne of d.together with other principles governing the usage of the word 'true' which we are intuitively inclined to regard as obvious and, indeed, as part of the meaning of the word 'true'-such principles as "A con.junction is true iff both cor~junctsare true"-as constituting an inforrnal theory. We tllc,ory cf truth: it is not a theory that we shall refer to this theory as our rlcri~lc~ consciously or explicitly avow. What the starred sentence shows us is that our naive theory of truth is inconsistent with manifestly observable eriipirical fact; specifically. the naive theory is inconsistent with the fact that "l'he starred sentence is not true'

=

theory of truth with a scientific thcory that is consistent with tlie evident empirical and mathematical facts. The therapy proposed is elective therapy. Symptoms of the difficulties that beset the naive thcory of truth were first noted by Epimenides (the Cretan who said that Cretans always lie) in the fourth century B.c.. and we havc not yet died from them. In fact, the evident inadequacies of the naive theory of truth cause remarkably little disruption in the way wc usc the word 'true'. In practice, we treat the rule that permits us to assert biconditionals

5 is true iff 6 for 5 a quotation narne of 4 , as a rule that admits exceptions. We restrict the rule on an rrrlhoc, basis. withholding assent fro111those rare instances of tlie rule which seem likcly to cause mischief. We somehow manage to restrict the rule ,just enough s o that. without impairing the usefulness of the notion of truth, we are able to avoid being tricked into accepting outrageous o r outlandish conclusions. Thus. you are likely to be disappointed if you try to beat a traffic ticket by telling the judge, "Your honor, if what I am telling you is true, I was only going 55.'' expecting the judge to reason as li)llows: What the dckndant says cannot be false. since if what she said were false, then. being a false conditional, it would have to have a true antecedent, so it would have to be true. S o what she says is true. S o we have a true conditional with a true antecedent. Hence we must havc a truc consequent, that is, the defendant must have been driving within the speed limit. 1 find the defendant not guilty.'

the starred sentence.

Charles Chiliara [I9791 has usefully distinguished two problems that arise in situations like this one. in which obvious preniisses lead us by seemingly impeccable reasoning to absurd conclusions: the diagnostic problem and the therapeutic problem. I shall return to the diagnostic problem in the final chapter, but for now my rcsponsc to the diagnostic problem is short and simple: theories that have observably false consecluences are incorrect; this rule applies to informal prescientific theories no less than to scientific ones. The naive theory of truth has an observably falsc conscqucnce, viz., 'The starred sentence is not true' # the starred sentence. Therefore, the naive theory of truth is incorrect.' C r ~ ~ though dc this diagnosis may be--it is on a par with the medical diagnosis. "You are a very sick man"-it is precise enough to indicate a plan of therapy. The therapeutic program is to replace our demonstrably incorrect prescientific

' Although this is the most straghiforward diagnosis. it 15 by no means the only di:rgnosis posslblc. The alternative is t o locate thc \ourcc of thc difficulty not in the naive thcory but in the classical logic by which we derive an absurdity from the naive thcory. A version of thls position will he discussed in chapter 4.

T o understand how we manage to restrict the naive rule so deftly, that is, to understand in fine detail our ordinary practice in using the word 'truc', is a philosophically interesting problem in ordinary-language metaphysics, but not a problen~1 wish to investigate here. The fact that it is possible to use the word 'true' coherently without possessing a coherent theory of truth may co111fort those who have no taste for theory, for it shows that an adequate thcory of truth is not required for brute survival. On the other hand, if we want to obtain a theoretical understanding of the connection between language and the world. it will be necessary to develop a satisfactory thcory oftruth. The naive thcory of truth is demonstrably not a satisfactory theory, since it has observably false consequences. So. if we want to obtain a theoretical understanding of the connection between language and the world, we must go beyond the naive thcory. The fact that i t is possible to get around in the world without having any theory of truth beyond the naive theory should not lead us to suppose that we ought to rest content with the naive theory, any more than the I

Thlr evariiple is adapted frorir Lob (1955. p. 1171

fact that it is possible to get around in the world without understanding relativity theory ought to pcrsuadc us to rest content with Newtonian mechanics. Ordinary language may be all right, but our ordinary theory of language is not all right. Toward developing a theory of truth freed of the evident flaws of the naive theory, a natural first thing to try is to suppose that sentences like the starred sentence, though syntactically well-formed. are senlantically defective. Declarative sentences arc typically used to express propositions, and the sentences are said to bc true or false according as the propositions they express are true or false. A sentence like the starred sentence, although constructed out of meaningful components in an unexceptional way, does not express a proposition. either true or false. This account, though appealing. cannot be right. Consider the sentence $

The sentence marked with a dollar sign does not express a true proposition.

We are to suppose that this sentence, though grammatically well-formed, does not express a proposition. It does not express a true proposition and it does not express a false proposition. But that the sentence marked with a dollar sign does not express a true proposition is precisely what the sentence marked with a dollar sign tells us. Thus. our theory is self-defeating, since it concludes that its own conclusions do not express meaningful propositions. We see here a dialectical pattern that we shall meet again. An account of the paradoxical sentences is advanced, but the account is turned against itself as the theory's own words are used to formulate a new and devastating version of the paradox.4 We see the simplest version of this pattern, if we take the prototypical paradoxical sentence to be the simple liar senteilce: This sentence is false. I t would appear, naively. that, if the simple liar sentence is true it has be false, and if it is false it has to be true. A natural response is to say that the simple liar sentence is neither true nor false: it has an intermediate truth value, perhaps, or no truth value at all. We might go on, if we wishcd, to say that the status of the simple liar sentence is like that of sentences containing denotationless proper names or sentences containing category mistakes. Although this response is intuitively satisfying as an account of the simple liar sentence, its futility is liar demonstrated when we focus our attention on what is called the slret~gthet~ed sentence:

This sentence is not true. I S we respond to the strengthened liar sentence just the way we did to the simple liar, by saying that the sentcnce is neither true nor false, then we will have to say, a ,fortiori, that the strengthened liar sentence is not true. But that the

' Cf.

Burge [1979, p. 911

strengthened liar sentence is not true is precisely what the strengthened liar sentence says, and we are back in the briar patch. This maneuver of responding to an account of the paradox by turning the account's own words against it will recur sufficiently often that it will be useful to have a name for it. We shall refer to it as the .srrengrhcned liar response. It is the aim of science to find our what is true. Were it the case that hurnan beings were perfect in knowledge and wisdonl. we would simply require, as part of scientific methodology: A satisfactory theory should never make claims that are ~ n t r u e . ~ But since we are limited as we are, such a rule would not be useful, for we would not know how to apply it. Let me propose a couple of other rules that we can apply, though fallibly: (PI) A satisfactory theory should never make claims that manifestly contradict clear observations. (P2) A satisfactory theory should never make claims that are, according to the theory itself, untrue. These principles do not uniquely determine the theory we are going to be developing, but they guide its development in important ways. These principles arise out of a belief that truth is an aim of scientific inquiry and that agreement with observed fact is a rnark of truth. It is hard to give an argument for this belief, for it is hard to find more basic principles on the basis of which to argue. That a successful theory should give results that conform to observation is, by now, fairly well-established, but that truth is an aim (though not the only aimb) of scientific inquiry remains controversial. One could perhaps argue on historical grounds that aiming for truth is a vital component of scientists' psychological motivation, and that, were it not directed toward the goal of truth, science would stagnate. To support such a contention would require a massive investigation that cannot be undertaken here. Here let me merely remark that, if we did not accept (P2). we would probably not find the notion of truth to be particularly interesting, useful, or important, so that our most likely response to the liar paradox would be to abandon the mischievous notion of truth altogether. We have already seen ( P l ) and (P2) in action. ( P l ) was what led us to acknowledge the untenability of the naive theory, since the naive theory has the observably false consequence that 'The starred sentence is not true' # the starred

' The way 1 shall bc using the terms. "

'untrue' will he synonymous with "not truc," and 'false' will he synonymous with "has a true negation." It may sometimes happen that the aim of getting the truth conflicts with some of the other aims of science. E . R . , the aim of getting a theory that is simple envugh to be useful. In such cases, no completely satisfactory theory is available: we do the beat we can.

scntence. A natural tirst response to the misl'ortunc that befalls the nalve theory is to say either that thc starred sentence does not express a proposition or that the starred scntence expresses a proposition that is neither truc nor false: but this response violates (P2). since it requires us both to assert the sentence marked with a dollar sign and to deny that the sentence marked with a dollar sign is true. ( P I ) and (P2) will guide us in developing a theory of truth which will be expounded at length in the chapters to come. Let rrle now give a sketch of the account. An initially attractive theory that we have already had to abandon tells us that the paradoxical sentences arc semantically defective. The rules of our language link an ordinary scntence like 'Toby's cat plays n l ( l h , j o r ~ g , q ' with somc situation. state. or event. and it is in virtue of this linkage that the sentence is either true or false. With the paradoxical scntences. the wheels are spinning out of gear, s o that the sentences are not linked to any situation, state, or event, and thus the sentences are not either true or false. This account, I want to say. is partly right and partly wrong. What is right about the account is the observation that the paradoxical sentences are semantically defective: what is niistaken is thc attempt to express this insight by saying that the sentences arc ncither true nor false. For an ordinary sentence. the rules of our language establish a link between the sentence and the world. and this link detern~ineswhether or not the sentencc is true. For the paradoxical sentences. no such link is established, so the rules of our language do not determine whether the sentences are true. It is one thing to say that it is not determined whether the paradoxical sentences are true, and is sonicthing quite different to say that it is determined that the sentences arc not true. Thus. if we say that the truth vali~eo f the strengthened liar sentence is undetermined. wc are not compelled thereby to say that the sentence is not true. and so we are not drawn into contradictions. Sentcnccs, I want to propose, fall into three categories: scntcnccs that the rules o f our language. together with the empirical facts, determine to be definitely true: sentences that the rules of our language. together with the enlpirical facts, determine to be definitely not true: and sentences that arc lelt unsettled. The transition from the trichotomy trucltalseineither true nor false to the trichotomy definitely trucidefinitely not trueiundetermined is rather undramatic. but its effect on the paradoxes is quite dramatic. In terms o S the Sormcr trichotomy. we car1 reason as follows: Suppose that the strengthened liar sentence is neither true nor false. Then it is true that the strengthened liar sentence is neither true nor false, and a fortiori it is true that the strengthened liar sentence is not true. Rut that the strengthened liar sentence is not true is just what the strengthened liar sentence says. S o the strengthened liar sentence is true after all.

If we substitute the

c l c y i ~ l i t rl

i ~ S~PrI ~ ~ P I ~ C . ~ , ,

This sentence is not definitely true. we get the following bit of argument: Suppose that the definite liar sentence is unsettled. that is, neither definitely true nor definitely untrue. Then it is definitely true that the definite liar sentencc is unsettled. and ( I jbriiori it is definitely true that the definite liar sentence is not definitely true. But that the definite liar sentence i \ not definitely true is just what the definite liar sentencc cays. S o the definite liar sentencc is definitely true after all. This argument is no good. From the hypothesis that a sentence is unsettled. it by no means follows that it has been settled that thc sentence is unsettled. Quite the contrary, if a sentence is unsettled, then we are free to adopt linguistic conventions that settle it. Of course. the observation that one particular tactic for recasting the strengthened liar argument in ternis of the definite liar sentencc has been thwarted docs not show us that there is not some other tactic that succeeds in getting a contradiction from the definite liar. For that, we need a consistency proof. which we shall get in chapter 8 (theorem 8.15). The linguistic rules for using the word 'true' leave it undetermined whether the paradoxical sentences arc true. In this respect, the word 'true' acts like a vague term. If Harry has only a little hair, the linguistic rules leave it undetermined whether 'Harry is bald' is true. I wish to exploit this similarity as vigorously as possible. Thus. 1 shall develop a for~nalmodel of the logic of vague terms. then use this formal model to give a theory of truth which treats 'truc' as a vague term. The formal model of the logic of vague terms, which is based upon the work of' Rudolf Carnap [ 19371, Bas van Fraassen [ 19661, and Kit Fine ( 1974). will have it that the meanings of vague terms are given by a system of meaning postulates. T o say that Harry is definitely bald will be to say that 'Harry is bald' is derivable. in a certain system of infinitary logic, from the meaning postulates together with certain precisely expressed statements of fact. 'I'hus. if the system of meaning postulates consists of the sentences 'Anyone with fewer than ten thousand hairs on his head is bald' and 'No one with more than twenty thousand hairs on his head is bald', then, if IIarry has five thousand hairs on his head, he will be definitely bald, whereas if he has forty thousand hairs. he will be definitely not bald. and if he has fifteen thousand hairs, the baldness question will be unsettled. I want to treat 'true' as a vague predicate. I do not intend to suggest by this that. in our ordinary usage, 'true' is simply a vague predicate like ordinary vague predicates. Ordinary vague predicates are predicates whose applicability is

