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

Cop! risht ( 1990 b) V a r ~ t ~ McGer

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hlc(icr. Vann. 1040- I'ruth. vayuene\\. and paradox. an e\\;I) o n the logic o f truth'

Vann McGee.

P - cn1. 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) 3 . Liar pat-adox I l'itlc

H1)171 . M i 7 1900 160-dcl0 89-17742

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Contents

Preface 0 . Our Project I . Formalized Versions o f the Semantic Antinomies 2. Logical Necessity 3. Tarcki's Solutions to the Liar Antinomy 4. Kripke and 3-valued Logic 5. Kripke's Construction and the Theory of Inductive Definitions 6. Rule-of-revision Semantics 7. Partially lnterpreted Languages 8. Truth in Partially Interpreted Languages 9. Definite Truth in Partially lnterpreted Languages

10. Toward a Semantics of Natural Language Bibliography Index

vii 1

18 3 1 67 87

107 127 148 158 196 209 223 23 1

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O

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~ehow intimately 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.rr pronouncement.; of the Pope. I have. in a sense. asserted the conjunction of all the Pope's o.r c~rthrrli-rr pronouncements. 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~lness of the ordinar) notion of truth. and if our scientifically reconstructed notion of truth is to continue to perform the ~ ~ s e f u l work that 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 so 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~plishing the 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 Engl~sh within 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.

The big philosophical cause this book aims ~~l t imate ly to promote is the unity 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 of truth for various fragments of a n a t ~ ~ r a l language; for exariiple we can develop 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 go 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 pa- tience 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~tly what 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. We may think of the biconditionals

is truc iff ct,

for € a cli~otation narne 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~juncts are true"-as constituting an inforrnal theory. We shall refer to this theory as our rlcri~lc~ tllc,ory cf truth: it is not a theory that we 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' = 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 impecca- ble 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 ~ ~ d c though 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 .

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 fro111 those rare instances of tlie rule which seem likcly to cause mischief. We somehow manage to restrict the rule ,just enough so that. without impairing the usefulness of the notion of truth, we are able to avoid being tricked into accepting outrageous or 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.'

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. Declara- tive 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 demonstrated when we focus our attention on what is called the slret~gthet~ed liar 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 (Pl ) and (P2) in action. (Pl ) 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, so 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~ines whether 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~e o 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 r l i ~ ~ r S P I ~ ~ 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~nal model 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~paratively easy 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\~entiotl T: 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 'Tr 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. We use the rnetal(lngicc~gc~ when we give a definition of truth for the o11jjcc.t larlguclgc~. '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. To 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 prob- lerns. 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~cal planet 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

"' 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.

I ' The locus c lus~icu~ for 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 building- block 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 unneces- sary 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."

To 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~ng In our mother tongue :rnd taking its words at face value." Quine's foregn~und languagc/bnckgrou~~d languase d~st~nct ion i< what we are calling the object langu:age/metala11puage d~st inct~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 ~ e that 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. We can specify the nieanings of the speaker's words, but not how or why they came to have those meanings. We 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) (3 language 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<."

The counterfactual

If our language had evolved differently. 'rabbit' might have referred in my idiolect to groundhogs.

is ambiguous in just the way that

If Mondale had been more persuasive, the President of the United States in I987 would have been a Democrat.

is anlbiguous. If, in spite of the indeterminacy problems raised by Quine and others, it turns

out to be possible to single out a correct translation from one language to another, we should expect to be able to derive (i) from the following three sentences, which are true in my idiolect when spoken by me:

( 3 language Y)[l \peak Y & ' rabb~t' IS a term oi 2 & n(V language y * ) (V tcrm T)(T 15 correctly translated froni Y* to a\ 'rabbit' -+

T refers In Y* to rabbits)] @(V language Y)(V term T of Y)(T 1s correctly translated from Y to 9 as

7)

(VY)(VT)(T I C a term of the language 'P -+ U(T 1s a term of the language

The semantical paradoxes are sometimes dismissed as mere curiosities, brain- teasers that amuse the technical-minded but need not trouble those with deeper philosophical concerns. This complacency has always surprised me. 1 should have thought that. inasmuch as no notion is more central to philosophy than the notion of truth, the fact we are unable to give even the rudiments of a consistent theory of truth for our own language would have been a cause for some alarm. I hope that, even if the solutions offered here are found to be without merit, this book will at least foster an awareness that the problems have merit.

The plan of the book is as follows. So far wc have only talked about the paradoxes generated by our naive understanding of truth, but there are similar paradoxes involving denotation, satisfaction, necessity, and knowledge. The connection of these other paradoxes with the liar paradox will be discussed in chapter 1 , which sets up the rormal apparatus for the rest of the book. The only one of thesc other paradoxes which will be discussed in any dctail will be the paradox about necessity, which will be discussed in chapter 2.

In trying to understand the logical problems that beset the naive theory of truth and the possible ways to overcome them, two articles are supremely important, Tarski [I9351 and Saul Kripke [ 19751. The first of these is discussed in chapter 3 and the sccond in chapters 4 and 5. Kripke's paper will be important not only for its own sake, but also because it provides the mathematical foundation for the later chapters.

The rule-of-rcvision semantics proposcd by Anil Gupta [1982]. Hans Herz- berger 119821, and Nuel Belnap [1982]. which we discuss in chapter 6, combine some of Kripke's ideas with interesting new insights.

In chapter 7 we introduce the basic model theory of partially interpreted languages. In chapter 8 we develop our theory of truth for the languages, and in chapter 9 we develop a theory of definite truth, using constructions from modal logic which will have been developed in chapter 2. It will emerge that our naive notion of truth has, in a sense, split into two notions, truth and definite truth.

We conclude with some rather speculative remarks about the prospects for applying our formal methods to natural languages.

Lying on the border between philosophical and mathematical logic, the book makes extensive use of mathematical methods. Let me say a word about what kind of mathematical training will be presupposed. For most of what we do. a sufficient background will be a standard beginning graduate or advanced under- graduate course in logic. We use basic definitions and results from model theory (definition of satisfaction; completeness, compactness, and Lowenheim-Skolem theorems), recursion theory (Godel incompleteness theorems; Church's thesis; representation of recursive and recursively enumerable sets within the language of arithmetic); and set theory (transfinite induction; simple ordinal and cardinal arithmetic; the well-ordering theorem). Occasionally more advanced or esoteric knowledge will be required, but these occasions will be easily isolated and can be skipped without loss of continuity. I promise to warn you about them.

Our official mathematical theory will be Zermelo-Fraenkel set theory with the axiom of choice (ZFC), though we shall not come close to using its full strength. In chapter 6 we talk in terms of proper classes, because Gupta. Herzberger, and Belnap talk in terms of proper classes.

I shall make a great deal of use of the theory of inductive definitions, but for technical reasons it has almost never been possible to proceed by simply citing textbook theorems. The textbook theorems never give quite enough information, so what we have used of the theory we have had to develop from scratch. Although this makes for more work, it has the advantage of making the book accessible to those who have had no previous acquaintance with inductive definitions. Although 1 should be pleased if the book should prove useful as an introduction to the theory of inductive definitions, it is by no means intended to serve as a textbook on the sub.ject: for that purpose, I defer to the excellent textbooks already available, Yiannis Moschovakis [I9741 and K. Jon Barwise [1975].

Formalized Versions of the Semantic Antinomies

Kurt Giidel 1193 1 1 showed. for a wide class of formal languages, how to reduce the theory of syntax to the theory of numbers. He did this by associating a numerical code with each expression, in such a way that basic syntactic operations, such as forming the co~~junc t ion or dis,junction of two li)rmulas, prefixing a quantifier to a formula, or substituting a term for free occurrences of a variable in a formula, correspond to recursive operations on tlie codes. Since. at the time (jiidel wrote, syntax was ~lnderstood only superficially. whcrcas number theory had been fruitfully studied f'or centuries, Godel's method constituted a signal advance in the study of syntax.

In order to take full advantage of Giidel's proposal, we shall suppose, for much of what we do. that we arc working with a countable language Y for the first-order predicate calculus with identity, and that we have fixed, without lingering over details, a numerical code r&l for each fi)rmula (r, in such a way that tlie nu~iierical operations corresponding to disjunction ((r(t1 V r$l) =

r(+ V d l ) ] ) , negation (lr(r,l = ri(r,l), existential quantification ( (3v) r+1 =

r(3v)(r,?), substitution (rq5-i \'IT = the code for the result of substituting the term r for free occurrences of 1. in 4). and so on are recursive.' Occasionally we shall add new symbols to Y : when we d o so. we shall suppose that the coding system has quietly been adjusted.

We want to look at languages that are able to describe their own syntax. Prirr i~ ,fiic.ie it would appear that, number theoretic methods being so useful in the study of syntax, a language that was well-equipped to describe its own syntax would have to contain the following three components: the ability to describe expression types and their syntactic properties. the ability to talk about natural numbers and their matheniatical properties, and the ability to talk about the coding relations. In practice, however, once we have set up the coding relations. there is nothing useful to be said about the expressions that cannot be said just as well in terms of thcir codes, and it is simpler and more convenient to talk about numbers and about the arithmetical properties that correspond to syntactic properties o f the expressions they encode. rather than to keep going back and forth between

' This pleasant notational convention of indicatiny an arithmetical operation by placing a dot under the symbol for the corresponding logical operatlor1 is due to Fefermati ll9h01.

expressions and thcir codcs. M1c "identif)." an cxprcssion with its code. so that, in order to be able to describe its own syntax, it will be enough for a languagc to be able to talk about numbers.

Before the expression types vanish from view. however. there is a philosophical point to be made. Language is a concrete phenon~enon in human social history, and yet the study of the syntax of language has become a part of abstract matheniatics. What it is important to realize is that the transition from concrete to abstract occurs not in the move frorn expression types to Godel numbers but in the move from expression tokens to expression types. Linguistics, as ordinarily practiced, is the study of expression types. One studies the rules of a language by examining the totality of expressions that are constructible by the rules. investigating their syntactic and semantic structure without restricting oneself to those expressions which it has served sonleone's purposes actually to bctokcn. We can, for example, describe a sentence as obtained frorn another sentence by a sequence of transformations without supposing that each of the intermediate transformations has resulted in a concrete utterance. A devout nominalist might try to develop a linguistic theory that worked solely with concrete expression tokens, classifying them directly into grammatical and semantic categories, but ordinary linguists are not constrained by nominalistic scruples.' Expression types are part of the everyday mathematical tools of the working linguist, and expression types are fully abstract objects.

Thc syntactic theory of expression types is, in fact, interreducible with number theory. Godel showed how to reduce syntax to number theory, and Quine [ 19461 has shown how to go the other way, using the syntactic operation of concatenation to encode arithmetical operations. Thus, we could, if we wanted, develop a syntactic theory that utilized number-theoretic methods but only talked about expressions, since we could replace talk about numbers by talk about their Quine codes. It is really only a matter of notational convenience whether we develop a theory that talks about both numbers and expressions, a theory that talks only about nunibers, replacing talk about expressions by talk about their Giidel codes, or a theory that talks only about expressions. replacing talk about numbers by talk about their Quine codes. Considerations of notational simplicity are, in this case, quite decisive, and we choose the number-theorctic option.

In order for a language to be able to describe its own syntax, it will suffice that the language be able to describe the natural-number system. It will be useful to specify a formal language to use in describing the natural numbers, so we take the lrirzg~tugr of tirithrnetic to be the language for the predicate calculus with

Wherea\ a demand thirt u e only talk about actual concrctc cxpl-c\slon token\ would be intolei-ably restrictive. thc \traightforward development of linguistics proceed uithout impediment if u c allow ~)ur\clvc\ to talk about /7o.s.sihl' \entente tokens. Chihara [I9X11 shahs that much of cla$$~cal mathemat~cs can be reproduced within this less austere norninall\rn.

identity whose nonlogical constant5 are the individual constant 'O' , a unary function sign 'S'. binary function signs '+' and '.', and a binary relation sign '<'. We name the numeral for a number by drawing a line over a name of the number; thus

- '0' abbreviates 'O', which is the numeral for 0. - ' 1 ' abbreviates 'S(O)', which is thc numeral for 1. - '2' abbreviates 'S(S(0))'. which is the numeral for 2.

and so on. !I? is the standard model of the language of arithmetic, whose universe is the set of natural numbers and whose arithmetical symbols have their usual meanings. We identify the set of natural numbers with w , the least infinite ordinal.

It will be useful to be able to classify the arithmetical formulas according to the complexity of their quantificational structure. The boirnded formulas, also known as the x:{ formulas and the I: formulas. constitute the smallest class of expressions which contains the atomic formulas and which contains (4 V $), (4 & 4). 1 4 , (3v)(v < T & +), and (Vv)(v < T + $), for each variable v and term T, whenever it contains 4 and $. The last two expressions are abbreviated (3v < 7)4 and (Vv < T)+, using the boirnded quuntijiers (3v < T) and (Vv < 7). A X j l , , forrnula is obtained from a :: formula by prefixing 0 or more existential quantifiers, and a I : + , formula is obtained from a 2:; formula by prefixing 0 or more universal quantifiers.

A relation on w is said to be 2:; or jj if it is the extension of a C:: or a jj formula. A relation is A: if it is both xj: and li. The 2; relations are closed under finite unions, finite intersections, bounded quantifications, and unbounded existential quantifications. The relations are closed under finite unions, finite intersections, bounded quantifications, and unbounded universal quantifications. The j: relations are the complements of the 2: relations. The A:: relations are closed under finite unions, finite intersections, complements, and bounded quantifications. The x:' relations are the recursively enumerable relations. The A: relations are the recursive relations.

We shall take negation, disjunction, and existential quantification to be the primitive logical operations. treating conjunction, the conditional, the biconditional, and universal quantification as defined. For official purposes, we take the variables to be 'v,', 'v , ' , 'vL1. and so on, but in practice we often use other letters as variables to avoid proliferation of subscripts. A .sentence of the formal language is a formula without free variables.

An interpretation (or moclel or strucrure) for a first-order language 2 will be a function !)I defined on {individual constants of 2} U {function signs of Y} U {predicates of 3) U ( '3 ') so that

!)1('3'), also written /!'[I, is a nonempty set. If c is an individual constant, 91(c), also written c'", is an element of 91('3').

Iff is an n-ary function sign. ?l(f), also written j ' l . is an rl-ary operation on "L('3').

If R is an rz-place predicate, !'I(R''',). also written R"'. is an n-ary relation on "l( '3'): in particular. '= ' is a binary predicate such that ?'i('=') = {<s.x>: s F: YI('3')).

Truth under an interpretation is detined in the usual way; we write !'I 4. It will be useful to have a specified axiomatization of the predicate calculus.

Following Quine [ 19401, we take the axioms of logic to consist of all sentences obtained by prefixing universal quantifiers to instances o f the following schemata (where '4' and 'I/)' are to be replaced by formulas, 'T' and 'p' by terms and 'v ' by a variable):

Tautologies. (Vv)(+ + $1 -+ ((Vv,& -+ (Vv)$) 4 -. ( V V ) ~ , where does not occur free in 4 (Vv)+ -+ 4 L'lT, where no free occurrence of 1. in 4 occurs within the scope

of a quantifier that binds a variable that occurs in T

(3v)4 l(VV)lC$ 7 = 7 7 = p -+ (4 ++ $), where $J and $ are atomic formulas that are just alike

except that T and p have been exchanged at some places

4 is derivable from r (in syn~bols. + 4) iff there is a finite sequence of sentences which ends in 4, each member of which is either a member of I', an axiom of logic, or obtained from earlier members of the sequence by modus ponens (from $ and ($+ 4) to infer 4 ) . The methods of Giidel 119301 show that 4 is derivable from r iff 4 is true in every model of I'.

Robinson's arithmetic R (from Tarski, Andrzej Mostowski, and Raphael Rob- inson [1953, p. 531) consists of the following axioms:

r 1 . p = r 1 . p - -

i n = p for each n # p - - (V.K)(X < n V x = n V n < X)

(Vx)(Vy)(.u < y - (1 x = y & (3z)(z + x = y)))

(Vx) 1 x < 0 - (Vx)(x < n + I -+ (X = 0 V x = T V . . . V x = II)

R is useful because of the following theorem: THEOREM 1.1 (Tarski, Mostowski, and Robinson). Every true 2: sentence is provable in R. For any n-ary recursively enumerable relation S there is a formula ~ ( I J , . . . . , v.) such that, for any k , , . . . , k, , , we have

4,. . . . . k,,=. F S iff K t- d) (F , . . . <)

(we say that qb ~tle~ilil! r.cy?r.e.ser7t.c S in R) . For any 11-ary rccursivc relation S, we can tind a formula JJ(Y,. . . . . 1,") such that. for any k , , . . . . k,,.

< k t . . . . , k,,> E S iff^ +ih(F. . . . , K) and <k,, . . . , k,,> 6- S iff' R + i$(K. . . . , K)

(we say that strong!\ i.cp-rsozt.~ S in R ) . For any 17-ary recursive function ,f we can tind a formula H ( i 1 , . . . . , I , , , , \> , , , ,) such that, for any k , , . . . . k,,.

R + cvy)ce(G. . . . , K,!) - !. = fik,, . . . . k,,)i

( W C say that H ,fi~r~c.rior~olly I . L ~ ~ Y ~ . W I Z ~ . Y ,f' in R )

For a proof see Tarski, Mostowski, and Kobinson 1 1953, pp. 56ffl. 'm R gives us a rudimentary number theory formulated within a language speciti-

cally designed for talking about natural numbers. We want to see how to reform- ulate R within other languages that can talk not only about natural numbers but about other things as well. For this purpose, we utilize the notion of a r r l a r i~~c~ intrrpr-rttrtion, due to Tarski, Mostowski, and Robinson 11953, pp. 20ffl. There arc ac t~~al ly three notions; we can relatively interpret a language. a theory. o r a model. Although the technique is perfectly general, we describe it here only t i~ r the case in which the theory is R and thc model is !I;.

We relatively interpret the language of arithmetic into a language 2; by finding formulas N(s), Zs), S(.a,y). A(.r.y,z), M(-u,\'.z) and L(-~.yi whose intended extensions are the set of natural numbers, {O}. the graph of the successor function. the graph of the addition operation, the graph of the multiplication operation, and the less-than relation, respectively. Once we have done so. we can translate a sentence of the language of arithmetic into a scntencc of Y by the following procedure:

First rewrite thc given sentence into a logically equivalent sentence in which the only terms that occur are variables. individual constants, and function signs with variables as arguments, and in which all atomic Ibrmulas have one of the forms T = I,, or 1', < L,! (where 7 is a term and 13, and v, are variables). Next replace 0 = v, by Z(lS,). S(ls,) = l., by S(r,,, l*,), vi + L; =

11, by A(\,,. l,,, 1:). . I,, = l*, by M(\',. 1;. 11,). and 1; < 1,) by L(v,, I!,) (changing bound variables if neccssary to avoid collisions),'

Finally. relativize the quantifiers by replacing (V~,)I,!I by (ttr,)(N(~.,) + 31)

T l i ~ \ theorem is proved in man) texthookb. hut with bonl~. \lightly different theory in place of R . For our purpose\. it will not matter ~f one of tlic\e othcr tl~eoric\ i b usetl in\read of R. In the future. I \hiill oftcn take 11 for granted. w~thout explicit ~ilention. that. In rriaking ~ubst~tut ions . step5 have been taken to avoid collision\ of bar~ahles.

and replacing (3v)+ by (3v)(N(v) & 4) (again changing bound variables, if necessary).

The relative interpretation of R into % will consist of the translations of the axioms of R , together with sentences that declare that exactly one object satisfies (N(x) & Z(x)) and that S(x,y), A(x,y,z), and M(x,y,z) all define functions on the extension of N(x); for example, the axiom for A(x,y,z) will be

A theory r formulated in 2 will be said to relatively interpret R iff each of the sentences in the relative interpretation of R is a theorem of I-. If this occurs, then every theorem of R will translate into a theorem of r.

Let 71 be a model of 2 such that the first-order theory of 91 relatively interprets R . We say that '21 relatively interprets 'J? iff the mapf: w + N" given by

f(0) = the unique member of N" f l Z" and f ( n + 1) = the unique member x of N" such that <f(n), x> is in S"

is onto. In this case f is an isomorphism between 92 and the restriction to N" of 91.

Throughout the present chapter, we shall suppose that we have settled upon some method for relatively interpreting the language of arithmetic into 2. The two assumptions-that the syntax of 3 has been arithmetized by choosing a well- behaved Godel coding and that the language of arithmetic has been relatively interpreted into %--ensure that 2 can give a good account of its own syntax.

Once we have fixed a relative interpretation of the language of arithmetic into 3, we shall often speak as if the language of arithmetic were literally included in 2. We shall speak of a sentence of the language of arithmetic as being true in a model of 9, when what we really mean is that the translation of the sentence is true in the model. We shall speak of a sentence of 2 being a logical consequence of R , when what we really mean is that the sentence is a logical consequence of the relative interpretation of R . If we have an arithmetical relation described in informal English, we shall sometimes speak as if the ordinary English phrase that describes the relation were a part of a formula of 2, when what we really mean is that the translation into % of the translation into the language of arithmetic of the ordinary English expression is a part of the formula of 2.

One such informally defined arithmetical relation is the function k, which takes a number n as input and gives the numeral for n , i, as output. If 'Bew' abbreviates an arithmetical predicate that expresses provability in some formal system, the strictly correct way to formulate the thesis that it is provable that every numerical substitution instance of a formula O(x) is provable would be to write

Such multiple overscoring produces expressions that are difficult both to write and read, so we shall normally write simply

R is a very weak theory: i t contains the bare minimum amount of information needed to obtain theorem 1.1. Toward the other end of the spectrum is Peano arithmetic (PA), which is strong enough so that nearly all our ordinary ways of reasoning about the natural numbers can be easily formalized in it. It consists of the following axioms:

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

p(m) = ~ ( ~ z ) ( H ( % . z ) & ~ ( z . L ! , , . . . . I, , ,))] if m is the Giidel number of a formula 8(x, i);

= 0 otherwise.

We see by Church's thesis that p is recursive.' It follows by theorem 1.1 that we can find a formula on U ( x , ; ) so that. for any m ,

R ( V Z ) ( ~ ( ~ , Z ) = = pol

Let $(v, , . . , v,,) be (3z)(9(r1J'(x, z)l ,z) & 4(z, v, , . . . , v,)). Then

[ > ( ~ P ( X , Z ) ~ ) = ~+(LJ, , . . . , v,,Y

and so

from which we derive

R I- (Vv,, . . . ( v ~ , , ) ( ( ~ z ) ( Y ( ~ ~ ( x , z ) ~ , z ) & $( z,Vl, . . . , bl,J) ++ 4(r$1, L > , , . . . 3 v,,))

that is

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 , 2 being 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~nent is 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. Witt- genstein'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:

TIIEOREM 1.4 (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) i r ( r h 1 ) + A (This is a theorem of R . ) (ii) r(rA7) - A (This is an instance of (4 ) . )

(iii) A (From (i) and (ii) by (2).) (iv) r ( r h 1 ) (From (iii) by (3) . ) (v) r ( rh1) - i A (This is a theorem of R . )

(vij 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 ( r41) ' 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 ( r41) ' 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, so 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. We 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 that the truth of a sentence depends in palt upon there being an appropriate correlation between what the sentence says and what the world is like. Like all philosophical doctrines. this is open to dispute.

' The applicability of thcorcm 1.4 to thc thcory of knowlcdgc was first notcd by Montague and Kaplan [1960].

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 , $(a) is true? As the theorem below shows, if we assume the Barcan formula,

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

a(+ + $) + (04 -+ O+) and m 4 + 1 04

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 (Vx ) (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 ~ymbol s havc 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 & 2 = y ) V (3w)(N(w) & ,X = S(W) &L z = r(vz)(f(G,j ,z) + 7(2))1)1)

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 r. 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 r also contains a , which says that not every result of prefixing TS to [a1 is T, we shall have our w- inconsistency.

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':

(v) 70- -+ (Vx)(N(x) -+ ( t l z ) (~ (x , r a l , z ) + ~ ( 2 ) ) ) (from (iv)) (vi) l a -+ ( V z ) ( ~ ( ~ , r c r ? ,z) -+ ~ ( z ) ) (from (v))

(vii) ~ ( 0 , r c r l ,rcrl) (from (ii))

"' From McGee [ 19851

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

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 il l

r: (ix) * i ( V - r ) ( N ( a j + ( V Z ) ( F ( . ~ , ~ ~ T ~ . Z ) -+ ~ ( 2 ) ) ) (from ( iv)) ( 1 + i r + V r + T om (ix) by (3))

x i ) T ' + N V F ( Y ) -+ Z ) (from (x) by (4ja))

(xii) ~ i ~ 3 v - ~ ) ( ~ ( - r i + (Vz)iF(.r,ru1,:) + 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) j i l ) -+

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))

( x i ) ) + N . -+ ~ F S ) + T ) ) ) (from (xiv) and (xvi))

V ) ( ' I -+ N ) ( F ( ) + ( j ) (from (xvii) (xix) ~ ( r v l ) -+ (+ (from (iv) and (xviii))

Consequently, r contains

(xx) rr (from (viii) and (xix))

We 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) (Vz) (~(E, r (+ l ,z) + ~ ( 2 ) )

is in r. Then so are these sentences:

( X X V ) T ( ~ ( ~ ; ) ( F ( ~ , ~ c T ~ . z ) -+ T ( z ) ) ~ ) (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

(xxviii) (V:)(F(~, ,ul ,z) - ~ ( 3 )

is in r. On the other hand. since (iv) and (xx) are in T. so is

S o 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

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

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

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.

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:

r47 is true iff at least one individual satisfies r$ & v = vl iff every individual satisfies r+ & v = 1.17

I' 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~c paradoxes is a deep problem to which, regrettably, the present work has nothing to contributc.

'' See Russell [ 19051.

r ( iv)$(~.! ) l denotes a i f f LI satisfies r(V.'.r)($(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.

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

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

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

For the philosophically interesting application, take 6 to be a formula that is intended to express the denotation relation for y. PROOF: NOW take ' ~ ( 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~plicated business.

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:

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 first- order 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 ~ r Y(t-,D) got from !)I by letting the extension of 'D' be the set of all ordered pairs <r( ix)$(x) l ,b> where r(i.r)$(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 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<, of Y(t..c.) as the model got from !'I by taking the denotation of 'c' to be the set of all ordered pairs < r ( ~ x ) $ ( x ) l ,b>

I" Where. for any set S. iP(S) is the set consisting of all suhscts of S. we detinc

v,, - O v,,+, = ?P(V,,) V , = ,?Uh V , ) . for A altlnit

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

l'lo k (vy)(u( .Y.r(~(.r)1,c~) - 44.v))

for each formula $(x) of ?Y(e ) . Use the self-referential lemma to find a formula <(.Y) of Y ( F ) SO that

R F (Vy)(<(!) - i ( i (y .r i( .~)l ,?.))

but also, because t?(j,,x,c,) represents the satisfaction relation on % F ) ,

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:

THEOREM 1.9. Let Y ( c ) and 2 ( e , T ) be, respectively, the first-order language whose only nonlogical symbol is the binary prcdicate ' F ' and the first-order language whose only nonlogical symbols are the binary predicate '8 ' and the unary predicate 'T'. Let \!I be the tnodel of ?Y(c) in which the universe is VSI and in which ' e ' is interpreted as the restriction to 1911 of the elementhood relation, and let !)[I be the model for Y( t . ,T ) got from ?I by letting the extension of 'T' be the set of sentences of - y ( e ) that are true under !)I. Thcn there is no forniula 6( .r ,y ,n of iY(e.71 such that, for each formula $(xi of Y ( F ) ,

Moreover, there is no forniula a(y,.r.T) of Y(e .71 such that, for each formula $(,I ) of :P(c),

!'I' + (v?.)(cr~y,r$(\)',n - $(!.)I Thus thc set of truths of Y ( E ) is definable within 2(t . .T) , but neither the denotation nor the satisfaction relation for X ( F ) is definable within Lf(e,T). PROOF: Suppose that there were a fortnula 6(.r.y.71 so that

Let Z(F , ( . ) be the first-order language whose only nonlogical symbols are the binary predicate 'c ' and the individual constant ' c ' , and let !)I, be the model of K , is the least uncountable ordinal.

Y(F.c) got from !)I by taking the denotation of ' c a ' to be the set of Giidel numbers of sentences of Y(E) true in 91. Let 6(x,?,(.) be the formula of Y ( E , c ) got from 6(x,~,71 by replacing each occurrence of ' T ( r ) ' by '7 E c'. Then we havc

for every formula $(x) of % ( E ) . Use the self-referential lemma to find a formula t(.u) of 2 ( ~ ) SO that

It follows from the separation axiom of ZFC that, Y ( w ) being an element of VXI, there is an element of 1911 which satisfies [(x). It follows from the extensionality axiom that at most one element of (911 satisfies ( (x ) . Hence,

$1 k (1 x ) [ (x ) exists

and

!)I k (VY)(.Y 6. (1x)t(-r) - ( y F %w? & ( v z ) ( 6 ( r ( l x ) t ( x ) l ,z,JJ) + 7 JJ E z ) ) )

Since 'c' denotes a set of Godel numbers, !)I, k c E Y ( w ) , and so

Now, since, in !)I,, the formula 6(x,,v,c) represents the denotation relation for definite descriptions of Y ( E ) ,

Hence,

Contradiction. Since denotation is definable in terms of satisfaction, the undefinability in 91' of

the satisfaction relation follows from the undefinablitiy in !)I' of the denotation relation..

We have seen that it is not, in general, possible to define satisfaction in terms of either truth or denotation and that it is not, in general, possible to define denotation in terms of truth. We can make it possible to produce such definitions, however, by extending our language, adding a new constant cr for each object a in the universe of discourse. Assuming that the function taking LI to a can be specified in the language, we can stipulate:

a satisfies r$(v)l iff r+(u)l is true. -

r (~v )$ (v ) l denotes a iff r(Vv)($(v) c* v = a)1 is true. (I satisfies r $ ( v ) l iff r( iv)(v = 6t $(v)) l denotes a.

Thus, in developing a sen~antical theory for the extended language, we can take truth to be our sole primitive sen~antical notion. We can autonlatically extend a theory of truth for the extended language into a theory of satisfaction and denotation for the extended language. Of course, we can cut back down to the original language at the end, so that we get a thcory of truth, denotation, and satisfaction for the original language.

Thus, our task will be to obtain a theory of truth for the extended language. Thinking of our task this way will makc the task a bit simpler. since denotation and satisfaction are more cumbersome to work with than truth." In what follows, we shall have almost nothing to say about either denotation or satisfaction. other than to point out that, for whatever we shall have occasion to say about truth, there are obvious corresponding things to be said about the other two notions.

Let me emphasize that I do not suggest that a natural language contains a name for every object or that it is humanly possible to extend a natural language by adding a name for every individual. On the contrary, the sentences of the extended language arc abstract entities that we introduce to help us describe the concrete phenomenon of human language.

We have no reason to suppose that the extended language will be countable, so we cannot use natural numbers as codes for the expressions of thc extended language. We may still be able, however, systematically to find objects in our universe of discourse to serve as codes for expressions of the extended language. If we are sufficiently careful, the coding will be well-behaved, so that we shall be able to produce Godel-style constructions without difficulty.