underdeternlined by the rules of our language, whereas, intuitively, our linguistic rules overdetermine the applicability of the word 'true' in conflicting ways. Ordinary English rules for determining whcn to apply the word 'true' present us with two kinds of problem cases. For some sentences, like the truthtcllcr scntcnce ('This scntence is true') and the sentence 'Harry is bald', the rules give no answer, and for other sentences, notably thc liar sentences, the rules give bizarre and conflicting answers. I propose that we adopt a reformed usage of 'true' which treats all thc problematic cases as unsettled. For the vast majority of sentences, the reformed usage will agree with traditional usage in declaring the sentences unequivocally either "true" or "not true." All the problem cases will be regarded as unsettled. In cases where traditional usage gives conflicting answers, the reformed usage will give no answer at all, treating all such cases on a par with cases of vagueness. Such a reform. 1 want to argue, will preserve those logical features of our everyday usage of 'true' in virtue of which the notion of truth is so useful to us, without succumbing to paradoxes and contradictions. If, contrariwise, wc attempted to eliminate vagueness as well as contradiction. replacing our traditional way of using 'true' by a reformed usage that was perfectly precise as well as perfectly consistent, the logical structure of our everyday usage of "true" would. I claim, be damaged beyond repair. To get a picture of how vague terms behave in English, we shall utilize mathematical structures called pnrtinlly interpreted latzguagrs. It is not intended to be a terribly accurate picturc; certainly no one would think of English as one of these languages. Our partially interpreted languages are vastly simpler than natural languages. This is why they are useful. They present important logical features of natural languages in simple contexts in which it is con~parativelyeasy to see what is going on. It is hoped that partially interpreted languages will be useful in understanding the logic of vague terms in much the way that familiar first-order languages are useful in understanding the logic of precise terms. To develop an adequate theory of truth for a natural language is a task of staggering difficulty, for natural languages are among the most intricate of the works of mankind. But it is not an impossible task. It would be an impossible task, if we restricted ourselves to theories of truth which refined and made precise the naive theory, for then we would obtain. at the end of all our labor, a theory that implied, absurdly, 'The starred sentence is not true' # the starred scntcnce Work like the present project, which aims to develop logical tools with which to talk about truth without becoming ensnared in paradox, is needed as a preliminary to the task of developing a theory of truth for a natural language. One feature that our partially interpreted languages share with ordinary interpreted first-order languages is that, once the interpretation or partial interpretation has been fixed, the context in which a sentence has been uttered does not enter

into the determination of the semantic status o f thc utterance. In this respect. the fomlal languages are quite unlike English, where we have sentences like 'The cat is on the mat', the truth value of an utterance of which will depend not only on the meaning of the words and the location of the world's cats and mats but also on contextual features that tell us what cat is being referred to, what mat, what time, and what spatial orientation counts as "on." For our purpose of trying to investigate the problems raised by the paradoxes in a simplified situation with as few complications as possible, this feature of our formal languages is a tremendous advantage, since the problems that arise when we try to understand how the truth value of an English sentence changes with its contcxt of utterance are particularly thick and thorny.' and since, as we shall see in the next chapter, even in our simple formal languages the semantic paradoxes hit us with full force. Because of their independence from context, it is legitimate to speak of the scntcnccs of our formal languages as being either true or false. We cannot normally do this with English sentences, since the same English sentence will often be true on one occasion and false on another. Thus, I spoke above of the sentence This sentence is false. as being paradoxical. But if the sentence is uttered while pointing to an arithmetical equation on the blackboard, the utterance will not be paradoxical. It would be more precise, rather than speak of a sentence as true. to say that the sentence as used by a certain speaker at a certain time is true, or that the proposition expressed by the sentence on a certain occasion is true, or that the statement made by a certain utterance of the sentence is true. But this added attention to detail leaves the problems of the paradoxes unresolved, as we can see from the following examples: As used by me now, this sentence is not true. This sentence is not now being uscd to express a true proposition. In writing this sentence now I am not making a true statement. For what we are doing here, always to make explicit the context dependence of the sentences we discuss would produce no real benefits and it would be a considerable nuisance. So 1 hope the reader will forgive me if I continue to suppress superfluous speaker and time parameters. Similarly, I shall continue to use the word 'sentence' as if sentences were only used to make assertions. In fact, we also use sentences to ask questions or issue orders or make promises, but we shall have little occasion to talk about these other sorts of speech acts. Contemporary philosphical discussion of truth has largely centered around the following proposal made by Alfred Tarski 11935, pp. 187fl: See Searle [1979, ch. 51

Cot1\~entiotlT: A formally correct definition of the symbol ' T r ' , formulated in the metalanguage, will be called an aclrqlratr tlefit1iriotl of [ruth if it has the following consequences: ( a ) all sentences which are obtained from the expression 's F Tr if and only if p' by substituting for the syrnbol '.r' a structural-descriptive namc of any sentence of the language in question and for the symbol 'y' the expression which forms the translation of this sentence into the metalanguage.

( p ) the sentence 'for any .r. if x c-

Tr then .r t. 5' (in other words ' T r

C

S').

Here Tarski is using the word 'metalanguage' to mark the distinction between the language we are speaking and the language about which we are speaking. W e use the rnetal(lngicc~gc~ when we give a definition of truth for the o11jjcc.tlarlguclgc~. 'S' refers to the set of sentences of the object language. There are two fundamental questions raiscd by convention T: (1) What constitutes a correct translation of the object language into the metalanguage? This problem has bccn a sub.ject of vigorous investigation, particularly after W . V . 0 . Quine 11960, ch. 21 showcd that there is no purely behavioral criterion of correctness. ( 2 ) Because of Epirnenides-type problems, if the object language is identical with the metalanguage, it will not be possible to give a definition of truth which is materially adequate in the scnsc of Convention T.' Is there nonetheless some reasonable sense in which one can give a theory of truth for a language within the language itself? Only the second question will concern us here. S o scrupulously shall we avoid the first question that we shall only look at situations in which either the object language is identical with the metalanguage or the ob.ject language is a part of the metalanguage. The ultimate aim of this effort is to obtain a theory of truth for thc vcry language I speak. That is, 1 would like, ultimately, to get a theory of truth for the dialect of English spoken by mc. This dialect is very nearly the same, presumably. as standard English, but, in any case, it is rny idiolect that I speak and understand, so it is my idiolect that I have available to use as my metalanguage, and so. when attacking the second question, it will be my idiolect that i shall have available to use as an object language. Solving both problems would provide us with a versatile and powerful method Exception$ can occur uith certain artihcial language\ uhosc ability to describe their own syntax i$ \everely restricted: \cc Gupta [ 1987. $111.

for obtaining theories of truth for natural languages. If l could solve the second problem, I could get a theory of truth for my own language. If I could solve the first problem. 1 could translate other languages into my own language. Then I could combine the two solutions. Given a scntcnce of a foreign tongue. I could first translate the sentence into my own language. then use the theory of truth for my own language to give the truth conditions for the translated sentencc. A sentence of the alien language will count as true, if and only if it translates into a true sentencc ol' my own language. Regrettably, it is not possible to attack the problems of giving a theory o f truth for my own language and of learning how to translate other languages into my own language independently. The problem is in obtaining the truth conditions for indirect speech reports. For the sentence 'John said that Maureen ate the last Moon Pie'. to be a true sentence of my idiolect, John must have made some statement that correctly translates into my idiolect as 'Maureen ate the last Moon Pie.' Thus. in order to know when I have got the truth conditions for my own sentence right. 1 have to know when I have got the translation right. The problem is even stickier when wc report the mental attitudes of creatures who lack spccch. such as infants. beasts. and the mute. T o give the truth conditions for these reports, 1 shall need solmething like a correct translation into my dialect of English of the sentences of the subject's language of thought. One tries to solve a difficult problem by breaking it down into simpler problerns. Thus. a promising strategy would be to begin by developing a theory of truth in which both the object language and the metalanguage consist of a fragment o f one's idiolect from which indirect speech reports and psychological attitude statements have been eliminated, postponing the problem of giving a theory of truth for the full language until after one has worked on the translation problem. Thus, one begins by working on a version of problem ( 2 ) that does not get entangled in problem ( I ) . It is hoped that the present work will bring this initial stage a little closer to completion. An advantage of examining the second problem before attacking the first is that it is a natural constraint on a successful translation that it should preserve the central semantic features of the object language. Thus, if 'chien' is a term of the object language which refers to dogs, the term of the metalanguage that translates 'chien' should also refer to dogs. Hilary Putnarn's 119751 "Twin Earth argument."\hows us that we need such a constraint, by showing that the fact that a

"

Putnani considers a hypothet~calplanet in which the visible features of the environment and the psychological 5tates of the inhabitants are just like those found on earth, yet. because of hidden differences bctwcen the environments. the worda of speakers on Twin Earth do not mcan the same as those of their counterparts on earth: by 'water' they do not refcr to water but to another kind of stuff which looks and tastes like water.

translation successfully reflects the psychological states"' of speakers of the object language is not enough to guarantee the correctness of the translation. We also need to make sure that the terms of the metalanguage refer to the same things as the terms of the object language which they translate. To apply this constraint, we need to know what kind of semantic properties to expect the object language to manifest; to know this, we shall need at least an outline of a semantic theory for the object language, and to get even an outline of a semantic theory, we shall need to solve problem (2). Our answer to (2) will, in part, determine how rich a semantic structure there is for the translation to preserve. Thus, on a classical view, nearly every term of a natural language has a determinate referent; the occasional nonreferring term, like 'the present king of France' or 'phlogiston', is regarded as an exception. In typical cases, reference is fixed by an appropriate causal connection between word and object. The theory to be advanced here. by contrast, postulates a wide range of terms for which no referents have bcen fixed. For a fragment of the language (thefiilly interpreted part), we postulate a classical, referential semantics, but for the rest of the language, the meanings of the terms are given by a system of meaning postulates that are not sufficiently powerful to specify a definite referent for each term. Thus. the principle that successful translation must preserve reference can only be meaningfully applied to the fully interpreted part of the language. Thus, on our account, the requirement that a successful translation should preserve reference is considerably less onerous than it is on a classical account. More generally, one would have expected the semantics of a natural language to be developed compositionally. Semantic values are assigned first to the members of a finite vocabulary, then to more complex expressions according to rules that prescribe the behavior of the logical connectives, until finally truth values are assigned to sentences." If we have such a semantics, it is natural to require that a correct semantics preserve the semantic values at every stage. On the present account, this requirement has no effect, since the semantics proposed is not compositional. For sentences outside the fully interpreted pan of the language, the truth of a sentence will consist in its being implied (in an enriched sense of implication which goes beyond ordinary deduction) by other sentences. Thus, the truth value of a sentence is determined not by the referents of the components of the sentence but by the position of the sentence within an implicational network of sentences. The position of the sentence within the implicational network will be determined in part by the sentence's syntactic

"' I'

We are using 'psychological states' in the narrow sense in which "no psychological state, properly so-called. presupposes the existence of any individual other than the subject to whom that state is ascribed" [ 1975. p.2201. The locus c l u s ~ i c ufor ~ compositional semantics is Frege's work. See. for example, [1891].

structure, but this determination will not procccd by establishing a semantic value for each of the syntactic components. In Donald Davidson's terms [1973, p. 2211, our semantics is constructed by the holistic method rather than the buildingblock method." One might suppose that we must have a compositional semantics in order to explain how it is possible for finite beings over a finite period of time to learn truth conditions for infinitely many sentences. But this is not so. The semantic theories we shall develop will be learnable because they are recursively-indeed, in many cases, finitely-axiomatizable. In the first stage of the overall program for developing semantics for natural languages, both the object language and the metalanguage will be taken to be a fragment of standard English from which indirect speech reports and propositional attitude statements have been excised. Or, more precisely, the object language and the metalanguage will both be a fragment of my idiolect of standard English. What 1 am doing here is describing, writing in my own idiolect, a program that, if carried to completion, would give me a theory of truth for my own idiolect. Reading this book and treating its words as words of your own idiolect, you read a description of a program that, if carried to completion, would give you a theory of truth for your own idiolect. In doing so. you are translating my idiolect into your own homophonically," thus acknowledging me as a mcmbcr of your own speech comnlunity. At the end of this program, neither of us quite gets a theory of truth for standard English, though we see how to get one. Roughly, we identify a community of English speakers by social and historical considerations, and we regard a sentence as true in standard English if it is true in the idiolects of a predominance of English speakers. For most purposes, dividing up a language into individual idiolects is unnecessary and a bit precious. I am writing and you are reading a book in English describing a program that, if taken to completion, would give us a theory of truth for English. We speak the same language; unless one has a special purpose in mind, there is no purpose in dwelling upon small differences. The special purpose here is to separate the problems that arise from question (2) from the problems that arise out of the indeterminacy of translation. Quine ([I9601 and ll9681) argues that the totality of a subject's dispositions to verbal behavior does not suffice to determine the referents of the subject's words. I'

"

The proposal that natural languages lack compositional semantics was advanced by Schiffer [I9871 on the basis of considerations disjoint from our concerns here. I was initially shocked by Schiffer's proposal, and I was even more shocked when I belatedly realized that I myself had been proposing a noncompositional semantics. That is, "Fido" is translated "Fido." The translation will not be entirely homophonic. There will be occasions when, faced with a choice between "The author's views are bizarre to the point of madness" and "The author is using some of his words eccentrically." you will charitahly choose the latter alternative.