Given a model !'I for the language 2 , we want to find an efficient Godel coding of a language Y, , which is just like 2 except that it contains a constant u for each element a of IYII. To - do so, we use finite sequences from I?lI, taking the Godel code of 0(;, . . . , a,,) to be the ordered pair < T O ( " , , . . . , v,,) l ,<a,, . . . , a,,>>, whose first component is the Godel number of a formula of 3.''

To describe the syntax of Y,, within %,, we shall not literally require that every finite sequence from /!'I1 be an element of I!)l(. What we shall require is that every finite sequence from 1911 be coded by an element of 1911. The paradigm here

" If we htay within the unextended language, we need the notion of satisfaction to describe how the quantifiers work. For the extended language. we only need the notion of truth, because quantification in the extended language can be regarded as substitutional. In the extended language, substitutional quantification (where (tlv)H(v) is regarded as saying that every substitution instance of B(v) is true) and objectual quantification (where (Vr.)0(1,) is taken to say that every object in the universe satisfies H(1,)) coincide. They coincide because in setting up the extended language we used objectual quantification in the metalanguage in stipulating that there should be a constant for every individual.

'"ome convention is needed to determine what sequence of variables < t , , , . . . . v,> to use. We want to arrange things so that, if one formula of the extended language is obtained from another by a pern~utation of the new constants, their Giidel codes will havc the same first component. We might. for example. always choose the numerically least possible formula.

is Giidel's coding o f a tinite sequence < ( I , , ( 1 : . . . . . (I , ,> of natural numbers by the single natural number 2"'" . 3"'" - . . . p::"' ' (where / I , , is the 11th prime), which enabled Giidel to use thc language of arithmetic to talk about finite sequences of natural numbers. "'

Thus. we need a c~oding sr~lzernc. a onc-one function that associates an element of 1?1/ with each finite sequence from ) ? I / . In order to be ablc to use Y:,, to talk about tinite sequences from /:)I/. wc require that thc set

Seq = {codes of finite ~equences}

and thc functions

I l ~ ( a ) = the Iengtli of the scquence coded by .I- . if .r codes a sequence = 0 otherwise

and

L J ( X , I Y I ) - the mth member of the sequence coded by s. if .r codes a sequence of length m or longer

= 0 otherwise

be definable in Y'; more prccisely. we require that Seq and the graphs of' Ih and y be extensions of formulas of Y . If this condition is met, !)I is said to havc a built-in coding scheme. If we have fixcd a coding schcma for !'I, we shall use the notation < a , , . . . , a,,> ambiguously for the tz-tuple and its code.

In order to be able to describe the syntax of Y,, within X,, we require two things. First, in order to be able to describe the unextended language YIP, we need to be ablc to relatively interpret \)r' into \![. Further, in order to describe the extension of Y to X,, !)( has to have a coding scheme built into it. Structures that meet these conditions were singled out for study by Moschovakis 11974, p . 221, who refers to them as ur~c~prahle structures. Because we are interested in languages that can describe themselves, we shall concern ourselves almost exclu- sively with acceptable structures. Let me list some basic facts about the set Seq and the functions Ih and y:

Ih is a function frorn the whole universe into the set of natural numbers. y is a function whose domain consists of all pairs <A,;>, with i a natural

number. ( V . r ) ( i S e y ( s ) -. Ih(.r) = 0 ) (V.v)(V natural nurnber i ) ( ( i S e c / ( s ) V i = 0 V Ilz(.r) < i ) + q(.r.i) = 0 ) (3x)(Sey( .r) & ICl(.i) = 0) (namely. the empty sequence < >)

"' This is no1 at all trivial, since exponentiation 1s not available in the language of arithmetic as a pr~mitive operation. Seeing how to codc finite sequences was, in fact. a majot- technical break- tht-ougli achieved by Gddel [I931 1.

(V.r)(V!)([Sey(.v) & SPY(!) & Ih(.r) = / / I ( ! ) & ( V natural number i ) (q(a , i ) = y ( y , i ) ) ] -+ ,I- = y) -

( 'dayl , . . . (V2r,,)(3!)(Seq(y) bt lh(!) = 1 & r/(y.T) = .r, & . . . & y(.i',il) = x,,).

Once wc have fixed a relative interpretation of the language of arithmetic into -'f and we have determined which formulas of 5f are to represent Sey , Ih, and ( I ,

these basic facts can be formalized in -y. 12et the axiom system R consist of the formalizations of these basic facts together with the relative interpretation of R . R. will play the same role in the study of Y.,, that R plays in the study of Y. Thus, the theorems we have proved about -Y using R can be easily nlodified into theorems about Y>!, using R. .

Using R to describe the syntax of Y, we are able to see that the basic laws of syntax contradict our naivc understanding of the laws governing truth, denotation, and satisfaction. By expanding our language, using K to describe the syntax of Y, , , we are able to consolidate these three problems, so that, if we obtained a satisfactory account of one of the three notions. we would havc a satisfactory account of all three. Necessity and knowledge are not reducible to truth in any straightforward way, so getting a satisfactory theory of truth would not automatically give us a satisfactory theory of necessity or knowledge. In this book, necessity will get separate treatment and knowledge no treatment at all.

We focus on these five concepts-truth. dcnotation, satisfaction, necessity, and knowledge-because they are central to our understanding of the relation between language and the world, but there are other versions of the semantic paradoxes that d o not employ these five concepts. One way to find such paradoxes is by looking at some of the characterizations of truth which philosophers have proposed. A true sentence has been said to be one that

agrees with reality:'" corresponds to the way things are; depicts an existing state o f affairs;" says to be the case something that is, in fact, the case;" says that the state of affairs is so-and-so and the state of affairs is indeed so-and-so;" says of what is that it is and of what is not that it is not;'"

'" C'f. Tarski [1043, p. 3431. " Cf. Tarski [1044. p. 3431. " Cf. Chihara 11979. p. 6051. '' Cf. Tarski [1935. p. 1551. '' This definition of truth \*as given by Ar~\totlc [M(~ttr,,h\.ti~.$ 101 Ih271. 'l'hc analogou.; dclinition

of falsity had been givcn by Plato [Sopl~i.sr 241aJ.

is such that, for some p , the sentence \ays that p and p;"

and so on. Each of these characterizations of truth generates a paradox, as we can see by examining the following sentences:

This sentence does not agree with reality. This sentence does not correspond to the way things are. This sentence does not depict an existing state of affairs.

And so on. This proliferation of paradoxes is unfortunate. A natural first response to the

paradoxes is to try to isolate the concepts and expressions that give rise to the difficulties. Once we have done so, we can concentrate our efforts upon repairing or replacing the offending concepts. in the meantime exercising special caution whenever the concepts are employed. Unfortunately, since it is possible to produce the paradoxes in so many ways, sometimes without using vocabulary that is obviously semantical, it does not appear that there is any simple syntactic test that will tell us which concepts and expressions we need to quarantine."

Ultin~ately, one would like to have a unified account. Such an account would either repudiate the characterization, "A sentence is true iff it depicts an existing state of affairs," perhaps on the ground that state-of-affairs talk is not to be taken literally, or else it would employ a sophisticated notion of depiction according to which

This sentence docs not depict an existing state of affairs.

is nonparadoxical. Such a unified account, if attainable at all, lies far in the future.

" One might be tempted to try to explain away the quantifiers 'for some p' and 'for all p' as substitutional quantttiers, treating '(3/7)((1. says that p ) & p) ' as saying that every result ot substituting a sentence for 'p ' in ' ( v says that p ) & 17' is true. The problern is that one of the substituents for 'p' is the sentence '(3p)iiv says that p) & p) ' itself. so that we find ourselves caught in an cndless circle, trying. as i t were, to construct a disjunction that contain5 itself as part of one of its disjuncts. The circle is vicious, as we can see by using the self-referential lemma to construct a sentence < so that

R F ( - -1(3~)((r[' says that p ) & p )

(First-order derivations from R will not be disturbed by adding another style of variable to the language.)

?6 The propensity of paradoxes to crop up even when the lanpuage we are using seems innocuous is even more widespread than these examples indicate. Chihara [1979, pp. 593-5953 shows that even the notion of eligibility to join a club can generate paradoxes. He considers the SecLib Club, whose rules stipulate that membership is open to those persons who arc secretaries of clubs that they are not eligible to join. The problem rises to the surface when SecLib's secretary, a certain Ms. Fineline, applies for mcmbership.

Logical Necessity

According to Leibniz [ 17 14, $#33-351,

There are . . . two kinds of truths, those of reasoning and those of fuc.1. Truths of reasoning are necessary and their opposite is impossible; truths of fact are contingent and their opposite is possible. When a truth is necessary, its truth can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary.

It is thus that in Mathematics speculative Tlz~orerns and practical Canons are reduced by analysis to Dejinitions. Axioms, and Postulates.

In short, there are sinlple iclecrs, of which no definition can be given; there are also axioms and postulates, in a word prirrzary pritlciples, which cannot be proved, and indccd have no need of proof; and these are idetzticcll propositions, whose opposite involves an express contradiction.

Leibniz's distinction between necessary and contingent truths should be carefully differentiated from the distinction between essential and accidental properties which figured prominently in Aristotle's understanding of how change occurs. The necessity of which Leibniz spoke is an epistemologically defined attribute of propo.sitions, whereas the distinction between essence and accident is a distinction among properties, a distinction that does not depend upon human knowledge, specch, and thought. The difference between the two notions of necessity is already seen clearly by Peter Abelard,' who describes it as the difference between attributing necessity to the thing or to what is said. Abelard's example is

Whatever sits sits necessarily.

If this is understood as an attribution of necessity to the thing (de re) , it makes the false assertion that to be sitting is part of the essence of those who sit, so that, if someone is sitting, it would not be possible for her to be in any other position. If, on the other hand, it is understood as an attribution of necessity to what is said (de dicto), it makes the true assertion that 'Whatever sits sits' is a truth of reason. The notion of essential property, which faded from prominence with the demise of Aristotelian physics, has been revived by Kripke 119721, who

' Sce Kncale [1962].

characterizes it as a notion of t~tr~ttrl?llj.sic~cll rlec.e.v.si!\., as contrasted with Leibni1.s notion of lofiic,trl tzec.essi~y. It is logical necessity, also known as ~tt(~l!tic.i!\', that will be our principal concern here.

These two notions of necessity by no rneans exhaust the ways we use the word 'necessary'. There are epistcniic necessities. physical necessities, moral necessities. practical necessities, political necessities. and necessities imposed upon us by rules of etiquette. We shall not concern ourselves with these other necessities.

Logical necessity is a property of propositions, an indication of their cpistemic status. In keeping with our customary policy of effacing the distinction between sentences and the propositions they cxpt-ess, we shall speak of logical necessity as a property of (the Giidel numbers of) sentences. Whereas the use of modal notions in logic dates from the time of Aristotle, the use of modal logic dates only from thc second decade of the present century. Modern modal logic has a tangled history that is summarized in Ruth Barcsn Marcus's remark' that modal logic was conceived in sin, the sin of confusing use and mention. The sin was committed by A. N. Whitehead and Bcrtrand Russell [1910J. who read '4 -t $' as '& implies $." As Quine points out lucidly ((19531 and [ 1961 I ) , this is a category mistake, since '-+' is a connective but implication is a relation between sentences. Whitehead and Russell are confusing parts of speech; '+' is what grammarians call a conjunction, whereas 'implies' is a transitive verb. On either side of '+' one puts a sentence, whereas on either side of 'implies' one puts a tzcrnw of a sentence.

C. I . Lewis (19181 saw that there was something badly wrong with Whitehead and Russell's account. since there are many cases in which a material conditional is true even though there is no plausible scnsc in which the antecedent implies the consequent, but Lewis failed to diagnose the problen~ correctly. He thought that 'implies' was a connective other than the material conditional, whereas we now see that 'implies' is not a connective at all.

As luck would have it. this compounding of errors had happy consequences. It led Lewis to introduce a new connective '3' with the property that (4 3 $) was to be true i f and only if 4 implied JJ. Lewis studied in some detail the logical properties of this new connective. together with the associated connectivcs representing necessity and possibility. 'n', defined by

04 iff (14 3 &)

and ' 0 '. detined by

0 iff i ( 4 3 14)

' Reported In C)uine [ 196 1 1. Actually. Wh~tcliead and Rus\cll uwd thc \)mbol '3' whcrc we arc uung .-'

These investigations of Lewis wcl-c quite 1'ruitfi1I. forming the basis for all subsequent work in modal logic. In discussing this work. it will be convenient to take '0' as the sole modal primitive and to treat '3' and ' 0 ' as defined:

We wish to employ Lewis' methods to investigate logical necessity. S o o& is to count as true iff is logically necessary. So what our modal logic will look like will depend crucially upon what we take logical necessity to be. If we take logical nece\sity to be strict logical validity. we get a very barren system. As far as pure logic is concerned, there is no reason why thc extension of 'is logically necessary' should be the set of logically necessary truths rather than. say, the set of all pelicans, and so no sentence of the form n4 will be either logically necessary or logically impossible. We can determine whether a formula of the rnodal sentential calculus is valid under the construal of necessity as strict logical validity by the following simple test: we can think of a given ti)rmula cf, as built up by means of the Boolean connectives from atomic sentences and from sentences of the form or/,. To see whether I$ is valid. replace each of the latter components by a new sentential letter. then use a truth table to check whcthcr the resulting forn~ula of the classical scntcntial calculus is a tautology.

In ordcr to get a nontrivial modal logic. we must en~ploy a richer notion of logical necessity, according to which there arc certain sentences whose truth is so basic to our way of thinking and talking that they have the same epistemic status as logical validities, even though they are not actually logically valid. The traditional doctrine, as found, for examplc, in Carnap 119471. is that there are certain sentences, such as 'For every .Y, .r is a bachelor i f fx is an unmarried man', whose only function is to stipulate the meanings of some of the words we use. Such sentences are true in virtue of linguistic conventions, rather than on account of what the world is like. The set of logical truths is the closure under logical consequence of the set of such meaning postulates.'

If our notion of logical necessity is generoils enough. we shall be able to count basic linguistic and mathematical facts among the necessary truths. Assuming that we can specify the set of meaning postulates, we can use the methods of Godel 1193 I ] to talk about the closurc under logical consequence of the set of meaning postulates. Thus, we shall be able to prove theorems about the set of necessary truths. and these theorems will themselvcs express necessary truths.

Aristotle's doctrine in Mertrl~lr\.\ic..\ % that thc essential properties of a thing are those properties which are contained in the dctinition of the thing sounds quite \imilar to the rnodern analytic philosopher's doctrine that thc logically necessary trulhs are the logical consequences of definitions. The similarity is mi.;leading. however. since the role of definitions is quite different in Aristotlc's philozophy and in analytic philosophy.

This program confronts two formidable obstacles. The first is that the traditional doctrines about analyticity have fallen into ill-repute. Traditional logical empiricists drew a sharp distinction between analytic truths, which were true in virtuc of linguistic conventions, which were entirely devoid of empirical content, and which could be known, incorrigibly and a priori, by anyone who had a command of the language; and synthetic truths, which were learned from experience and which always had a clear, and preferably testable, empirical content. Much of epistemology rested upon this fundamental distinction between matters of language and matters of fact, until it was pointed out by Quine [I9511 and Putnam [ 19661 that the philosophical burden was more than the distinction could bear. The task of creating a scientific language and the task of creating a scientific theory are inextricably intertwined parts of the same enterprise, and there is no way of predicting, in the rough-and-tumble of scientific investigation, which parts of our theories will be treated as definitional and which as empirical.

In the present chapter, my response to this difficulty will be simply to ignore it. I shall suppose that it is possible to distinguish a class of sentences whose principal role is is to express linguistic, mathematical, or logical truths, but I shall have no occasion to assume that the sentences so distinguished are completely free of empirical consequences, that they are known u priori by anyone who understands the linguistic conventions in virtue of which they are true, or that they are immune from revision or rejection, nor to assume that the distinction is free of arbitrariness. My own view, which I shall not defend here, is the same as Putnam's: H. P. Grice and P. F. Strawson [I9561 are right in arguing that there has to be something to our intuitive distinction between the analytic and the synthetic, but Quine is right in arguing that the distinction cannot do the work that classical epistemologists have wanted it to do.

Ignoring the first obstacle, we shall concentrate upon the second, which is the problem raised by theorem 1.4. Once the use-mention confusions that deceived Whitehead, Russell, and Lewis have been cleared up, it becomes clear that implication is a relation between sentences and that logical necessity and possibility are properties of sentences. To these properties and relations we ought to be able to apply the apparatus of quantification theory, just as we can apply the apparatus to other properties and relations. We ought to be able to treat 'is necessary' as a predicate and to treat ' "Horses are animals" is necessary' as having subject-predicate form. But if we do this we shall find, according to Montague's theorem, that the basic laws of modal logic contradict the basic laws of arithmetic.

The standard response to the analogous problem raised by theorem 1.3, the liar antinomy, is to say that the set of true sentences of a given language cannot be defined within that language itself; we shall discuss this response in the next chapter. We cannot happily make the same response here. If, following Leibniz, we identify the necessary truths as those statements from whose denials one can

derive an express contradiction, and if we take derivability to be provability in some explicitly describable system of rules, it will follow that the set of necessary truths is a recursively enunierable set. But if the set of necessary truths is recursively enumerable. theorem 1 . I shows us how to explicitly define it.

There is a formula of our language5 2 that enurrierates the necessary truths of 2. Thus, the standard philosopher's response to paradoxes of self-reference, which is to claim that the formulation of the paradox employs notions that are only definable in the metalanguage, is not available to us here. We must confront the paradox directly; there is no escaping into the metalanguage.

Although we can define within :f? the set of necessary truths of 2, Montague's theorem shows that wc cannot consistently describe the set of necessary truths of 2 in the ways that we are naturally inclined to. The premises of Montague's theorem give a system of closure conditions. It is intuitively obvious that the set of logically necessary truths satisfies these closure conditions. But Montague's theorem shows us that, in spite of being intuitively obvious, the thesis that the closure conditions are satisfied is false. We want to see which of the intuitions that support the closure conditions can reasonably be maintained and which should be abandoned.

In supposing that we can treat logical necessity as derivability in some explicitly specifiable system of rules, I am idealizing. While the philosophers' technical notion of logical necessity is a good deal less vague than the everyday concept of necessity, it is nonetheless far from precise. There will inevitably be borderline cases, in which it is unclear whether a given biconditional ought to be regarded as a definition or as a deeply held tenet of empirical theory. There is not, in fact, a single formal system such that our informal notion of logical necessity coincides with derivability in that formal system. Instead there is a whole family of formal systems derivability in each of which is a reasonable candidate for how to make our informal notion precise.h

We are assured by Church's thesis that, once we have settled, somewhat arbitrarily, upon a particular, speciticd system of rules of deduction as our way of making precise Leibniz's informal notion of reducing a necessary truth to identical propositions, the precisionized set of logically necessary truths will be recursively enumerable. It follows by Craig's [I9531 theorem that there is a recursive set F such that the logically necessary truths are the first-order consequences of r. A proof of Craig's theorem, which tells us that, for any recursively

As before. wc take 'L to be a countable first-order language into which the language of arithmetic has bccn relatively interpreted in some fixed fashion cind for wh~ch a system of G(idel numbering has been established. Later on, in chapter5 7 arld 9, we shall dcvclop an account of the logic of vague terms. according to which, roughly. a sentence contalnlng a vague term will counl as definitely true if it comes out true under any reasonable policy for making the vague term precise. Thus we may think of our results here as definitc truths regarding our vague, ~n fo rn~a l concept of logical necessity.

enumerable set A of sentences which is closed under tirst-order conscquence, there is a recursive set I' such that A is the set of tirst-order consequences of I'. is as follows: since A is nonempty and recursively enumerable, there is a total recursive funct ion, / ' s~~ch that A is equal to the range of,/: Let I' consist of all the sentences (3\,,,)(11 such that ;I/I; = J l r r ) . 1' is recursive. since to test whether s is in I ' you first check whet her.^ has the form 1(31.,,)411 and. if so. calculate whether

. f l t l ) = rg1. Since the precisionized set of logically necessary truths is recursively enumera-

ble. it must he the extension of some Z',' forrnula of the language of arithmetic. I t is. in fact, the extension of many different 2:' li)rmulas. but there is one p~rt icular 2:'formula that stands out as especially simple, natural. and mathematically tractable. Assuming that we have a recursive set of axioms l', whether gotten by Craig's theorem or in some rnorc direct manner. let us take 'Bc\t,(.t)' to be the forrnali;ration of the statement that there exists a finite sequence of sentences ending in .r, each of which is either an element of I', an axiom of logic, o r obtained from earlier n~embers of the sequence by t?~odus l~orzrrls. 'UCIL,' was introduced by Giidel [ 193 1 1 to enumerate the set of theorems of a certain highcr- order version of Peano arithmetic.

We know that the extension of 'Rr>~t" is the set of logically necessary truths. 1 would like to propose that we go a step tarther and take 'BcII'(.v)' as a fo rmal 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, that the formal properties of 'B~N, ' are not delicately sensitive to which set of 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) -+ 10 = 0' is equivalent to ' iB r1 t l ( r~0 = 01)'. which asserts the consistency of T. and so, according to Godel's second incompleteness theoren~ ( 193 1 1 , ' i ~eu~( 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, so that mq5 means the same as Bertl(r$l), we find that some instances of the schenia

are not neccssary. We 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$l)is a theorem of 1' U (14). Now ' i B ~ ~ ~ t ~ ( r ~ 1 ) ' asserts the consistency of I' U {1$}. But, according to GBdel's second incompletcncss theorern. 1' U { i d } will be able to prove its own consistency only if r 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

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 detini- tions. We suppose that we have fixed a language for the modal sentential calculus with the connectives 'V', '1'. and '0' which has an ample supply of sentential 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,)* = (4" v +:". (-14)s = ~ ( 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~pleteneaa theorem is an unpublished but wcll-known discovery of Kripke. Liib gavc a more d~rect arfurnent using the self-referential lemma.

Axiom schemata:

All tautologies. O(4 + $1 +(O$ + O$)

OCUQ, - 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 ic to r io~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:

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 a s that are $1-valid without being r-valid.

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 - UdJ, I "[n(Ud, -+ d) WI 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~iiulas will 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.

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

(V modal sentence & ) ( 3 interpretation :+)(#I is not a theorem of G +

4:': is not a theorem of I')

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

(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 T- consistent.

The proof that we d o not have full compactness for either r-consistency or 31- consistency 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'), Con(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 !'I- consistent. 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 ( ( O " T & i O " " ~ ) + i ~ ) : all 11).

A typical finite subset of A has the form A =

" 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 d i \ jo~nt fronl D. For the construction of ~ c h sets. \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~npleteness theorern), it is not hard to see that an interpretation * that sets p* rc/~al tnM. ( 0 "T_& 1 O " ' ' T ) (where '\u' 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 & 1 0 "" T ) + p))* is true in \'[I. Then, because the same 2:' sentences are true in !'f and in !I?, B = {n: ~ e w ( r ( ( 0 "T & 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 "+IT) + 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 "1- validity 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:

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 1 ) is 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 r, and so 4 is true in every

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

possible world. Thus. if every xy 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 x': sentence 4, if $J is true in \'I. then Brw( r$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

is not r-valid.

PROPOSITION 2.4. For any sentence 4 and world $1, ~ e w ( r 4 1 ) is 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 1 ) is not true in 91 and try to find a world %, accessible from 91. in which 4 is false. To be a world, 2: must be a model of I . and to be 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' (6: Bew(r81) is true in "I}, it is enough to find a model of

{ O : Bew(r01) is true in 91) U { i 4 } 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) + Bew(r0 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 essen- tialism" 11953. p . 1741.

We have seen that, propcrly understood. involvment at the second level can be entirely benign. u& can be taken to mean the same as ~ e ~ . , ( r & l ) , SO that at the 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~dtit77e for Bot~zo is 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 de ditto, thus rendering the third grade of modal involvement no more perilous than the second. If '~(z 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 ~ M " abbreviate 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:

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. To 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.

We 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 am saying something that might bc of interest to the F.B.l-is

as contrasted with the innocuous

I believe r(gx)(x is a spy)1 .''

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 = &,I)* = (v = kt,)

(4 v q,)* = 4:s v $:" (TO)* = -,(d,*) ((31~)4)* = (3v)(&*)

- ( n $ ( ~ ~ , . . . , v,,))* = ~e\t , ( r ( tb(? , . . . . . {,)*I)

Even if we use the us 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 'a 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 r, 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. On 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 'a is an F' from T, we must think of r as a purely abstract

" 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 i t 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 possible world % is accessible from \)I iff there is a function j I!)(/ + 191 such that, for any formula +(v,, . . . , v,,) of the language of arithmetic, for any a , , . . . , a,, in if ~ e w ( r 4 ( & , . . . <)I) is true in !'I then 4(fm, . . . ,,fo) is true in 3. Then, for any sentence 4 of the language of arithmetic, Bew(r41) is true in \!I iff 4 is true in every world accessible from !)I.

PROOF: We want to see that, for any formula 4 ( v , , . . . v,,) of the language of

IJ Quine could see some point to specific judgments that a particular sentence was or was not necessary, but he could see no meaning or purpoae in ~terating modal operators; he complained [ I953a, p. 1741 about the "idle and cxcessive elaboration of laws of itcrated modality" in the modal sentential calculus. This was in 1953, bcfore Lob's theorem. In view of thc extensive and fruitful investigations of the logical laws of iteratcd modalities Lob's theorem provoked. one would not make the same complaint today. One does want, however. to make the same complaint about the modal predicate calculus. In our present state of knowledge, we have some undcrstanding of particular modal judgments but no understanding of iterated rnodalitiea.

- 1 . arithmetic, for any c r , , . . . , a,, in I$II , Brrr~(r4(;, . . . , u,,) 1 ) IS true in !'[ iff, for any world '8 and function J, if % is accessible from \'I via f , then 4vm, . - . . , ,ffLI,,)) is true in %. For this. it is enough to show that, if B P I V ( ~ + ( ~ , . . . , ' I , , ) ? ) is not true in $I, then

- T U (8 E Y.,,: ~ewl(r01) is true in 91) U {14(%, . . . , a,,))

is consistent. The proof of this is just like the proof of proposition 2.4.. Assuming that the axioms of I- are all true in !)?, then, for any world $1, if

every world is accessible from $1, then, in particular, 9? is accessible from !)I via some function f ; and it is easy to check that f is an isomorphism between ?I and '3?. Conversely, if t'l is isomorphic to 9>, then, for any world %{, % is accessible from t'l via the function that takes the nth member of t'L to the nth member of %. Thus 9; is uniquely characterized. up to isomorphisn~, by the fact that every world is accessible from 91.

The functionf'assigns to each member of It'll a counterpart in 1%). This notion of a counterpart relation is not quite the same as that developed by David Lewis 1 19681, for Lewis defines the counterpart relation individual-by-individual, picking out, for each individual a in /!)[I, the individual (or individuals, in case of ties) in which most resembles ( I , whereas for us the counterpart relation is defined globally, seeking out the best overall fit between the individuals of !)I and the individuals of 9. This global approach avoids some anomalies of Lewis's approach which arise because Lewis's approach, while it secks to ensure that the counterpart relation preserves as many as possible of the properties of each individual, makes no effort at all to ensure that the counterpart relation preserves relations among individuals. Thus, one would think, intuitively, that, Sonia being my child, it is not possible that instead I should have been Sonia's child. But on Lewis's account it may well be possible for me to have been Sonia's child, since there may be some world in which the individual who most resembles me happens to be the child of the individual who most resembles Sonia.

~ The relative merits of global versus local approaches to counterpart theory is

I an interesting problem in metaphysics, but to discuss it would take us too far from our concern here, which is the modal model theory of arithmetic.

In developing the nonmodal model theory of arithmetic, one particularly useful fact is this: assuming that the axioms of T are true in 91, is isomorphic to an initial segment" of every model of T. We can modify our possible-world semantics to incorporate this feature into our modal model theory by requiring

I that a world '!I be isomorphic to an initial segment of every world accessible from

I ?I; we restrict ourselves to countable models:

" Let 91 and % be models of the language of arithmetic with %+ a submodel of I)[. We say that %+ is an initial segtnent of 91 iff, whencvcr a c !'[I, h E /%+I, and <u.h> t: <". we have u E 1%1. If this occurs. then every X: sentence of Psi, that is true in %I will bc true in :'I.

PROPOSITION 2.6. Now take a possible world to be a countable model of r and say that a possible world % is accessible from !)I iff there is a function ffrorn 1911 onto an initial segment of /%\I so that, whenever Bew~(r+(:, . . . , - a,,)?) is true in "1. &(fo, . . . ,flN,,)) is true in \?i. Then. for any sentence 4 of thc language of arithmetic, B e ~ ' ( r d 7 ) is true in ?)I iff & is true in every world accessible from \)I.

The proof, a straightforward application of the extended omitting types theoremLh is omitted. The crucial step in the proof is thc observation that

is a theorem of PA.. Once again, 9i, assuming it is a model of T, is characterized up to isomorphism

by the fact that every world is accessible from 9;. Principles of logical inference which we learned for the modal sentential

calculus are upheld for the modal predicate calculus. Thus, if we take quantified G to be the deductive calculus whose axioms are the instances of the axiom schemata we gave for the predicate calculus in chapter 1, together with the axioms for G given earlier and whose rules are tno(luLsponerzs, necessitation, and universal generalization (from 6 to infer (Vv)H). we find that all the theorems of quantified G are r-valid. Among these theorems is the converse Barcan formula,

The Barcan formula,

will not be r-valid, so long as I- is consistent. since Gadel 11931 1 showed that, for any consistent recursively axiomatized extension of PA, there will be unprovable generalizations each of whose numerical instances arc provable.

The axioms of quantified (;' consist of the instances of the axiom schemata for sentential G', together with a11 sentences DO, for 8 an axiom of the predicate calculus, and the rules of quantified G' are modu.~pot~ens and universal generalization. Thc theorems of quantified G' properly include the theorems of quantitied G," and they arc all %-valid.

What we would like to do now is to give completeness theorems analogous to Solovay's results. We cannot do so. We cannot give a complete axiomatization of the set of r-valid modal formulas, bccause the set is not recursively enumerable. For the same reason, we cannot give a complete axiomatization of the set of %-valid formulas. The theorems below require supcrcrogatory knowledge of

I h Theorerrl 2.2.1.5 of Chang and Keisler 11'9731. '' To see the inclusion. tirst thow. by inductLon on the lcngths of proofs in quantilicd C , that, for

any theorem d, of quantified G, nb is a theorem of quantiticd G' . then usc the Fact that (a4 + d) I S an axiom of cluant~lied G'.

recursion theory and of the model theory of arithmetic. Readers whom cithcr training or taste disinclines to pursuc such results arc invited to skip to the next chapter; thc theorems skipped will not be used elsewhere.

THEOREM 2.7 (Artemov. Boolos. and McGee)." The \ct o f '9-valid 11 . formulas 1s complete , In thc sct of truths of arithmet~c.