N o amount of behavioral evidence will dcternmine whether the subject's word "gavagai" ought to be translated "rabbit" or "rabbit stage" or "undetached rabbit part" or "rabbithood locally manifest." Quine concludes (rather too abruptly. in my view) that there is no fact of the matter whether the subject is referring to rabbits o r rabbit stagcs. Once the subject's dispositions to verbal behavior are accounted for, there are no further grounds. either visible or hidden, for saying that one theory of reference for the languagc is better than another. The discussion takes on a more urgent tone when it is noted that indeterminacy begins at home. There is no principled basis. says Quine, for prcferring the hypotheais that I mean by 'rabbit' what my neighbors mean by 'rabbit' to the hypothesis that when 1 use the word 'rabbit' I refer to what normal English speakers call "rabbit parts." T o discuss the mcrits of Quine's argument would take us far from our concerns here. For present purposes. what is important to realize is that the indeter~iiinacy Quine describes does not arise if one looks only within one's own idiolect." Within my own idiolect,

'Rabbit' refers (in my idiolect) to rabbits is definitely true. It is true because of the rneaning postulates for the word 'refer': I d o not need to confirm it by examining my own behavior. As we shall see below, our naive understanding of the notion of reference is undermined by the paradoxes just as our naive understanding of truth is, but it is not so deeply undermined as to dissuade us from 'Rabbit' refers to rabbits. According to the theory wc shall be developing, the sentence

V

l h e sentence marked with a heart is not true in my idiolect.

does not have a definite truth value, even within my own idiolect. Thus, an indeterminacy underlies the liar paradox. and, unlike the indeterminacy Quine describes, i t can he found even within a single language. There is no fact of the matter whether "true" refers in my o ~ w idiolect to the sentence marked with a heart. Thus. reference of a foreign term is doubly inscrutable. The translation of a foreign term into our native tongue is underdetermined. and, once we have settled upon a translation manual, it may turn out that the doniestic term that translates the foreign term is one whose referent is underdetermined. I'

As Quine says 11968. p. 201 1. "In practlce we end the regress of background languages. in discussions of reference, by acquiesc~ngIn our mother tongue :rnd taking its words at face value." Quine's foregn~undlanguagc/bnckgrou~~d languase d~st~nction i< what we are calling the object langu:age/metala11puage d~stinct~on.

Quine 11968. pp. 300f] likens tlic I'act that it is possible to pin down the referent of a tern1 in one's own idiolect by stipulating By 'rabbit: 1 shall r c k r to rabbits to the fact that one can pin down the position and time of an event once one has laid down a coordinate system. The indeterminacy of translation corresponds to the fact that there is, in nature. no preferred coordinate system. Rut within a particular coordinate frame. the position of an event is uniquely determined, just as, as far as the considerations Quine adduces are concerned. within one's own languagc. the referent of a term or the truth value of a sentence is uniquely deterniined. But, in fact. the truth value of the starred sentence remains undetermined even within one's own language. Sentences like the starred sentence are evcnts whose location remains unspecitied even after we have ti xed a coordinate system. They are. to extend the metaphor. singularities in the coordinate metric. The study of the paradoxes takes us beyond Quine's ontological special relativity, where within a particular coordinate system everything is smoothly Euclidean, to an ontological general relativity." L,et me c m p h a s i ~ ethat an individual's idiolect is not a private language in the sense of Ludwig W-ittgenstein (19.53. $3268-3701. It is the variant of a public language which is spoken and understood by a particular speaker. Indeed. it will make no difference, as far as our fornial development is concerned, whether we are talking about standard English or some speaker's dialect of English. By isolating a particular speaker's language from its social and historical context. we are getting a one-dimensional picture of the language. W e can specify the nieanings of the speaker's words, but not how or why they came to have those meanings. W e d o this purposely, in order that we can focus our undistracted attention on the internal logical structure of the languagc. REMARK 0.1. The observation that "'Rabbit" refers in my idiolect to rabbits' is true in my idiolect because of the rrleaning of the word "refers" raises an interesting problem. suggested by Hartry Field [1986]. The claim would appear to imply that '"Rabbit" refers in my idiolect to rabbits' is logically necessary, o r as nearly logically necessary as it can be, given the presence of the indexical "my." Yet, if our language had evolved a little differently, I would have spoken a language in which "rabbit" referred to groundhogs. This puzzlc can be solved by paying careful attention to the scopes of modal operators. The following sentences are perfectly compatible. and apparently true: (i) (3language Y)(I speak Y' & n('rabbit' refers in if to rabbits)) (ii) V ( 3 language Y)(I speak Y' & 'rabbit' refers in 3 to groundhogs) I'

As if anticipating thiq metaphor. Gaifman [I9871 refer\ to the genuinely paradoxical sentences as

"hl;ick hole ( ~ P ( X , Z=) ~~)+ ( L J ., , .

4(z,v, , . .

. , v,)).

Then

. , v,,Y

and so

from which we derive

R

I-

(Vv,, . L>,,

. .

( v ~ , , ) ( ( ~ z ) ( Y ( ~ ~ ( x ,& z )$( ~ ,z,Vl, z) .

.

. , bl,J) ++ 4(r$1,

. . . v,,)) 3

that is

and all sentences obtained by prefixing universal quantifiers to instances of the following induction axiom schema:

The induction axiom schema tells us that every nonempty definable set has a least element. If we add new symbols to the language and allow these new symbols to appear in the induction axiom schema, we increase the store of definable sets, and so we get stronger versions of PA. Unless otherwise stipulated, 'PA' will always refer to the original version, in which only symbols from the language of arithmetic are allowed into the induction axioms. The power of Giidel's arithmetization of syntax is seen in the following theorem, which is a cornerstone of modcrn logic. Most of the results in this book can be regarded as corollaries to this basic result:

THEOREM 1.2 (Giidel's Self-referential Lemma). For any formula 4(x,v,, . . . , v,,) of Y , one can find a formula +(L#,, . . . , v,,) so that R + (VIP,,. . . (Vv,,)(+(v,, . . . , v,,) ++ $(r+l, v , , . . . , v,?)). Notice that the variables here are unrestricted; they do not have to range only over numbers. PROOF:Let p be the arithmetical function given by

As an immediate corollary, we have our formalized version of the liar a n t i n ~ m y : ~

THEOREM 1.3 (Tarski-Epimenides). Let r(x) be a formula of 2 . There is no theory consistent with R that entails all instances of the schema

PROOF:Use theorem 1.2 to find A so that We get the philosophically interesting application of theorem 1.3 when we take r(x) to be a formula whose intended meaning is "x is a true sentence of 3." The theorem shows that speakers of 2 cannot consistently adhere to the naive theory of truth. Either their language lacks the means for talking about truth at all, or else the naive theory of truth, as expressed in their language, is inconsistent with basic arithmetic. Of course, in reality there are no speakers of 3 , 2being a first-order language. In talking about speakers of 2 , we are employing Wittgenstein's method of trying to understand our use of language by performing thought experiments involving This method of seeing that a function is recursive by first observing that the function is computable and then appealing to Church's thesis will be our standard procedure. ' From Tarski [1935, pp. 247ffl. Tarski's accomplish~nentis not so trifling as the presentation here suggests, since when Tarski wrote only special cases of theorem 1.2 had been proved.

speakers and cultures whose languages are much simpler than our own. Wittgenstein's method is particularly applicable here, since adding to the expressive power of the language could only make matters worse for the naive theory; increasing the expressive power of a language by introducing new operators and connectives cannot make an inconsistent theory consistent. 2 is the minimum we need to get the liar antinomy. S o long as our language is one in which we can carry out first-order deductions, in which we can d o arithrnetic (either directly o r viu a coding), and in which we can describe basic syntax Giidel codes, we get the paradox. Because we can always replace talk about numbers with talk about their Quine codes, the requirement that our theory I' be able to relatively interpret Kobinso~l's arithmetic amounts to no more than a requirement that we be able to give a moderately detailed theory of syntax. Thus, theorem 1.3 shows that the naive thcory of truth is inconsistent with the basic laws governing syntax. This takes us beyond the observations we made in chapter 0 , which only showed that the naive thcory of truth was inconsistent with observable empirical facts. Richard Montague 119631 isolated what was essential to Tarski's construction to obtain a stronger result: TIIEOREM1.4 (Montague). Let

r be a sct of sentences which

( 1 ) contains the axioms of R; (2) is closed under first-order consequence; (3) contains r ( r 4 7 ) whenever it contains 4 ; and (4) contains all instances of the schema

Then

is inconsistent.

PROOF: Taking A as above, the following sentences are in T: (i) (ii) (iii) (iv) (v) (vij

-

i r ( r h 1 ) + A (This is a theorem of R . ) r(rA7) A (This is an instance of ( 4 ) . ) A (From (i) and (ii) by (2).) r ( r h 1 ) (From (iii) by ( 3 ) . ) r(rh1) i A (This is a theorem of R . ) IA (From (iv) and (v) by (2)..j

Tarski's theorem points out a basic difficulty that confronts us when we attempt to interpret ' r ( r 4 1 ) ' as 'r4l is true'. What Montaguc has shown is that we encounter the same basic difficulty when we attempt to interpret ' r ( r 4 1 ) ' as 'r41 is necessary'. It is natural to think of necessity as a property of sentences; it is a property possessed by those sentences which express necessary truths, and lacked by those sentences which express contingent truths and by those sentences

which express falsehoodc. If the kind of necessity we have in mind is logical necessity or analyticity. then the necessary scntenccs will be those sentences which are deducible from meaning postulates. Unless the set of meaning postulates is extravagantly complicated, there will be an arithmetical predicate r(s) whose extension is the set of Godel numbers of necessary truths, and we can interpret the modal formula a& as r ( r & l j. What logical features would we expect the set of necessary sentences to possess'! We would certainly expect that the basic axioms of arithmetic express necessary truths, and we would expect the set of necessary truths to be closed under first-order consequence and the rule of necessitation (fr-om 4 to infer 04). and to contain the instances of the schema ( u 4 -+ 6).But these expectations are contradictory. Semantics, according to Tarski [I 936. p. 40 1 1. concerns itself with the connection between expressions of a language and the objects and states of affairs which those expressions refer to. By this definition. logical necessity is not a semantical concept, since, at least on the traditional conception, a sentence is logically necessary solely in virtue of the definitional and grammatical connections among the expressions out of which the sentence is constructed. quite independent of the objects to which those expressions refer. Nevertheless. necessity intuitively implies truth and truth is a sernantical concept.' This indirect connection with semantics is enough to ensure that necessity is afflicted by the selnantic paradoxes. Like necessity, knowledge intuitively implcs truth. This observation leads us to fear that the concept of knowledge also falls prey to the paradoxes. This fear is borne out. Montague's theorem is not directly relevant to the attempt to interpret ' r ( r 4 1 ) ' as is known', since the set of known truths is not closed under first-order consequence. The proof of Montague's thoerem is directly relevant, however: let us imagine that we are engaged in a process of rigorous reasoning, s o that we are careful only to assert things we are sure we know. Now we are able to assert (i). because we can deduce ( i j from basic laws of arithmetic. W e are able to assert (ii), because (ii) is an instance of the first principle of episten~ology, is known

+

(1,

W e derive (iii) from (i) and (ii). Now we reflect that we have obtained (iii) by rigorous deduction from secure premisses; we conclude that we know (iii), that is, we conclude (iv). But from (iv), together with ba\ic laws of arithmetic, we are able to derive the negation of (iii).' That truth is a sernantical concept is a consequence of the traditional doctrine sentence depends in palt upon there being an appropriate correlation between says and what the world is like. Like all philosophical doctrines. this is open ' The applicability of thcorcm 1.4 to thc thcory of knowlcdgc was first notcd Kaplan [1960].

that the truth of a what the sentence to dispute. by Montague and

We would not expect that, whenever we know every instance of a universal generalization, we know the generalization itself, for we may not be able to gather all our knowledge together into a single thought. Similarly, if we take logical necessity to be derivability from a set of meaning postulates, we will not suppose that a universal generalization is logically necessary whenever each of its instances is. We know from Godel's tirst incompleteness theorem that it is sometimes possible to prove every instance of a general law without being able to prove the law itself. In other words, if '04' is taken to mean 'r41 is logically necessary', we shall not expect the Barcan formula

to be valid. Even if we take 04 as an attribution of metaphysical necessity, as in Kripke [1972], the Barcan formula will remain highly doubtful. Even if every individual who happens actually to exist is essentially 4. it remains possible that in some other world there exist individuals who do not exist in the actual world and who, in that other world, are not 4. On the other hand, if we take '04' to mean 'r41 is true', the Barcan formula will become very plausible indeed. A universal generalization is true if each of its instances is true. What else could it mean to say that a generalization (Vx)$(x) is true other than that every object in the universe of discourse satisfies $(x)? And, assuming that for every object a in the universe of discourse there is a name a, what else could it mean to say that every object satisfies $(x) other than that, for every a , is true? As the theorem below shows, if we assume the Barcan formula,

$(a)

as part of our theory of truth, together with the modal principles

a(+ + $) + (04 m 4 + 104

-+

O+)

and

and a slightly stronger version of the closure conditions (1),(2), and (3) of theorem 1.4, then our theory of truth will be w-inconsistent.' Unlike Montague's theorem, this result does not assume either direction of the principle that naively characterizes truth,

"

A theory 1' is w-inconsistent iff there is a formula $(x) such that i ( V x ) ( N ( x ) -,$(.r)) is a theorem of i- and yet, for each n , $ ( n ) is also a theorem of r: such a theory cannot have any models in which the arithmetical ~ y m b o l shavc their usual meanings. tiodel [I9311 constructed the first example of a consistent, w-inconsistent theory by adding to a system of basic axioms for arithmetic the ncgation of a sentence that asserts its own unprovability.