Before starting the proof, there is a notational problem that needs to be cleared up. I originally introduced to represent the Gtidel number of thc result of prefixing rl 'S's to '0'. Later on I took, for a given model 91 and cr c /? ' I) , to be a standard name for a taken from outside the original language. Now, if u happens to be one of the standard integers of !)(, this notation will bc ambiguous. Most of the time, we shall bc concerned with situations in which 9; is relatively interpreted into !'L. so that all of 91's integers will be standard. In such situations, the ambiguity scarcely matters, since the two notational systems are readily interde- finable. At present, howcver, we want to look at nonstandard rnodels of arithmetic, so we had best avoid the ambiguity. Thus, where ili is the language of arithmetic, $1 is a model of if, and a , , . . . , ' I , , are mcmbcrs of I?'lI, let

represent the result of substituting the standard name of u , for v , . . . . . and the standard name of a,, for v,, in the formula 4 ( v , , . . . , v,,) of 2. 'n' will continue to denote the rcsult of prefixing n 'S's to '0' and '.? to represent the function taking rz to n. PROOF: The set of !)?-valid formulas is { [ d l : (V*)(* is a finite function defined on thc subformulas of + which can be extended to an interpretation -+ 4" is true)}; clearly this is in the set of arithmetical truths.

To prove completeness, Ict S be a set that is ': in the set of arithmetical truths. Thcrc is a number rn so that, given some nicely behaved standard listing of the oracle Turing machines,

S = {k: supplied with an oracle for the set of arithmetical truths and given k as an input, the n~th oracle Turing machine does not halt}.

To show completeness, we need to find, in a uniformly effective manner, a modal formula t,, for each k , so that k C: S iff ti is %-valid.

Fix k. Let H(x.V) be the formalization of the following open sentence into the language of arithmetic with the additional unary predicate 'V':

Supplied with an oracle for V and given input k, the mth oracle Turing machine halts after fewer than x steps of a computation that only asks the oracle about numbers less than x.

I' Artenlov [I9861 showed that the set of !I?-valid formulas lies outside the arithmetical hierarchy. The rnore specific result is from Boolos and McQee [19X7].

(We do not include explicit parameters for tn and k when writing 'H(x,V)', because m and k will remain fixed throughout the argument.) Let me use '2' to represent the language of arithmetic and 'X,.' to represent the result of adjoining the new unary predicate 'V' to 2.

Thus, k e. S iff i(3x)O(.r,{arithn1etical truths}). So, if k & S, there will be a number .r, so that C)(x,,,{arithmetical truths}). Our plan is to construct our modal formula 5, so that, if k & S, our counterexample to the \3?-validity of 5, will be an interpretation * that translates arithmetical predicates homophonically ('<'* =

'<', etc . ) and takes V* to be an arithmetical predicate such that, for each y < .x,,, V*(y) i f fy E {arithmetical truths}.

We need to find a way of constructing an arithmetical predicate so that, for a given number x,,, the numbers less that x,, which satisfy the predicate will be the arithmetical truths < x,,. Since there is an algorithm for testing whether an atomic predicate of the language of arithmetic is true, we know by theorem 1 . 1 and Church's thesis that the set of true atomic sentences is strongly representable in PA by a formula of the language of arithmetic. Using this representation, we take the phrase 'V is a truth predicate for y' to be an abbreviation for the following formula of Y,.:

y is a sentence of Y & r (vz)[the sentence z is a substitution instance of a subformula of y +

[(z is atomic & (V(Z) * z is true)) V (3v)(3w)(z = (v \/ w) (Viz) - (V(V) V Viw)))) V (3w)(z = TIMI & (V(Z) - iV(w))) V (Z is an existential sentence & (V(z) * (3w)(u3 is a substitution instance

of z & Viw)))]]

PA cannot be finitely a x i o m a t i ~ e d , ~ ~ but we can find a theorem of PA which entails a substantial fragment of PA; for definiteness, let us say that p entails all of z\,-PA, the fragment of PA we get by only allowing Z:,, formulas into the induction axiom schema.'"

We have, for each sentence + of the language of arithmetic,

p F V is a truth predicate for r+l + ( ~ ( r + l ) ++ 4) I '" See Ryll-Nardzcwski [1952]. I 211 Though there is no formula of Y: whosc extension is the satisfaction relation for Y , we can find

a formula Sat:',, of iP whose extension is thc restriction to L':,, formulas of the hatisfaction relation i for Y'. p will be a conjunction consihting of all the axiom5 of PA othcr that the induction axiom schema. together with enough instances of the induction axlom schema to prove all instances of 1 the schema I for Z::,, together with the sentencc that says that, for each I:',, formula F, if thcrc exists a number t h a ~ bears at:',, to y, then there exists a least number that bcars Sut':,, to V .

This is proved by induction on the complexity of aubforn~ulas of 6; the induction is carried out in the metatheory, not in p.

For each sentence 4 of the language of arithmetic, we can find a formula ~ ( x ) of the language of arithmetic so that

q is a truth predicate for r47

(i.e., the result of substituting 7 for 'V' in 'V is a truth predicate for r41') is a theorem of p. Do this by taking ~ ( x ) to be the disjunction of all formulas

( 3 9 , ) . . . (3",J(,Y = rqJ(c, . . . , <)l & (V(x) * $ ( \ I , , . . . , ~ 1 " ) ) )

for $(v,, . . . , v,,) a subformula of 4. Define 'V is a truth predicate below x' to be an abbreviation for '(Vy)((~l is a

sentence & y < x) + V is a truth predicate for y)'. For any natural number n , we can find an arithmetical formula q(x) so that

p c 7 is a truth predicate below n simply by letting x be the conjunction of all sentences with Godel numbers less than t~ and finding +q so that

p F q is a truth predicate for rX1

Let us agree, for the duration of the chapter, to treat the language of arithmetic as formulated using the predicates 'Z(x)', 'S(x,y)', 'A(x,~l.z))' and 'M(x,y,z)' in place of 'x = 0', 'S(x) = y', 'x + y = z', and 'x . y = 2'. We do this for ease of translation back and forth between the language of arithmetic and the modal language.

At last we arc ready to construct our formula 5,. It is the conditional:

[ p & (Vx)(Vy)(Vz)(A(x,~.,z) ++ nA(x,y.z)) & (v.x)(Vy)(Vz)(M(x.y, z) c* oM(x, v, z))l -+

(vx)(V is a truth predicate below x -+ 1O(V.x))

CLAIM 2.8. 5, is %-valid iff k E S

PROOF: (3) Suppose that k & S. Then there is a number .u,, so that O(x,,,{arithmetical truths)). Find an arithmetical predicate ~ ( x ) so that

\3? q is a truth predicate below

Given input k, the mth oracle Turing machine will do the same thing, whether it is supplied with an oracle for {arithmetical truths) or for {.r: q(x)), since the machine only asks the oracle about numbers < x,,, and below x,, {arithmetical truths} and {x: ?(x)) agree. Hence,

Let * be the intcrpretation that translates arithmetical formulas homophonically and takes 'V(x)'"o be $1). Then 91' it,": :so 5, is not 9;-valid. (c) The principal result we need is Tennenbaurn's theorem," which tclls us

that any model of p with universe w whose addition operation and multiplication operation are both recursive will be isomorphic to !);. The idea of Tennenbaum's proof is as follows: suppose that ?'I is a model of p with universe w that is not isomorphic to \li. Then, whcre C and L) are disjoint. recursivcly inseparable, recursively enumerable sets. we can find a non-standard element cr of I \ ' ( / so that, for cach 11 F C , :)lk (IS, divides .r) [a] and, for each n c: D , \'I (I-?, docs not divide .r) ( ~ 1 ; here p,, is the nth prime. If the addition and multiplication operation on '!I were recursive, { r z : :'l (I-?,divides.r) [ N ] } would bc a recursive set separating C and D.

Suppose that 5, is not 9;-valid. Let * be a counterexample to the %-validity of 6,. Define a model 9 of YL by

and so on. Clearly, for any sentence x of the language of arithmetic,%b x iff Y i k x*.

Let f be the function that rnatchcs up a member o f 19iJ with the corresponding member of (%I. Thus,

For any N , 6, and c , we have, becausc ((V-r)(Vy)(Vz)(A(x,y,:) ++ oA(x,y,z)))* is true in !I?,

<ci.h,c> F A'" iff < N . ~ . c > F A*'" iff <t i , b.c> is provably in A*"'

Thus, A"' is the graph of a recursive total function. Similarly, M"' is the graph of a recursive total function. It follows by lennenbaum's theorem that the reduction of % to the language of arithmetic is isomorphic to \Ji, that is, that the range of f ' is all of I$+/.

Expand \li to a model 9?+ of 2, by setting

" Fro111 Tennenhaum (19591. For more rccent, sharper results sce McAloon [I9821

Thcn f becomes an isomorphiarr~ from !I;+ onto \li. Now

!)i (3.1-)(V is a truth predicate below .I- & O(V,.\-)I"'

So. because of the way q\ was defined using *. ?; 1; ((3s)(V is a truth predicate bclow .r & H(V,.r))

and, because $4 is isomorphic to 3;+,

!I>+ k (3.r)(V is a truth predicate below .I- & O(V,s))

Consequently, for some .r,,. v"" is a truth predicate below .r,, and

Supplied with an oracle for v"'- and given input k , the mth oracle Turing machine halts after fewer than s,, steps of a computation that only asks the oracle about numbers < .r,,.

Since the members of v"' ' less that .r,, are the true arithmetical sentences less than x,, we have

Supplied with an oracle for the set of arithmetical truths and given input k , the nlth oraclc Turing rnachine halts after fewcr than .r,, steps of a computation that only asks the oracle about numbers < x,,.

that is, k 6. S.8.

THEOREM 2.9 ( ~ a r d a n ~ a n ) . " Let r be a recursively axiornatized extension of PA that does not entail any false 2:' sentences. Then the set of r -val id modal formulas is complete '2'.

PROOF: The set of r-valid modal formulas is I

( r47 : (V*)(* is a finite function defined on the subformulas of that can

bc extended to an interpretation -3 (3y)(y is a proof from I- of 4")))

S o the set of r-valid formulas is 1. Let R be a ternary recursive function. To show that the set of r-valid formulas

is complctc, we want to see how to find, in a uniformly effective manner, a modal Formula 5, for each k , so that 5, is r-valid iff (Vx)(3y)R(x,y.k). Using the notation of the last proof, 5, will be the following formula:

[ p & (V.x)(Vy)(Vz)(A(x,).,z) ++ oA(.r,y,z)) Bi (V,x)(Vy)(Vz)(M(x,y,z) - nM(x,y,:))l -+

(V,x)(V is a truth predicate below .r + ( 3 y ) ~ ( x , y , - ) )

'' Vardanyan []9X5]. George Boolob very kindly providcd mc with an English translation of Vardan- yan's paper.

CLAIM 2.10. 5, is r-valid iff (Vr ) (3y )R(x , y , k ) PROOF: (3) Suppose that i ( V x ) ( 3 y ) R ( x , y , k ) . Take m so that ( V y ) i R ( r n , ~ l , k ) . Let * be an interpretation which interprets the arithmetical formulas homophonically and which lets V* be an arithmetical formula such that

PA + V* is a truth predicate below m Let 91 be a model of U {true :' sentences}. Then 91/= i t , * , so that 5, is not 1'-valid. (G) Suppose 5, is not r-valid. Let 91 be a model of T and * an interpretation

such that !)IF -15 ," . Define a model % by:

and so on. For any arithmetical sentence 6, ! ) I /= +* iff */= 6. In the proof of the last theorem, we defined a function f: w -. w so that:

f (0) = (ib)(\31' (nZ(x))': [ b ] ) f ( t z+l) = (1b)(g) k (oS(x,y))* [f(rz),b])

Formalizing this recursive definition within the language of arithmetic, we get a Z:' formula F(x,y) so that

Now, v"" is definable in \'I; it is {c: ?Ik (3y ) (F(x . y ) & (V (y ) ) " ) [ c ] } . On the other hand, the set of sentences true in !'I is not definable in $1, and so v"" f {sentences true in ?I). In fact, according to theorem 1.3, there is a sentence A of 2 so that

F"' is the graph of a monomorphismf from 91 into the reduction to the language of arithmetic of %. There is no difficulty in applying the proof of Tennenbaum's theorem to 9I in place of \3?, showing that j ' i s , in fact, an isomorphism. If we expand 91 to a model !'I+ for LtV by stipulating v"" = {a F 1\'1/:f(u) E v ' ~ } , f will be an isomorphism from !)I+ onto %. Now. since

91 ( ( 3 x ) ( V is a truth predicate below x- & i ( 3 v ) ~ ( x . y , E ) ) ) * I we have, because of the way % was defined using *,

I

% /= (3x ) (V is a truth predicate below x & -1(3y)~(.r ,y ,k)

and so, since ?+ and 91 + are isomorphic, I

!)I + ( 3 x ) ( V is a truth predicate below x & i ( 3 v ) ~ ( x , y , k) Hence, for some a F (911,

!'I + /= (V is truth predicate below x & ( ~ v ) i ~ ( x , y , k ) ) [ a ]

Hence, -

!)I+ V is not a truth predicate for rhl

So,

? r + k x s m [ u j

and, for some rn 5 rA7,

Now, any ': arithmetical sentence true in !'I is true in 9i, since 9i is isomorphic to an initial segment of 91, and sentences are preserved under taking initial segments. Hence,

This proof is interesting because of the way it uses theorem 1.3. We normally think of theorem 1.3, the liar paradox, as a deeply disappointing result. Perhaps our grandchildren will view it as a felicitous result, i t having helped guide humanity past the folly of the naive theory of truth; but, from our present perspective, it appears to be a profoundly negative result. But the proof of claim 2.10 shows that, in spite of its negative thrust, the theorem can have positive applications within classical mathematics. This way of taming the liar paradox to do useful work was discovered by Robert Vaught 1 1 9671. As a simpler application of the same technique, we show that the one positive result from proposition 2.3 cannot be transferred to the modal predicate calculus.

PROPOSITION 2.11, Let r be consistent, recursively axionlatized extension of PA. Then the modal predicate logic of 1'-consistency will not be countably, or even recursively, compact.

PROOF: Let h be PA U {'(Vl)(Vy)(Vz)(A(x,?.,;) * oA(s,\~.z)) ' , ' (V, t)(Vy)(Vz) (M(x,v,:) - ~ M ( X . ~ , Z ) ) ' } U {'V is a truth predicate below n': all n) . We saw in the proof of theorem 2.7 that. for each finite subset A of A . we could, by

interpreting the arithmetical predicates honiophonically and making an appropriate choice of V*, find an interpretation :': such that. for cach 6 E A , 6:': is a theorem of PA. Thus. every finitc subset of A is r-consistent.

Take any interpretation *. and suppose that :'I is a model of I' U {H : ! : : 0 is an axiom of PA) U {'((vt-)(Vj3)(V:)(A(,r,y,:) t. nA(.r,!,z)))":', ' ((V.r)(Vy)(V:) (M(.x.j,,z) - oM(x,y .z ) ) )* '} . Define a model t; for .Y', by setting

j2y = )!)I) z'?' = Z*:" S': = S*" "?I ":+:\)I

and so on. Find a 2:' arithmetical formula I..(.r.!) so that

Letf be the function whose graph is F'", and let 91 + be the expansion of 'l: to a model of 2, got by letting v"' ' equal {tr E I!)(/: ~ ( L I ) E v'!'}. By the method used in the proof of claim 2.10, one can show that f is an isomorphism: !I + - 23 and that v"' ' is definable in \'I. It follows by theorem 1.3 that. for sorne , I , 'V is a truth predicate below i' is false in :'I +. Because 23 is isomorphic to !I +, ' V is a truth predicate below n' is false in 23. and so, because of the way 'li was 1 defined, ' (V is a truth predicate below n):~' is false in $1. Thus, !)I is not a model of (6" : 6 e A}, and indeed r U (6": 6 e A} has no n1odels.m

Our discussion of the modal logic of logical necessity will inspire us, I hope, with a healthy methodological holism which will serve us well in the subsequent chapters when we try to understand the logic of truth. In particular, it should make us realize that the failure of the intuitively obvious principle

rC#J? is true iff d, I

does not automatically preclude the dcveloptnent of a coherent theory of truth, any more than the failure of the intuitively obvious principle

precludes the development of a coherent theory of logical necessity. The success of Solovay and others in solving Montague's paradox should give us courage for the attempt to solve the insolubilitr."

2' Less optimistic lhan wc, mediaeval logicians referred to the various versions of the liar antinorny as thc "irrsolrtbilia": see Kneale and Kneale [ 1962. pp. 227ffl.

Tarski's Solutions to the Liar Antinomy

Of the work that has been done on the liar antinomy, possibly the most profound and certainly the most influential has been that of Tarski ( 1 19351 and [1944]). Tarski presents two proposals for policies that would enable us to talk about semantics without becoming ensnared in antinomies; they are the subject of the present chapter.

Both of Tarski's proposals require that, in giving a theory of truth for a language, the language one is speaking about should be poorcr in expressive power than the language one is speaking. In general, there need be no particular connection between the language one speaks and the language about which one speaks. One can speak about any language one chooses, utilizing any language one knows how to speak. The object language and the metalanguage can be identical or they can be entirely disjoint o r they can partially overlap or either can be a proper part of the other. However, says Tarski, one docs require a special connection between the object language and the metalanguage if one intends to give a theory of truth for the object language. Tarski's preferred policy for avoiding antinomies is to require that, in giving a theory of truth, the metalanguage rnusl be essentially richer than the object lurzguuge in expressive power.

Tarski never says precisely what he means by saying that one language is essentially richer than another, but he does give examples. One example is that second-order arithmetic, which has variables ranging both over natural numbers and over sets of natural numbers, is essentially richer than first-order arithmetic, which only has variables ranging over natural numbers. We can see how much greater the expressive power of second-ordcr arithmetic is by observing that in second-order number theory one can, in effect, talk about real numbers, since one can identify a real number with the set of places in its binary decimal expansion where Is appear. Thus, in first-order number theory, one can do arithmetic. whereas in second-order number theory, one can d o geometry, calculus, and analysis as well. In general, (n+ !)st-order number theory is essentially richer than nth-order number theory.

As another example, we may take the metalanguage to be the language of set theory, understood in the usual way, with variables ranging over all sets and ' F '

designating the elcmenthood relation, and the object language to be a first-order language that has been interpreted by specifying. within the language of set

theory. a set, which is to be the universe of discourse of the object language, and functions and relations on that set, which are to interpret the function signs and predicates of the object language. Within the object language. the only individuals we can talk about will be elements of the specified set, whereas in the metalanguage we can talk about the whole universe of sets. By any measure of the size of sets we can think of, the universe of set theory is immensely larger than any particular set. For any particular set, there are very much larger sets, and the universc of set theory is very much larger than any of these. Thus, in speaking the object language, one restricts one's attention to a very tiny fragment of what there is.

Tarski showed, by giving examples, that for a clearly specified interpreted formal language 2, built up from a finite vocabulary, one can give, within an essentially richer metalanguage, an r.rplic.it dejnition of the true sentences of Y. That is, one can produce an open sentence r ( x ) of the metalanguage, containing no semantic terms, such that the definition

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. ' I V ~ S a 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 -

(VX) (X is a p - +(x))

is a nominal definition of the tern1 p and 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 (Vx ) ( x is a p - O(.r)). But it is, at worst. a pedagogical defect if the theory treats (Vx)(x is a p - +(x)) as a definition and (Vx ) ( x is 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)(x is 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 first- order 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 ) l , rr1 + rpl = rr + pl , and rr1 : rpl = r ~ . ~ l , we first find a formula that specifies the denotations of closed terms:

Den(x.y) =,, ( 3 Q ) [ Q is a finite set of ordered pairs & (Vv)(Vw)(<~' ,w> F Q + ( v is a closed term & [ ( v = 'ol & M' : 0 ) V (3t)(3u)(<t,u> E Q & v = S( t ) & w1 = S(u ) ) V (3q)(Zr) (3 t ) (gu)(<q,r> F Q & <t,u> F Q &

v = q + t & w = r + u ) V (3q)(gr) (g t ) (gu)(<q,r> E Q & <t,u> F Q &

v = q: t & w = r.u)j)) & <x,y> E Q ]

Because we can use natural numbers as codes for finite sets of natural numbers, letting the code for the finite set S be 3.2', we do not really need the second- order 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 first- order 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:

Tr(x) ++ ( V R ) [ [ ( V y ) ( R ( y ) -, y is a first-order sentence) & (Vy)(Vz) (y and z are closed terms + (R (y 5 z) ++

( 3 v ) ( 3 w ) (Den(y,v) & Den(z,w) & v = w ) ) ) & (Vy)(Vz) (y and z are closed terms + (R (y < z) -

( 3 v ) ( 3 w ) (Den(y,v) & Den(z,w) & v < w ) ) ) & (Vy)(Vz) (R(y V 2 ) ++ ( R ( y ) V R(z ) ) ) & ('d sentence y ) ( R ( l y ) e i R ( y ) ) & (V variable v)(Vy)(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~ethod we 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:~ction as 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~able assignment iff it is satisfied bq every variable assignment.

speak these languages, we cannot use them to give an definition of truth for I

English. We can speculate that, someday in the future, human beings, by an awesome

I

I intellectual feat, will teach themselves a language essentially richer than present- day English in expressive power. But even these superhumans of the future,

i 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 expres-

i sive power. Tarski accepted this conclusion with remarkable equanimity. He is content to

1 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

I 'element' and 'molecule' but need not contain names of linguistic objects. This

1 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:

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

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,

1 according to Tarski, universal, and we cannot speak any language essentially I 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

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:

(Vy)(Tr(y) + y is a sentences of the object language). (Vy ) (Vz ) ( j and i are closed terms -+

(Tr(y y z) - (3v)(3w)(Detz(y,v) & Den(z,w) & v = w ) ) ) . (Vy)(Vz) (y and z are closed terms -,

(Tr(y < z) - ( 3 ~ ) ( 3 w ) ( D e n ( y , v ) & Den(z,w) & v < w)) ) . (Vy)(Vz)(Tr(y V z) * (Tr(y) Tr(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

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 (Vx)(R(x) t, R1(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 ' (Vx)(Tr(x) - Tr 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

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 = ", (VR)[ [ (Vw) (R(w) -+ w is an ordered pair) & R(<O. 1 >)

& (Vu)(Vv)(R(<u, v)> ++ R(<S(u),v.x>)] I + R(<y,z>)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

I 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. I

In applying Frege's method to convert an implicit to an cxplicit definition, we I 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, i f our object language is a first- order 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 'Tr ' , 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 1

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 i ; = 31 =

( V ~ ) ( V Z ) ( I ~ ~ ( ~ . ? E :I) - .v E :)

(Vy)(Vz) (Tr(y V z) - (Tr (y ) V TI-(,-)) (V sentence ! ) ( ~ r ( ? ~ ) - ~ T r ( y ) ) (V variable ~l) (Vx)(Tr( (? t , )y ) - ( 3 z ) T r ( y v 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(Tr) to be the conjunction o f the six axioms, taking y to be the formula got from y(Tr) 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.

The true sentences of :fie, the original language of set theory without the added 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 G ~ set theory, unlike number theory, moving to a second-order logic docs us no good. Where y(Tr) is the conjunction of the six sentences that implicitly define the true sentences of Yv, we write

' I f , alternatively, we take our second-order definition to be 'Tr(.r) - ., (3:) ( y ( z ) & .r R :)', we conclude ' (V. t ) lTr( .x) ' , which is equally absurd.

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 (Giidel- Bemays 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 ~ ~ t s , consist 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 r.w = 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~quely specifies the referents of 'true sentence of the language of physics' mod~rlo the 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,

we had best give up a11 pretense that philosphical inquiry is rational. Philosophical terms do not refer. Philosophical beliefs are not true, indeed i t 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 I 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 self- understanding.

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

I 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 1 convey to a friend the information that l- is not wholly accurate? I

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 truth, 1 can succinctly convey the information that is inaccurate simply by

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

I 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 -+ x 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<n and giving axioms implicitly defining 1 'Tr,,' as a truth predicate for 2,, We may even extend this process into the

I transfinite, continuing as long as we have names for the ordinals to use as subscripts.

As we shall see in chapter 5 , this procedure increases the expressive power of the original object language substantially. It does not, however, bring us closer to the goal of obtaining a theory of truth for English. To understand how the English language works, we need be an account that encompasses the language as a whole; an infinitely fragmented account will not do. To be sure, we can get

I

8 2,,. r which was first investigated by Carol Karp [1964]. differs frorn ordinary first-order logic in that, while ordinary logic only allows ua to form the conjunction or dis~unc!ion of two formulas I

at a time, in %,,,, we can form thc conjunction or disjunction of an arbitrary set of sentences, finitc or infinite. Thus thc set of true sentences of the objcct language is the cxtension of the dissunction of the set of all sentences of the form (x = r$' & 6). for 6 a sentencc of the objcct language.

' The importance of the notion of truth as a means for, in cffect. producing infinite conjunctions and disjunctions was emphasized by Quine [1970].

"I This objection to the redundancy theory was advanced by Tarski [1944, %16].

about in the world even if our semantic theory is infinitely fragmented, but then again we can get about in the world with no semantic theory at all.

The infinitely fragmented account serves neither the practical purposes that the ordinary unitary notion of truth already serves nor the theoretical purposes that we would like it to serve. For practical purposes, we would like to use the notion of truth as a means for asserting or denying a totality of sentences that we can name but cannot list. But in order to assert a set of sentences A of the language with the 'Tr,'s by saying that the members of A arc all Tr,,, for appropriate n . we have to have an upper bound on the subscripts of 'Tr' that occur in A . But such an open bound will not usually be available, for cases in which we cannot list the members o f A are typically also cases in which we cannot determine an upper . .

bound on the subscripts in A. For theoretical purposes, we would like to be able to use the notion of truth

to express significant generali~ations about the relations of the expressions of the language to the world and to each other. Significant generalizations suggest themselves, for example,

But these generalizations cannot be legitimately formulated anywhere in our sequence of languages, for they use a variable as the subscript of 'Tr'. I f we were to allow variable subscripts, treating 'Tr,(x)' as a binary relation, we would be able to use therorem 1.2 to produce a sentence A so that

and, once again, we would be mired in paradox. Tarski claims to have shown how to give, for a wide class of formalized

languages, semantic theories that are "in harmony with the postulates of the unity of science and of physicalism" [1936, p. 4061; in this he has succeeded admirably, showing how to utilize the logical resources of higher-order languages to explicate the semantical features of lower-order languages. It is sometimes claimed (though not by Tarski himself) that Tarski has shown how to give a general account of semantics which is in harmony with the postulates of the unity of science and physicalism, but this is surely excessive. Tarski himself disavowed any attempt to give a semantics for natural languages, and if we nonetheless try to use his methods to investigate the semantics of a natural language, what we get, far from a unified science, is a disjointed and fragmentary account that can describe the parts of language used in chemistry or physics but cannot describe the language used in linguistics or philosophy. If we accept the constraints Tarski's policy imposes, we must either relinquish the hope for a unified science or else dismiss semantics (except as applied to formalized languages) as a pseudoscience.

laws of physics and biophysical bridge laws. we could derive all the truths of biology. We would rcquire an utterly detailed physical description, and we would not be able to derive "Larvac of Agatlzyrnus urTxrza feed upon Aguve palrrzeri and Aguve c,hpsantha" (unless we were willing to abide a spccies-by-species list of names), but we would be able to derive detailed descriptions of the lives of butterflies. Although this doctrine plays an important role in reductionist metaphysics, surcly we should not assess the work of a flesh-and-blood biologist by the success of his theory in the hypothetical situation in which biology is a science of pcrfect information, but by its success in the rcal-life situation in which data are scarce and hard to come by. In the real-life situation, the lepidopterist can do no better than systematically to list observcd species.

Semantics, it seems to me, is more like lepidoptery than like chemistry, for the connections between words and objects studied by semanticists are accidental. Although undoubtedly there are important general laws governing the reference relation, it is surely too much to expect that general laws will tell us that 'inflammable' means "flammable" rather than "not flammable" or that 'egrc- gious', derived from a Latin word meaning "outstandingly good," means "outstandingly bad." It is a defect of Tarski's account qua semantic theory that it fails to inform us of the general laws governing the reference relation." But, except perhaps in an eschatological limiting case, knowing the general laws governing the reference relation would not enable us to determine the precise reference of every single term of the language. and we need to determine the precise reference of cvery single term if our theory is going to be materially adequate in the sense of convention T. Although a theory along the lines proposed by Field might be, in some ways, conceptually more satisfying than Tarski's theory, it would provide too little information to satisfy convention T. To get a detailed account, in semantics just as in lepidoptery, we must give a list.

It might be responded that I have misapplied the analogy. Tarski claimed to show how to develop the semantics of a single language, and an analogous investigation in lepidoptery would be aimed not at describing all the butterflies in an entire continent, as Howe aims to do, but at understanding a single specimen in detail. Here the situation looks more promising. since we can derive the macroscopic properties of the insect from the molecular structure of its nucleic acids, thus reducing biological features of the organism to chemical features. We do not find such a derivation in Howe's book, but that is because lepidoptery has only begun to absorb the impact of the recent revolution in molecular biology. Within the forseeable future, it should be possible to determine the genetic structure of each species of butterfly and to explain the life cycle of the species in terms of this genetic structure. To be sure, we shall not be able to infer from

" It is not so clear, however, that this is any defect in Tarbki's account qua dejinition of truth, and Tarski's expressed aim was to provide a definition.

the genctic structure that A . trn.sna lives in the eastern Sonoran desert and feeds upon Agave. but we shall be able to deduce that it lives in a desert climate and feeds upon succulents. The molecular lepidoptery of thc future will reduce present-day naturalistic lepidoptery to chemistry.

Unfortunately, focusing our attention upon how the life cycle of a single insect depends upon its genetic structure does nothing to improve Field's position. The difficulty is in specifying the genetic structure of the particular specimen we are examining. Although there are general laws that constrain what genetic structures are possible, these general laws stop far short of determining the genetic structure of a particular individual. Were it not the case that the general laws of nature drastically underdetermine the genetic structures of particular butterflies. the order Lepidopteru would not display the splendid diversity it in fact displays. To describe the genetic structure of a particular insect. we cannot do better than to give a nucleotidc-by-nucleotide description of the DNA molecule. Thus, the basis of the genetic description of the insect is a nucleotide-by-nucleotide list that plays much the role in molecular lepidoptery that Tarski's word-by-word vocabulary list plays in his semantics. The derivation of the overall biological features of the organism from its DNA structure by means of general biological laws is analogous to the derivation of the overall semantic features of a language from the references of the simple terms by means of general semantic laws. We may think of molecular lepidoptery as describing a space of possible genetic structures of butterflies and as depicting the overall biological features of the possible butterfly that occupies each position in that space; only a few of the hypothetical possibilities are realized in actual butterflies. To deterimine the position of a particular butterfly within the genetic space, we must specify in exact detail the molecular structure of the DNA. For this purpose, we can do no better than to give a list.

The conclusion Field wants to draw from his objection to Tarski's methodology is that Tarski's methods fail to provide a physicalistically acceptable understanding of semantics. Although we have found little merit in Field's methodological complaint, we have good reasons to accept Field's conclusion. A reason that has been stressed here is the restriction of Tarski's method to languages that are poorer in expressive power than the language we speak. Another reason, closer to Field's concerns, is that Tarski's methods only give a semantic theory for the special case in which the object language is included within the metalanguage. To have a satisfactory understanding of semantics, we would have to see how to give truth conditions not only for our own language but for a wide diversity of languages, and to do this we would need to solve a problem Tarski completely sets aside, how correctly to translate other languages into our own language.

We nlay speculate that future semantic theory will describe a space of possible human languages and will give the truth conditions for the sentences of the language at each position in that space. If we wish to determine truth conditions for a language with the degree of precision envisaged by convention T, we must

specify the language in precise Ltail: we nlust distinguish English from the variant that is just like English except that 'infianlmable' means "not flammable," and for this purpose we cannot expect to do much better than to give a term-by- term list. Even so, we would likc to have some sort of social and historical explanation of how we came to speak the language we speak. It would not do to answer the request for such an explanation by pointing out that the referents of the terms of our own language are determined by the simple schema

refers in this very language to T

This response would be unsatisfactory in just the way that it would be unsatisfactory to ask your colleagues to gather for a meeting, then to respond to their question-"Why are we heres?''-by pointing out that the rules for using indexicals ensure that every utterance of 'We are here now' is true.

The upshot of our discussion of Field is to agree that Tarski has left important questions unanswered and so failed to provide us with a satisfactory foundation for a general semantic theory, yet to doubt whether, given the aims Tarski intcnded to accomplish, these omissions should be considered a defect in Tarski's work.

Kripke and 3-valued Logic

Kripke [I9751 has significantly advanced our understanding of the problems raised by the paradoxes by applying to these problems the methods of the mathe-

! matical theory of inductive definitions. Kripke develops an account according to I which the paradoxical sentences are neither true nor false, utilizing the strong

! 3-valued logic of S.C. Kleene [1952, $541 to describe the logical properties of

I such non-truth-valued sentences. The construction Kripke dcvelops is an ex-

1 tremely versatile mathematical tool that can be fruitfully used for a variety of

I philosophical purposes. I As Kripke emphasizes [1975, p. 771, his results do not depcnd crucially upon I the choice of the Kleene 3-valued logic as the method for handling truth-value

gaps. A variety of logics for languages with truth-value gaps have been proposed, and analogues to the results Kripke obtains using the 3-valued logic can be obtained for most of these other logics.' But whercas the mathematical results will be the same, their philosophical significance may vary. The discussion in this chaptcr is intended to apply only to the particular version of Kripke's construction which employs the strong Kleene 3-valued logic.

The idea behind Kripke's construction is that the paradoxical sentences are defective, in much the way that sentences that contain denotationless proper

I names and sentences that contain category mistakes have been thought to be I defective. Unlikc semantically well-formed sentences, these defective sentences

are neither true nor false. One uses a 3-valued logic to describe how these 1 defective sentences interact with normal sentences. i Given a countable first-order language 2 and an acceptable structure !'I for 2, I

we form the language Y' by adjoining the single new unary predicate 'Tr' to 2. We expand '?( to a rlassiccrl model (!'I ,E) of 2 ' by picking a subset E of (t ' II , which is to be the extension of 'Tr'. We get aparrinl model ( ! ' ( , (E ,A ) ) by picking two disjoint subsets E and A of It'll. The extension E is to consist of those things to which the predicate 'Tr' definitely applics, while the anti-extcrzsion A is to consist of thosc things to which thc predicate 'Tr' definitely does not apply. There

' An historically Important example of a method for handling truth-value gaps which is not amenable to Kripke's techniques is the 3-valued logic of tukasiewitz (19201. We ahall encounter another example in chapter 8.

may be members of /!'I/ which are in neither E nor A; for such things it remains unsettled whether 'Tr' applies. Whether a sentence is true in the partial model is determined by the following rules:

An atomic sentence of YiP.,, is true (falsc) in (?L,(E,A)) iff it is true (false) in !'I . An atomic sentence Tr(r) is true in (!'I,(E.,A)) iff r" E E, and it is false iff 7'1' F A . A disjunction is true in the partial model iff one or both of its disjuncts are true, and it is false iff both disjuncts are false. A negation is true in the partial model iff the sentence negated is false and false iff the sentence negated is true. (3v)$(v) is true in (!'I,(E,A)) iff, for some n E /?I/. $(;) is true in (?l,(E,A)), and it is false iff, for each a in I?Il, $(a) is false.

If 'Tr' occurs only positively within 4, that is, if every occurrence of 'Tr' in 4 is within the scope of an even number of negation signs, then 4 will be true in the partial model (!'I,(E,A)) iff 4 is true in the classical model (!'I,E). In particular, a sentence of Y\!, will be true in (!'I,(E,A)) iff it is true in :'I.

If A should happen to be the complement of E-that is. if A U E = /?I[-then the sentences of %': true in the partial model (\'I,(E,A)) will be precisely the sentences true in the classical model (?I,E).

Conjunctions, conditionals, biconditionals, and universal quantifications are defined in terms of disjunctions, negations, and existential quantifications in the usual way. The behavior of the Boolean connectives is nicely summarized in the following truth table:

As Kripke remarks [1975, p. 65111, '1' should not be thought of as representing a third truth value. intermediate between truth and falsity, but rather as indicating the absence of a truth value. Kleene introduced his logic to describe the operation of decision procedures in arithmetic. In response to a question of the form, "Is n in S?," 'T' represents an affirmative answer and 'F' represents a negative

e i answer. "?' indicates that no answer has yet been obtained. without judging the

question whether an answcr will be obtained sometime in the future. The system is designed so that, once we have assigned a truth value 'T' or 'F' to a sentence, future investigations will not cause us to revoke that assignment. In other words, the system is monotone, so that if(UI.(E,A)) and (!'I,(F.B)) are partial models with E C F and A C B, any sentence that is true (false) in (\!I,(E.A)) will be true (false) in (!'I,(F,B)).

i There is no reason why predicates other than 'Tr' should not be partially defined. It is only for simplicity that we focus our attention on the case in which

I -.

'Tr' is the only predicate that is not fully defined. Theorem 1.3 tells us that we cannot find a classical model (91,E) such that

! E = {sentences true in the classical model (?I,E)}

Kripke's fundamental observation is that one can find partial models that have the analogous property.

I DEFINITION. A $xed point is a pair (EVA) such that

E = {sentences true in the partial model (:'(,(E,A))} A = {members of (!'1( that are not sentences} U

{sentences false in the partial model (!'(,(E,A))}

THEOREM 4.1 (Martin, Woodruff, and Kripke). There is a fixed point. In fact, thcre is a fixed point (E.,A,) which is least in the sense that, if (@,A) is another fixed point, then E, E and A, A.

PROOF: Define, for each ordinal a ,

E,, = {sentences true in the partial model (:'I.( E,, pU,, AP))}; A,, = {nonsentences} U

{sentences false in the partial model (:'I,( ,,U," EB, pUuAB))}.

Set E. equal2 to LJoK Ecx and set A% equal to goKAcr. Evidently, the Ems and the Aes are nonstrictly increasing, that is, if /3 5 a , then El, C E, and A,, C A, It follows that there exists an ordinal K with E, = Ex and A, = A,. Hence,

Ex = E, + , = {sentences true in (I'I,(E,,A,))) = {sentences true in (!'I,(E,,A,)))

Similarly.

A, = {nonsentences) U {sentences false in (PI,(E,,A=))I

Talk about OR, the class of all ordinals, is not to be taken literally. 'a E OR' is short for 'a is an ord~nal.'

so that (E,,A,) is a fixed point. A transfinite induction shows that, if ( E , A ) is a fixed point, then, for each a , E, C E and A, C A.. HISTORICAL REMARK 4.2. The existence of a fixed point was first demonstrated by Robert Martin and Peter Woodruff [1975], who worked with a different system of 3-valued logic. also due to Kleene. Martin and Woodruff used Zorn's lemma to show that the set of dis.joint pairs (E,A) such that E C {sentences true in (\'l,(E,A))} and A C {nonsentences} U {sentences false in (\'l,(E,A))} has a maximal element, and then they showed that this niaximal element is a fixed point.

Kripke's proof, which was obtained independently, gives a bit more information about the stucturc of the fixed points, giving, in particular. the existence of a least fixed point. Kripke's proof is especially important because of the intimate connections with the mathematical theory of inductive definitions which we shall explore in the next chapter..

A grave and obvious difficulty stands in the way of any attenlpt to apply theorem 4.1 to the study of English. 'The difficulty is the strengthened liar response. The Kripke construction is intended to codify an intuition that the paradoxical sentences, being semantically defective, are neither true nor false, and that the antinomies arise because we treat these defective sentences like normal truth-valued sentences. The fundamental thesis motivating Kripke's theory, that the liar sentencc A is neither true nor false, can be expressed by a sentence of 2', namely,

But this fundamental thesis of Kripke's theory is itsclf, according to Kripke's theory, semantically defective and so untrue.

Kripke's response to this problem is to invoke the object languagclmeta- language distinction, this time with a curious twist. The twist is that, unlike Tarski's theory, where, the object language being part of the metalanguage, an object language sentence will have the same truth value whether it is regarded as part of the object language or part of the metalanguage, in Kripke's theory whether a sentence is true may depend on whether it is regarded as part of the object language or part of the metalanguage. Whereas no sentence that has a truth value yuc~ sentence of the object language has a different truth value qua sentence of the metalanguage, a sentence that has no truth value quu sentence of the object language will be truth-valued when it is construed as a sentence of the metalanguage. Thus, in the object language Y' the sentence,

is truth-valueless, whereas, in the metalanguage in which classical 2-valued logic is used to prove theorem 4.1, the sentence is true. Similarly,

the formalization of the English sentence, 'Evcry true sentence is a true sentence', is truth-valueless in the object language, true in the metalanguagc.

Kripke regarded this disparity between the status of a sentence in the object language and its status in the metalanguage as reflecting stages in our linguistic development. He says,

Such seniantical notions as "grounded." "paradoxical," etc. belong to the metalanguage.' This situation seems to me to be intuitively acceptable; in contrast to the notion of truth, none of these notions is to be found in natural language in its pristine purity, before philosophers reflect upon its semantics (in particular, upon the semantic paradoxes). I f we give up the goal of a universal language, models of the type presented in this paper are plausible as models of natural language at a stage before we reflect on the generation process associated with the concept of truth, the stage which continues in the daily life of nonphilosophical speakers [1975, p. 801-11.

Kripke presents no evidence to support the contention that the construction given in theorem 4.1 represents a stage in the development of English speakers, and, in the absence of such evidence, the contention does not seem to be very plausible. If the contention were correct, we would expect nonphilosophical speakers of English to be unwilling to assert

Every true sentence is a true sentence.

since, according to the logic they employ, the sentence is not true. But, in fact, whatever hesitation English speakers feel about asserting

Every true sentence is a true sentence

comes not because they think the sentence is dubious but because they think it is too obviously true to be worth stating. One would suppose that a child learning the logical rules of English would naturally incline toward the simple rules of classical 2-valued logic rather than to the more complicated rules of _?-valued logic and that it is only at a much later stage of her development, after philosophical pranksters have drawn her attention to certain perplexing special cases, that the speaker begins to doubt the universal validity of the law of the excluded middle. Thus, in the absence of definite psychological evidence, it appears likely that the order of developnlent is just the opposite of what Kripke describes. It is the classical 2-valued logic that is employed in the daily life of nonphilosophical speakers, and speakers employ the 3-valued logic, if they employ it at all, as a

' A sentcncc is said to be grortnded if it is assigned a truth bulue in the least tixcd point. A sentence that does not receive a truth value in any fixed point ia purudo.rica1.

highly sophisticated response to problems that are only visible upon philosophical reflection.

The Kripke construction does not appear to bc helpful in understanding ordinary speakers' usage of the word 'true'. Quite frankly, the philosophical explica- tion of folk semantics does not seem to me to be a particularly interesting or =-

useful task, in any event. The more important task, it seems to me, is to establish the logical basis for a consistent semantics of natural language. The problem of giving voice to the our preanalytic intuitions about truth is comparatively less important, just as understanding popular misconceptions about space and time is comparatively less important than understanding the actual geomctry of space- time.

As a vehicle for developing a consistent semantics for a natural language, the construction given in theorem 4.1, taken at face value, does not make it past the starting gate. The problem here is the same problem we encountered with Tarski's preferred solution. The construction of theorem 4.1 requires that the universe of discourse of 2' be a set. The universe of discourse of English is not a set. Therefore, the construction cannot begin to give a theory of truth for English. All theorem 4.1, taken at face value, gives us is is a class of artificial languages, unavoidably essentially poorer than English in expressive power, which are able to give a rudimentary account of their own semantics. The account these artificial languages give of their own semantics is not a very good one, since all difficult questions, such as what to say about the liar sentence, are referred to the metatheory.

Theorem 4.1 does not have to be taken at face value. As noted earlier, the I Kripke construction is a remarkably versatile philosophical tool. The construction !

can be utilized in a variety of ways in attempting to overcome the restrictions imposed by Tarski's preferred policy for avoiding antinomies. We shall now examine two straightforward methods for applying Kripke's construction in attempting to develop the semantics for a language within the language itself. Further applications will be found in chapters 5 and 8.

Both of the applications of the Kripke construction which we shall be discussing in this chapter make use of the following result of Solomon Feferman 1 19821:

DEFINITION. KF, the Kripke-Feferman axiom system, consists of the following sentences of 2 :

(Vx)(Tr(x) -+ x is a sentence of 5$,) (Vx,) . . . ( V , Y , , ) ( T ~ ( ~ ~ ( Z , . . . , <)I) - $(xl, . . . , x,,)), for

+(xi, . . . , x,,) an atomic or negated atomic formula of Y . (Vx,) . . . (Vx,,)(Tr(r~r(r(%, . . . , c)7) * Tr(r(x,, . . . , x,,)), for

r(xI, . . . , x,,) a term of Y ' (Vx,) . . . (V~ , , ) (Tr (~ lTr ( r ($ , . . . , $17) cf T r ( l r ( x , , . . . , x,,)),

for r (xI , . . . , x,,) a term of 3'.

(Vl)(Vy)(Tr(x V yj cf ((Tr(x1 V Tr(y))j (Vx)(Vy)(Tr(~l(x \! .v)) cf (Tr ( lx ) & T O Y ) ) ) (VX)(T~(T~ 1 x 1 cf Tr(x)) (V variable v)(Vy)(Tr((?v)y) - (3x)Tr(v Vi-;)) (t/ variable v)(Vy)(Tr(l(Jv)y) - (Vx)Tr(l(v vik))) i (3x)(Tr(x) & Tr(7x))

THEOREM^.^ (Feferman). For E a subset of /?I(, the classical model (YI,E) is a model of KF iff the partial model (!)i,(E,{nonsentences) U (4 : 1 4 E

E))) is a fixed point. Also, for each sentence 4 of Y$,

is a theorem of KF U R,

PROOF: The proof is quite straightforward, although the details are fairly lengthy. First, assume that the classical model (!)l,E) is a model of KF, set A = {nonsentences) U (4: 1 4 F E), and prove, by induction on the complexity of 4, that, for each sentence 4 of 2:,,

4 is true in (?(,@,A)) iff 4 F E

and

4 is false in (\!I,(E,A)) iff 4 E A

Next, verify that, if (?I,(E,A)) is a fixed point, then each of the axioms of KF is true in (!'l,E). Finally, use induction on the complexity of + to show that, for each formula $(x,, . . . , x,,) of 2',

and

are both theorems of KF U R., ,. (This induction is conducted in the metatheory, not in KF U R,,.).

Our first proposal for using Kripke's construction to overcome the limitations imposed by the usual object languageimetalanguage policy is the proposal that we repudiate classical logic and henceforward work entirely within the object language with its 3-valued logic. According to this proposal, the unabashed use Kripkc makes of classical logic in developing his construction must be regarded as merely a heuristic measure. Out of habit, we continue to use classical logic while we are getting our bearings in a nonclassical world. But, once we get a firm handhold, we shall kick away the ladder, and thenceforward we shall use only the weaker 3-valued logic.

In speaking of the repudiation of classical logic, I am being contentious.

Kripke himself hotly denies any suggestion that the abandonment of classical logic is being contemplated. He says.

I have been amazed to hear my use of the Kleene valuation compared occasionally to the proposals of those who favor abandoning standard logic "for quantum mechanics," or positing extra truth values beyond truth and falsity, etc. Such a reaction surprised me as much as it would presumably surprise Kleene, who intended (as 1 do here) to write a book of standard mathematical results, provable in conventional mathematics. "Undefined" is not an extra truth value, anymore than-in Klcene's book-11 is an cxtra number in sec. 63. Nor should it be said that "classical logic" does not generally hold, any more than (in Klccne) the use of partially defined functions invalidates the commutative law of addition. If certain sentences express propositions, any tautological truth function of them expresses a true proposition. Of course formulas, even with the form of tautologies, which have components that do not have truth values may have truth functions that do not express propositions either. (This happens under the Kleene evaluation, but not under the van ~ raassen .9 Mere conventions for handling terms that do not designate numbers should not be called changes in arithmetic; conventions for handling sentences that do not express propositions are not, in any philosophically significant sense, "changes in logic" [1975, pp. 64fl.

Thc word 'proposition' is used in two different ways..' On one way of speaking, propositions are inherently bearers of truth values, and it makes no sense to speak of a proposition that has no truth value; one might speak this way, for example, if one identifies a proposition with a set of possible worlds. On the other way of speaking, propositions are the objects of mental attitudes. Thus, for example, if I express a childlike faith in authority by saying, in all sincerity,

Everything President Reagan says is true.

then (assuming paradoxical sentences lack truth values) what 1 have said will be true, false, or truth-valueless, depending on what President Reagan says. Thus, if we identify propositions as the bearers of truth values, we shall say that, depending upon what President Reagan says, I have expressed a true proposition, a false proposition, or no proposition at all. On the other hand, no matter what President Reagan says, in saying what 1 said, I expressed a belief, be it true, falsc, or paradoxical. Thus, if we identify propositions as the objects of mental

Van Fraassen's methods will be described in chapter X . Sec Pollock [1982]. In the discussion here, I am taking it for granted that there are propositions although this is a dubious thesis.

attitudes, we shall say that, depending on what President Keagan says. I have expressed a true proposition, a false proposition. or a truth-valueless proposition.

I f propositions are taken to be the objects of mental attitudcs, then Kripke's contention that "conventions for handling sentences that do not express propositions are not, in any philosophically significant sense, 'changes in logic,"' is very likely correct. 'Curious green ideas sleep furiously' is an example of a gramn~atically well-formed sentence that docs not express a thought that a human being might believe or disbelieve. Such sentences are far removed from the logician's central concerns, and conventions for handling them may indeed be regarded as arbitrary. But the paradoxical sentences do not fall into the same category as 'Curious green idcas sleep furiously'. Paradoxical sentences can be believed or disbelieved, so if one regards propositions as the objects of mental attitudes, one will not deny that the paradoxical sentences express propositions.

I f . on the other hand, one rcgards it as inherent in the notion of proposition that propositions have truth values. then one will find it natural to deny that paradoxical sentences express propositions. But then one would not regard the differences in the way various logical systems handle propositionless sentences as trifling, for such differences distinguish logical systems that permit one to deduce untrue conclusions from true premisses from logical systems that reliably yield true conclusions.

One of the aims of logic is to teach us how to reason well by showing us patterns of inference which are reliable. Two primu facie requirements that a logical system must satisfy in order to secure this goal are the following:

The patterns of reasoning sanctioned by the system must be reliable, that is, they must never permit us to infer an untrue conclusion from true premisses.' It must be possible for human reasoners to learn the patterns of inference and to follow them.

Thus, for example, a rule like

From 4 and (4 + $) one may infer 4, provided 4 is true

is reliable, but it is not learnable, since we have no way of telling whether the restriction 'providcd 4 is true' is met. Notice that, if propositions are taken to be the bearers of truth values, the condition that a logical system never permit the deduction of a sentence expressing an untrue proposition from sentences expressing true propositions is not enough to guarantee reliability, since it leaves open the possibility that we might acquire untrue beliefs on the basis of valid inference from true premisses; these untrue beliefs would not be expressed by propositions.

(' In chapters 8 and 10, I shall propose a different standard of reliability, according to which a pattern of infcrcnce ought always to lead from detinitcly true premissea to definitely true conclusions.

If we try to reconcile classical logic with the doctrine that the paradoxical sentences lack truth values, we run afoul of one or the other of our two requirements. If we adopt the classical rules without restrictions, we find that we are able to infer untrue premisses from true conclusions. If we restrict the rules by adjoining to each of them the caveat 'provided thc conclusion is truth-valued', for example,

I From (VxI(4i.r) -t $ ( X I ) one may infer ((Vx)(+(x) & 8ir)) -t iVx)($(x) & I

O(x))), provided the conclusion is truth-valued. 1 we violate the learnability requirement. I

! If we regard logic as a science that tells us how we can reliably obtain true beliefs

as the conclusion of arguments from true premisses, then, if we accept the view that ii

paradoxical sentences are truth-valueless, we nus st indeed abandon classical logic. If we regard logic as a more ethereal science, whose concerns with structural rela- I tions among the bearers of truth values are only remotely connected with how we should reason or what we should believe, then we may maintain that our logic is I

I still classical. My usage here of the word 'logic' will reflect the former view, though I hope to avoid any quarrels over how to use the word correctly.

Supposing that we adopt the Kleenc semantics, what will be the rules of inference which will take us reliably from true premisses to true conclusions'? So far, we have only looked at cases in which the only partially defined predicate is 'Tr ' , but we can talk generally about a 3-valuecl interpretution of a first-order language, given by specifying a nonempty set to be the universe of discourse of the interpretation, by specifying a member of the universe for each individual constant and an n-ary function on the universe for each n-place function sign,' and by specifying two disjoint sets of n-tuples to be the extension and antiextension of each n-place predicate; we assume that the extension and antiextension of '=' are the identity relation and its complement. Such an interpretation determines, for each sentence, whether its truth value is true, false, or undefined, and we may identify the valid inferences as the ones for which there is no 3-valued interpretation under which the premisses are true and the conclusion untrue.

DEFINITION. A sentence 4 is a 3-vulued consequence of a set of sentences r iff + is true under every 3-valued interpretation under which all the members of r are true.

To prove an argument valid, we can use the following straightforward modification of the classical system of natural deduction given by Benson Mates [1965, chapter 71:'

Here I am assuming that there are no nondenoting closed terms, cvcn though potentially important applications of 3-valued logic occur in cases wherc this assumption does not hold. To allow nondenoting terms would significantly complicate our discussion.

' Other deductivc calculi fo r thc 3-valued logic are discussed by Feferman 119821 and by Kremer (19881.

DEFINITION. A -?-valued derivation is a finite sequencc o f sentences, written on successive lines according to the following rules, such that, whenever one writes a sentence on a line, one also indicates a finite set of sentenccs that is to be the prerniss set of that line:'

If y is a member of 1 , you may write y with r as its premiss set. If every member of h has been obtained with r as its premiss set, and if 4 has been obtained with A as its premiss set, you may write + with r as its premiss set. If you have obtained 8 with r U ($1 as its premiss set and you have also obtained 6, with r U {$) as its premiss set, you may write 8 with r U {(+ V +)} as its premiss set. You may write (+ V 41) with {+) as its premiss set or with {$) as its premiss set. You may write i(+ v $) with ( 1 4 , i $ ) as its premiss set. You may write either i+ or i$ with {i(+ v $)} as premiss set. You may write + with {1i+) as its premiss set. You may write ii+ with ($1 as its pre~niss set. You may write 1C, with ( 4 , 1 4 ) as its premiss set. If you have obtained 8 with r U {$(c)} as premiss set, and if the individual constant c does not appear in T, 4, or 8, then you may write 8 with r U {(3v)$(v)} as premiss set. You may write (3v)$(v) with {$(T) ) as premiss set, for any closed term 7.

If you have obtained ( 8 V i $ ( c ) ) with 1 as premiss set, and if the individual constant c does not appear in T, in 8, or in $, then you may write (8 v i ( 3 v ) $ ( v ) ) with r as premiss set. For any closed term T , you may write ~ $ ( r ) with { i ( 3 v ) $ ( v ) ) as its premiss set. You may write T = 7, for any closed term r , with the empty set as pren~iss set. If + and $ are atomic or negated atomic sentences that differ only in that 7 and p have been exchanged at some places, then you may write + with {T = p, 4) as premiss set.

DEF~NITION. A Sentence 6 is derivable from the set of sentences r in 3- vcrlued logic iff there is a 3-valued derivation of 6 with a premiss set that is included in r. THEOREM 4.4. 6 is a 3-valued consequence of l- iff 6 is derivable from r in 3-valued logic.

" 1 assume that thcrc are infinitely many individual constants in the language: if not, add them

PROOF: We check soundness by inspecting the rulcs. verifying that each of thc rules preserves the relation of 3-valued logical consequence. Completeness is proven by a Hcnkin construction. Assume that 6 is not derivable from F. Where K is thc cardinal number of the set of sentences of the language, extend the language by adding K new individual constants. and enumerate the sentenccs of the cxtended language as <$,j: /3 < K > ; it is clear that adding these new constants

will not affect either derivability or logical consequence. Write Y A if some disjunction of sentences in A is derivable from Y. For each P 5 K , we dcfine sets of sentences r, and A, as follows; the 17,s consist of sentences we are trying to make true, whereas the A,s consist of sentences we are trying to refrain frorn making true:

r',, = F and A,, = {a). If $r U {$..) s 8.. and $.. is not a negakd existential sentence, then ,, = r , " a n d A ,,,, = A . . If r,, U {$,,) %- A,, and $, = i (3v)O(v) , then let c be the first individual

constant which does not appear in T,, A,,. or $<,, and set , , = r, and Act + I = A,z u {hO(c)}. If ITce U {$,.) h Au and $<? is not an existential sentence. set rcx + , = r, U {JJ,,} and A, + = A,. If r, U {$,I ++ A,, and $, = (3v)O(v), let c be the first individual constant that does not appear in I',, A,, or $,,and set T, + , = I;? U {$~v,O(c)) and A,, + I = A,. r, = r, and AA = A, for A a limit.

By induction, ITuA% A,. for each a. Thus, TK + AK, though if r is any proper

extension of rK, 1' 9 A,. Now define, for each closed term T,

and define a partial model Y1 by:

(91( = {IT]: 7 a closed term). (." = [(-I, f"' ([7,1, . . . 3 IT,,^) = [f(71, . . . , 7")l.

< L T , ] , . . . , [T,,]> t. extension of R iff R(7,, . . . , T,,) E TI. <[r l l , . . . , IT,,]> e anti-extension of R iff i R ( r , , . . . , T,,) E r,.

(The last of the rules of inference ensures that this is well-defined.) We can show by induction on the complexity of sentences that, for each sentence X , x is true (resp., false) in !?I iff x (resp., TX) is in rK. Thus, all the members of r are true in !)[ but 3 is not..

PROOF: Just like the proof for classical logic. If T is inconsistent by _?-valued logic, then there is a 3-valued derivation of i c = c. from T. Thc finite subset of r which forms the prcmiss set of l c = c in this derivation is inconsistent by 3- valued logic..

In theorem 4.4 we are given the logic in terms of which, according to the proposal under discussion, a correct theory of truth ought to be formulated; we are not, however, given a correct theory of truth. The theory of truth is still given in the classical metalanguage. But it surely does us no good to be told that the reason that we have been troubled by the paradoxes is that we have been attcmpt- ing to formulate the theory of truth within classical logic and to be given the logic in which the theory of truth ought to be formulated, if we are not also given the properly formulated theory of truth. We need to be given the theory of truth for the 3-valued language, formulated in the 3-valued logic. Until this is done, we cannot even begin to implement the proposed revolutionary change in our way of thinking.

William Reinhardt [ 1986l"' has used Feferman's axioms to give an ingenious solution to this problem. Keinhardt begins by giving a two-language solution. The object language is the language 2', with the predicate 'Tr' regarded as only partially interpreted. The metalanguage is again the language TI, this time interpreted classically. Our metatheory consists of those theses expressed in the language 2 which we are willing to accept as axiomatic, whatever they are, together with Feferman's axioms. It is this metatheory which tells what sentences of the object language to accept as true. We shall accept a sentence 4 of the object language if Tr(r41) is a theorem of the metatheory, and we shall reject 4 if i T r ( r $ l ) is a theorem of the metatheory. But now we can eliminate the metatheory. We can take our theory, formulated entirely within the object language, to be the set of sentences 4 such that ~ r ( r 4 1 ) is a theorem of the metatheory. Assuming that the original metatheory was recursively axiomatized, this theory will be given as a recursively enumerable set of sentences. We can treat the metatheory as an uninterpreted calculus, useful only as an aid in producing the axioms of the object theory, or, if we prefer, we can dispense with the metatheory altogether and substitute some purely mechanical method of generating the axioms of the object theory. We can even use Craig's method to give a recursive set of axioms whose 3-valued logical consequences are precisely the sentences 4 such that ~ r ( r 4 1 ) i s a classical logical consequence of the metatheory. Although this can be by no means regarded as the final solution to the problem of how to give a self-contained theory of truth for the 3-valucd object language-it is troubled by the familiar difficulties that afflict the purported applications of Craig's theorem

"' I have cnjoyed several opportunities to discusc with Reinhardt his paper and 11s connections w ~ t h the theory being proposcd here. I have found Reinhardt's insights t o be invaluable.

COROLLARY 4.5. The 3-valued logic is compact.

in the philosophy of scienceH-it is. at least, a promising approach toward a solution.

Assuming that it is indeed possible to formulate the theory of truth for the 3- valued language within the 3-valued language in a satisfactory way, three further difficulties are encountered by the proposal that we abandon classical logic in favor of the 3-valued logic: OB.IECTION 1. DIFFICULTY OF LEARNING THE 3-VALUED LOGIC. The first obstacle is simply how difficult it would be, in practice, for us to use the 3-valued logic in place of the familiar logic. Classical logic has served us well since earliest childhood, yet we are asked to abjure it in favor of a new logic in which many familiar and hitherto unproblematic modes of inference are forbidden. So restrictive are the new rules that, according to Feferman 11984, p. 2641, "nothing like sustained ordinary reasoning can be carried on" in the 3-valued logic. In view of the prodigious effort that would be required for us to make such a drastic change in our ways of thinking, we shall perhaps be forgiven for our laziness if we are reluctant to make the change.

These judgments about how difficult it would be to accustom ourselves to utilizing the new logic are rather speculative. The system of formal rules given in theorem 4.4 is not terribly dissimilar from familiar systems of natural deduction, and it is conceivable that, with sufficient practice, we could train ourselves so that the new ways of thinking would seem as simple and natural to us as classical reasoning seems now.