THEOREM 1.5."' Let I' be a set of sentences which ( 1 ) contains the axioms of R, together with the assertions that the successor function is one-one and that zero is not a successor; (2) is closed under first-order consequence; (3) contains ~ ( r 4 1 whenever ) it contains 4; and (4) contains all instances of the following schemata:

Then

is w-inconsistent

PROOF: Use theorem 1.2 to find a formula F(x.y,:) of the language of arithmetic so that (i) (Vx)(Vy)(Vz)(F(x,y,z)* [(x = 0 & &L z = r(vz)(f(G,j,z) + 7(2))1)1)

2

= y)

V (3w)(N(w) & ,X

=

S(W)

is a theorem of R . Using the facts that the successor function is one-one and that zero is not a successor, we derive (ii) (Vy)(Vz)(F(O,y,z) * z = y) (iii) (Vx)(N(x) + (Vy)(Vz)(F(S(x),y,z) ++ z

=

r(~z)(F(k,;,z ) + ~(z))'))

A rough English translation of 'F(n.y,z)' is "z is the Godel number of the result of prefixing n 7's to y." Now use theorem 1.2 to find a sentence a so that

is a theorem of R. cr says that not every result of prefixing 0-or-more TS to ra7 is T. We want to see that a is in Once we have done so, we can use (3) to prefix more and more TS to r d , SO that, for each n , I' will contain the sentence that says that the result of prefixing n TS to rcrl is T. Since also contains a , which says that not every result of prefixing TS to [a1 is T, we shall have our winconsistency. ICJ says that every result of prefixing 0-or-more TS to u is T. In particular, i c r implies that the result of prefixing 0 TS to a is T, that is, l a implies ~ ( r c r l ) Formalizing . this argument, we show that the conditional i a -+ ~ ( r a l ) is in r by showing that the following sentences are in 1':

r.

r

(Vx)(N(x) -+ ( t l z ) ( ~ ( x , r a l , z+ ) ~ ( 2 ) ) ) (from (iv)) (vi) l a -+ ( V z ) ( ~ ( ~ , r c,z) r ?-+ ~ ( z ) ) (from (v)) (vii) ~ ( 0 , r c r,rcrl) l (from (ii)) (v)

"'

70-+

From McGee [ 19851

(viiii l u 3 ~ ( 6 ~ 1(from ) (vi) and (vii))

(xxviii) (V:)(F(~, ,ul ,z)

Ncxt, we usc the closure conditions in clauses (3) and ( 4 ) to show that the conditional ~ ( r d+ ) v is in T. by showing that the following sentences arc ill

is in

-

r. On the other hand.

~(3)

since (iv) and (xx) are in T. so is

r:

* i ( V - r ) ( N ( a j ( V Z ) ( F ( . ~ , ~ ~ T-+~ .~Z( )2 ) ) ) (from (iv)) i r +V r +T om (ix) by (3)) xi) T ' N V F ( Y ) -+ Z ) (from ( x ) by (4ja)) (xii) ~ i ~ 3 v - ~ ) ( + ~ ( (Vz)iF(.r,ru1,:) -ri + T ( Z ) ) ) ~ ) + TT(-(v.~)(N(.~-) -+ (VZ)(F(-V,~CT: .z) + ~(:ji j l ) (by (4)b)) (xiii) ~ r ( - i ~ . r ) ( N i .+ r ) (VZ)(F(X,,C~,zi + T(Z) jil) -+ V . T V (1 ,+ T j (by ( 4 ) ~ ) ) x i ) ( ' 1 V r N +( V Z ) F ( . , ) 7 (froln (xi), (xii), and (xiii)), (XV)(Vx)(N(x) -, ~(~(x).-crl.,(~z)(F(i,rv1,z) -+ T ( Z ) ) ~ ) )(from (iii) (xvi) (V.r)(N(x) (Vz)(F(s(s),-rrl ,z) + T(z))) .+ . + T V F ) +T (from (xv)) (xi) ) N . ~ F ) + T ) ) ) S (from (xiv) and (xvi)) V )( ' I N ) ( F ( ) + ( j ) (from (xvii) (xix) ~ ( r v l )-+ (+ (from (iv) and (xviii)) (ix)

( 1

+

+

+

+

-

-

-

+

-+

So

is o-inconsistent.. We shall see below (rcmark 6.9) that the conclusion of the theorem cannot be strengthened to say that r' is simply inconsistent. The liar paradox and its variants by no mcan cxhaust the antinonlies that arise out of self-referential applications of semantical concepts. Thus. our naive understanding of the notion of reference gives rise to a numbcr of paradoxical constructions. It will be useful to regimcnt ordinary usage a little bit, breaking up the ordinary notion of reference into two notions: denotation. which applies to singular terms: and satisfaction, which applies to general terms. Thc satisfaction relation has a complicated structure hccause of the varying nurnbers of variable places that different general terms possess: but, for now, let us restrict our attention to satisfaction as a relation between individuals and one-place general terms. Intuitively. we would expect the satisfaction relation to meet the condition

-+

Consequently, (xx) rr

r contains

(from (viii) and (xix))

(V\)(y satisfies 'x does not satisfy x'

W e intend to usc mathematical induction to show that, for each natural number 1 1 . the sentence '(V=)(F(n,rrrl,z) -+ ~ ( z ) ) is ' in r. For the case n = 0, observe that the following sentences are in r: ixxi) ~ ( r c r l ) (fro111 (XX)by (3)) (xxii) (Vz)(F(6,rcrl,z) -+ z = r(+l) (from (ii)) (xxiii) ( V ~ ) ( F ( O , ~ ( + ~ ,+ : ) ~ ( 2 ) ) (from (xxi) and (xxii)) Now suppose, as inductive hypothesis, that (xxiv) ( V z ) ( ~ ( E , r ( +,z) l +~(2)) is in

For example, my housecat Quijon satisfies '.r is a good mouser' iff Quijon is a good mouser. But our intuitions here cannot be correct, since substituting '.u does not satisfy x' for '4(x)' yields

r. Then s o are these sentences:

( X X V ) T ( ~ ( ~ ; ) ( F ( ~ , ~ c T ~T. (zz)) ) ~ ) (from (xxiv) by (3)) ( V, r l + z ~) ~ z + ( 2 )) ) (from (ijj)) (xxvii) (Vz)((F(S(k),-fl1.i) + ~ ( 2 ) ) (from (xxv) and (xxvi)) -+

It follows by mathematical induction that, for cach

17,

the sentence

-

y does not satisfy x)

which implies, absurdly,

-

'x does not satisfy .r' satisfies 'x does not satisfy x' 'x does not satisfy .r' does not satisfy 's does not satisfy .r'"

If we think of denotation as a relation bctwccn closed terms and individuals, we shall not be prcscntcd with any unpleasant surprises. Thus, the relation correlating a closed term of the language of arithmetic with thc number it denotes is a perfectly harmless recursive relation. When we encounter paradoxes is when we inquire into the denotations of definite descriptions. There are only countably many English definite descriptions, and s o there must be ordinal numbers that are not denoted by any definite description. But it looks as if the definite description 'the least ordinal not denoted by any English definite description' ought to name the least such ordinal. This contradiction, due to Julius Kiinig [1905], has a finitary version, due to G. G. "

This antinomy comes from Grelling and Nelson [I9081

err^:" There are only finitely many English definite descriptions containing fewer than forty syllables, so there must be natural numbers that are not denoted by any definite description containing fewer than forty syllables. But it would appear, absurdly, that the definite description 'the least natural number not denoted by any English definite description containing fewer than forty syllables' succeeds in naming the least such number in only thirty-three syllables. Another paradox of denotation, due to Jules Richard [19051. draws our attention to the phrase

r(iv)$(~.!)l denotes a iff

LI

satisfies r(V.'.r)($(s)++ x

=

v)1

Truth is definable in terms of denotation:

r41 is true

iff r(iv)(v = 0 & $)l denotes 0

These definability relations enable us to establish that our naive theory of reference is inconsistent by deriving the inconsistency of the naive theory of reference from the inconsistency of the naive theory of truth.

the number r between 0 and 1 such that, for every rz, the nth digit in the binary decimal expansion of r is equal to 1 iff 0 is the nth digit of the binary expansion of the number named by the alphabetically nth English definite description that denotes a real number.

COROLI,ARY 1.6. Let a(y,x) be a formula of 2. There is no theory consistent with R that entails all instances of the schema

This phrase should appear somewhere, say at the kth position, on an alphabetical list of English definite descriptions that name real numbers. Yet the number named by the phrase has to differ at the kth decimal place from the number named by the kth definite description. " These paradoxes show that our naive conceptions of truth, of satisfaction, and of denotation are all afflicted with inconsistencies. In view of the intimate connections between the three conceptions, it is not surprising that all of thcm should be inconsistent, if any of them is. In describing these connections, let me make use of the following standard notation: '(ix)(x is a so-and-so)' will be used to symbolize the definite description 'the so-and-so'. Thus, intuitively, for any y , '(ix)(x is a so-and-so) will denote y iffy, and y alone, is a so-and-so." Naively-that is, without taking the paradoxes into account-it would appear that truth and denotation are both definable in terms of satisfaction, as follows:

For the philosophically interesting application of this corollary, take cr to be a formula that is intended to express the satisfacion relation for Y . PROOF:If we take ' ~ ( z ) to ' be an abbreviation for ' ~ ( 0( ,z & rx = 01))', we see that a theory that implied R together with all instances of

r47 is true iff at least one individual satisfies r$ iff every individual satisfies I'

"

r+ & v

would imply all instances of the schema

.

which means, according to theorem 1.3, that the theory must be inconsi~tent.'~

COROLLARY 1.7. Let iS(x,y)be a formula of 2 . There is no theory consistent with R that entails all instances of the schema

& v = vl =

1.17

See Whitehead and Russell 11910. p. h l ] . One cannot help being struck by the intimate connection between these senlantic paradoxes and the antinomies that afflict naive set theory. Thus, the contradiction in naive serrlantics discovered by Grelling and Nelson is derived virtually word-for-word from Russell's [I9021 paradox about the set of all sets that d o not contain themselves. if we substitute 'general term' for 'set' and 'satisties' for 'is an clement of'. The cloae relation between KBnig'a paradox and Burali-Forti's paradox about the order type of the ordinals is likewise clear. When we examine Richard's paradox, u e see that its principal ingredient is the diagonal argument used in Cantor's theorem that, for any set S , the power set of S has more members than S. But this is the theorem used by Cantor [ 18991 to obtain the paradoxical result that the power set of the universe ha? more members than the universe.

To obtain a proper understanding of the connection between the semantic paradoxes and the set theoret~cparadoxes is a deep problem to which, regrettably, the present work has nothing to contributc. '' See Russell [ 19051.