Luckily we do not need to resolve these speculations. Taking advantage of the fact that we are not asked to relinquish classical modes of inference in all our reasoning but only in our deliberations about truth, we can devise a system of inference that is tailor-made for the situation in which we have a classical language to which a nonclassical truth predicate has been adjoined. This specialized system of inference is, we shall find, no harder to learn or employ than classical logic. In this system, which is due to Reinhardt [1986], we employ all the modes of inference of classical first-order logic, without restriction, taking as our premisses Feferrnan's axioms together with those sentences of the language 2 which we are willing to accept as axioms. What is nonclassical about our new system of reasoning is what we are willing to accept as a finished deduction. Classically, we would regard a sentence 4 as proved if there is a derivation of 4 from the premisses. In the new system, we regard the sentence 4 as proved only if there is a derivation of ~ r ( r 4 1 ) from the pren~isses. Modifying classical logic in this specialized way completely allays the misgivings I expressed about the difficulty of adopting a strange and somewhat clumsy nonclassical logic. There is nothing awkward about Reinhardt's logic. There are no unpleasant new rules we have to learn and

" See Hempel [I9581

I

no familiar rules we have to abjure. The only difference between the new system and the old is that now we do not get to write 'Q.E.D.' as often.. OBJECTION 2. UNAVAILABILITY OF SCIENTIFIC GENERALIZA.I.IONS. According to Aristotle (Metuphysic,~ A), we can distinguish the knowledge of the scientist from the knowledge of the artisan by the fact that the scientist knows universal laws whereas the artisan knows only particular cases. If that is so, then we find, much to our dismay, that, if we adopt the 3-valued logic we relinquish the possibility i

of scientific knowledge, for in the 3-valued logic universal generalizations are virtually impossible to come by.

Consider Jocko. Jocko is a tiny fictional creature that lives right on the border between animals and plants. Jocko has many of the features we regard as characteristic of animals and many features we regard as characteristic of plants. Jocko's animallike characteristics are those we expect to find in protozoa, so that Jocko is also on the border between protozoa and nonprotozoa. It is natural to say that Jocko is neither in the extension nor in the anti-extension of 'animal' and that Jocko is neither in the extension nor in the anti-extension of 'protozoon'; if

I

that is so, then

(Jocko is a protozoon --, Jocko is an animal)

will be neither true nor false. Hence,

(VX)(X is a prc)tozoon -+ x is an animal)

will be neither true nor false. Jocko's story is fictional, but it is a realistic fiction. We have no good reason

to suppose that, in partitioning the organisms into plants and animals, nature made a clean break. It would not surprise us in the least to discover some actual creature in the position of the fictional Jocko, equipoised between protozoa and plants. But if we do not have any good reasons to suppose that there is no creature in the position in which we have imagined Jocko, then we do not have any reason to suppose that

(VX)(X is a protozoon -+ x is an animal)

is true. The generalization

All protozoa are animals.

becomes highly suspect. 'All protozoa are animals' is not an accidental generalization. It is a basic

taxonomic principle that is about as secure as a law of nature could ever be. To forbid the assertion that all protozoa are animals is to outlaw science.

Things get even worse. Because Jocko is neither in the extension nor the anti- extension of 'animal', the conditional

(Jocko is an animal - Jocko is an animal)

is exiled to the gap between truth and falsity. Hence,

(VX)(X is an animal - x is an animal)

is untrue; the 3-valued logic does not permit us even to assert

All animals are animals.

One might respond to this objection in the same way we responded to the first objection, by insisting that 'true' is to be the only predicate accorded a nonclassical treatment. so that, for sentences not containing the predicate 'true', our customary practices are undisturbed. Since most of our scientific reasoning does not make essential use of the notion of truth, most of our scientific reasoning is happily undisturbed by our change of logics.

Although this response contains the damage, it does not eliminate it. The science of zoology, for instance, does not make essential use of the notion of truth, so it will not be disrupted by our change of logics. On the other hand, the change of logics ensures that we cannot have a scientific semantics, since even the most harmless semantic generalizations are forbidden to us. We cannot assert, for example:

If a conjunction is true, then both conjuncts are true Since

lies in the gap," so does

Indeed, since

Tr(rh1) -+ Tr(rh1)

is untrue, so is

so we cannot even assert

Every true sentence is a true sentence..

OBJECTION 3. DEGKAUATION OF METHODOLOGY. The third difficulty with the proposal that we repudiate classical logic in favor of the 3-valued logic is the one that I regard as the most telling. It is based upon an admonition of Field [I9761 that our methodological standards in semantics ought not be any lower than our methodological standards in the empirical sciences. We shall contravene this

" Where A is the liar scntence constructed in theorcni 1.3

i admonition if we attcmpt to cover up the deficiencies of our naive theory of truth by abandoning classical logic.

i Imagine that we have a genetic theory to which we are particularly attached, perhaps on political grounds, and that this theory tells us that, if a certain particular

i DNA molccule has an even number of nucleotides, then all fruitflies are brown; that, if that particular molecule does not have an even number of nucleotides, then all fruitflies are green; and that fruitflies are not all the same color. It would

! surely be absurd to respond to this circumstance by saying that our cherished I

genetic theory is entirely corrcct and that the appearance of contradiction arises because we have failed to realize that classical logic does not apply when we are doing genetics. What we have to say instead is that the genetic theory has been refuted. This is not to say that the theory must be abandoned altogether-perhaps some simple modification would repair it-but it is to say that the thcory can no longer be wholeheartedly accepted as it stands.

As preposterous as it would be to be to respond to the embarrassment faced by the genetic theory by saying that classical logic no longer applies when we are doing genetics. it would be no less preposterous to respond to the liar paradox by saying that classical logic no longer applies when we are doing semantics. The liar paradox refutes the naive theory of truth. It is our duty to come up with a better theory of truth. It is a dereliction of duty to attempt to obscure the difficulty by dimming the natural light of reason.

Actually, the situation is somewhat worse if we want to give up classical logic in semantics than if we want to give up classical logic in genetics. In genetics, we have a huge body of empirical data that our theories are attempting to explain. We can imagine this body of data by its sheer bulk pushing classical logic aside; Putnam at one point [I9681 was pushing the line that something of the sort had happened with quantum mechanics. Now, the pressure to abandon classical logic in semantics does not come from an overwhelming body of linguistic data but rather from our metaphysical intuitions about truth. In metaphysics, we scarcely have any data. All that we have to take us beyond our preanalytic prejudices is our reason, and now we are asked to modify the rules of reason so that they no longer contravene our preanalytic prejudices. In the end, the role of reason in metaphysics will be merely to reconfirm whatever we have believed all along.

This is a devastating objection, but it is not quite a conclusive objection. I have argued that, in a one-on-one conflict between classical logic and our metaphysical

! intuitions about truth, classical logic ought to prevail. I f other sciences should enter into the fray, the outcome will no longer be so certain. As of now, our only motive for adopting the 3-valued logic is that it appears to vindicate some of our naive intuitions about truth. There are other general reasons for dissatisfaction with classical logic, but these reasons do not point us specifically in the direction of Kleene 3-valued logic. I t may happen that. sometime in the future, adoption of the 3-valued logic will be found to be advantageous in some science other than

i semantics, so that classical logic will be pressured from more than one side. This has not happened yet, but if it should happen it will become necessary to reevaluate our position..

So far we have looked at two possible methods for applying Kripke's formalism to obtain a theory of truth. The first, preferred by Kripke himself, uses 3-valued -**:

logic in the object languagc and classical logic in the metalanguage. The second uses only the .?-valued logic. Now let us look at a proposal that uses only classical

I logic. The proposal is that we take the Kripke-Feferman axioms, understood in a perfectly ordinary, classical way. at face value as a theory of truth.

\ We should distinguish the principle of bivalence (the semantic thesis that every sentence is either true or false) from the law of the excluded middle (the logical schema ( 4 V 14)). If, as seems natural, we identify the falsity of C#J with the truth of 1 4 , we may write the principle of bivalence as "(V sentence x)(Tr(x) v Tr(l.u))." The proposal, then, is that we maintain full classical logic, including the law of the excluded middle,'.' but we give up the principle of bivalence.

This proposal does not fall prey to the methodological difficulties that afflicted our other proposals. On the contrary, the methodology of this new interpretation I

of Kripke's construction is exemplary: it is an admirable illustration of what one would like a theory of truth to look like.

Kripke characterized the fixed points model theoretically, by giving a set of models; to do this he required an essentially richer metalanguage. Feferman characterized the fixed points proof theoretically, by giving a set of axioms, and no metalanguage was required. One still needs the model-theoretic construction to prove that the Kripke-Feferman axioms are consistent, but we commonly use theories for which we are unable to give consistency proofs.

Feferman's axioms are as simple, natural, and graceful as any axioms we I

could hope to find. What is more, the theory is informative, showing us how the truth conditions for complex sentences depend upon the truth conditions for their simple components. A principal reason why the explicit definitions of truth given by Tarski are so useful is that they show us how the truth conditions for the sentences of a language are determined by the meanings of the finitely many building blocks out of which the sentences are constructed. Feferman continues Tarski's program by giving a similar analysis for a language without bivalence.

What Feferman has given us is a procedure by which we can give a theory of truth for any interpreted first-order language that can describe its own syntax in a recognizable way. We may, if we like, use the procedure to give a theory of truth for a substantial fragment of English. We can identify a first-order structure

" Noticc that, if we add the rule,

You may write (4 V -14) with the empty premias set.

to the rulcs of theorem 4.4 we get a sound and complete system of axioms for the class~cal predicate calculus.

within ordinary English discourse. There are English connectives that can reasonably be paraphrased as truth-functional connectives and English quantihers that can reasonably be paraphrased as first-order quantifiers. The technique for doing this is what we learn in introductory logic classes. Although one can quibble about the accuracy of the paraphrase in troublesome cases, on the whole the procedure works quite well. In this way, English, or rather the fragment of English consisting of sentences used to make assertions, can be regarded as a first-order language. Of course, quite a bit is lost in translation. Although we can exhibit part of the logical structure of English by thinking of English as a first-order language, a great deal of the logical structure remains hidden. For example, 'Agatha believes that, if two people really love each other, then it does not matter what the world thinks, they can find happiness together' is analyzed as having simple subject-predicate form, attributing a certain unary predicate to Agatha. But this does not show that the first-order analysis is wrong, only that it is shallow.

But although English can plausibly be regarded as a first-order language, it cannot plausibly be regarded as an interpreted first-order language. Certainly if we take an interpreted language to be one for which an appropriate function on {'3'} U {constants, function signs, and predicates of the language) has been specified. English is not an interpreted language, because an interpreted language has a set as its universe of discourse and English discourse is not restricted to any particular set. Even in the less formal sense in which a language is interpreted if we know what its terms mean, English is not an interpreted language, for the meaning of an English term is often determined by the context in which the term is used.

Even so, we should be able to find a very substantial fragment of English constructed from nonsemantic terms whose references are context-independent. We can treat such a fragment as our language Sf i , and we can form the language 2' by adding the adjective 'true' to 2. Now we can use Feferman's method to give an axiomatic theory of truth for a substantial portion of the English language. This theory can be used as a basis for future semantic research. As our understanding of English grammar advances, we shall come to understand grammatical constructions more subtle than the crude constructions the first-order analysis represents. As we do so, we can add new axioms to the Feferman axioms to give the truth conditions for these more delicate constructions.

As we can see, the Kripke-Feferman axioms have a great deal to recommend them as a theory of truth. In spite of these splendid virtues, I am not entirely satisfied. The difficulty I see is that, if we adopt the Kripke-Feferman axioms, we shall be unable to understand the connection between truth and proof. We normally suppose that the reason we prove things is that we wish to acquire true beliefs and we suppose that proving something gives us very good grounds for accepting that it is true. Of course, we recognize that we are susceptible to error, so that sometimes we produce apparent proofs of claims that are not true, but we

suppose that this can only happen if our alleged proof is somehow defective. We would not simultaneously accept a proof as valid and assert that the proposition proven is untrue. A proof is as good a guarantee of truth as we can hope to obtain; that is why we prove things.

If we accept the Kripke-Feferman theory, this simple connection betwecn truth and proof will be broken. In the Kripke-Feferman theory, we can prove things that are, according to the Kripke-Feferman theory, untrue. An example is the sentence ( A V i A ) , where A is the liar sentence. The sentence ( A V 11) is a tautology, so naturally we can prove it in KF. but we can also prove i ~ r ( r ( 1 V I A ) ~ ) in KF. But we do not see why we prove things, if proving something gives us no reason to suppose it is true. And we do not see why we cherish truth, if we are willing to countenance deductions whose conclusions we believe to be untrue.

According to our naive conception. the notion of truth is crucial to understanding how people reason. The aim of rational inqxiry is to find out what is true, and we reason the way we do, utilizing the modes of reasoning we employ, because we believe that these methods will get at the truth. The notion of truth is also crucial to understanding how people talk; if we are honest, we assert only things we believe to be true, and we expect our fellows to do likewise. If wc adopt the Kripke-Feferman axioms, the notion of truth will no longer occupy this central role. When we accept the axioms, we modify our naive conception of truth; according to this modified conception, the notion of truth no longer has a crucial role to play, since it is perfectly all right to accept and to assert statements that are not true. What we get is an elegantly axiomatized theory of a notion that no longer has any particular importance.

In brief, my objection to the Kripke-Feferman theory is that the theory violates principle (P2), the principle that a satisfactory theory should not make claims the theory itself regards as untrue. But although the Kripke-Feferman theory does not yet constitute a satisfactory theory of truth, it is a significant step in the right direction.

Kripke's Construction and the Theory of Inductive

Definitions One of Kripke's principal aims in developing the construction given in theorem 4.1 was to produce "an area rich in formal structure and mathematical properties" [1975, p. 631. In this chapter, we prove some theorems that demonstrate how thoroughly Kripke has succeeded in this aim. These results will be applied extensively in later chapters.

We begin by making an assumption, followed by some definitions:

ASSUMPTION 5.1. In this chapter, we shall assume that Y is a first-order language without function signs built up from a finite vocabulary, and we shall assume that '!I is an acceptable structure for -y.

An occurrence of a predicate R within a formula 0 is positive (izegative) iff it occurs within the scope of an even (odd) number of negation signs. Ajrst-order positive inductive dejnitiorz is a sentence

where 0 is a formula of the language obtained from 2 by adding the single new n-place predicate 'R' and where 'R' occurs only positively in 0. An n-ary relation X on IYII is ajfixeclpoitzt of the definition iff the definition is true in the structure (?I,X), which is the model we get from 91 by taking the extension of 'R' to be X.

LEMMA 5.2 ( ~ o s c h o v a k i s ) . ~ A first-order positive inductive definition has a least fixed point.

PKOOF: Define, for each ordinal a , -

r; = { < o ~ , . . . , a,,>: r:) t= o(;, . . . , a , , , ~ ) }

r; = &JOR r;. Because R occurs only positively in 8(v1, . . . , v,,. R), the function that takes S to the extension of 8(vl , . . . , v,,, R) in (91,s) is monotone. Therefore, the Tis are a nonstrictly increasing sequence of subsets of It'(/". Cardinality considerations tell us that there is an ordinal K with ra = rz We have

' See Mvschovakis [1974, p. 81 or Barwise [1975, pp. 200fl

so that ri is a fixed point. A transfinite induction shows that. for each a , T); is included in every fixed point.. EXAMPLE 5.3. Consider the following first-order positive inductive definition:

(Vx)(R(x) c* [x is a sentence & [x is an axiom of logic V (3y)(R(y) & R(y -T, x)]])

The fixed points of this inductive definition are all sets of sentence\ that are closed under first-order consequence. The least fixed point, which is obtained after o stages, is the set of logically valid sentences. The largest fixed point is the set of all sentences..

DEFINITION. A relation Y ldl(" is inductively definable2 or simply inductive over !'I iff there is a formula +(x,, . . . , x,,,R,, . . . , R,,) of the language obtained from Y, , by adding new predicates R,, . . . , R,, and there are relations XI , . . . , X , on 1911 so that

1) each X, is the least fixed point of some first-order positive inductive definition over 91;

2) each R, occurs only positively in &s,, . . . , .r,,,R,, . . . , R,,,); and 3) Y is the extension of +(x,, . . . , x,,,R,, . . . , R,,) in (?(,XI, . . . ,

X!").

If the formula 4 ( x l , . . . , x,,,R,, . . . , R,,,) is parameter-free, that is, if it does not contain any of the constants (I used in extending 2 to X,,, X is parcimeter-free inductively definable. If (!'(I" - Y is inductive, Y is coinduc- five. If Z is both inductive and coinductive, Z is hyperelementurv.

Inductive and hyperelementary relations have been studied extensively, starting with the fundamental investigations by Kleene of the hyperelementary relations on 3i.' Parameter-free inductively definable sets are a more specialized concern arising out of peculiar interests we have here. Here we are particularly concerned with theories that are learnable. A theory formulated in 9?!!, using the

Moschovakis's original definition, though simpler than the definition given here, is less convenient for our prcscnt purposes. Moschovakis defines an 11-ary relation Y to be inductive iff there exists an n+m-ary rclation X and b , , . . . , b,,, E ]\'I1 such that X IS the fixed point of a first-order positive inductive detinition over !)I and Y = {<x,. . . . , .x,,:>: i . r , , . . . . x,,, h , . . . . , b,,,> E X ) . Theorem ID. I of Moschovakis [I9741 shows the two characterization'i of the inductive sets to be cqu~valcnt.

' Hyperelementary relations on !Ji arc called h?pc~rari/hmr/ic.trl. Rogers [1967, ch. Ih] gives a pleasing summary of Kleene's theory.

parameters a will not count as learnable, even if it is finitely axiomatized, because 3,, is not a human language, either natural or artificial, but a mathematical abstraction. If we have a finitely axiomatized theory from Y,,, we may attempt to give a concrete version of the theory by adding to 2 new individual constants for the finitely many individuals named in the theory, but in order to do this we must be able to specify the individuals the new constants are to denote, whether by describing the individuals, by pointing to them, or by some other means, and there may not be any effective procedure for doing this. In general, we shall count a theory as learnable only if it is recursively axiomatizable in a language without individual parameters.

EXAMPLE 5.4. A first-order positive inductive definition

has a greatest fixed point. This greatest fixed point will be coinductive, since its complement is the minimal fixed point of the first-order positive inductive definition

EXAMPLE 5.5. The set of true sentences of 2,, is hyperelementary. To see this, recall from chapter 3 that the set of true atomic sentences of %,, is explicitly definable in X, essentially by giving a list. In a similar way, the set of true negated atomic sentences is explicitly definable. The set of true sentences is the minimal fixed point (indeed, the only fixed point) of the following first-order positive inductive definition:

(Vx)(T(x) c* [x is a sentence of 2,,, & [.r is a true atomic or negated atomic sentence V (3y)(3z)(x = (v V z ) & (T(y) V T(z))) V (3~) (3z ) (x = 3 y V z ) & T O y ) & T(7z)) V ( ~ Y ) ( X = 1 1 y & T(v)) V ( 3 variable v)(3y)(x = (3v)y & (3w)T(y L1l~)) V (3 variable v)(3y)(x = ~ l ( 3 v ) y & (Vw)T(ly v l~ ) ) ] ] )

Obviously, the set of true sentences is a fixed point of this inductive definition. To see that it is the only fixed point, let S be a fixed point and show that

(V formula 4 of 2)(V sentence IC, of 2,, that is a substitution instance of E S - IC, is true).

by using the following, frequently useful, characterization of the formulas of 2 :

The formulas of 2 constitute the smallest class of expressions which contains the atomic and negated atomic formulas;

contalns ((I, V $). 1 14. and (3\7)1$ whenever ~t contalnj (I, and $; and contalns i ( q 5 V 4) and 1 ( 3 ~ , ) 4 whencvcr it contatns 14 and 1 4 .

This shows that the set of true sentences is inductive. To see that it is coinductive, observe that its cornplemcnt is the extension of

( 1 ) is a not a sentence V ( 3 y ) ( T ( y ) & ?. = 7 ~ ) )

in (!'l ,{true sentences)) .H

EXAMPLE 5.6. E,, the extension of 'Tr' at the least tixed point of the Kripke construction, is parameter-free inductive. It is the least fixed point of the following first-order positive inductive dctinition, where Derz is the function, definable in Y. that takes a closed term of Y:>, to the individual it denotes:

(V-r)(F(a) c* [.r is a sentcnce of Y:, & [x is a true atomic or negated atomic sentence of Y!,, V ( 3 closed term r ) ( s = r ~ r ( r ) l & E(Den(r) ) ) V ( 3 closed term r)(s = r i ~ r ( r ) l & E ( ~ D r n ( r ) ) ) V ( 3 y ) ( 3 z ) ( x = (.v V z) &k ( E ( y ) V E ( z ) ) ) V (3? . ) (3z) ( -~ = l ( . v V & E ( 1 y ) & E ( 1 : ) ) V (3.v)(x = 1 & ~ ( y ) ) V ( 3 variable 1,)(3y)(.x = ( 3 v ) y & (3w)E(y 1'1,;))

V ( 3 variable 11)(3y)(x = i ( 3 v ) y & ( V I V ) E ( ~ ~ vi;))]])

We shall see below (corollary 5.11) that E , is not hyperelementary.. DIGRESSION 5.7. WOODRUFF.~ SYSTEM. If ( E , A ) is a fixed point of the Kripke construction, then E is a fixed point of the inductive definition given above. The converse docs not hold, however. There are some fixed points E of the inductive definition such that, for some sentence I/), and riI/11 are both in E. If we set A = {nonsentences} U {rq5l: r 1 4 1 E E), we shall find that ( ? ( , ( E V A ) ) does not count as a partial model, since our definition of partial model requires that the extcnsion and the anti-extension of 'Tr' be disjoint.

We can liberalize the Kripke construction by allowing extensions and anti- extensions to overlap. This involves replacing the 3-valued logic with a 4-valued logic in which, in addition to truth-value gaps (sentences that are neither true nor false), there will be truth-value gluts (sentences that are both true and false). There are four truth values; as well as being simply true, simply false, or neither true nor false, a sentence can now be both true and false.

The rules for determining truth values of compound sentences of 2$ in terms of the truth values of their simple components are unchanged from classical logic: A dis,junction is true iff at least one if its disjuncts is true, and false iff both its disjuncts arc false. A negation is true iff the sentence negated is false, and false iff the sentence negated is true. An existential sentence is true iff some of its

substitution instanccs are true. and false iff all its substitution instances are false. What distinguish the 2-, 3-, and 4-valued logics are the possibilities they allow for the classilication of atomic sentences.

By augmenting the class of possible nlodels. we diminish the class of valid inferences; for example, disjunctive syllogism (from (4 V 11) and i d , to infer 4) is valid in 3-valued logic but not in 4-. If we modify the deductive calculus discussed in theorem 4.4 by deleting the rule of reductio ud ~lbsurclum,

You may write with (4. 1 4 ) as its premiss set.

we get a system that is sound and complete for 4-valued logic. This liberalization of Kripke's system has been studied by Woodruff [ 19841,

who obtains a number of elegant results about the resulting system, notably the theorem that the lattice of fixed points is complete and self-dual." The extensions of 'Tr' in fixed points of Woodruff's system are precisely the fixed points of the inductive definition given in example 5.6. They are characterized by the axiom system got by deleting the axiom " 1 ( 3 x ) ( T r ( x ) & Tr(1-r))" from KF..

In exploring the connection between the Kripke construction and the theory of inductive definitions, we shall need to make use of the self-referential lemma 1.2, which tells us that, for any formula +(x,v, , . . . , v,,) of If', we can find a formula $(v , , . . . , v,,) of 3' such that the biconditional

is a first-order consequence of the first-order theory of !'L. It does not follow from this that the biconditional (#) will be true in the partial models (!'I,(E,A)). Although the first-order theory of ?I is true in every partial model ( \ ! I , (E ,A) ) , the first-order consequences in 2' of that theory need not be, since it might happen that both sides of the biconditional (#) should lack truth values. What will be true, as we shall now see, is that, for each partial model (\'I,(E,A)) and for each

(true 1 a , , . . . , cr,, E 1!)1(, $(lT, . . . , 6) will be in (! ' t , (E,A)) iff +(r$l ,

undecided

- , . . . , ) is { } in (!)L,(E,A)).

undecided

If ( E , A ) and ( F . 8 ) are fixed points, we say ( E . A ) 5 ( F . B ) iff E F and A C B. To say that we

have a complete lattice is to say that every nonempty set of tixcd points has a lcast upper bound and a greatest lower bound. To say the lattice is self-dual is to say that there is a bijection h from the set of fixed points onto itself such that ( E . A ) 5 ( F . B ) iff h((l.'.B)) 5 hi (E ,A) ) . h will be given

by

h( (E .A) ) = (l!'[I - A, 1911 - E ) )

I DEFINITION. TWO formulas &v,. . . . , L,,,) and $(I!,, . . . , L;,) of Y:, are i

g u ~ ~ - r q ~ ~ i v a l c i ~ t (with respect to !'I) iff, for any a , , . . . , tr,, in 1?)1/ and any 1 disjoint subsets E and A of 1!)i/, (b(c. . . , <) is true, false, or undecided ! in (!'I,(E,A)) according as $(a;, . . . . <) is true, false, or undecided. I

< .:

PROPOSITION 5.8 The construction of theorem 1.2 gives us, for each for- ' mula +(.r, v, , . . . . v,,) of .yi,. a formula JJ(v,, . . . , I-, ,) so that $ ( v , . . . . ,

I v,,) is gap-equivalent to $~(r$I,v,, . . . , v, , ) .

In proving this result, we make use of a handy technique from Feferman [1982]: I

LEMMA 5.9 ( ~ e f e r i a n ) . Let TL be the language obtained from 2 . by adding two new unary predicates 'Tr' and 'Fa' . Given a formula 6 of X:,. I

I

let H + be the formula of 2' got by replacing each negative occurrence of 'T~(T) ' by ' ~ F U ( T ) ' . If (!)I ,(E,A)) is a partial model of 3 ' , let (91 ,E,A) be the classical model of LYL got from !)I by letting the extension of 'Tr' be E and the extension of 'Fa' be A. Then, for any sentence 6 of Y,',, 6 is true I

in (!)i,(E,A)) iff OL is true in (\'l,E,A).

PROOF: By induction on the complexity of 0.. PROOF OF PROPOSITION 5.8: We follow the notation of the proof of theorem 1.2 in detail. We have

- - t,b(n,, . . . , a,,) is false in (\'I,(E,A))

- iff (3z)(8(r!Y(x,z)1,z) & q%z,u,, . . . , z)) is false in (!)I,(E,A)) iff (Vz)(%r8(x,z)1,z) + ~ + ( z , < , . . . . <)) is true in (!)I,(E,A)) - iff I(Vz)(%rY(x,z)l,z) + i $ ( z , z , . . . , u,,))]. is true in (91,E,A)

- iff (Vz)(g(r8(x,z)1 ,z) * l ic$(z,K, . . . , u,,)]') is true in (Yl ,E,A)

[because 'Tr' does not occur in Y] -

iff [hc$(r$l,c, . . . , u,,)]' is true in ( ~ I , E , A ) [because ( ~ z ) ( ~ ( r 9 ( x , z ) l ,z) ++ z = r$1) is true in (t'l ,E,A)]

- iff l + ( r $ l , c , . . . , a,,) is true in (P~,(E,A))

- iff +(r$1,(1,, . . . , a,,) is false in (!)I,(E,A)).

- The proof that $ ( c , . . . , u,,) is true in (!)[.(EVA)) iff c$(r$l,c, . . , <) is true in (91.(E,A)) is similar..

Something to notice about the proof of the self-referential lemma is this: if 'Tr' occurs only positively in the given formula (b. then 'Tr' occurs only positively in the constructed formula $. Sentences in which 'Tr' occurs only positively are worth noticing because, as noted earlier, such a sentence will be true in (91 ,(E,A)) iff it is true in the classical model (!'1,E).

Example 5.6 shows that E, is an element of the class of inductive sets. The following theorem shows that E, is a universal member of that class:

THEOREM 5.10 ( ~ r i p k e ) . ' For any (parameter-free) inductive set C. there is a (parameter-free) formula y(x) of Xi, such that, for any u ,

a F C iff rY(i)l F E, .

PROOF: It will clearly suffice to prove this for the special case in which C is the minimal fixed point of a first-order positivc inductive definition

To simplify notation, we look at the special case n = 1 . We prove the theorem by matching up the stages in the construction of r1,

with the stages in which E, was constructed in the proof of theorem 4.1. Where 'z' is a variable that does not occur in 6, form a formula q(x,z) of 3' from O(x,R) by replacing each occurrence of 'R(v) ' by 'Tr(zx/;)', and use the self-referential lemma to find a formula y(x) that is gap-equivalent to q(x,ry(x)l). Thus, y(x) is gap-equivalent to the result of substituting '~ r ( ry ( i ' ) l ' for 'R(v)' in 6(.r,R). Show by transfinite induction that, for each a and x,

x t. rS; iff ry( i ) l t. E ~ .

COROLLARY 5.11. E, is not coinductive over 91.

PROOF: If Ex were coinductive, there would be a predicate v(x) such that, for any x, x is in the complement of E, iff rv(i) l is is E x . Use the self-referential lemma to find a sentence 5 that is gap-equivalent to v(r(1). We have

r(1 F the complement of E, iff rv(r51)l F E, iff rtl E Ex..

DEFINITION: For a E E,, let la1 = the least ordinal a such that a E E,,.

This norm induces a pre-well-ordering" of E x . The following technical lemma shows that this pre-well-ordering can be specified within 2':

LEMMA 5.12. There is a formula L of 2' such that, for any a and 6,

' From Kripke [1975, p. XI]. In other words, the relation < on E , given by

.r < y iff 1x1 < Ivl ia well-founded [i .e. , every nonempty subset of E, has a <-leaat element] and sat~sfiea

(V,r)(Vy)(Vz)((.r < J & J < z ) -. .r < :) and (V~r ) (vy ) (V~) (x < -+ (x < .v V J z))

Every well-founded relation that satisfies these conditions is induced by an ordinal norm.

rL(a.h)l e E7 ~ f f tr E E, and ( h 4 E , or b e E with (a1 < ( b ( ) ; and r ~ ( a , b ) l E A, iff b c E, and (cr 4 E, or tr F: Ex wlth lal 2 lbl.

PROOF: I apologize for this proof. 1 do not know of any way to prove the existence of the desired formula other than actually to exhibit the formula, but the length of the formula makes the exhibition rather unpalatable. In defining L ( i , b ) , we look at all the possibilities for what a and h might bc: they might be nonsentences or atomic sentences or negated atomic sentences or disjunctions or negated disjunctions or so on. The formula L((r,h) amounts to a case-by-case examination of all these possibilities. Use the self-referential lemma to find a fortnula L(>r,y) that is gap-equivalent to:

{ [ ( x is a sentence of y,, & l r ( x ) ) V (-r is a sentence o l the form i T r ( r ) & den(^) is not a sentence of y,,)] &

[!I is not a sentence of YG V ( y is a sentence of Y,, & T r ( 7 v ) ) V ( y is a sentence of the form Tr(7)) V ( y is a sentence of the form l T r ( 7 ) & Detl(r) is a sentence of Y; ) V is a sentence of the form ( z v w ) & Tr(rL(k,;) l) & ~ ~ ( r ~ ( k , 6))) V ( y is a sentence of the forni' ~ ( z V W ) & ( ~ r ( r ~ ( i , ~ $ ) l ) v

T r ( r L ( i , l $ ) l ) ) ) V ( y is a sentence of the form 7 l z & ~ r ( r ~ ( f , t ) l ) ) V ( y is a sentence of the form ( 3 v ) z & ( V ~ ) ~ r ( r ~ ( k , z ) l ) ) V ( y is a sentence of the form l ( 3 v ) z & ( ~ W ) T Y ( ~ L ( ~ , ~ -))I}

V {x is a sentence of the form Tr(rr) & ~r ( rL (Derz (a ) , j ) l ) &

[ y is not a sentence of Y',; V ( y is a sentence of 2,,, & T r ( 1 v ) ) -- V ( y is a sentence of the fornl M I ) & T r ( r ~ ( D e r z ( o ) , ~ e n ( r ) ) 1 ) ) V ( y is a sentence of the form ~ l T r ( r ) & Den(r) is a sentence of Y,:, --

& Tr(rL(Den (a) , l D e n ( r ) ) l ) ) V ( y is a sentence of the form ( z V w) & T r ( r L ( i , i ) l ) & Tr(rL(k ,w) l ) ) V ( y i a sentence of the form ~ ( z v w) & (Tr(rL(i,+)l) v

T r ( r L ( k , l $ ) l ) ) ) V ( V is a sentence of the form 7 1: & Tr(rL(k, l;) l)) V ( y is a sentence of the form ( 3 v ) z & (Vw)Tr(rL(k,-11)) v ( y is a sentence of the form l ( 3 v ) z & ( 3 ~ ) ~ ~ ( r ~ ( . ? , l - ) l ) ) ] }

V {x is a sentence of the form i T r ( r r ) & Den(a) is a sentence of 21, & - .

T r ( r L ( l ~ e t z ( c r ) , j ) l ) &

[ y is not a sentence of V ( y is a sentence of Z:,, & T r ( 7 y ) ) -- V ( y is a sentence of the form Tr ( r ) & T r ( r ~ ~ ( ~ e r . l ( u ) , D e n ( ~ ) ) l ) ) V ( y is a sentence of the form 7 T r ( r ) & Den(7) is a sentence of --

2;t'J, & ~ r ( r l ( l ~ e n ( u ) , ~ ~ e n ( r ) ) l ) ) V ( y is a sentence of the form ( z V W ) & Tr ( rL ( i , $ ) l ) &

~ r ( r ~ ( k , G ) ? ) ) V ( y is a sentence of the form -I(: V w ) & ( ~ r ( r L ( k , ~ i ) l ) V

Tr( r L ( k , l MI)^))) V ( y is a sentence o f the form 1 7 2 & Tr(rL(k , f - ) I ) ) V ( y is a sentence of the form (3v ) z & ( ~ w ) ~ r ( r ~ ( i , = ) l ) ) v ( y is a sentence of the form 7 ( 3 v ) z & ( 3 w ) T r ( r ~ ( k , ~ m ) l ) ) ] )

V {,r is a sentence of the form ( z V w ) & (Tr(rL(Z',T)l) V Tr(rL($,$)l))}

V {x is a sentence of the form l ( z V w) & T r ( r ~ ( ~ i - , $ ) 7 & ~ r ( r L ( l $ , j ) ) )

v {x is a sentence of the form 1 l z & Tr( rL(1 , f ) l ) )

V {x is a sentence of the form ( 3 v ) u & ( 3 w ) T r ( r L ( m , j ) l ) }

V {.r is a sentence of the form 1(31x)u & ( ~ \ t ' ) T r ( r L ( l m , $ ) 1 ) } &

~ ( ~ w ) ( v s ) T ~ ( ~ L ( ~ w , r ~ ~ ( ~ ~ ) l ) l ) V y is not a sentence of :fi, V ( y is a sentence of 2,, & T r ( 7 y ) ) V ( y is a sentence of the form Tr(r) & (Vw) T r ( r ~ ( l - , ~ e n ( . r ) ) 1 ) ) V ( y is a sentence of the form l T r ( r ) & Den(r) is a sentence of %:;, & (Vw)Tr( r~(1m) ,1)Detz(r)) l)) V ( y is a sentence of the form ( z V w ) & Tr ( rL (k , t ) l ) & Tr(rL(k,G)?)) V ( y is a sentence of the form l ( z v w) & ( ~ r ( r L ( k , ~ 1 ; ) 1 ) V

~ r ( r ~ ( k , l $ ) l 1)) V ( y is a sentence of the form 1 l z & ~ r ( r L ( k , i ) l ) ) v ( y is a sentence of the form (3 t ) z & (V.s)Tr(r~(k,*)l)) V ( y is a sentence of the form ~ ( 3 t ) z & ( ~ s ) T ~ ( ~ L ( I , ~ ~ ) ' ) ) I } .

V { [ x is not a sentence of 3; V ( x is a sentence of Y!,, & T r ( l x ) ) ] & ~r(rL( .? , $7) & (y is a sentence of y : , + ~ r ( 1 ~ ) ) )

It is straightforward, though tedious beyond endurance, to verify that this works.. If C is a recursively enunlerable set of natural numbers, C can be represented

as {x: (3rn)-y(i.m)}, for some bounded formula y. This determines a natural pre-well-ordering on C, got by stipulating that x is below y in the pre-well- ordering iff the least rn such that y ( i , m ) is less than the least n such that y()-,;). In a similar way, an inductive set D can be represented as {x: ( 3 a ) ( r 6 ( i ) l e E,,}, for some formula 6 of 9?(,, and a natural pre-well-ordering on D is determined by stipulating that .r is below y in the pre-well-ordering iff the least a such that r6(;)7 is in E,, is less that the least p such that r6 ( i ) l

is in E,. The similarity between these constructions is the basis fix a far- reaching structural analogy between the recursively enumerable sets and the inductive sets. To illustrate this analogy, we look at a couple of classical results about recursively enumerable sets, together with their analogues for inductive sets.

LEMMA 5.13 (REDUCTION PRINCIPLE FOR RECIJRSIVELY ENUMERABLE SETS).' For any recurs~vely enunlerable sets C and D , there arc recursively enumerable sets C' and D' such that

C' c c D ' C D C' U D' = C U D and c1 n D' = 0

PROOF: Say C = {I: (3m)y(x,rn)} and D = {x: (3n)6(i,n)}, where y and 6 are bounded. Let

THEOREM 5.14 (REDUCTION PHINCIPI,E FOR INI)UCTIVE SETS).' For any (parameter-free) inductive sets C and D, there are (parameter-free) inductive sets C' and D' such that

C' c c D' c D C' U D' = C U D and ct n D' = 0

PROOF: Say C - {x: (3a)(ry(;)1 E E.)} and D = {x: (3p)(r6(i) l E E,)). Let

C' = {x: r(y(i) & ~ ( r y ( x ) 1 , r ~ ( x ) l ) ) l e: Ex) D' = {x: r(6(x) & i ~ ( r y ( x ) l , r 8 ( x ) l ) ) l F Ex).

COROLLARY 5.15 (SEPARATION PRINCIPLE FOR (: SETS).' For any two disjoint 7 sets F and G, there exists a recursive set that includes F and is disjoint from G.

PROOF: Apply theorem 8.13 with C = w - G and D = w - F. The recursively enumerable set D' that we obtain will be the complement of C', and so C' will be a recursive set that includes F and is disjoint from G..

Theore111 5-XVll of Rogers 119671. Theorem 3A.4 of Moschovakis [1974J.

' This is problem 5-33 from Rogers [1967].

COROLLARY 5.16 (SEPARATION PRINCIPLE FOR COINDUCTIVI.: SETS)."' For any two disjoint (parameter-free) coinductive sets F and G, there exists a (parameter-free) hyperelementary set that includes F and is disjoint from G.

PROOF: Corollary 5.16 is got from theorem 5.14 in just the way that corollary 5.15 was got from theorem 5.13 ..

Theorem 5.10 shows that E, is a universal member of the class of inductive sets, in the sense that, for any inductive set C, questions about membership in C can be reduced to questions about membership in E,. We now want to show something stronger, namely, that the ordered pair (E,,A.) is a universal member of the class of disjoint pairs of inductive sets. This result will be applied extensively in chapter 8. We begin, once again, by proving an analogous theorem for recursively enumerable sets:

LEMMA^.^^ (Smullyan and ~u tnam) . " Let A , B , C, and D be recursively enumerable sets with

A n B = 0 C f l D = @ {theorems of Robinson's R) C C; and {sentences refutable in Robinson's R} C D

Then there is a 2': formula v(x) such that, for any x,

x F A iff rv(2)l E C and x E B iff rv(x)l E D

PROOF: Take bounded formulas a(x,y), P(x,y), y(x,y) and 6(x,y) so that

A = {x: (3y)a(x,y)) B = {x: (3y )~(x ,y ) ) C = {x: (3y)y(x,y)) and D = {x: (3y)6(x,y))

Now use the self-referential lemma to find a C:/ formula v(x) so that

' I ' Theorem 3A.5 of Moschovakis [ 19741. " Theorem G in the Supplement to Smullyan [IYbl]. Different from Smullyan's proof, the proof

given here is designed for easy adaptation to inductive sets.

We verify that x E A iff rv(i)7 6 C: (=)) Suppose that A e A but rv(s)1 4 C. Then lor some y, a(;.\.) IS true. For each 2 < y. ~ ( ; . i ) is false, since x ( B, and y(rv(;)1.;) i \ false, since ru(i)l 4 C. Therefore, v(*) is a true 2: sentence, so that rv(;)l e {theorems of R ) C C. Contradiction. (e) Suppose that rv(;)I E C but r ( A Then, for some z,, we have y(rv(;)l ,z). S 0

For each .v r z,, a(;,?) is false and so. every true bounded sentence being provable in R ,

- id) R (QY 5 :,,)%i;~) & (n < y h p ( ; , ~ ) & (vz < ? ) ~ ~ ( r ~ ( ; ) i , ~ ) l .

Putt~ng (c) and (dl together, we get -

(el R 1(3~)1a(\-.\-) (VZ < ~ ) l p ( i . ~ ) & (K < y)ly(rv(i)~,z)l.

From (b) we derive

Now, rv(;)l ( D. so 6(rv(i) l , y) is false for every y, so, since every true bounded sentence is provable in R ,

Combining (0 and (g), we get

(e) and (h) give us

But then rv(i) l E {sentences refutable in R} C D . Contradiction. The proof that x E B iff rv(;)l E D is similar.

(+) From the assumption that x e B but rvi;)l 4 D , one can derive that R F

TUG). But this gives the impossible result that rv(f)l E {sentences refutable in R} C D. (a Now. assume that ru(;)l E D but i 4 B. For each z, pi;.;) and y(rv(;)1,;) are both false. and so (3g)(~(ru(;)l,\-) & (VZ<~)T~(; ,Z) & ( V ~ < ~ ) l ~ ( r ~ ( ; ) l

is a true Z',' sentence. But this gives the result that ru(;)l c. {theorems of R } C C'. which is i~npossible. since C and D are disjoint..

I THEOREM 5.18. Let A, B, C. andD be (parameter-free) inductive sets with

A n B = 0 c n ~ = 0 E, C C and A, C D

I

Then there is a (parameter-frce) formula v(x) such that. for any x,

x E A i f f rv(;)l E C and I

x F B iff rv( i ) l F D

PROOF: Take (parameter-free) a , P, y , and f i so that, for any a,

x E A iff ra(;)l e E , x E B iff rfi(X)1 E E , x F C iff rY(X)l E E x and x r-: D iff r6(;)7 E E,

Now use thc self-referential lemma to find a formula v(x) that is gap-equivalent to

[ff(x) & ? ~ ( r p ( i ) l . ra(i)-) & l L ( r y ( r v ( i ) l ) l , ra(.i.)')l v 1 rv(2)i ) & TL( rp(i)l, ra( rv(;)l ) & TL( ry( rv(.i)i )I, r6( rv(.~jl 11 11

The verification that v(x) has the desired properties mimics the corresponding verification in theorem 5.17 ..

As the first application of theorem 5.18, we obtain a stronger version of , corollary 5.1 1 :

THEOREM 5.19. There arc no two disjoint coinductive sets E and A so that

I (E,A) is a fixed point of the Kripke construction. If, following Woodruff, we allow E and A to overlap, there will be coinducti\~e fixed points, but even if we allow overlap there will be no hyperelementary fixed points.

I

I PROOF: First we show that, if truth-value gluts are forbidden, then there are no

coinductive fixed points. Suppose, on the contrary, that (E,A) is a coinductive fixed point, with E and A disjoint. By the separation principle (corollary 5. lh) , there is a hyperelementary set C that includes E and is disjoint from A. By theorem 5.18. there is a formula y(x) so that, for any .r,

each occurrence of Tr(7) by ~ h . 5 ) . Then S is fully defined in (:)I, (E,. A , ) ) by P ( x ) . (2) 3 (1): Obvious..

The so-called Tarski hierarchy of languages 14 the sequence Y,,, Y', , Y,. . . ot interpreted languages got by setting

X,, = interpreted by ?)L -2,, , , = the result of adding to Y,, a new predicate Tr,, whose extension is

the set of (Giidel codes of) true sentences of Y,,

2,, = Y , , , Interpreted by !'I %,,, , = the result of addlng to Y(x a new pred~cate Tr,, for each u e E, w ~ t h

la1 = a. strpulating that the extension of TI , 1s to be the 4et of true sentences of Y,,

2, = *UA Y,, for A a l~rnlt

We would like to extend this construction into the transfinite. but doing so is a matter of some delicacy, the sticky point being the availability of a suitable system of notation for the transfinite ordinals. Let us imagine that we have extended the Tarski hierarchy to get an interpreted language YI, for each /3 r K .

Then to determine whether ' ~ r , ( r ~ r , , ( r ~ -- 07)1)' is a well-formed formula of g,, we have to know whether a < P. In order for Y,, to be able to describc its own syntax, the set {<rTr,(x)1,r~r,(x)l>: a < /3 5 K ) has to be definable within YK, so that the order relation on the ordinals less than K is recoverable from the system of Godcl codes. Thus, the establishment of a Godcl coding for Zi requires the development of a definable system of notation for the ordinals less that K , so

THEOREM 5.22. The set of true sentences of Y,,',,,,,, is parameter-free inductively definable over !)I.

1

In provlng t h ~ s , we make use ot the following lemma, u h ~ c h shows how one inductive definition can r ~ d e p~ggyback on top of another inductive defin~tlon.

that the problem of extending the Tarski hierarchy into the transfinite becomes intertwined with the intricacies of ordinal notations.

The Kripke construction gives us a particularly convenient system of notation for the ordinals less that o(31); for a E E,, we can treat as a notation for lal. This enables us snloothly to extend the Tarski hierarchy up to o(!'l). We assume I

! that we have defined within ?I a function that takes a E It'll to the Godel code of a predicate Tro with as subscript." The T~lc~ki-Kripke hierarchy consists of the following sequence of interpreted languages:

I' Only the Tr,,s for n F E, will actually be interpreted. but introducing T,;, for every u cnablea us the have thc Godel codes ready in advanct. available when wc need them.

LEMMA 5.23 (MOS~.HOYAKIS PIC;(;YBACK I A ~ h i ~ ~ ) . l h If Y is the least fixed point over ("I. X) of a first-order positive inductive definition

whcre S and R occur only positively in H and where X is (pararneter- free) inductively definable over ? I , then Y is (parameter-free) inductivcly definable over :)I.

PROOF: Simplifying slightly, we may suppose that we have formulas 4 ( x . ~ , R ) and $(x,R), in which R occurs only positively, such that. where Z is the least fixed point of

there exists an element b of \?t \ such that

X = the extension in (?I ,Z) of +(,Y.&,R)

Let W be the least fixed point of the fi>llowing first-order positive inductive detinition:

(Here '~,(x,Q(x,y,G))' refers to the result of substituting 'Q(T,~,G)' for each occurrence of 'R(T)' in $(x,R). 'y' is supposed not to appear in JJ or 0.) Y will be the extension in (91,W) of Q(x,b,?).

- If X is parameter-free inductively definable, we may leave out the parameter h.. PROOF OF THEOREM 5.22: Let YLi be the language got from by adding a

unary predicate Tr,? for each a E \!'[I. We introduce this uninterpretcd calculus because we can describe the syntax of y,, within Y , whereas we cannot fully describe the syntax of ~ , , , , , , within 2 because we cannot specify which potential subscripts of 'Tr' name members of E x . The set of true sentences of Y,,,,, is the least fixed point in (?(,Ex) of the following parameter-free first-order positive inductive definition, whcre L is the formula constructed in lemma 5.12:

(Vx)(V(s) ~f [.r is a sentence of yi & [x is a true atomic or negated atomic sentence of z,, V (X is a sentence of the form b r , ( a ) l & Tr(y) &

Den(a) is a sentence of YL; & (Vz)(Tri occurs in Den(a) + L(z,y)) & V(Den(cr1))

"' This lemma is roughly the same as theorem 1C.l of Moschovakia [1974].

V (X is a sentence of the form r i ~ r , ( c r ) ? & IDen(a) is not a sentence of Y , V iDen(u ) is a sentence of 2, & (3 z ) ( rTr , occurs within Den(cr) & T r ( r h ~ ( t , ; ) l ) ) ) V V ( ~ D e n ( c r ) ) ]

V (X has the form ( y V z ) & ( V ( v ) V V ( z ) ) ) V (1 has the form 1(?: V :) & V(,?) & V(T z ) ) ) V (x has the form 1 ~v & V ( y ) ) V (X has the form (3v)y & ( 3 w ) V ( y viH+)) V ( X has the form 7 ( 3 v ) y & ( V w ) V ( l y vi;))]]).m

One way to measure the expressive power of an interpreted language is to note what sets of individuals are nameable in the language; this measure can be applied whether the language is interpreted by a classical model or by a partial model. The following corollary shows that. by this measure, the language Y:, interpreted by the partial model (?)I,(E..A.)) has precisely the same expressive power as

~',,,, 1:

COROLLARY 5.24. A subset of 191 is the extension of a formula of 4,,1,1 iff it is fully definable in (?f.(E,,A,)).

PROOF: The left-to-right direction is an immediate consequence of theorem 5.2 1 ( ( I ) 3 ( 2 ) ) and theorem 5.22. For the right-to-left direction, note that, if lo1 =

p+ I , then E, will be the extension of ~ r , ( r~ r , ( r0=01)1) in jp),!,,,, and apply theorem 5.21 ( ( 2 ) 3 ( 5 ) ) ~

Key to understanding the everyday usefulness of the notion of truth is the observation that we can use the word 'true' to form, in effect, the conjunction or the disjunction of a set of sentences without being able to list the members of the set. Thus, if 1 express a skepticism about the accepted foundations of set theory by saying, "Not all the axioms of ZFC are true." I am. in effect, asserting the dis.junction of the denials of the axioms of ZFC. I could not achieve the same effect without utilizing the notion of truth (or some other semantic notion), since the axioms of ZFC are infinite in number, and, indeed, there is no finite system of axioms which is equivalent to ZFC." By utilizing the notion of truth, we are able to simulate operations of infinite conjunction and disjunction.

Of course, we cannot literally have infinite conjunctions or disjunctions in any spoken or written language. But we can describe abstract mathematical languages that contain these infinitary operations. The most prominent of these is the language Y,<", whose formulas are given by the following stipulation:

Every atomic formula of 2,, is a fi~rnlula of 2,,. If 4 is a formula of 2 x , , , so are 1 4 and ( 3 ~ 0 4 .

l 7 See Jech 11978, pp. X9fl

If S is a set of formulas of 2,- which contain among them only finitely many free variables, then the disjunction o f S , W S , is a formula of 2,,,. Nothing is a formula of y,,, unless it is required to be by the clauses above. lX

The truth conditions for the sentences of Y,, are the natural generalization of the truth conditions for 2,,. W S is true iff at least one member of S is true.

Although the introduction of the notion of truth increases the expressive power of the language g,,, it does not give us anything like the expressive power of %=,. Y,, enables us to form the disjunction of an arbitrary set of sentences, whereas the notion o f truth only enables us to form the disjunctions of definuble sets of sentences. If we can name the set S, we can form the disjunction of S by saying,

Some member of S is true.

Thus, to get a model of the way the introduction of a truth predicate increases the expressive power of a language, we look at the fragment of Y,,, which we get if we are only permitted to form disjunctions of definable sets of formulas. If our introduction of a truth predicate follows the format recommended by Kripke. we shall want to look at the fragment of Yti,, whose Godel codes are given by the following stipulation:

DEFINITION. The Kripke-dejnable fragment of Ym,, is formed as follows: If 4 is an atomic formula of 2,,, r41 is in the Kripke definable fragment. If x is in the Kripke-definable fragment, so is < T i 1 ,x>. If x is in the Kripke-definable fragment, so is <r(3v,) l ,x>. If S is a subset of the Kripke-definable fragment, if u(x) is a formula of 3: which fully defines S over (? ( , (E , ,A , ) ) , and i f the members of S have among them only finitely many free variables, then <rwl , ru (x )?> is in the Kripke-definable fragment. Nothing else is in the Kripke-definable fragment. (Here r i l , r (3v , ) l , and r w 1 are intcgers chosen as Godel codes for '7' , ' ( 3 v , ) ' , and ' W ' . )

Thus, we are able to describe the syntax of our fragment of 3," within Y' by identifying a set of sentences with a formula of $, which fully defines it. Members of the Kripke-definable fragment are codes of formulas of 2,',,, and the formulas can be recovered from their codes as follows:

'91 is customary to admit uncountably many variables into ZL,, but so long as one is only interested in the truth conditions for sentences (as contrasted with the batisfaction condit~ons for formulas), the extra variables do not do any useful work.

If 4 is an atomic formula of Y,,. cfec.odc( rbl) = 4. rfec.ocle(< r -11 .x> = the negation of derode(.r). ckc,ocle(<r(3vt)1,.~>) = the result of prefixing (3v,) to dc.rode(,r). clecode(< Twl. ru(,r)l>) = the disjunction of. the irnage under dcc.ocie of

the extension of v(x) in (!)I , (E , ,A,)).

THEOREM 5.25." The set of true sentcnces in the Kripke-definable fragment of 9?je,c,, is parameter-free inductively definable.

PROOF: The proof is so similar to proofs we have already looked at that wc only give a sketch. Let S U B ( ~ , L , , M : , ~ ) be the relation on /\'[I which obtains iff x is a member of the Kripke-definable fragment. v is a variable, w is a term. and z is the member of the Kripke-definable fragment got by substituting MJ for free occurrences of v in x. SUB can be obtained as the lcast fixed point of a certain parameter-free first-order positive inductive definition over ( ' ! ( ,Ex) . Thc set of true sentences in the Kripke-definable fragment can be obtained as the least fixed point of a parameter-frec first-order positive inductive definition over (91,Ex.SUB). Use lemma 5.23 to put these inductive definitions together..

COROLI.AKY 5.26 A subset of I \ ! [ / is the extension of a formula in the Kripke-definable fragment of Y, , iff it is fully definable in (\'l,(E,,A,)).

PROOF: The left-to-right direction is immediate. To get the right-to-left direction, wc show that if S C (!'I/ is hyperelementary, then S is the extension of some formula in the Kripke-definable fragment. Take a formula p(x ) of 9; so that the set consisting of all formulas rx = ):I such that y E S is fully defined in (:)l,(E,,A,)) by p(x ) . Then S is the extension of < r w l , r p ( x ) l > in the Kripke-definable fragment ..

Readers familiar with the theory of admissible sets will recognize the image under d ~ c o d r of the (true) sentences in the Kripke-definable fragment as the (true) sentcnces of thc admissible fragment Y .,," n HYP.,,.

Gupta1 [ 19821 and Herzberger [ 1982al. independently of one another, and Relnap [1982], in response to Gupta's work. have developed systems of what Herzberger has called "naive semantics." He called his system so because he intended it to capture the intuitive understanding of the meanings of self-referential sentences which we have before we introduce any apparatus from formal logic. The three systems are closely similar to one another formally. I propose to describe them by first describing Gupta's system and then saying what modifications the other two systems require.

As in the last chapter, we begin with a countable2 first-order language Y, interpreted by a first-order structure 91, and we expand Y to Y' by adjoining a new predicate 'Tr'. We no longer require that 91 be acceptable or that its vocabulary be finite, although we do still require it to contain arithmetic. Our problem is to find the extension of 'Tr'. Gupta recommends that we pick the extension by many applications of a rule of revision which enables us to pick better and better candidates for the extension of 'Tr'. He says:

Idealizing somewhat, we can say that underlying our use of words such as 'rcd' there is an application procedure that divides objects into two classes, those objects to which the word applies and those to which it does not apply. . . . In contrast, 1 am suggesting that underlying our use of 'true' there is not an application procedure but a revision procedure instead. When we learn the meaning of 'true' what we learn is a rule that enables us to improve on a proposed candidate for the extension of truth. It is the existence of such a rule, I wish to argue, that cxplains the characteristic features of the concept of truth [1982, p.2121.

Our first candidate for the extension of ' T r ' , U,,, is a pure guess. As our second candidate, we takc the set U , of all sentences true in the model (%,U,,). As our third candidate, we take the set Liz of all sentences true in the model ( ' ! l ,U,) . And so on. When we get to w . we reassess our original choice: if, after a certain point,

' Gupta read a draft of this chapter, and I have enjoycd several conversations with him. Thcse have proven to be extremely valuable. In addition, 1 had some helpful discussions with Belnap. All our results will have straightforward analogues for uncountable languages.

a sentcnce has turned up consistently in the U,,s, we put it in U,. If. after a certain point, a sentence has been consistently outside the lJ,,s, we exclude it from U". Otherwise, we stick to our original guess. We continue this process into the transfinite. The precise definition is this:

DEFINITION. A Gupta sequence is a sequence <I/,?: a e OR> of sets of sentences satisfying the following conditions:

U,,, , = {sentences true in the model (Y1 ,U,,)). U, = {sentences eventually in <U, : a < K > } U

(U,, - {sentences eventually outside <Ua: a < ~ > ) ) - - ,,g<t'g<.xc: u (U" - C! n (Sent- Up)). <"- .K LI.:@I:K

for K a limit, where Sent = {sentences of Y ' )

Gupta's idea is that, as the process continues, our choices for the extension of 'Tr' keep getting better and better. They cannot, however, keep getting better forever. Cardinality considerations show us that eventually we must reach a stage such that, from that stage on, no new candidates will ever appear, and every candidate that ever appears is bound to reappear endlessly. Once we reach this stage, none of the candidates for the extension of 'Tr' we look at will be any better than any of the others; they are all "best candidates" for the extension of 'Tr' .

Each Gupta sequence induces a classification of the sentences of 3' into three categories: Some, like ' T r ( r i ~ r ( r 2 + 2 = 51)1)', are eventually true, that is, they are in U, for all sufficiently large a . Others are eventually false. Still others, like the liar sentence, 'This sentence is false', keep flip-flopping. For some sentences, like the truthteller, 'This sentence is true', what category the sentence winds up in depends upon our initial choice of U,.

We are not really interested in the consequences of any particular initial guess U,, since we have no reason to prefer any choice of Uo over any other. What we are interested in is properties sentences have independent of any particular initial guess. Categorizing sentences according to their eventual behavior with respect to different starting sets, we ger a sevenfold classification. So that we can have examples, let us take A to be the liar sentence constructed in the proof of theorem 1.3, and let us take T, and T , to be truthtellers, so that

and

and so that T,, and T , are independent of one another? There arc three pure cases:

1 ) Stably true: sentences eventually true for all initial guesses, for exam- p l e , ' ~ r ( r 2 + 2 = 41)'.

2) Stably false: sentences eventually false for all initial guesses, for example, ' l ~ r ( r 2 + 2 = 41)'.

3) Par~ldo,xicul: sentences neither eventually true nor eventually false for any initial guess, for example A .

1 There are four mixed cases:

4) Weakl? smble: eventually true for some initial guesses, eventually false for others. Either eventually true or eventually false for each initial guess. For example, 7,).

5 ) Eventually true for sorne initial guesses, but not others. Eventually false for no initial guess. For example. (T,, V A).

6) Eventually true for no initial guess. Eventually false for some initial guesses, but not all. For example, (T,, & A ) .

7) Eventually true for some initial guesses, eventually false for others, and. for others still, neither eventually true nor eventually false. For example, (7,) & (71 V A)).

Notice that, to describe this classification of the sentences of 2', one requires the resources of a metalanguage essentially richer than 3'. Thus. Gupta's work does not directly advance our purpose here, which is to see how to develop a theory of truth for a language within the language itself. Gupta's purpose is not our purpose; Gupta's purpose is to explicate preanalytic intuitions. We shall see, however, that the work of Cupta and the others casts an indirect light upon the prospects for our project.

A Herzberger sequence is like a Gupta sequence except that, for K a limit, it will have

U , = {sentences eventually in <U,,: a < K>) = u " u,j

,,'CK acp.h-

As an example, let us take A,, and A, to be independent liar sentences, and let us suppose that A,, is in U , but A , is not. In the Gupta sequence beginning with U,,, A,, and A, will continue to have different truth values at every stage. In the

' One w a j to get such sentences is to use the self-referential lemma to tind a fen-mula u(u) of Y L so that

R c (V.r) (u(.x) - ((.u = 6 & ~ r ( ~ c r ( c ) ' ) ) V (.u = 7 & ~r( ' (v(7) ' ) ) ) . -

and set T,, = ( ~ ( 0 ) and i, = (u( I ) . A similar technique gives us the two liars that we use -

below.

Herzberger sequence beginning with li,,. A,, and A , will havc different truth values only at the finite stages. Thcy will both be outsi(ie U.,. and, for all infinite a. they will be either both inside or both outside U,,. 'Thus. in Herzberger's system. but not in Gupta's. (A,, t. A , ) will be stably true.4

Thc Beltzap seyrrr17ce.s include the Gupta sequences, the Herzberger sequences, and more besides. A Belnap sequence is any sequence <Utp: a u OR' of sets of sentences such that

U,, , , = {scntcnces true in thc model ('?I ,Un)) . Zl,, for K a limit, is a set of sentcnces that includes all the sentences

cventually in <Uct: tu < KB, and excludes all the sentences eventually outside <UC7: a < K >

Thus. whereas in Herzberger's system (A,, - A , ) is eventually true and in Gupta's systcm it is either true at every stage or false at every stage, in Belnap's system there is thc further possibility that the sentence keeps switching at limit stapes. Thus, in Herzberger's and Gupta's systems. but not in Relnap's. (A,, t. A , ) - Tr(rh,, t. A,?) will be stably true.5

The Hcrzberger sequence beginning with the empty set is also a Gupta sequence and a Belnap sequence. Thus. a sentence that is. say, eventually true in every Belnap sequence will be eventually true in at least some Herzberger sequences and eventually true in at least some Gupta sequences. It follows that no sentence that is placed into one of the pure categories by one of the systems is placed into either of the other pure categories by either of the other two systems. Where the systems differ is that one of the systems can place in a pure category a sentence that one of the other systems places in a mixed category, or two of the systems can place the same sentence in different mixed categories. The relationships among the pure categories are given by the following theorem:

THEOREM 6.1. The set of sentences in Herzberger's sys-

tem includes the set of sentences system, which

stably true includes the set of sentences

inclusions are proper.