For the philosophically interesting application, take 6 to be a formula that is intended to express the denotation relation for y. PROOF:NOWtake ' ~ ( z ) to ' be an abbreviation for '6((Lx)(rx= 01 & z), 0)'. Any theory that implied R together with all instances of

would imply all instances of the schema

which means, according to theorem 1.3, that the theory must be inconsistent.. We can define both denotation and truth in terms of satisfaction, and we can define truth in terms of denotion, but we cannot, in general, reverse these "

I am grateful to Shaughan Lavine for pointing out these simple proofs to me. I had thought proving corollaries 1.6 and 1.7 was a much rnore con~plicatedbusiness.

delinability relations. We cannot. in general. define either denotation or satisfaction in terms of truth. nor can we define dcnotation in terms of satisfaction. Intuitively. this is what we would expect. I f we know what sentences are tnle. we shall know which definite descriptions have denotations. but we need not be able to specify to which individual a given dcnoting definite description refers; we shall know which general terms are satisfied, but we shall not know which specilic individuals satisfy a given general terrn. If we know which individual. if any, a given definite description denotes. we shall be able to say, for any indiilidual that happens to be named by somc delinitc description, what general terms that individual satisfies. Rut what about those individuals which are not named by any detinitc description'? There is no reason to suppose that we should be able to specify what general terrns those nameless individuals satisfy. Let me now give a specilic example showing that, as expected, we cannot generally define satisfaction in terms of denotation:

where r(is)$(.r)l is a definite description in X ( e ) that. under '1, denotes b. Let a ( y , . ~ , c be . ) the formula of 2 ( c , c )got from a ( . ~ , n . , D by) replacing each occurrence of ' D ( 7 , p ) ' by ' < ~ , p > F c ' . Then we have

TIIEOREM 1.8. Let y ( e ) and :p(c.D) be, respectively, the lirst-order language whose only nonlogical symbol is the binary predicate ' e ' and the firstorder language whose only nonlogical symbols are the binary predicates ' F ' and 'D'; I presume that the language of arithcmetic and the notation for ordered pairs have been relatively interpreted into Y ( c ) in one of the standard ways. Let \'I be the model of Y ( e ) in which the universe is VsI and in which 'c' is interpreted as the restriction to )!)I] of the elenlenthood relation,'" and let t'l" be the rnodel l i ~ rY(t-,D)got from !)I by letting the extension of 'D' b e the set of all ordered pairs < r ( i x ) $ ( x ) l , b > where r(i.r)$(x)l is a definite description in Y ( e )that, under 91, denotes b. Then there is no forrnula a ( y , x , D )of 2 ( c , D ) such that, for each forniula $(x) of

Contradiction.. We next show that it is not generally possible to define either denotation or satisfaction in terms oftruth by presenting the following example, loosely derived from Richard's antinomy:

Y(t.), Thus the denotation relation for 2 ( e ) is definable within X ( c , D ) , but the satisfaction relation for 2 ( c ) is not definable within X ( c , D ) . PROOF: Suppose that there were such a formula. Let Y ( c , c ) be the first-order language whose only nonlogical syrnbols are the binary prcdicate ' e ' and the individual constant 'c'. Now the the extension in 1'1" of ' D ' is a countable subset of VsI x VsI, and every countable subset of VsI x VsI is actually an element of V,,. Hence, we can define an interpretation \!I~t" 1 would like to propose that we g o a step tarther and take 'BcII'(.v)'as a f o r m a l i ~ a tion of '.\ is logically necessary'. This will enable us to employ Godel's methods to investigate, precisely and in detail, the logical properties of logical neccssity. 'BPI\,'will depend upon 1'. which will. in turn. depend on how we choose to make our infomral notion of logical necessity precise. It will emerge, however, ' not delicately sensitive to which set of that the formal properties of ' B ~ N ,are sentences we take I- to be. We do, however, require that Peano arithmetic be relatively interpretable into 1'. Once we have settled upon a formula 'Bc~t,'to represent the set of logically necessary sentences. we can sce which of the conditions of theorern 1.4 needs to be relinquished. Condition (4) tells us that all instances of the schema Be\tt( r&l)

+

4

are to be counted as neccssary. Since the logically necessary sentences are precisely the consequences of T, it follows that

is a consequence of I'. But 'Rr1r3(ri0 = 01) -+ 1 0 = 0' is equivalent to ' i B r 1 t l ( r ~ 0= 01)'. which asserts the consistency of T. and so, according to Godel's second incompleteness theoren~( 193 1 1 , ' i ~ e u ~ ( r - = d )07)' will be a consequence of I- only if I- is inconsistent. Thus, if we interpret the modal operators by taking 'n' to represent logical necessity, s o that mq5 means the same as Bertl(r$l), we find that some instances of the schenia

are not neccssary. W e can say more. M. H . l i b 1195.51 shows that the conditional

will be 3 theorem of T only if its consequent 4 is a tlreoreni of T. Thc proof of this is a straightforward rrrodification of the argument just given:' Suppose that ~ p \ c , ( r $ l )-+ 4 is a theorem of T. Then -i~rrcj(r$l)isa theorem of 1' U (14). Now ' i B ~ ~ ~ t ~ asserts ( r ~ 1 )the ' consistency of I' U {1$} But, . according to GBdel's second incompletcncss theorern. 1' U { i d }will be able to prove its own consistency only if U { i d }is inconsistent, that is, only if 4 is a theorem of T. 'The proof of Lob's theorem can be formalized within T. and so all instances of the schema

r

are neccssary . We can, in fact. give a precise characterization of the modal schemata like ( L ) that have the property that all their instances arc necessary. 'To give this characterization, which is due to Robert Solovay 119761, we need some detinitions. W e suppose that we have fixed a language for the modal sentential calculus and '0'which has an ample supply of sentential with the connectives 'V', '1'. letters.

DEFINITION. An itlterl7r~t~itiotz of the modal language is a map ": associating a sentence o f 3 with each modal fornrula such that

(4

" 3,)*

(-14)s

(4" v +:". = ~ ( 4 " )and . =

(04)" = Bcll!(rq5:*l).

I~EFINITION. A modal formula Q, is T-\,cllid iff, for every interpretation

4" is a

logical consequence of

*,

T.

Thus, in our technical usage, necessity and validity are quite different things. Necessity is a property of sentences of Y, but validity is a property of modal formulas. It is the property a modal formula has if every sentence of 2 you get by interpreting the formula is necessary.

DEFINIIION. G is the modal deductive system given by the following axioms and rules: This derivation of L(ib'a theoreni from the second incon~pleteneaatheorem is an unpublished but wcll-known discovery of Kripke. Liib gavc a more d~rectarfurnent using the self-referential lemma.

Axiom schemata: All tautologies.

O(4

-

+

OCUQ,

$1

+(O$

+

O$)

4) -. ud,

Rules: Modus ponetzs. Necessitation. There is an algorithm for deciding whether a modal formula is a theorem of C .

THEOREM 2.1 (Solovay). Assuming that I' does not entail any false C: arithmetical sentences, the 1'-valid modal formulas will be precisely the theorems of G. For a proof see Solovay [ 19761, George Boolos 1 19791, or Carl Smorynski

1 l985J.. Notice the stability of this result. For any of a very wide variety of choices of our theory T, we shall get the same system of r-valid modal formulas. The research culminating in Solovay's results shows us that the modal logic of logical necessity is not at all what we would have expected it to be. The results are interesting, anlong other reasons, because of what they teach us about the methodology of semantics. Consider this principle:

(N) All instances of the scherna

04- 4 are necessary. Before the work of Godel and Montague, principle (N) might well have been regarded as so conceptually secure as to be inviolable. The principle, it might have been argued, was part of the meaning of the word 'necessary'. If someone denies the principle, even though he tells us that by '0'he means "it is necessary that," we would not know what he was talking about. Even though he uses the word 'necessary', he could not be talking about necessity; he must be using the word in a deviant way. Now it is not required by the meaning of the word 'necessary' that necessity should be a property of propositions; it might be an attribute of properties, of states of affairs, or of something else. On the other hand, it is certainly not required by the meaning of the word 'necessary' that necessity should rzot be a property of propositions. The concept which Leibniz referred to as necessity and which we have been referring to as logical necessity is, without doubt, a legitimate notion of necessity. I daresay that no one has ever responded to the quoted passage from the Monadology by saying, "This fellow Leibniz must be using the word 'necessary' in a queer way ." Thus, on at least one concept of necessity, necessity is an attribute of propositions.

There is a derivative notion of necessity as an attribute of sentences. A sentence is necessary just in case the proposition it expresses (in normal usage) is necessary. 'Bachelors are unmarried' is necessary; 'Buchanan was unmarried' is not. The partitioning of sentences into necessary, contingent, and impossible is just as apt for the sentences of an interpreted formal language as for the sentences of a natural language. Logical necessity, regarded as a property of sentences of a formal language, can be investigated, precisely and in detail, using the methods of Godel [1931]. When this is done we discover. much to our surprise, that principle (N) fails. We find ourselves trapped between three conflicting theses: [ I ] No legitimate notion of necessity can fail to satisfy principle (N). [2] Logical necessity is a legitimate notion of necessity. 13) Logical necessity does not satisfy principle (N).

Let me emphasize that the third principle is not a part of the meaning of the words 'logical necessity', except in a very dilute sense in which we regard even the most recondite logical consequences of the meanings of our words as parts of their meanings. Far from being evident to the ordinary speaker, thesis 131 is a deep and surprising result. What Solovay's theorem shows us is that if we abandon thesis [ I ] , as I recommend, our thinking about modalities is not reduced to incoherence. Nor do we find, as Montague fears, that "virtually all of modal logic . . . must be sacrificed [1963, p. 2941. On the contrary, if we abandon our preconception that thesis [ I ] has to hold and we investigate logical necessity systematically, following the logic where it leads us, we obtain a particularly rich and elegant modal logic. If we simply repudiated principle (N), without anything new to take its place, we would indeed be left with a system so weak that it would be useless, so weak that it could scarcely be recognized as a modal logic. But, in fact, we do not give up principle (N) without compensation. When we give up principle (N), there emerges a new and powerful modal principle, Liib's (L), which our dogmatic insistence upon principle (N) had obscured from view. What this suggests is that we should take a more holistic view of how the meanings of our terms are determined. Principle (N) is indeed part of the meaning of the word 'necessary', but other features of the way we use the word also enter into its meaning, in particular, Leibniz's idea that what are necessary are the truths that can be established by unaided reason. When different principles governing the usage of a word come into conflict, as they do most dramatically in Montague's theorem, there are no inflexible rules to determine which principles will emerge v i c t o r i o ~ s . ~ So far, our formal investigations into the logic of logical necessity have proceeded at a purely syntactic level. The definition of r-validity was formulated This point of view has been advanced forcefully by Quine. See, e.g.. [1951].

entirely in terms of dcducibility in a f'ormal sy5tem. without considering what the tcrnms of the formal language might rel'er to o r what sentences of the formal language might bc true. 4 was said to be r-valid iff. for every interpretation :+. 4" is provable. If we now take the formal languagc If to be interpreted by giving a first-order model 1' of T , we can define a new notion of !'I-validity:

( V modal sentence & ) ( 3 interpretation 4:': is not a theorem of I')

I)EFINITION. G ' is the modal deductive aystern given by the li)llowing axioms and rules:

Axiom schemata:

- -

00 for 0 a tautology UIU(ct, 4) ( 0 4 "[n(Ud, d) WI -+

UdJ,

I

-+

04 + 006 u4- 6

Rule: Modics pot~elzs

(;' is a proper extension of G. Like G , it is decidable

THEOREM 2.2 (Solovay). If !)( is a model of r U {true :' arithmetical sentences), then the 31-valid for~iiulaswill be precisely the theorems of G'. Again see Solovay [ 1976 1 , Boolos [ 19791, or Smorynski [ 1985 1 for a proof..

DEI~.INITION. A set A of modal formulas is 1'-consistc?nt iff there is an interpretation * such that (6": 6 E A) is first-order consistent with I-. A is !)(-consistent iff, for some *, all the members of 16": 6 t. A) are true in ?I. DEFINITION. A method for classifying sets of sentences as either consistent or inconsistent is (countabl\.) c-ornpclc.1 iff, for every (countable) set of sentences A , if every finite subset of A is countcd as consistent, then A itself is countcd as consistent.

is not a theorem of G

+

The result can be strengthened by providing a single interpretation that works for every choice of d :

DEFINITION. Let \)I be a structure fix 'f.A rnodal f'ol-mula d, is !'I-valid iff. Ihr each interpretation *. 6'" is true in !)I. Since we know that there are sentences of Ythat are true without being provable, it will not surprise us that there are rnodal f o r m ~ ~ l that a s are $1-valid without being r-valid.

:+)(#I

(3 interpretationt)(V rnodal sentence $)($ is not a theorem of G -,

4t

is not a theorem of

r)

For a proof see Sniorynski [ 1985, pp. 153f].' The proof relies heavily upon the countability of the modal language. Suppose that A is countable and that every finite subset of A is 1'-consistent. and let { a , , . . . , 6,,) be a finite subset of A . Since ( 6 , . . . . , a,,}is r-consistent, 1 ( 6 , & . . . & 6,,) is not r-valid. and so i ( 6 , & . . . & a,,) is not a theorem of C . Hence, ( 6 , & . . . & a,,)? is consistent with r . It follows by the compactness of the predicate calculus that {a+: 6 r A) is consistent with r, so that A is Tconsistent. The proof that we d o not have full compactness for either r-consistency or 31consistency is easy: let A be { i o(p,, p,): cu and p are distinct countable ordinals). Since we can find an infinite set of sentences of 2 no two of which are provably equivalent-for example C o n ( r ) . C o n ( r U {Con(T}), C o n ( r U {Con(l'), C o n ( F U {Con(l')}))), . . . (where Con(.Il) is the natural sentence asserting that 11 is consistent)-every finite subset of A is both T- and !'Iconsistent. On the othcr hand, A is neither r-consistent nor '?(-consistent, since, the language :P being countable, for each interpretation * there must exist a f p with p,:+ = pB*, SO that (o(p,, tt p,))* is provable. Finally, to see that ?(-consistency is not countable compact, let C and I!) be disjoint, recursively inseparable recursively enumerable sets."' Where O n T is the result of prefixing 11 ' 0 ' s to the tautology T . let A be the following thcory:

-

':

-

{0((0"T & i o " " ~ + ) p ) : 11 E C) U { n ( ( O " T & i 0 ' " ~ ) 11,): 11 e D) u {o((o"T & ~ V " + ~ T ) + ~ I ) + - + ~ U ( (&Oi "OT " " ~ ) + i ~ )all: 11). A typical finite subset of A has the form A

=

PROPOSITION 2.3. Let $1 be a model of r U {true :' arithmetical sentences}. Neither r- nor !)I-consistency is compact. r-consistency, however, unlike !)I-consistency, is countably compact. PROOF: (This requires some results not proved here.) The surprising result is the countable compactness of r-consistency. It is obtained as a direct corollary of the following theorem: Fix a countable language for the modal sentential calculus. Solovay's result is that