' This example is discussed by Gupta [1982, p. 2311 and by Belnap (1982. pp. 110fl. Tlris example is presented by Belnap [ 1982. p. I I]. who aays it was discovered b )~ Gupta

PROOF: The examples we just looked at show that the sets of stably true sentences do not coincide for any two o f the three systems. That the sets of stably false sentences do not coincide is shown by considering the negations of these exani- ples. That the sets of paradoxical sentences do not coincide is shown by considering the following two sentences: (A,, V A,), which is paradoxical in Herzberger's systen~ but not in Gupta's, and ((V natural number n) 7r1 ' ( r ( (~ , , ++ A , ) ++ Tr( r (~ , ,

A , ) I ) ~ ) & A,,) (where the notation 'Tr'" is defined by: ~; ' ( r$ l ) = +, Ti '+ (r+') = ~ r ( r ~ r " ( r + i ) l ) , which is paradoxical in Gupta's system but not in Belnap's. Thus. we know that the inclusions. if they obtain at all, are proper.

stably true That the set of sentences in Gupta's system includes the set of

paradoxica

in Belnap's system follows immediately from the fact

that every Gupta sequence is a Belnap sequence. So all we have to show is that stably true

the set of sentences in Herzberger's system includes the set of

We prove this in a sequence of three lemmas:

LEMMA 6.2. If <U,,: a E OR> is either a Gupta sequence or a Herzberger sequence, then {a < 0 , : U , = UuI} is cofinal in

PROOF: Given y < w , , pick a countable ordinal K,, bigger than y that is so big that any sentence that is either eventually in <U,,: a < w,> or eventually outside it has already settled down at U,,,. Our aim is to form an increasing sequence <K,: tz < W > of countable ordinals so that, for any sentence i / J . if + is neither eventually inside nor eventually outside <U,: tu < w , > , then $ will be neither eventually inside nor eventually outside <U,,,: 17 < a>. To do this, we form a list, ordered by w, of all the sentences that are neither eventually inside <U,,: a < w, > nor eventually outside it in such a way that every sentence that is neither eventually inside nor eventually outside <U,: a < w,> appears on the list infinitely many times. Given K,, if the nth item on the list is the (2j)th occurrence of the sentence 4. pick K, ,+ , > K,, so that $ is in U,,,+,. If the 11th item on the list is the (2j + I )st occurrence of +, pick K,,, , > K,, so that 11 is not in U , , + ,. Thus

" Here w , is the least uncountable ordinal. The lemma says that there arc arbitrarily large countable

ordinals a with U,, = U,,,. The ordinal ar~thmetic we use here and in the other theorems in the chapter can be found in many standard textbooks. After all thcse yeara, the best source on the subject is still Sierpihkii [19581. Those who find ordinal arithmetic unpleasant may skip to the

end of remark 6.9 without loss of continuity.

at the even stages we ensure that + is not eventually outside <UK,,: n < w>, and at the odd stages we ensure that + is not eventually inside <UK,,: t1 < ,>.

Let K . ~ = K,,. K, is countable. and U = Uw,.. K w

DEFINITION. g J U ) = the ath term of the Gupta sequence beginning with 0 , and h.(U) = the a th term of the Herzberger sequence beginning with u .

LEMMA 6.3- each y 2 1. for each 8. hi., ?,,, (u ) = hw,+ , (u ) , and RiwI y , t ~ ( U ) = gw, +a(U) .

PROOF: We prove the lemma for Herzbergur sequences; the proof for Gupta sequences is similar. For any u and a. if h,,(U) = h.ol(U), then, by induction, h i = h U . So it will be enough to show that h,, ,(U) = h W l ( U ) . We may assume y > 1, since the case y = 1 is trivial.

We prove this by induction on y . We shall use the result from lemma 6.2 that there is a countable ordinal a with h,.(U) = hk j I (U) .

First of all. let us suppose that y is a successor, say y = 6 + I . We have

Y(U) = h<,,l ,, , l ,(U) - - h ~ w l ~ R ~ + w l ( u )

- - hUl ,,,(U) [by inductive hypothesis]

= hcr+,,(U) = h,,(U)

Now, SUppose that y is a limit, and suppose that $ is in h , ,(U). There is a 6 . I 6 < 7. so that $ & , , ,n, h,.(U). Take any P with a < P < o,. There

' O ' . W , y exists a 5 , 0 < 6 < oI SO that /3 = a + 5. We have

Thus, $ & wg ,,h,(U) c h, , , (W. For the opposite inclusion, suppose that $ & h w l ( U ) . Use lemma 6.2 to find a

countable ,b so that $ E ,_Q ,,hJU) and so that h,(U) = h, , , , (U) If w , < c < w l . y then there exist 6, 1 4 6 < y , and 5 < w , so that u = (w l .6 ) + 6 . We have

- - / z i W , n l+c (U) [by inductive hypothesis] = h,(W

LEMMA 6.4. For each ordinal K , hw,+K(U) = g.(h,,(U)).

PROOF: 'The proof is by induction on K. The zero and successor stages are obvious, so we may assume that K is a limit. We have

g.(h,,(U)) = {sentences eventually in <g,(hwl(U)): P < K>} U h W l ( U ) - {sentences eventually outside <g,(h,,(U)):

P < K>) = (sentences eventually in <h,,,, ,,(U): P < K>)

U h W l ( U ) - {sentences eventually outside <h,, +,(U):

P < K > I

[by inductive hypothesis] = h w l t , ( U )

U h,,(U) - {sentences eventually outside <hWl+,(U): /3 < K > I

Thus to show that g,(h,,(U)) = h,, +,(U), it will be enough to show that h w l ( U ) - {sentences eventually outside < h w l , ( U ) : /3 < K>} is included in hWl ,,(U), for which it will certainly be enough to show that hWl(C') is included in h, ,+, (U) . The proof breaks into two cases, depending on whether K is countable. CASE I. K is countable. We need to show that, if we assume that $ is not eventually in <h,,,,+,(U): P < K > , we can conclude that $ is not eventually in <hlj(U): P < w,>. Take 6 < o , . We need to show that there is a y , S < y < w , , so that $ 4 h,(U). We know that we can find a /3 < K such that + 4 hUl+,(U). Lemma 6.2 tells us that there is a countable ordinal a > 6 with h,(U) = h,,(U). Thus we may set y equal to a + P. CASE 11. K is uncountable. Then we can find an ordinal 6 2 1 and a countable ordinal /3 such that K = (w l .6 ) + P. We compute

hWl t,iW = - h,*l+,i,l.,,t,,i~) - ( ,

= hWl+,(U) [by lemma 6.31 C h,,(U) [by case I , if P # 0; trivially, if /3 = 01.

PROOF OF THEOREM 6.1 (continued): We are going to show that any sentence that is stably true in Gupta's system is stably true in Herzberger's. Suppose that the sentence $ is not stably true in Herzberger's system. Then there is a starting guess U such that $ is not eventually in <h,(U): a s OR>. Thus, $ is not eventually inside <h,,+,(U): a e OR> = <g,,(hwl(U)): a e OR>, so $ is not eventually inside the Gupta sequence beginning with h,,(U), and hence $ is not stably true in Gupta's system.

The proof that any sentence that is paradoxical in Gupta's system is paradoxical in Herzberger's is similar.

The proposition that every stable falsehood of Gupta's system is stably false in Herzberger's system is derived from the corresponding fact about stable truths by taking negations ..

COROLLARY 6.5. For any starting set U. h,",(U) = {scntences eventually in <h,(U): 6 E OR>), and g,,,,(U) = {sentences eventually in <g,(U): 6 e OR>) U (U - {sentences eventually outside <g,(U): 6 c OR>}

PROOF: We can use lemma 6.2 to find a countable ordinal a such that h,(U) = h,,>,(U) and w c h that every sentence in ha,,(1J) is in <,.g , ( U ) Then. for each \entente 4.

6 is eventually in <h,JU): P E OR> iff (3 ordinal y r I)(V 6 3 w,.y)(+ c h,(u)) iff (3 ordinal y 2 I )IV 6 2 y)tV countable P)(d F h,", ,, + ,(u)) lff ( v countable E h_, + ,(U)) [by lemma 6.31 iff (V countable /3)(4 c h, . ,(u)) lff 4 & Iz,,,,(U)

The demonstration for Gupta sequences is similar.. From Corollary 6.5 it follows that

(V set of sentences U ) ( 3 countable ordinal a ) ( V sentence $) (if $ is eventually in the Herzberger sequence <U,j: /3 E OR> starting with U, then t,!~ is in U, for all ,6 > a ) .

The same holds for Gupta sequences. We might hope to reverse the order of quantification, finding a single countable a. depending only on the underlying model ?'I, that would work for every U and $; the least such a might serve as a measure of the complexity of ?I. It turns out that we cannot find such an a.

PROPOSI'TION 6.6. We can construct a sentence r ( 0 ) such that, for any countable ordinal K > 0 , there is a set U , of sentences with the property that, although a ( 0 ) is eventually in the Herzberger sequence <UU: a e OR> beginning with U , (which is also a Gupta sequence and a Belnap sequence), a ( 0 ) is not in any U, with a 5 K.

Thus, although the level at which ~ ( 0 ) settles down must surely be countable, we can, by an artful choice of U,,, make thc level at which it settles down be as high a countable level as we like. PROOF: Use the Godel self-referential lemma to find formulas r(x,y) and a(?.) such that

R + (V natural numbers x and y)(r(x,y) - Tr(rT(.;.f)l)) R + (v natural number ?)(a(?.) - (V natural number x)

(r(x,y) -+ Tr(ra( . i ) l ) )

Given a countable ordinal K , let us pick a well-ordering of w of ordcr type K + I with 0 as its greatest element. We take the set U,, to be {rr(m,n)1: m is less than

n in the well-ordering we have selected). Then u ( 0 ) is in the sequence <U,,: u

& OR> after the ~ t h level, but not earlier.. A I-Ierzberger sequence or a Gupta sequence is a proper class; thus, the central

notions of the semantic theory, the notions of stable truth and so on, arc defined by means of quantification over propcr classes. Now, because of lemmas 6.2 and 6.3 and corollary 6 .5 , a Herzberger sequencc or Gupta sequence can be completely described by saying what it does at countable levels, so that the quantification over proper classes is inessential. For Belnap sequences this is not so clear, and Belnap asked [1982, p. 1091 whether the seniantic notions of his theory could be defined within the language of set theory. An affirmative answer is contained in the following theorem:

THEOREM 6.7. Let us call an w,-length initial segment <U,: u < o,> of a stably true

Belnap sequence an w,-Belnap sequence. A scntence is paradoxical

eventually inside Belnap's sense iff it is

neither eventually inside nor eventually outside every o,-Belnap sequence.

The corresponding result for Gupta and Herzberger sequences is an immediate consequence of corollary 6.5 . PROOF: We prove that a sentence is stably true (in Bclnap's sense) iff it is eventually in every w,-Belnap sequence. The other two proofs are similar. (3) Suppose that the sentence 4 is not eventually in the w,-Belnap sequence <U,.: a < w , > . Pick a countable limit ordinal /3 so large that any sentence

that is eventually { ~ ~ ~ s ~ ~ e } < u C v : a < w l > is { b ~ ~ ~ ~ c } u.. for all a 2 -

/3. Every ordinal can be uniquely written in the form (w, .y )+6 , where 6 is countable. Define a Belnap sequence <V,: a F OR> by setting V,,, ,,+,equal to Upis. Because 4 is not eventually in <U,: a < w , > , we can find a countable ordinal { such that 4 & Up+,. Therefore, for each y, d, & V ,,,,, ,,+,. Thus, 4 is not eventually in <V,: a e OR>, and so it is not stably true. (e) Now let us suppose that 4 is not stably true. There is a Belnap sequence <U,: a F OR> such that { a E OR: 4 & U,,} is unbounded. Thus {limit ordinals A: for some finite n , 4 & U,+,,} is unbounded. Since we cannot write an unbounded class of ordinals as a countable union of bounded sets of ordinals, there must be a finite n such that {limit ordinals A: 4 & U , +,,) is unbounded. Hence, we can find limit ordinals A , and A?, A , < A?, such that 4 & UAI+,, and such that U,, = U,,.

Let us consider a first-order language containing two unary predicates '0' and 'S' and two binary predicates 'L' and 'U ' . Let us specify an interpretation of

this language by declaring:

'o"?' = {ordinals A?) ' ~ ' 2 ~ = Sent ),q '07?A " .s,?+ 'L'" = the less-than relation on '0"' 'u'" = {<a,$>: a 5 AL and $ F UUJ

We can use the Lowenheim-Skolem theorem7 to find a countable elementary submodel of % which contains within its universe of discourse all of the sentences of 2' as well as the ordinals X, and A,. An elementary submodel of a well-ordering is still a well-ordering, and so the structure <'o", 'L"> is isomorphic to a countable initial segment of OR via an isomorphism that we shall call f. The sequence <U,-I,,,: a <.f(A,)> is the beginning of a Belnap sequence. We want to extend it to a Belnap sequence by stipulating that, after the f(Al)nd term the sequence will endlessly repeat the cycle i U , - ~ , ~ , : f l A , ) 5 a s,f(h2)>. To make this precise, we define an operation I / / I taking ordinals to ordinals less than f(Az), as follows:

IIOII = 0 l l f f + Ill = l l f f l l + 1 For y a limit, define limsup(<llal): a < y>) to be the least 6 such that ( 3 a

< r)(VB)(ff 5 p < y + IlPll 5 6). If limsup(<(lal(: a < y>) <f(A2), let IIyJJ = limsup(<llall: a < y>). If limsup(<)(cu(/: a < y>) = ,f(Az), let llyll = f(Al).

Now we define, for each ordinal a ,

It is straightforward to verify that <V,: a E OR> is a Belnap sequence, and so <Va: a < o,> is an w,-Belnap sequence.

We want to see that 4 is not eventually inside <V<,: a < w , > . Recall that 4 U . Notice that, since the finite ordinals are in both 1%) and 61, we must

have f(k) = k for each finite ordinal k, and, in particular.f(n) = n. Looking at how / / 1 ) was defined, we see that, for each countable ordinal [ 2 1, 11Cf(A2)-[) + nl = A h , ) + n, and so ( a < o , : llall = A h , ) + n) is unbounded in o , . If 1JaJ = AX,) + n, then

Thus. we can find arbitrarily large countable ordinals a such that 4 & V,,.. REMARK 6.8. APPI,ICATION OF THEOREM 1.5 TO RULE-OF-REVISION SEMAN- TICS. The set of sentences that are stably true in Herzberger's sense will satisfy all the conditions of theorem 1.5 except for condition (4)c); so will the set of sentences eventually in the Herzberger sequence <ha((/): a s OR>, for each starting set U. For the set of sentences stably true in Gupta's system or in Belnap's system, conditions (4)a) and (4)b) will also fail. To see this, use the self-referential lemma to find sentences A and 6 so that

Upon successive applications of the rule of revision, A describes a cycle of length two: true, false, true, false, true, false, true, false, . . . , while 6 describes a cycle of length four: true, true, false, false, true, true, false, false, . . . . Let U include A, i h , and ( A -+ a ) , but exclude 6. Then, for K a limit, none of the four sentences A, i A , ( A + 6), and 6 will be either eventually inside or eventually outside <g,,(U): a < K > , and so A, i h , and ( A + 6) will be in g,(U) and 6 will be outside g,(U). Therefore, g,+,(U) will exclude both (Tr(rlA1) -+ i ~ r ( r h 1 ) ) and (Tr(r(A -+ 6) l ) -+ (Tr(rA1) -+ Tr(r61)). so that neither (Tr(riA1) + - ~Tr(rhl)) nor (Tr(r(A + 6)l) + (Tr(rh1) + Tr(r61)) is stably true in Gupta's system. Consequently, according to theorem 6.1, the sentences will not be stably true in Belnap's system either.

If we suppose, as seems reasonable, that our ordinary usage of 'true' tacitly presupposes the principles that a sentence and its negation are not both true and that the set of truths is closed under modus pnnens, then these examples provide evidence that Herzberger's system represents ordinary usage more faithfully than Gupta's or Belnap's. To defend Gupta's or Belnap's proposal, one might claim that, when they are dealing with sentences suspected of paradox, ordinary speakers no longer regard (4)a) and (4)b) as reliable. It is hard to say what would count as decisive evidence here.

In defending their systems over Herzberger's, Gupta and Belnap do not look at particular examples but at broad theoretical con~iderations.~ The aim of their endeavor is to develop the idea that our usage of 'true' is governed by the rule of revision. Now, the rule of revision does not tell us what to do at limits, and if we want the rule of revision to be our governing principle, we should adopt a limit rule that adds as little as possible to what the rule of revision tells us. In Herzberger's system, they would say, we do not see the effects of the rule of revision alone. We see the effects of the rule of revision together with a policy of declaring disputed sentences untrue rather than true..

' Theorem 3.1.6 of Chang and Keisler [1073]. ' See especially Belnap [1982, p. 1051. In trying to understand the issues here. I benefited greatly from conversations with Gupta and Belnap.

REMARK 6.9. APPLICATION OF RULE-OF-REVISION SEMANTICS TO THEOREM 1.5. We can use the methods of rule-of-revision semantics to show that we cannot strengthen theorem 1.5 to say that any set of sentences which satisfies

I conditions ( 1 ) through (4)c) is simply inconsistent. Indeed. we do not get a sinlple , inconsistency even if we strengthen condition ( 1 ) to require that l- contain all of

true arithmetic. Let us take our language 2 to be the language of arithmetic, and I

let us take our model 9t to be the standard model 92 of 2. Consider the axiom system, which I shall call A, consisting of all sentences obtained by prelixing 'Tr's and universal quantifiers to instances of the following schemata:

4 * Tr(r41). for $I an atomic formula of Y Tr( r($I V $1' ) - ( ~ r ( r41) V Tr( r$l 1) T r ( r i 4 1 ) - i T r ( r 4 1 ) Tr( r(3 v)1$1) - ( 3 ~ ) T r ( r 4 7 )

Here 'Tr(r41)' is to be understood in such a way that all variables that occur free in 4 occur free in Tr(r41). If 4 is $(I,,,\.,, . . . , I,, ,), ~ r ( r 4 1 ) will be Tr(r+(G,;, . . . , iy).

The set of first-order consequences of A U {true arithmetical sentences} satisfies conditions ( 1 ) through (4)c) of theorem 1.4. Yet A U {true arithmetical sentences} is first-order consistent, as we can see by observing that, for any U, the model <\3>, h,(U)> satisfies all axioms of A i n which fewer than k 'Tr's have been prefixed to instances of the given schemata. This tells us that every finite subset of A U {true arithmetical sentences} is first-order consistent, and so, by the compactness theorem, A U {true arithemetical sentences} is first-order consistent..

So far, our discussion of the three systems has been purely formal. Let us now ask how the formalism is to be interpreted philosophically, examining briefly what each of the three authors has to say.

Belnap thinks that Gupta's theory contains important philosophical insights, and he thinks that these insights are captured better by Belnap's formalism than by Gupta's own, a judgment with which Cupta himself has come to agree.' But, in spite of these philosophical insights, Belnap does not cornmit himself to the theory. On the contrary, he expresses a preference for a rival theory of truth, the account of Dorothy Grover, J. L. Camp, Jr., and Belnap 119751, although he does not elaborate his reasons for the preference.

Herzberger explicitly disavows any intention of developing a consistent theory of truth; he wants to study our inconsistent naive theory. Citing Chihara 11979], he says that his purpose is diagnostic rather than therapeutic. He says that his proposal

See the postscript on p. 234 of the reprint of Gupta [I9821 in Martin [19X4]

begins with languages of an ordinary two-valued kind in which semantic parddoxes are virtually inevitable. Rather than try to eliminate those paradoxes, 1 want to consider the experiment of positively encouraging them to arise and watching them work their way out. This approach, which I call naive semunrics, is a deliberately nondirective exercise. The idea is to stand back and let the paradoxes reveal their inner principles (1982, p. 4791.

Herzberper wants to make explicit those semantic principles which we implicitly utilized while we were still innocent of semantic theory. Herzberger surely does not mean by this that ordinary speakers of English custo~narily make use of transfinite methods in evaluating their sentenccs. Every day we meet persons who exhibit thorough mastery of the ordinary use of 'true', even though they are entirely unschooied in Cantor's methods. Herzberger's contention must be something weaker. along these lines: there are certain procedures that we tacitly employ in evaluating the sentences we encounter in ordinary life. We normally apply these procedures only in rather homely, finitary contexts, but there is nothing to prevent our attaching the apparatus of the everyday semantical rules to the powerful machinery of set theory, obtaining results that go far beyond the finitary. Herzberger docs not claim that his system gives either the rules we in fact use or the rules we ought to use, but that it gives the rules we would use if we utilized the methods we use now more efficiently.

It is difficult to assess the correctness of this claim, since we have not been told the rules we actually use. If Herzberger's thesis were broken into two parts- here are the rules we actually use, and this is what results from letting that system of rules be driven by the machinery of set theory-we could evaluate each part separately. As it is, the thesis that the rules we customarily employ, whatever they are, are such that they extend to Herzberger's system is an interesting psychological speculation, but not a doctrine we have very good reason to believe.

Gupta bases his account on the fundamental distinction between two kinds of concepts. Ordinary concepts, such as red, blue, and sum. we learn by learning an application procedure, which enables us to separate those objects to which the concept applies from those to which it does not. Other concepts, notably the concept of truth, we learn by learning a rule of revision, which enables us to improve upon a proposed candidate for the extension of the concept. Gupta explicitly disavows the claim that we consciously apply the rule of revision when we use the word 'true1-"That would be an absurd piece of sociology," he says 11982, p. 2331-but hc apparently thinks that the rule of revision plays some unconscious role in the use of 'true'. There is a familiar distinction between learning how to apply a rule and learning how to state it. Thus. children learn how to apply the rules of grrmmar, (or they learn how to produce grammatical sentences, but they do not learn how explicitly to state the rules, nor are they

consciously aware of them. Perhaps Gupta's contention is that children learn how to apply the rule of revision in the way that they learn how to apply the rules of grammar. But such a contention is not very plausible. The rules of grammar describe recursive operations that can be applied in a finite number of steps. There is nothing recursive or finite about the rule of revision. Indeed, it is not so clear that the rule of revision should properly be called a rule at all, since it does not present a procedure to be followed; it only describes an abstract mathematical function.

Let me suggest a much simpler account, according to which children first learn to apply the word 'true' by learning the rule of application described by schema (T). The principal merit of this account is that it makes out the child's task in

.. --a

learning the word 'true' to be a very simple task, and so accounts for the ease with which children are able to learn the word. We arc not required to suppose that the child can carry out transfinite inductions or that she has other extravaeantlv

L , complicated conceptual apparatus at work. Another merit is that the account explains the way we react to the antinomies. If the rules for using the word 'true' which we learned as children were consistent, we would not be startled and disconcerted by the antinomies. We would regard the antinomies as anomalous. perhaps, but not as paradoxical. But we do, in fact. find the paradoxes disturbing. One observation Gupta cites in favor of his theory is that it accounts for the fact that the liar sentences jar our intuitions in a way that truthtellers do not (1982. p. 2321. But this observation is nicely accounted for by the simple account that says that our naive theory of truth is given by schema (T). The liar sentences jar our intuitions because they show that the naive theory is inconsistent; the truthtellers merely show that it is incomplete. The account of language acquisition that says that we learn to use the word 'true' by adopting a rule that permits us to assert the instances of schema (T) is an extremely simple theory that accounts for a great deal; its attractions are hard to resist. As a matter of fact, this theory, or something like it, is presupposed by Gupta's theory. To apply the rule of revision in going from Urn to U,, , ,, we have to determine what sentences are true in the model (91,U,), and to assess what sentences are true in (?(,U,J, we make use of the classical theory of truth, which accepts schema (T). Thus, we have to have a firm grasp upon the classical theory of truth before we can begin to comprehend the rule-of-revision theory. l o

Gupta's theory looks to be a lot more plausible if we take it to be, not an account of how children acquire the use of the word 'true', but an account of the use of 'true' at a much more sophisticated stage in the development of a nonphilosophical speaker. This is the stage at which, having been shown a liar sentence, the speaker realizes that the simple rule that enabled her to assert all

"' These criticisms of Gupta's account of how wc learn to u ~ c the word 'tme' were suggested to me by Chihara.

i instances of schema (T) is unsatisfactory, but at which she also realizes that, whatever the problem is, the vast majority of her uses of 'true' are not affected by it. She therefore finds some way of restricting schema (T) so as to avoid the dubious applications while keeping the rest. Now, the nonphilosophical speaker surely does not hit upon anything as elaborate as the sevenfold classification Gupta produces. But the nonphilosphical speaker is able to classify the vast majority of sentences she encounters as unproblematically either true or false, and she is able to make intuitive distinctions among the problematic cases, seeing that the truthteller sentences, while they have something of an air of paradox about them, are not viciously paradoxical in the way the liar sentences are. Gupta can claim that his classification merely refines this intuitive classification. Particularly striking is the fact that we are able to make decisive intuitive judgments about the truth and falsity of a wide range of convolutedly self-referential sentences that narrowly skirt the edge of paradox. For such sentences, Gupta's classification agrees with our intuitive truth assignment remarkably well.

Although the rule-of-revision semantics are not very credible as accounts of the psychological mechanism that underlies ordinary speakers' usage of 'true', the theories are remarkably accurate in describing ordinary speakers' attributions of the predicate 'true' in particular cases. Of course, there are many sentences which arc assigned truth values by the semantic theories but which are so complicated that ordinary speakers find them baffling. Also, there are sentences whose truth values ordinary speakers cannot assess because they lack the needed factual data. But if we look at sentences for which ordinary speakers are able to make clear judgments based upon correct factual information, we find that the judgments of ordinary speakers and the judgments of the rule-of-revision semantics almost always agree. Let us look at a couple of examples: EXAMPLE 6.10. Imagine that Xochitl, a reliable and trusted friend, warns me about Yolanda, saying

Not everything that Yolanda tells you will be true.

Suppose, moreover, that Yolanda tells me exactly two things:

Wooden matches are insoluble. Nothing Zaida tells you will be true.

Zaida, in turn, tells me exactly three things:

The fish in Lake Anza swim backward to keep the water out of their eyes. In Ohio, the ratio of the circumference to the diameter of a circle is 4.0. The apartments on Elm Street have rats as big as raccoons.

On the basis of Xochitl's warning, together with the known fact that wooden matches are insoluble, 1 conclude that at least one of Zaida's statements is true,

from which. in turn. I conclude that the apartments on Elm Street have rats as big as raccoons, so I do not rent an apartment on Elm Street.

Let as see how this reasoning is represented in the rule-of-revision semantics, For present purposes. i t does not matter which of the three systenls we use. Let our starting set be U. and let us conveniently ignore the fact thilt the sentences we are looking at contain indexical elements. and so cannot properly be said to be true or false. If we assume, contrary to what we want to show. that it is not the case that the aparttnents on Elm Street havc rats as big as raccoons, then we see that each of Zaida's three statements will be false in the underlying nonsemm- tic alodel. and so none of them will be in any U<. with o r I . This it~lplies that

Nothing Zaida tells you will be true.

will be true i n each ("I.U.), with o 2 I . and a o both of Yolanda's statemen. will be in Up for each p r 2. Hence,

Not everything Yolanda tells you will be true.

will be outside U, for every y 2 3, and so Xochitl's statement is stably false, contrary to our assumption of Xochitl's complete reliability.. EXAMPLE 6.11. The same as example 6.10. except that this time Zaida makes a fourth statement, namely,

The residents of Elm Street are constantly menaced by wildcats.

We can no longer conclude that the apartments on Elm Street have rats as big as raccoons, but we can still conclude the following:

Either the apartments on Elm Street have rats as big as raccoons or the residents of Elm Street are constantly menaced by wildcats.

and this will still be enough a) prevent me from renting an apartment there. The reasoning is similar. If the conclusion were false, then all four of Zaida's statements would be false in the underlying nonsernantic model, and so Xochitl's statement would be stably false.. EXAMPLE 6.12 (from Gupta [1982, p. 2 101. A says:

(a I) Two plus two is three. (a2) Snow is always black. (a3) Everything B says is true. (a4) Ten is a prime number. (as) Something B says is not true.

B says:

(b l ) One plus one is two. (b2) My name is B .

(b3) Snow is sometimes white. (b4) At most one thing A says is true.

We would ordinarily reason as follows: (a3) and (a5) contradict each other, so at most one of then] is true. Since ( a l ) , (a?), and (a4) arc obviously false, this implies that at most one of A's statements is true. that is, that (b4) is true. Since (bl), (b2), and (b3) are obviously true, this i~llplies that all of B's statements are true, that is, that (a3) is true and (a5) false.

This reasoning is reproduced in the rule-of-revision semantics as follows: (a3) and (a5) might perhaps both be in U,,, but they cannot both be in U, . ( a l ) , (a2). and (a4) cannot be in U , either, since they are false in the underlying nonsemantical model. So at most one of A's statements is in U , . Therefore, (b4) is true in ( % , U , ) , and so (b4) is in UZ- ( b l ) , (b?), and (b3) are all also in U,, since they are true in the underlying nonsemantical model. And so (213) is true in (!'(,I/?) and (as) is false in (\'(,I/?). The construction stabilizes at this point, so that (a3), (bl) , (b2), (b3), and (b4) are all stably true, whereas ( a l ) , (a2). (a4), and (a5) are all stably false.. EXAMPLE^.^^. Where K is an ordinal, let $1 be an acceptable structure containing the ordinals < K such that 'OR'"' is the set of ordinals < K and '<"" is the order relation ,on the ordinals < K . Using the self-referential lemma, find a formula w(x) so that

An intuitively correct induction shows that, for each a < K , u(&) is true. The rule-of-revision semantics concur with our intuitions; for any starting set U , a(&) will be in U, for every y > a , and so a(&) will be stably true..

That it is possible to give a systematic description of ordinary speakers' attributions of the predicate 'true' is a remarkable fact. One might have feared that, since ordinary speakers learn to use 'true' in non-self-referential contexts, their usage of 'true' in self-referential contexts would be so chaotic that it would resist systematic treatment. This work of the rule-of-revision semanticists shows us that this fear is unfounded, and that it is possible to give a very elegant description of ordinary speakers' use of 'true'.

The rule-of-revision semanticists intended their theories as representations of !

ordinary usage. If we understand this intention as an intention to produce a theory that accurately predicts the judgments of ordinary speakers as to the applicability of 'true' to particular cases, we see that they have succeeded very nicely.

In the present work, we are not interested in describing ordinary usage. On the contrary, our attitude in the present work can be summed up in the slogan "Ordinary usage be damned"; we are interested in replacing ordinary usage by a reformed usage that is scientifically respectable. Thus, it would appear that, for

our purposes, the fact that it is possible systen~atically to describe ordinary speakers' attributions of 'true' is, at best, an interesting curiosity.

Appearances are deceiving here. Although we are not interested in depicting the theoretical intuitions of ordinary speakers or in explicating ordinary-language metaphysics, that the theory we develop generally agree with ordinary usage about that applicability of 'true' in particular cases is essential to the success of our project. Whereas we are willing to reject the naive theory, we do not wish to repudiate the naive practice. Thus, in developing our theory, we submit to the following fundamental constraint:

ORDINARY USAGE REQUIREMENT. A successful theory of truth ought to agree with ordinary usage about the applicability of 'true' in a wide range of particular cases.