"

"

Sniorynski gives two proofs. one due to Franco hlontapna and Albcrt Vi.;ser and the other due to S . N A r t ~ m o v .Arnon Avrun. and (3corgc Boolos. C and D arc dis~oint.recursi\.ely enumerable sets \uch that thcrc i\ no recursive set that includes C and is di\jo~ntfronl D . For the construction of ~ c sets. h \ce Roger5 (1967. p. 941.

where 6 ,D , and E are f nite. By making use of the fact that, for each n. 0( 0"T 10'' IT) is true in I'! (a fact which follows directly from Godel's second inco~npletenesstheorern), it is not hard to see that an interpretation * that sets p* r c / ~ atnM. l ( 0 "T_& 1O " ' ' T ) (where denotes a long disjunction) will take each member of h to a sentence true in 91. On the other hand, A as a whole cannot be satisfied. For suppose that there were an interpretation such that. for each 6 in A. 6* is true in !'I, and let B = {n:(u((0 "T & 10 "" T ) + p))* is true in \'[I. Then, because the same 2:' sentences are true in f!' and in !I?, B = {n: ~ e w ( r ( ("T 0 & 1 0 " '' T -+ p ) * l )is true in !'I) = {n: Bew((( 0 "T & 1 0 " ' ' T ) + p ) " ) is true in 9)) is recursively enumerable. Similarly, the complement of B = {n: Bew(r((0 " T & 1 0 " + I T ) + I[?)*) is true in 91) is recursively enumerable. But this means that B is a recursive set separating C and D, contrary to hypothesis. Notice that, by Craig's theorem, 4 can be axiomatized recursively, so that "1validity is not even recursively compact.". In addition to the notion of necessity expressed in the quotation at the beginning of the chapter, Leibniz had another way of characterizing necessity which he regarded as equivalent: necessity is truth in all possible worlds. Kripke 11959, 19631 took Leibniz's characterization at face value, developing possible-worlds semantics for modal logic. Kripke liberalized Leibniz' conception by introducing a relation of accessibility between worlds. To be counted as necessary in a world w , a sentence need not be true in cvery world; it only need be true in every world accessible from w. What statements are necessary will vary from world to world; the actual determines the limits of the possible. If we want to obtain a possible world semantics for the notion of logical necessity we have been developing here, the natural way to proceed will be to take a possible world simply to be a model of T.We shall not want every world to have access to every other. It may happen that, even though 6 is not a theorem of r, a model ?[ of I will contain a nonstandard "proof' of 0, in which case we shall want models of 1 8 to be inaccessible from !'I. 91 regards 0 as logically necessary, so it does not regard any models of -10 as representing genuinely possible situations. The precise notion we need is the following: &

'\u'

"

DEFINITION. Where we take a world to be a model of T, a world 3 is accessible from world ?I iff, for each sentence 0 , if ~ e w ( r O 1is) true in ?I, then H is true in %. Assuming that r does not entail any C',' sentences that are false in \I?, we see that, if every 2:' sentence that is true in ?I is true in !It, then, for any sentence 4 , if Bew(r41) is true in ?I,then 4 is really a theorem of and so 4 is true in every

r,

"

'That is, there is a recursive !'I-incons~stent sct every finite subset of which is \'I-consistent

xy

possible world. Thus. if every sentence true in ?'I is true in ?li. then every world is accessible from \'I. Conversely, suppose that every world is accessible frorn 91. For any sentence 4, if $J is true in \'I. then B r w ( r $ J l )is true in \)I (since all instances of the schema (4 -+ Berv(r41)).with 4 2:. are provable in PA). Hence, 4 is true in every world accessible from !'I, and so 4 is true in 9i. Thus, we sce that a world \!I has access to every world iff every C': sentence true in !)I is true in 9i. In particular. if every 2';sentence true in !l' is true in 9?, then Yl has access to itself. But not every world has access to itself; this is because

x':

is not r-valid.

PROPOSITION 2.4. For any sentence 4 and world $1, ~ e w ( r 4 1is) true in \'I iff 4 is true in every world accessible from !'I. PROOF:The left-to-right direction is immediate. To get the right-to-left direction, let us assume that ~ e w ( r 4 1is) not true in 91 and try to find a world %, accessible : must be a model of I . and to be from 91. in which 4 is false. To be a world, 2 accessible frorn !'I, % must be a model of {H: Bew(r01) is true in !)(). Thus, we want to show that

r U (8: Betv(r61) is true in !'I) U ( 1 4 ) has a model. Since I' of

(6: Bew(r81) is true in "I}, it is enough to find a model

{ O : Bew(r01) is true in 91) U

{i4}

By the compactness of the predicate calculus, if there is no such model, then there exist sentences $, $?, . . . $,, in (6: Bew(r01) is true in ?l) so that

.

is valid. Hence,

is true in 91. By n applications of the fact that the conditionals B e ~ > ( r+ $ 01) -+ (Berv(r$l) + B e w ( r 0 l ) )

are theorems of I' and so true in \!I, we conclude that Bew(r47) is true in 91, contrary to assumption.. Quine [I9531 distinguishes three grades of modal involvement, differentiated by the noxiousness of their metaphysical commitments. At the first and most innocent level, one treats necessity as a semantical predicate one applies to sentences to indicate their logical or epistemic status. At the intermediate level, one treats 'necessarily' as an operator one attaches to sentences to produce new

sentences, and at the deepest level one treats it as an operator one applies to open sentences to produce new open sentences with the same free variables. At this deepest level, one is plunged into "the metaphysical jungle of Aristotelian essentialism" 11953. p. 1741. We have seen that, propcrly understood. involvment at the second level can l )that , at the be entirely benign. u& can be taken to mean the same as ~ e ~ . , ( r &SO second level one expresscs the same information expressed at the first level. using a notation that perspicuously exhibits the logic of provability. What are the prospects for continuing this process by giving a syntactic treatment of quantified modal logic? The results here are mixed. The principal obstacle Quine sees to treating the essential attributes of a thing as those attributes the thing can be proved to have is that what attributes a thing can bc proved to have will depend on how the thing is named. Thus. the actor who played the principal human role in Bc~dtit77efor Bot~zois provably an actor. where as the President of the United States in 1987 is not provably an actor. As Kaplan (1969, # 181 points out, we can solve this problem if we can somehow introduce standard names that refer directly to their bearers without introducing extraneous contingencies. If N is the standard name for ( I , we can treat the statement that a is essentially F' as equivalent to the statement we make by prefixing the operator 'n' to the sentencc :' is an F ' . In this way, we can reducc necessity de re to necessity d e ditto, thus rendering the third grade of modal involvement no more perilous than the second. If is an F)' can, in turn. be explicated as ' ''Z is an F" is provable (in sorile appropriate axiomatic system)', then the Aristotelean jungle will have been completely tamed. In constructing Y!,, out of Si', we developed the technical apparatus we need to implement Kaplan's proposal. Let !)I be the actual world, the intended model of Y ;let 1' be a theory expressed in X,,;and lct ' B ~ Mabbreviate " an open sentence of It':),which represents provability in r. Fix a language fol- the modal predicate calculus with identity that has no individual constants o r function signs but has a infinite supply of n-place predicates. for every tz. An irlterprctation will be a map :k taking modal formulas to formulas of ,lJ' so that:

'~(z

4" is a formula with the same free variables as &. If d, and ( / I are atomic formulas consisting of the came predicate followed by different variables, d,* and d ~ *will be just alike except for a corresponding change of Sree variables (and, il'necessary, a change of bound variables to avoid collisions). (V = = (v = kt,) &,I)*

(4 v q,)*

=

4:sv

$:"

as contrasted with the innocuous

I believe r(gx)(x is a spy)1 .'' Even if we use the u s as standard names of the referents of de re beliefs, we shall not get enough of the n s into to permit the treatment of the statement that LI is essentially F as a claim that is F' is a theorem of 1'. Consider the fact that, on the received view, every material body is essentially a material body. This means that, for each material body b , '6 is a material body' is a consequence of T.Now, in general, for any individual constant c , if O(c) is a consequence of then, unless the generalization (Vx)O(x) is a consequence of T, the constant c must appear in T. Now, '(Vx)(x is a material body)' is not a consequence of l', since not everything is a material body, and so, for each material body b, the individual constant h must occur in 1'.But not every material body is the referent of a de re belief. O n some accounts, to have dc re beliefs about a thing, I must be in direct causal contact with it. and, on other accounts, I must have an appropriately vivid name for it,'' but, on no account, d o I have de re beliefs about each material object in the universe. In order to succeed in thc proposed explanation of the essential F-ness of u as the derivability of is an F' from T, we must think of r as a purely abstract

'a

r,

'a

(TO)* = -,(d,*) ((31~)4)* = (3v)(&*) ( n $ ( ~. ~. ,. , v,,))* = ~e\t,(r(tb(?,..

r-validity and "1-validity can be defined just as they were for the modal sentential calculus. This technical apparatus is helpful in understanding quantified modal logic in just the way a tractor would be helpful to a farmer who had no land: the machinery does the farmer no good, for he has no land to use it on, but if he ever gets some land, the tractor will come in handy. What we need in order to put the machinery to use is the theory 1'. Now. the theory 1' is certainly not a theory that we believe, if by believing a theory we mean holding the axioms true. T o hold a sentence true means that one would sincerely assent to a token of the sentence. and the sentences of %!, are not the sort of thing that can have tokens. Sentences of Y are or can be realized by concrete tokens, but the extra sentences in 2!,[ are purely abstract entities without concrete realizations which we introduce to fill out the logical structure of possible concrete tokens. W e might want to extend our ordinary way of talking by admitting sentences of 2,, as possible objects of belief, perhaps to help us talk about de re belief attributions. Thus, the de re reading of ' I believe someone is a spy'-the sense in which I a m saying something that might bc of interest to the F.B.l-is

.

.

. {,)*I) -

" "

This example 1s discussed in Quine [ 19561 This is Kaplan's [I9691 proposal.

object. rather than a concrete theory that someone might actually believe. This need not be an impediment to the syntactic treatment of quantified modal logic, since metaphysical necessity, on the standard view, does not depend on human speech or thought. The obstacle is that wc do not know what basic metaphysical facts are to be included in T.What we have to guide us are a few striking examples, primarily from Kripke [1972]: gold is essentially metallic but contingently yellow; one's parentage is essential but one's birthplace accidental; and so on. Although these examples are helpful in making specific modal judgments, they are no help at all in understanding the laws of modal logic. For that. we need know the computational complexity of r and the logical structure of "Bew": we need a global understanding of the nature of necessity, not just specific exa~nples.As it is, we have an attractive logical framework but nothing to hang on it.'" Whereas the development of a general syntactic treatment of quantified modal logic must await further advances in metaphysics, therc is an important special case for which the development can proceed at once. This is the case in which the language is the language of arithmetic, the theory I- is a recursively axiomatized extension of Peano arithmetic, and the intended model is 92. For this special case, the intended model already contains a standard name for each object in the universe of discourse, so there is no need to go from 2 to to:,, . The development here is a straightforward extension of the work on the modal sentential calculus. Although it is obviously a very special case, it is also obviously a very important special case, so we shall look at it in some detail. Much of the development of the modal sentential calculus can be carried over without incident. Thus we can develop a possible world semantics for quantified modal logic as follows:

PROPOSITION 2.5. Take a possible world to be a model of T. Say that a + 1 91 possible world % is accessible from \)I iff there is a function j such that, for any formula +(v,, . . . , v,,) of the language of arithmetic, if ~ e w ( r 4 ( & ,. . . F Q + ( v is a closed term & [ ( v = 'ol & M' : 0) V (3t)(3u)( E Q & v = S ( t ) & w1 = S ( u ) ) V ( 3 q ) ( Z r ) ( 3 t ) ( g u ) ( < q , r >F Q & F Q & v=q+t&w=r+u) V ( 3 q ) ( g r ) ( g t ) ( g u ) ( < q , r >E Q & F Q & v = q : t & w = r.u)j)) & E Q ]

is a materially adequate definition of truth in the sense of convention T, which requires that every biconditional got by substituting a sentence of -2 for C$ in the schema

be a consequence of the definition. By 'consequence' here, Tarski refers to the consequences of some unspecified nonsemantical theory, formulated in the metalanguage, whose acceptance is taken for granted. The notion of a materially adequate definition is the same as the mediaeval conception of a nominal, as contrasted with a real, definition. A nominal definition merely picks out the right extension, whereas a real definition gives the essence of the thing defined. The inevitable example is that 'featherless biped' g. 'IV~Sa nominal and 'rational animal' a real definition of 'human being'. Tarski regards it as a minimal requirement on a satisfactory definition of truth that it be materially adequate in the sense of convention T , and he shows how to meet this minimal requirement. Of course, a definition that meets this minimal standard might nonetheless be, in many ways, an unsatisfactory characterization of the thing defined. It might be woefully uninformative and a poor guide to future research. But such defects need not force us to repudiate such a definition. Suppose that

--

is a real definition. Then it will be a serious flaw in a theory of ps, if the theory does not tell us that ( V x ) ( xis a p O(.r)). But it is, at worst. a pedagogical defect if the theory treats ( V x ) ( xis a p +(x)) as a definition and ( V x ) ( xis a p %(x)) as a theorem. rather than vice versa. (If modal considerations enter into our metatheory, we might be able to distinguish the two types of definition modally; O(Vx)(xis a p O(x))will be true, but n(V,r)(x is a p + ( x ) )might not be.) To illustrate Tarski's method, let us take our object language to be the firstorder language of arithmetic and our metalanguage to be the second-order language of arithmetic. Using syntactic operations S , and : so that ~ ( r r 1 = ) r ~ ( r ) lrr1 , + rpl = rr pl , and rr1 : rpl = r ~ . ~we l ,first find formula that specifies the denotations of closed terms:

Because we can use natural numbers as codes for finite sets of natural numbers, letting the code for the finite set S be we do not really need the secondorder quantifier to define the denotation relation. Replacing talk about finite sets by talk about their codes, we may think of 'Deiz(x,y)' as a formula of the firstorder language of number theory. We use this first-order formula 'Den(x,y)' in giving our second-order definition of the set of first-order truths:

3.2',

Tr(x) ++ ( V R ) [ [ ( V y ) ( R (y, ) y is a first-order sentence) & ( V y ) ( V z ) ( yand z are closed terms + ( R ( y 5 z) ++ ( 3 v ) ( 3 w )(Den(y,v) & Den(z,w) & v = w ) ) ) & ( V y ) ( V z ) ( yand z are closed terms + ( R ( y z) ( 3 v ) ( 3 w )(Den(y,v)& Den(z,w) & v < w ) ) ) & ( V y ) ( V z ) ( R ( yV 2 ) ( R ( y )V R ( z ) ) ) & ('d sentence y ) ( R ( l y )e i R ( y ) ) & (V variable v ) ( V y ) ( R ( ( ? ~ l ) y ) (3 closed term t)R(y vlt))l -. R(x)l

<

++

-

-

-

It is easy to check that this is a materially adequate definition. Here, because we are talking about an infinite set, the quantification is irredeemably second-order. Although this is a specialized example, the technique it illustrates is quite general. It can be applied to a wide range of interpreted formal languages. The

sine qua nnn for its application is that one have available the scmantic resources of an essentially richer metalanguage.' Given this requirement, the prospects for applying Tarski's method to give a definition of truth for a natural language are poor, indeed. Such a definition would have to be formulated in a metalanguage essentially richer than the natural language, and there is every reason to believe that there is no such language. According to Tarski 1 1935, p. 1641,

A characteristic feature of colloquial language (in contrast to various scientific languages) is its universality. It would not be in harmony with the spirit of this language if in some other language a word occurred which could not be translated into it; it could be claimed that 'if we can speak meaningfully about a thing at all, we can also speak about it in colloquial language. ' It may well be that Tarski overstates his case. Translators oftcn complain that, in translating one natural language into another, it is sometimes possible to give only a crude approximation of the meaning of the original text; speakers of one language use words whose meanings are not part of the conceptual repertoire of speakers of the second language. But these difficulties are not one-sided, as they would have to be if they were going to form the basis for a contention that the first language was richer than the second; one has difficulty translating English into Urdu, but one has just as much difficulty translating Urdu into English. Even if the difficulties were one-sided, they are not pervasive enough to support a contention that one language is essentially richer than the other. Natural languages are roughly equal in expressive power. Turning to formalized languages, we find that even very powerful formal languages, such as the language of set theory (in its standard interpretation, with its variables ranging over all sets), are poorer in expressive power than natural languages. We can see this by observing that we can readily translate sentences of the language of set theory into English; indeed, it is by learning such translations that we learn what expressions of the language of set theory mean. Mathematicians have described abstract formal languages containing expressions of infinite length, some of which have very impressive expressive power. But since we cannot

'

One special feature of the language of arithmetic which we are exploiting here is that, in thc language of arithmetic under the intended interpretation, every individual is denoted by some closed term. If our object language lacks this special feature. therc are two w a y we may proceed. The rr~ethodwe have favored here is first to extend the language by adding a name for every individual-advancing frorn Y to .Y:,,-then to define tnlth for the extended language. The nrethod Tarski hinrself employs is to define satisf:~ctionas a relation between variable assignment5 (functiorrs that atsociate an individual with each variable) and fornlulas. then to say that a sentence is true ilT it IS satisfied by at least one var~ableassignment iff it is satisfied bq every variable assignment.

speak these languages, we cannot use them to give an definition of truth for English. We can speculate that, someday in the future, human beings, by an awesome intellectual feat, will teach themselves a language essentially richer than presentday English in expressive power. But even these superhumans of the future, though they could give a definition of truth for present-day English, would not be able to use Tarski's method to give a definition of truth for their own tongue. If we adopt Tarski's preferred policy for avoiding antinomies, we cannot obtain a semantics for the natural languages we speak; we can only obtain semantic accounts of languages essentially poorer than those languages in expressive power. Tarski accepted this conclusion with remarkable equanimity. He is content to observe that investigations in a specialized science, such as chemistry, do not require a language with the full expressive power of natural languages. Chcmical investigations can be conducted in a restricted language that contains terrns like 'element' and 'molecule' but need not contain names of linguistic objects. This restricted language will be amenable to Tarski's methods. "There is," Tarski says, "no need to use universal languages in all possible situations. In particular. such languages are not needed for thc purposes of science (and by science hcrc 1 mean the whole realm of intellectual activity)" [1969, p. 681. Tarski continues:

I

I

I i

i 1

I

1

The situation becomes somewhat confused when we turn to linguistics. This is a science in which we study languages; thus the language of linguistics must certainly be provided with the names of linguistic objects. However, we do not have to identify the language of linguistics with the universal language or with any of the languages that are objects of linguistic discussion. The language of linguistics has to contain the names of linguistic components of the languages discussed but not the names of its own components; thus, again, it does not have to be semantically universal. [1969, p. 681

1 I

This is hard to understand. If linguistics is to use Tarski's methods to study the senlantics of natural languages, the language of linguistics has to be essentially richer than the natural languages in expressive power. But natural languages are, according to Tarski, universal, and we cannot speak any language essentially richer than a universal language. Thus, it appears that linguistics has to be restricted, so that it is only permitted to talk about narrowly circumscribed fragments of natural languages. But if science is indccd to encornpass the whole realm of intellectual inquiry, why is it not permissible to inquire scientifically about the meanings of the tcrms of the very language we speak? Tarski gives a second method for obtaining semantic theories. Although still not enabling us to obtain a semantics for a natural language, the new method

enables us to get semantic theories for languages only slightly poorer than natural languages. It is to this second method that we now turn. Tarski's second proposal (1935. $51 is that, rather than give an explicit definition of truth, we give a theory that irnlrlicill~defines the set of true sentences. We form the metalanguage from the ob.ject language by adjoining the single new predicate ' T r ' , and we take as our theory of truth the set of all biconditionals

for a sentence of the object language. It is obvious that this implicit definition is materially adequate in the sense of convention T. Tarski regards this approach as unsatisfactory for two reasons. The first reason is that the axiomatization obtained "would be a highly incomplete system, which would lack the most important and most fruitful general theorems" 11935, p. 2571 For example, for each sentence 4, the theory will enable us easily to prove

informative definition. Using it. one can prove, for example, that the rules of first-order logic are sound and complete, in the sense that a schema is a theorem of first-order logic if and only if all its substitution instances are true. Tarski's second objection 11935, p. 2581 is that a proper implicit definition ought to be categorical in the following sense: a theory r ( R ) , implicitly defining a predicate R will be categorical iff from T ( R )and T ( R 1 one ) can derive ( V x ) ( R ( x ) t , R 1 ( x ) ) .Neither of the proposed implicit definitions of truth is categorical in this sense. But the categoricity condition is surely too strong, since, according to Beth's theorem (theorem 2.2.22 of C.C. Chang and H.J. Keisler 11973]), whenever the condition is met the concept defined will already be explicitly definable. But although no implicit definition that cannot be made explicit meets the categoricity condition, our implicit definition of truth comes rather close to meeting it. If we let r ( T r ) consist of the six axioms given above for implicitly defining the truths of arithmetic, then from r ( T r ) and 17(Tr'),together with the Peano axioms strengthened by ullowirzg the predicates 'Tr' and 'Tr" to uppear in instances of' the induction axiom schema, one can indeed derive ' ( V x ) ( T r ( x ) T r l ( x ) ) ' Thus, . it would appear that the origin of the difficulty is not the inability of the semantic theory to pick out the referents of semantic terms, but rather the inability of number thcory to pick out the referents of arithmetical terms. Observe the connection between our implicit first-order definition and our explicit second-order definition. If we let 'y(Tr)' abbreviate the conjunction of the six axioms of the implicit definition, the explicit definition is

!

-

We cannot, however, prove the generalization (V sentence x)(Tr(x)V T r ( 1 x ) ) since this generalization depends upon infinitely many axioms and no proof can use more than a finite number of premisses. This objection points out a serious defect in the particular implicit definition of truth that Tarski considered, but there are other ways of implicitly defining truth which are not subject to this objection. To illustrate this. let us again take our object language to be the language of arithmetic. We form the metalanguage by adding the new unary predicate 'Tr' to the object language. Recall that 'Den(x-,y)' is an abbreviated formula of the object language. Our theory of truth consists of the following six axioms:

-

( V y ) ( T r ( y )+ y is a sentences of the object language). ( V y ) ( V z ) ( jand i are closed terms -+ (Tr(y y z) (3v)(3w)(Detz(y,v)& Den(z,w) & v = w ) ) ) . ( V y ) ( V z ) ( yand z are closed terms -, (Tr(y z) ( 3 ~ ) ( 3 w ) ( D e n ( y ,& v )Den(z,w) & v < w ) ) ) . (Vy)(Vz)(Tr(yV z) * ( T r ( y ) T r ( z ) ) . (V sentence y ) ( T r ( l y ) i T r ( y ) ) . ( V variable v)(Vy)(Tr((?v)y)e ( 3 closed term l)Tr(y vlt).

<

-

Here two of the axioms--one for each predicate in the object language-give the truth conditions for the atomic sentences, and the following three axioms indicate how the truth values of compound sentences are determined from the truth values of simpler sentences. This is a materially adequate definition. It is also a quite

This technique for turning an implicit definition into an explicit higher-order definition is due to Frege [1879],and it is available to us whenever we have a finite system of axioms implicitly defining a function or relation on a set. A typical example of Frege's technique is got by turning the usual recursive definition of cxponentiation x0 = 1 x5i'J= into an explicit second-order definition: x' = z =

", ( V R ) [ [ ( V w ) ( R ( w-+) w is an ordered pair) & R() & (Vu)(Vv)(R(

I

++

R()]

+ R()l

Godel [I9311 took the process a step further, showing how to turn the recursive definition into an explicit first-order definition; here we use the notation for finite sequences which we developed at the end of chapter 1:

---

Godel was able to do this because any particular value of the exponent function depends upon only finitely rnany previously obtained values of the exponent function. We cannot apply the same method to turn the second-order dcfinition of truth into a first-order definition, because the truth-value of a quantified sentence will depend upon the truth-values of infinitely rnany substitution instances. In applying Frege's method to convert an implicit to an cxplicit definition, we do not have to have a higher-order logic; a first-order language in which we can talk about sets will often do just as well. Thus, if our object language is a firstorder language 2,built from a finite vocabulary, that is interpreted by a model \'I, we can give, within the language got from Y by adjoining the new unary predicate ' T r ' , a finitely axiomatized theory that implicitly defines the true scntenccs of T!%i,,. The technique for doing so is just the method we used above for the language of arithmetic. Let y(Tr) be the con.junction of the axioms, and let y ( z ) be the formula got from y(Tr) by replacing each occurrence of a subformula of thc form 'Tr(p)' by ' p E 2 ' ; hcre 'z' is to be a variable that does not occur in y(Tr). We can now give an explicit definition of the true sentences of Y!,,,as follows:

Thus, we see that, whenever our object language is a first-order language with a first-order model, built from a finite vocabulary, Tarski's second method will not accomplish anything that could not be done just as well by Tarski's first method. The extra versatility of Tarski's second method only becomes evident whcn we look at languages like the language of set theory, whose variables are not restricted to any set. Such languages are not interpreted, in the technical sense, but they can still be meaningful, and we can describe what the symbols of the languages mean by saying how to translate the symbols into English or some other familiar language. If we extend such a language by adding a new constant for each of the individuals that the object language talks about, the grammar of the resulting language will be perfectly ordinary, even though the sentences of the resulting language will not form a set. We can give an implicit definition of truth for the extended language, just as we did for the language of arithmetic. For example. if Y is the language of set theory and il', is the language obtained from Y by adding a new constant ii for each set u , we can take our theory of truth for 2, to be the following: (v.v)(Triy)-+ v is a sentence of Y , ) ( V y ) ( V ~ ) ( ~ r (=r i31 ; =

( V ~ ) ( V Z ) ( IE~ :~I)( ~ . ?.v E ): ( V y ) ( V z ) ( T r ( yV z) ( T r ( y )V TI-(,-)) ( V sentence ! ) ( ~ r ( ? ~ ) ~ T r ( y ) ) (V variable ~ l ) ( V x ) ( T r ( ( ? t , ) y ) ( 3 z ) T r ( yv i ; ) )

I

I

I

-

It is easy to scc that this characterization of truth is materially adequate in the sense of convention T. If we attempt to turn this implicit definition into an explicit definition as we did before. taking y ( T r ) to be the conjunction o f the six axioms, taking y to be the formula got from y ( T r ) by replacing each occurrence of 'Tr(p)' by ' p t. z', and writing

we find that, since there is no set that contains all the truth sentences of %,-the true sentences of Y,, form a proper class-there is no sct that satisfies y ( z ) , and so our definition implies, absurdly.

1

the original language of set theory without the added The true sentences of :fie, constants, do form a set. Although we have several methods for implicitly defining this set, none of them gives rise to an explicit definition. We can define the true sentences of 2 as those true sentences of Yv which happen to be sentences of 2 , but the detour, getting to the true sentences of 2 by way of the true sentences of 2,, makes this implicit definition inexplicable. Similarly, we can define the true sentences of 2 in terms of the satisfaction relation on 2, but since the satisfaction relation on if is a proper class, the same problem arises. We can avoid any disruptive detours through proper classes by implicitly defining the true sentences of 2 directly, taking our theory of truth to consist of all biconditionals

for C#J a sentence of 2 . But now a new problem arises; since the theory is not finitely axiomatized, we cannot form the conjunction y(Tr). As we would anticipate from theorem 1.3, none of these attempts to produce an explicit first-order definition is successful. F Gset ~ theory, unlike number theory, moving to a second-order logic docs us no good. Where y ( T r ) is the conjunction of the six sentences that implicitly define the true sentences of Yv, we write

'

If, alternatively, we take our second-order definition to be 'Tr(.r) conclude '(V.t)lTr(.x)', which is equally absurd.

- .,(3:)

( y ( z ) & .r

R

:)',

we

But this gets us no further than its first-order counterpart. If we understand the second-order variables in the usual way, as ranging over all sets formed from those individuals over which the fi rst-order variables range, we find that the second-ordcr variables range over all sets of sets. But every sets of sets is already within the range of the first-order variables, so that the second-order variables get us nothing new. Moving to a second-order logic accomplishes nothing: we still get

The second-order definition will prove to be successful if, making the technical distinction between sets and classes, we take the second-order quantifiers to range over classes of individuals. Assuming a sufficiently powerful theory of classes (GiidelBemays is not enough), we shall indeed find that the second-order definition,

is materially adequate as a definition of the first-order truths of Y,, showing us that the language of class theory is essentially richer than the language of set theory. But this is not a philosophically satisfying resolution, since we encounter the same old difficulties when we attempt to give an explicit definition of truth for the language of class theory. For a more philosophically interesting example, let us take our object language to consist of the first-order language of set theory together with the basic vocabulary of physics. Under the intended interpretation, the variables range over physical objects and sets built up from physical objects.' Let me refer to this language as "the language of physics"; there may be some question about whether this appellation is apt, but it seems clear that the language of physics should include the basic physical vocabulary together with a mathematical vocabulary, and what I say here will not be sensitive to precise detail^.^ We can give an implicit definition of truth for the language of physics just as we did for the language of set theory. The reason this example is philosophically interesting is that it provides a prima,facie counterexample to a doctrine we may call linguistic physicalism, the doctrine that every (genuine) property can be described within the language of physics. So characterized, the doctrine utilizes the excessively vague notion of a property being describable in a language, but it has a tolerably precise consequence, viz.,

' Thus, wc let U,,, the set of so-called ~ r r ~ l r r n r ~consist ~ t s , of all physical

'

objccts. U,, , consists of the physical objccts together hith all subsets of ti,, and U, for A a limit IS ,,UhU,.. Our intended universe of discourse consists of everything that is in any of the U,,a, for cu an ordinal. Set theory with urelements is discussed in Barwise [1975. # I . l j and in Field [1980. ch. I]. Field [I9801 argues that, by artful coding constructions. cnough mathernatica to do science can be relatively interpreted into a language in which we only talk about physical objects. If that is so, we can take the language of physics to be a language that only talks about physical objects. For what we say here, thia change will makc no difference.

Every scientifically legitimate general term is coextensive with some open sentence of the language of physics. One might, I suppose, object to this alleged consequence of linguistic physicalism on the grounds that not every scientifically legitimate term needs to refer to a genuine property. Thus, one could claim that, although every property is expressed by a term of the language of physics, there exist scientifically legitimate terms that do not refer to properties and are not expressible in the language of physics. It is not so clear what the physicalistic basis might be for this distinction between those scientifically legitimate general terms which express genuine properties and thosc which do not. In any event, this would not be a happy objection for the physicalist to make, since the watered-down physicalism that results from it would be neither interesting nor important. Once the physicalist admits that there arc legitimate scientific terms that are not reducible to the language of physics, physicalisn~becomes nothing more than a quaint sect advocating an obscure restriction on the use of the word 'property'. The philosophically interesting version of linguistic physicalism accepts the thesis that every scientifically legitimate term is coextensive with some term of the language of physics. The primafacie counterexample to this thesis is the term 'true sentence of the language of physics'. This is a scientifically legitimate term, presumably, yet we know from theorem 1.3 that it is not coextensive with any term of the language of physics. It is open to the linguistic physicalist to respond by insisting that the expression 'true sentence of the language of physics' is somehow illegitimate. But it is by no means evident what is illegitimate about the expression 'true sentence of the language of physics', other than that it is an embarrassment for physicalism. We are able to use and understand the term, and, what is more, we are able to uniquely specify what it refers to by giving an implicit definition.' Moreover, it appears that the physicalist must himself employ the notions he wants to castigate as illegitimate. For the physicalist wants to claim that every scientifically legitimate term is coextensive with some term of the language of physics. But if 'true' is forbidden, 'coextensive' must likewise be forbidden, since truth can be defined in terms of coextensiveness, as follows: r $ l is true =, rx = x & $1 is coextensive with

rw .

=

xl

Thus, physicalism itself would appear to be one of those extrascientific metaphysical doctrines which physicalists want to eschew. A closely related difficulty is this: physical realism is, on one prominent account,' the doctrine that the terms

'

Morc prcciscly, the implicit definition un~quelyspecifies the referents of 'true sentence of the language of physics' mod~rlothe assumption that the referents of the terms of the language of physics are uniquely determined. This characterization of realism is due to Richard Boyd. and it is cited with approval by Putnam (1975, p . 731 and van Fraassen (1980, p. XI.

of the language of physics refer. But one cannot consistently hold that there is a genuine relation of reference, yet deny that there is a genuine property of truth, since, as we saw in chapter 1 , truth is definable in terms of reference. So the linguistic physicalist is forced to disavow physical realism. To understand the significance of this counterexample to linguistic physicalism would require a substantial philosophical investigation. Are we witnessing a deep difficulty with the physicalist tendency in philosophy or a superficial difficulty caused by an infelicitous expression of the tendency? Can we find an alternative formulation of linguistic physicalisnl which is frce of these logical problems'! To investigate these questions would take us too far afield. The purpose in raising thc questions is merely to reinforce the view that investigations into thc logic of truth have more than merely technical interest. Returning fro111 digressions, let us turn to a question that is central to our prescnt inquiry: To what extent is Tarski's second method useful in understanding the semantics of natural language'? We cannot use the method to give a semantics for an entire natural language, since we do not possess even an inessentially richer metalanguage. We can, however, use the method to get a theory of truth for a substantial fragment of a natural language. We work backward, starting with English as our tnetalanguage and carving out our ob.ject language from the metalanguage by excising all the semantical terms. Our semantic theory will be the naive theory, restricted so that it only applies to the object language. If we follow this plan, we shall not get a semantic theory for English. but we shall get an attractive semantic theory for the nonsernantic parts of English. There is no simple test for determining what counts as a semantic term, but among the notions that get booted out of the object language are truth, reference. necessity, and knowledge. One cannot imagine four notions more central to philosophical inquiry. One could conduct investigations into the natural sciences within the object language; one could even undertake investigations into the social sciences, other than linguistics and perhaps psychology, without serious disruption. But one could not begin to do philosophy within the object language. Epistemology and metaphysics would be entirely off limits. Fragments of ethics, aesthetics, and action theory would survive, but some of the central questions ("Are ethical judgments true or false'?" "Are truth conditions enough to give the content of an agent's beliefs and desires'?") would be excluded. It is only a slight exaggeration to say that we get the object language from the metalanguage by cutting out the language of philosophy. The doctrine that the language of philosophy needs to be singled out for exclusion from the domain of discourse to which semantical predicates can be applied is a doctrine whose acceptance ought to occasion considerable embarrassment among philosophers. We ordinarily suppose that the aim of a rational inquiry is to acquire true beliefs about the ob,jects to which the terms of our discourse refer. But if that is so, then, if we accept the exclusion of philosophical language from the realm of discourse in which terms like 'true' and 'refers' are applicable,

I

I I

I

we had best give up a11 pretense that philosphical inquiry is rational. Philosophical terms do not refer. Philosophical beliefs are not true, indeed it is improper even to ask whether they are true. Unlike scientific discourse. philosophical discourse is beyond truth and falsity. There is more at stake here than just the dignity of philosophy. At issue is the possibility of a unified scientific understanding in which human thought and action are no less intelligible or more mysterious than the planetary orbits. If we adopt the proposed solution, we shall find that within the object language we are unable even to describe human thought and action. We can describe and explain the motions of inanimate objects. and we can describe human institutions and behavior inasmuch as they are treated as meaningless, but intentional human activities, such as speaking, believing, willing, and acting, will be indescribable and inexplicable. Within the metalanguage we can obtain fragmentary descriptions of human thought and actions. We can describe intentional human activities that are directed toward inanimate objects, but thought about thought and talk about talk will remain indescribable and inexplicable. Thus, if we accept the limitations imposed by Tarski's proposal for avoiding antinomies, we forfeit one of the highest aspirations of the human spirit, the aspiration to selfunderstanding. Even though Tarski's second method does not give us a theory of truth for English, it does exhibit an important feature of the English usage of 'true', a feature so useful that we would expect any successful theory of truth for English to exhibit it. Let r be a set of sentences of the object language that I know to be not wholly accurate; r might, for example, be a report of the outcomes of games in my softball league which I know to be inaccurate because the total number of games won in the league is, according to r, different from the total number of losses. How can 1convey to a friend the information that l- is not wholly accurate? One method would be to assert the disjunction of the negations of the members of r, but this may be impracticable, either because a list of the members of r is not ready to hand or because r is intractably long. By utilizing the notion of is inaccurate simply by truth, 1 can succinctly convey the information that asserting, "Not every member of r is true." Without using the notion of truth or some other semantic notion, in order to convey the information that 1' is not wholly accurate, 1 need to be able to list the members of T. Using the notion of truth. I only need to be able to narne the set r. In general, if we are only interested in local, sentence-by-sentence properties of our systems of belief, we shall have no great need of semantic notions. If is a set of sentences of the object language which we can conveniently list, we can, in effect, assert that all the members of r are true by asserting the conjunction of r, and we can deny that all the members of r are true by denying the conjunction. On the other hand, to describe global properties of our systems of belief which go beyond sets of sentences we can readily list, explicit semantic notions are required. Thus, if r is a nameable set of sentences of the object

language, we can. in effect, assert the conjunction of 1' by saying, "(Vx)(.r c r is true)," and we can assert the disjunction of r by saying, "(3x)(x E I'& .r is true)." If D is a nameable set of sets of sentences, we can even, in effect, assert the conjunction of the disjunction of thc members of D by saying "(VA) ( A e D + (36)(6 e A & 8 is true))." We can even apply this method where the sets of sentences involved are infinite, thus simulating a fragment of the infinitary language YX,,,.' This is a reason why the notion of truth is so precious to us: it is one of the means by which finite minds are able to apprehend the infinite.' We now see what is wrong with the doctrine, suggested by Frege [1915], that the notion of truth is superfluous, since to say that a sentence is true tells us nothing more than the sentence itself tells us. The observation that we may replace 'r41 is true' by 4 and 'r41 is not true' by 7 4 enables us to eliminate the word 'true' from contexts in which truth is attributed or denied to a sentence that is named by a quotation name. But, in contexts in which truth is attributed or denied to a sentence or set of sentences for which quotation names are not available, the notion of truth is indispensable."' Using Tarski's second method, we construct the metalanguage out of the object language by giving an implicit definition. We see that the metalanguage constucted, though vastly weaker than the higher-order languages Tarski's first method requires, is a significant advance in expressive power over the original object language. We may, if we like, continue the process. Let 2,be the original object language, and give axioms implicitly defining 'Tr,,' as a truth predicate for Y,, Let 2, be the metalanguage thus produced, and give axioms implicitly defining 'Tr,' as a truth predicate for Y , ,treating the previous metalanguage as the new object language. Continue, letting X,, be the language obtained from the object language by adding 'Tr,'s for k