'The vagueness of this fornlulation is unavoidable. There are two motives for adopting thc ordinary usagc requirement. The first

is simply that the notion of truth, used the way we ordinarily use it, is a useful notion. Part of the notion's apparent usefulness is its value as a theoretical principle for explaining the connection between language and the world; but this appearance of usefulness is under suspicion, since we are talking about the use of 'true' as the central theoretical term of a bankrupt theory. But the usefulness of the notion of truth also has a practical side. We use the notion of truth to convey information that we could not readily convey by other means. We are only able to do this because there is widespread agreement among ordinary speakers about the conditions under which a particular given sentence is to count as true. The ordinary usage requirement is intended to ensure that our reformed usage of 'true' will continue to serve the practically useful purposes that our unreforrned usage serves.

The second, philosophically more interesting, reason for adopting the ordinary usage requirement is that we need the requirement to support the contention that the new notion we are developing and designating by the word 'true' will be a genuine notion of truth. We are replacing our old usage of the word 'true' by a new usage of 'true'. In order for it to be correct to say that, in doing so, we are replacing our old conception of truth by a new conception of truth (rather than that we are misusing the word 'true'), there must exist strong links between our old and new usage, and the ordinary usage requirement is crucial to providing such links. As we progress from our naive understanding of semantics to a scientifically reformed account, there will be a radical shift in the theoretical conceptions we associate with the word 'true'. But our old and new usage will be bound together by a body of common practices in applying the word in particular cases. Were it not for this body of common practices, what we would obtain would be, not a proposal that we adopt a reformed conception of truth, but a disguised proposal that we abandon the notion of truth altogether.

When a new theory takes the place of an old one, the lines of conceptual I I inheritance are often blurred. Adherents of the new theory will use old terms in I new ways. The concepts to which the old terms are newly applied will be similar 1 to the old concepts. but they will not be the same, so that, in employing the old

i terms, instead of inventing an entirely fresh vocabulary, we will be drawing I ! parallels that are, in some ways, illuminating, in other ways, misleading. Because

the consonance between the old and new usage is a matter of degree, questions of the legitimacy of conceptual succession will not always have definitive answers.

i In the absence of definitive standards, we can still say this much: if a proposed new way of using the word 'true' conforms to the ordinary usage requirement, then the proposed new usage will have a serious claim to legitimacy as a notion of truth.

Our current usage of 'true' is constituted by a wide range of principles, judgments, and practices. lf we want to obtain a consistent understanding, we must abandon the high-level theoretical principles that characterize current usage, but we shall be able to preserve a wide variety of our everyday judgments and I

practices, and it is in virtue of these that our new theory will still be a theory of truth.

~ e f o r e encountering the antinomies, we would have thought that schema (T) was so central to our understanding of the meaning of the word 'true' that any theory that denied (T) could not be a theory of truth. But now we see that schema (T) has to be relinquished. In a similar way, before the advent of relativity theory, we would have thought that it was inherent in the way we use temporal language that temporal precedence is a binary relation between events, rather than a ternary relation between two events and an observer. After the revolution, we use the same words-'time', 'space', 'mass', 'energy', 'momentum'-as before, but not with quite the same meanings. The envisaged revolution in semantics will provide a new way of using the word 'true' which will stand to the current usage in just the relation that relativistic usage of 'time' bears to prerelativistic usage.

The rule-of-revision semanticists aim to describe our ordinary conception of truth. Our aim here is not to describe our ordinary conception but to change it, but meeting the ordinary usage requirement is essential to achieving that aim. The rule-of-revision semantics satisfy the ordinary usage requirement extremely well, and that is why their results are valuable to us in our task here. They show us that the ordinary usage requirement is not out of reach, and they set a high standard by which to judge our own efforts to meet the requirement.

Before we end our discussion of the rule-of-revision theories, let me comment briefly upon an illuminating passage in which Cupta discusses the methodology of his work. What Cupta wants to understand is the notion of truth in natural languages, yet his research consists in investigations into the semantics of formal languages. In explanation, Gupta says the following:

In order to give the meaning oftrll paradox-free sentences containing "true," we would need not only to arrive at a general understanding of what meaning is and what form a theory purporting to givc the meaning of. sentences should take (something that is much disputed in present-day philosophy), but we would also need to give an account of a great many perplexing linguistic phenomena such as indexicals, vagueness, modality, rtc.. If we are even to begin to work towards a description of our concept of truth we need to set aside some of thesc problems or, at least, niake a tentative decision about their solution [I 982, p. 1771.

Gupta makes several simplifying assumptions, which he likens to idealizations in the natural scicnees. The first of them is the following:

I assume that the language (or the languagc fragment . . . ) under study I

has, apart from the concept of truth, none of the other complicating factors: indexicals, vagueness, ambiguity, intensional constructions, truth-value gaps, etc. Thus I assume that apart from the concept of truth the language

I

can be viewed as a classical first-order quantificational languagc. This assumption is satisfied only by very limited fragments of natural languages, I

I

but 1 believe that it removes little that is of immediate interest to us in our study of the interaction of truth and self-reference [1982, p. 1781.

I i In most respects, the methodology Gupta dcseribes here is exemplary. Nearly

always, in investigating a complicated phenomenon, one is best off first trying to understand the phenomenon in the simplest context in which one finds it. Simplifying assumptions are not to be deplored but eagerly to be welcomed; I

I without them we shall find ourself forever lost in a maze of complexities. This I methodological precept, like most methodological advice, is fallible. It might I

turn our that, evcn though indexicals, say, do not play any essential role in the formulation of the paradoxes, they do play an essential role in their solution, so that by directing our attention to situations in which indexicals do not appear we direct our attention away from the solution to our problem. But even if this is so, a likely way to discover that it is so is by attempting the simplified approach and coming to realize its limitations.

What is disturbing about Gupta's remarks is not what Gupta says but what he does not say. What Gupta says is that he is making the assumption that the I

language (or language fragment) he looks at has the syntactic form of a first-order language. What he neglects to say is that, apart from the truth predicate, the language he is looking at has the semantic form of an irzterprrted first-order language. The nonsemantic part of the language is interpreted by specifying, using the c-relation from set theory, a first-order structure consisting of a set. which is to be the universe of discourse, and other sets, which are to scrve as the extensions of the predicates. The semantic theory is then developed by producing

a transfinite sequence of expansions of the first-order structures. Gupta's construction depcnds vitally upon the availability of the language of set theory as an essentially richer metalanguage. Without utilizing such an essentially richer metalanguage, we could not even describe Gupta's construction, much less carry it out. In an interpreted first-order language, our universe of discourse is some particular set, and we can manipulate the set, combining it with other sets using the methods of set theory. In English, or in the fragment of English about which Gupta says he wishes to talk, our discourse is not restricted to any particular set, so we cannot use the methods o f set theory to manipulate models of English. This is a fundamental difference, yet it is a difference Gupta overlooks.

Gupta assumes, correctly 1 imagine, that the absence from first-order languages of such things as indexicals and propositional attitude constructions is not relevant to the study of the interaction between truth and self-reference. But without even mentioning it he makes a further assumption that is wildly implausible, namely, that the availability of a metalanguage that is essentially richer than the object language under scrutiny is not relevant to the study of the interaction between truth and self-reference.

Gupta wants to obtain an understanding of the notion of truth as it applies to English. To get such an understanding, one must solve the problem of how to present the semantics of a language within the language itself. Looking at simplified languages is fine, but if we can only give the semantics of our simplified language within an essentially richer metalanguage, the fundamental and difficult problem of how to give the semantics for a language within the language itself will still remain before us. Gupta does not appear to have recognized this problem, much less to have solved it.

The most common misfortunes to befall philosophical theories are that they be rejected and that they be ignored. A different misfortune has befallen Tarski's doctrine about how to cope with the antinomies. It has been accepted too well. Logicians take it for granted, without question, that it is their right and their duty, in discussing the semantics of a language, to make use of an essentially richer metalanguage. They do not acknowledge or even recognize that, in adopting this policy of Tarski's, they have committed themselves to a potent philosophical doctrine with powerful philosophical consequences. Like the proverbial eye- glasses that we cannot see because they are on our face, Tarski's doctrine has been accepted so thoroughly that it has become invisible.

Partially Interpreted Languages

Kripke wanted to treat 'true' as a predicate whose meaning had been only partially specified. To describe the logical structure of languages some of whose predicates were only partially defined, he utilized Kleene's 3-valued logic. In this chapter, we shall look at a different way of analyzing the logical structure of languages with partially undefined predicates, by employing a notion of partially interpreted languages which arises out of the work of Carnap. Later on, we shall see how to utilize this notion in developing theories of truth.

Camap [I9361 thought that it was particularly important to be able to recognize the observational consequences of a scientific theory, for he felt that precisely what distinguished scientific discourse from the chatter of metaphysicians was that the former has a clear observational content. Disposition terms, such as 'soluble', 'malleable', and 'conducts electricity', occur frequently in science. What is their observational content'? We can observe whether or not a thing that is put into water dissolves, but we cannot observe whether or not a thing that is not put into water has a potency to dissolve. A natural suggestion for a definition of 'soluble' is that a thing is soluble just in case it dissolves if it is placed into water. This will not do, since by this standard a wooden match that is never placed into water would count as soluble, since it satisfies the material conditional, "If x is placed into water, then x dissolves." One response would be to use an "If-then" construction that has something more than the force of a material conditional. saying, "If x were placed into water, x would dissolve." Carnap would regard this as a step backward, since the subjunctive conditional does not have any clear, observationally described truth conditions.

Carnap proposed to solve this problem by taking the term 'soluble' to be only partially defined by the so-called bilateral reduction sentence

(VX)(X- is placed into water + (x is soluble - .T dissolves)).

This stipulation tells us, of the things that are placed in water and dissolve, that they are soluble, and, of the things that are placed into water without dissolving. that they are insoluble. We leave it unspecified which, if any, of the objects that are never placed into water are soluble.

The bilateral reduction sentence leaves the applicability of the predicate 'soluble' indeterminate in a much wider range of cases than our ordinary usage does.

We would normally feel no hesitancy in declaring a wooden match "insoluble," even if it is never placed into water. We pronounce the wooden match "insoluble," because we have seen numerous other wooden objects placed into water and we have observed that none of them have ever dissolved. This pronouncement has more than speculative interest, since it enables us to make predictions. If we do not know whether a particular wooden match is going to be immersed in water, we can nonetheless use the judgmcnt that wooden ob,jects are insoluble to make aconditional prediction that, if the match is placed into water, it will not dissolve.

In conformity with ordinary usage, we can narrow the range of indeterminate- ness of the term 'soluble', so that wooden matches are declared "insoluble" whether they are placed into water or not, if, in addition to positi~lg the bilateral reduction sentence, we make the further specification:

Two bodies of the same substance are either both soluble or both insoluble.

Whereas the original bilateral reduction sentence can be regarded as purely stipulative, this additional specification has empirical consequences. Specifically, it implies

If two bodies of the same substance are each placed into water. then either both will dissolve or neither will.

A formal apparatus for talking about languages some of whose predicates are only partially defined is the following: A parriul interpretatiorz is a pair (\!I,T), where \!I is an ordinary first-order structure and l- is a first-order theory in a language extending the language of 91. The language of \!I is fully interpreted, but the rest of the language is only partially interpreted. In the example we have been looking at, the language of '?I will include the predicates 'is placed into water', 'dissolves', and 'is the same substance as', but it will not include 'soluble'. The theory r will consist of the sentences, '(Vx)(x is placed into water 4 (x is soluble - x dissolves))' and '(Vx)(Vy)(x is the same substance as y -t (x is soluble tt is soluble))'. If some object of the same substance as Lump A is placed into water and dissolves, then 'Lump A is soluble' will be definitely true. If some object of the same substance as Lump A is placed into water without dissolving, then 'Lump A is soluble' will be definitely untrue. If nothing of the same substance as Lump A is ever placed into water, then it will remain undetermined whether 'Lump A is soluble' is true or not.

Although Carnap does not give a formal semantics for partially interpreted languages, we may do so. A partial interpretation of a language should partition the sentences of the language into three classes: those which are definitely true, those which are definitely untrue, and those whose truth values remain undetermined. There are two natural ways of making this classification which I wish to discuss.

DEFINITION. A sentence 6 is dejinire(v true under thc partial interpretation (?l,I') in the model-theoretic sense (in syn~bols (!)l.I') 1 4 ) iff 4 is true in every expansion of ?)I to a model of I-. 4 is dejinite!y unrrue in the model- theoretic sense iff (91 ,r) 1 - 1 4 .

Here we are using technical terminology taken from Chang and Keisler [1973, $I . 3 ] An e.rponsiot7 of a first-order model is obtained by adding new symbols to the language and specifying the references of these new symbols, leaving the meanings of the old symbols unchanged and leaving the universe of discourse unchanged. An expansion is not to be confused with an extension of a model, which is obtained by leaving the language unchanged but adding new individuals to the universe of discourse, specifying which of the new individuals are to be in the extensions of the predicates of the language and specifying the valucs to be taken by the function signs with new individuals as arguments. If 8 is an expansion of !'I, ?)( is a reduction of %, and if % is an extension of ?)I. !)I is a submodel of 8.

( w ) 14 iff. once we have fixed !'I as the interpretation of the fully intepreted part of the language, 4 will come out true under any method for assigning references to the remaining symbols which makes the sentences in r all come out true. We may think of the language of 91 as the fragment of our language con~posed of symbols whose meanings have been fully specified, and we might think of the expansions of :'l as representing the hypothetical future interpretation that we will get when the meanings of all thc terms have been co~npletely specified. r represents features of that hypothetical future model to which our present-day usage commits us.

To give the other notion of definite truth we need a definition:

DEFINITION. Let 3 be a countable language extending the language of the first-order structure 91, and let 4 be a sentence and T a set of sentences of 9,,. An 91-logical derivation of d, from r is a well-ordered sequence of sentences of Y , , which includes 4 , each member of which is either

a member of r; an atomic or negated atomic sentence of the language of 9I that is

true in \!I; an axiom of first-order logic; obtained from earlier members of the sequence by modus ponens; or obtained from earlier members of the sequence by the 91-rule (From

$(a), for each ci in /?I/, to infer (thv)$(v)).

DEFINITION. 4 is definitely true under the partial interpretation (91,r) in theproqf-theoretic sense (in symbols, (!)I,T) 1 4 ) iff there is an 91-logical derivation of I$ from r. 4 is riefinitely untrue in the proof theoretic sense iff (?l,r) t 14 .

The basic connection between the two notions of definite truth is given by the following theorem:

THEOREM 7.1 (Henkin and orey). ' If (21 ,T) 1 +, then (91 ,T) 1 4. If 1!11 is countable, then the converse holds.

PROOF: The first statement is proven by a straightforward induction on the lengths of proofs in 91-logic. To get the converse, suppose that ]!'I] is countable and that ( 3 , r ) 4 . We would like extend r to a maximal first-order consistent set A, of sentences which contains ( 1 4 ) U {atomic and negated atomic sentences true in 91) and which contains an instance I/,(;) of every existential sentence (3v)$(v) it contains. If we have such a set, we can define an expansion '8 of 91 by letting

r. = the unique a s /PI1 such that the sentence (r. = a) c A,. - - R'" = {<a,, . . . , a,,>: R(u,, . . . , a,,) F A,,,}; and - -

f u ( a , , . . . , a,,) = the unique b such that (f(y, . . . , a,,) = b) E A,.

It is easy to verify that, for each sentence 0, 0 is true in $3 iff 6, s A,. To form A. we first set P equal to the set of 9l-logical consequences of T, and

we take A,, to be r U ( 1 4 ) . Extend A,, to a maximal first-order consistent set of sentences by adding in new sentences one by one, making sure that each time we add an existential sentence (3v)$(v) we also add an instance $(a). More specifically, we enumerate the sentences of the language as O,,, O,, t12, . . . , and we

i form A,,, , as follows:

If A,, U {O,,) is first-order inconsistent, set A,,, , equal to A,, U {TO,,}. If A,, U {tl,,} is first-order consistent and H , , does not begin with an existential quantifier, then A,,, , = A,, U {tl,,}. If A,, U{O,,} is first-order consistent and O,, has the form (3v)$(v), form A,, , , by taking a so that A,, U {$(;)} is tirst-order consistent and letting A,,,, =

A,, U {d,,, $(;)I. To see that we can find such an a , suppose that. on the contrary, for each a , A,, U {$(;)I is inconsistent. Then where 6 is the - conjunction of the finitely many sentences' in A,, - T, we have, for each a , (6 + i $ ( a ) ) eT. ~ i n c e T is closed under the !'(-rule, (Vv)(6 -+ 1 $ ( v ) ) is in r, so that A,, U {(3v)$(v)) is inconsistent. Contradiction.

I Now let A,,, be ,u, A,,.. COUNTEREXAMPLE 7.2. The countability condition in theorem 7.1 is needed, since if (?I/ is uncountable we can have (\!(,I.) 1 + but (\!I , f ) 4. To see this,

, ,. , , ,

let 91 be an uncountable acceptable interpretation of a countable language X',

' This result was originally obtained (for the special case 1'1 = !I>) as a corollary to thc omitting types

theorem. Sec Chang and Keisler [ I 973, proposition 2.2.131. - ' This is where the proof breaks down for l!'ll uncountable. For u transtinitc, A,, - would be

infinite.

and let Y 2 bc obtained from i l l by adding the unary function sign ' f ' . Let l- consist of the formalization in 2' of

(VA-)(fix) is a natural number) I and let $I be I Since I!)[[ is uncountable, (!)t,T) 1 4 .

I claim that (YI,T) d. Suppose, for reductio ad absurclum, that there is an !'I-logical derivation of 4 from T. Let

P = {<r$l,r~?> : $ occurs before H in the derivation) I Let (%,Q) be a countable elementary submodel of the structure ( ! ' f , ~ ) . ~ Then Q is a %-logical derivation of 4 from r. Thus. (8.1.) t + and so (%,I') 1 4 . But this is impossible, since, I%/ being countable, ?8 can be expanded to a model of I' U { 1 4 ) by taking the extension of 7' to be a one-one function from I%/ into the natural numbers..

Both our notions of definite truth are rather far removed from Carnap's concerns, which were epistemic rather than semantical. Carnap wanted to know how scientific laws are confirmed. Assuming the model ?I is infinite, we cannot confirm a statement by directly verifying that it is definitely true, whichever notion of definite truth we use. To verify that (!'I,r) 1 4 , we would have to examine the uncountably many expansions of 71 to a model of T. To verify that (%.I') (- 4 , we would have examine the infinitely many steps of a proof by ?I- logic. In describing proofs by t)[-logic, we hypothesize a rule of perfect induction, which enables us to affirm all and only those generalizations each of whose instances we can affirm. In actual practice, we have no such infinitistic methods i available to us; we are reduced to using the highly fallible methods of ordinary scientific induction. We can think of proof by !)I-logic as a kind of ideal to which ordinary inductive reasoning aspires; it is the form our scientific reasoning would take if our powers of observation were unlimited, so that we could affirm or refute every atomic sentence of the observation language, the language of ?1, and if our methods of induction were perfect, so that we could establish all and only those generalizations each of whose instances could be established. Thus, although Carnap does not talk about either notion of definite truth, the proof- I theoretic notion of definite truth is perhaps closer in spirit to what Camap was doing. 1

Such a submodel exists by the Liiwcnheim-Skolem theorem (theorem 3.1.6 of Chang and Keisler [1973]).

The model theoretic notion of definite truth is perhaps more natural, but the proof-theoretic notion is more tractable mathernatically, primarily on account of the following result:

THEOREM 7.3. Assume that :'I is an acceptable structure for a language 2 built up from a finite vocabulary. that 2" is a countable extcnsion of 2, and that I' is a set of sentences of 9;. I f I' is (parameter-free) inductive over !'I, then (4: (!)I .T) 1 4) is (parameter-free) inductive over 91.

PROOF: We expand '!I to a model 91" by adding to the language of 91 a new unary predicate 'G' whose extension is I'. In t'l", {+: (!'(,I7) 1 4) can be described as the least fixed point of the following first-order positive inductive definition:

(Vx)[R(x) c* {x is a sentence & [G(x) V x is an atomic or negated atomic sentence true in '!I v x is an axiom of logic V (3v)(R(j) & R(Y ? x)) V (3 variable r1)(3j)(x = ( V I , ) ~ & (Vz)R(y \ ' I t ) ) ] } ]

The proof that {4: (%,I') 4) is contained in every fixed point of the inductive definition is a straightforward induction on the lengths of proofs in 91-logic. In proving that (4: (\![,I') t 4) is a fixed point of the inductive definition, we use the well-ordering theorem of Emst Zermelo [I9041 to show that, if there exists, for each a in I?[[, a proof of $(a) in '!(-logic, then we can string the proofs together in a well-ordered sequence to get a proof by the \!(-rule of (Vv)+(v).

The theorem follows by the Moschovakis piggyback lemma 5.23.. Because of Theorem 7.3, the proof-theoretic notion of definite truth is admira-

bly well-behaved mathematically. On this account, the proof-theoretic notion will be the notion of definite truth we primarily use. (Sometimes I shall use the unmodified phrase 'definite truth' when 1 think that whatever I am saying is true of both notions.)

Carnap thought of partial interpretations as a method for extending the language of science, alongside the more familiar methods of explicit and implicit definition; new terms are partiully implicitljl dejned by the theory I'. Frank Ramsey [1929a] wanted to apply this formal apparatus on a much grander scale. Ramsey thought of the set of sentences I' as the entire corpus of a scientific theory. Only observa- t

tional terms are contained within the fully interpreted part of the language. The theoretical terms of the scientific vocabulary are partially implicitly defined by the scientific theory. DIGRESSION 7.4. THEORETICAL ENTITIES. Ramsey wanted to use the apparatus of partially interpreted languages to explicate the meanings of theoretical predicates, but we can also use these methods to help explain our talk about theoretical

English into Portuguese, it is reasonable to ask whether the Portuguese text that results means the same thing as the English original. I t is not reasonable to ask the same question if we have translated the English text into an interpreted first- order language, because terms of an interpreted first-order language do not have meanings in the ordinary sense. They have extensions, but not meanings, so the

If + is definitely untrue under (3.r)). Tr(r41) is definitely untrue under (\!l ,r) .

If 4 is neither definitely true nor definitely untrue under (",I7). Tr(r+l) is neither definitely true nor definitely untrue under (? l , r ) .

most we can ask is that the translation be extensionally correct. (Here we the .%' under 'Tr'. since there is only one language in

Similarly, the most we shall demand of a translation into a partially interpreted I view .) first-order language is that it be extensionally correct. Let us imagine that we the next chapter, we shall see that this condition of material adequacy can

have translated a zoological discourse into a partially interpreted first-order lan- be fulfilled.

guage, and let us consider how our translation treats the term 'protozoon'. Our ordinary use of the term 'protozoon' is a little vague. There are many things that are definitely protozoa, according to ordinary usage, and many things that are definately not protozoa, but there may be a few things that are right on the border. Our translation will be materially adequate (or extensionally correct) if ' a is a !

- protozoon' is definitely true if u is definitely a protozoon, 'a is a protozoon' is definitely untrue if a is definitely not a protozoon, and ' a is a protozoon' is neither definitely true nor definitely untrue if it is unsettled whether a is a protozoon. (Of course, if our formalized theory adjudicates some cases that ordinary usage leaves unsettled, this will not necessarily mean that the formalized theory is bad zoology, only that it is a bad representation of preexisting ordinary usage.)

We are particularly interested in the case in which the text we wish to translate is a body of linguistic discourse. Let us suppose that we are given a first-order language 2, partially interpreted by (YI,T). We can give a semantic theory of 3 in ordinary English, according to which some sentences of Y are definitely true, some sentences of 2 are definitely untrue, and some sentences remain unsettled. Let us imagine that we have translated this English description of the semantics of Y into a formal language 2*, partially interpreted by (@,A) , using the formula

I

'Tr,' to represent the English phrase 'true scntence of 2'. For this translation to be materially adequate, Tr,?(r+l) must be a definitely true sentence of Y* whenever 4 is a sentence of 2 that is definitely true, ~ r , ( r + l ) must be a definitely untrue sentence of Y* whenever 4 is a sentence of X that is definitely untrue, and the truth value of Tr,(r$l) must remain unsettled whenever the truth value of 4 remains unsettled.

For the special case in which 2 and X* are the same, this gives us an analogue of convention T:

A partial interpretation ('$1 ,T) of a language Y gives a materially adequate partial implicit definition of truth for Y iff, for each scntence 4 of 2,,, we have

If 4 is definitely true under ('$l,T), Tr(r41) is definitely true under (?x,r).

Truth in Partially Interpreted Languages

In the last chapter, we dcveloped the notion of a partially interpreted language, and we devised a criterion of material adequacy for a theory of truth for such a language. What we want to do now is to see how this criterion of material adequacy can be met.

The ultimate aim of our endeavor is to sce how to develop a theory of truth for a natural language. We cannot directly apply our work on partially interpreted first-order languages to the study of natural languages because, obviously, natural languages are not partially interpreted first-order languages. Even if we restrict our attention to a strictly regimented fragment of a natural language which can be regarded as having a first-order syntax, we shall still not be able to apply our results on partially interpreted first-order languages directly even to the study of the fragment, for a reason to which we keep returning. A partially interpreted first-order language has a set as its universe of discourse, whereas in English, or our regimented fragment of English, we are not restricted to talk about any particular set.

Our procedure is indirect. We develop the semantics of partially interpreted first-order languages, making full use of the fact that in studying these languages we have available to us an essentially richer metalanguage. In the end, we hope to come up with formal techniques that we can apply even when we do not have available an essentially richer metalanguage. Developing the semantics of partially interpreted languages is analogous to developing an aeronautical design by working with model airplanes in a wind tunnel. We try things out on a scale model before attempting to implement a full-size design.

In order that our semantics for partially interpreted languages serve our purposes, we impose three requirements. The first is the analogue to convention T that we obtained in chapter 7:

MATERIAI. ADEQUACY CONDITION. A theory r gives a materially adeql*ate account of truth under the partial interpretation (!)[,I') iff, for each sentence 4, we have

(91,r) 4 iff (91,r) f ~ r ( r 4 1 )

and

(?(,r) 1 7 4 iff (?(.r) t T T T - ( ~ ~ - ' ) I

Second we havc a vague but important requlremcnt from chapter 6:

ORDINARY USAGE REQUIREMENT. A successful theory of truth ought to agrce with ordinary usagc about the applicability of 'true' in a wide range of particular cases.

Finally we have a fundamental methodological constraint:

REQLIIKEMENT OF .THE INTEGRITY OF THE LANGUAGE. It must be possible to givc the semantics of our language within the language itself.

Thc requirements are dissimilar in character. The third requirement is intended to hold open the possibility that the methods we develop can be applied to natural languages. If, in developing the theory of truth for a languagc, we required the services of an essentially richer metalanguage, that possibility would be closed off. Unless we mect the third requirement, all we can hope to get will be yet another method for describing the semantics of a formal language within an essentially richer metalanguage. The third requirement makes it reasonable to hope that our methods can bc used to get a semantics of natural language. The first two requirements are intended to ensure that the semantics we get will be worth having. As we shall see below, meeting the first requirement gets us a good part of the way toward meeting the second.

We begin by seeing that the first requirement can always be met:

I)EFINITION. Let (91,r) be a partia! interpretation of a Janguage 2 and let 1' > r be a theory in a language 9 including y. ( ? ( , r ) is a conservative extension of (\!I ,T) iff evqy expansion of \!I to a model of I- can be further expanded to a model of T.

If (?I,T) is a conservative extension of (!'(,I'), then, in going from r to r , we succeed in partially implicitly defining the new symbols in Y without saying anything new about the referents of the symbols in if.

THEOREM 8.1 (Kripke). Let \!I be an acceptable structure and 1- a theory In a language 9 extending the language of \!I such that the pred~cate 'Tr' does not occur in 2. Then there IS a theory I-, such that (\!(,I',) is a conservative extension of (91 ,r) and I', is a materially adequate theory of truth for (!)I,r,).

PROOF: We may assume that there is at least one expansion of 91 to a model of I-, since otherwise we could take IT, simply to be I- U {'TO = 0').

We extend the language of 2 by adding a new predicate 'Tr', and we define I',, for each ordinal u, to be

I' U {'The extension of "Tr" is a consistent set of sentences'}' u {~r(F4'): (Yl,,&J" r,J 1 4 )

Let r, = UglxI',. Because <r,: CY t. OR> is a nonstrictly increasing sequence of sets of sentences, there must be an ordinal K with T, = T,.

We show by induction that, for each a , (YI,T,) is a conservative extension of (91,T). Assume that, for each /3 < a , (YI,T,J is a conservative extension of (?t,T). Then for each ,B < a, T, is consistent, and so pUaT, is consistent. If % is an expansion of YL to a model of T, % can be further expanded to a model of T, by setting the extension of 'Tr' equal to U r p . Thus (?l,r,) is a conservative

,-:a extension of (!)l,r), and so, by induction, (YI,T,) = (Yl,T,) is a conservative extension of (Y1,r).

We need to show that r, is a materially adequate theory of truth for (9l,T,). Suppose that (!)I,T,) b 4 , and let % be an expansion of '?I to a model of T.

Further expand 3 to a model Ci of T, by setting 'Tr" equal to {$: (91,T,) $1. Then 'Tr(r41)' is false in G, and so (YI,r,) Tr(r41). Hence, by theorem 7.1, (?)l,rx) $- Tr(r41).

Now suppose that (!)l,T_) 1 4 . Then (!)I,T,) 1 4 , so that 'Tr(r41)' is in T,,, = r, and (?)I,T,) 1 ~ r ( r 4 1 ) .

Putting '14' in place of '4' in the argument just given, we see that, if (91,r,) f 1 4 , then (\'I,17,) 1 ~ r ( r i 4 1 ) . But (YI,T,) 1 ( T r ( r i 4 1 ) -+ ~ T r ( r + l ) ) , SO, if (?l ,rX) 1 1 4 , then (91,rx) t ~ T r ( r 4 7 ) .

Finally, suppose that (!)l,T,) $- 14. Then {$: (91 ,T,) $1 U (4) is a consistent set of sentences, so that, if 23 is an expansion of \!I to a model of T, we can further expand 23 to a model K of T, by setting '~ r "? ' equal to {$: (?)t,r,) t +) U ($1. Since i ~ r ( r 4 1 ) is false in 6, (?l,r,) f i T r ( r + l ) , and so (Yl,T,) b i T r ( r 4 1 ) . ~ DIGRESSION 8.2. KRIPKE'S SYSTEM WITH SUPERVALUATIONS. When Kripke devised the construction used in the proof of theorem 8.1, he was working in a rather different context. He was looking at alternatives to the 3-valued logic for evaluating truth in a partial model (Vl,(E,A)). Among others, he considered the following three options, derived from van Fraassen [1967]:

( u , ) 4 is true (resp., false) in (YI,(E,A)) iff, for each set U of sentences with E C U C Sent - A, 4 is true (resp., false) in the classical model (YI,U).

( a? ) 4 is true (resp., false) in (YI,(E,A)) iff, for each consistent set U of sentences with E C U C Sent - A, 4 is true (resp., false) in the classical model (t'I,U).

(u3) 4 is true (resp., false) in (?I,(E,A)) iff, for each muximal first-order

' That is, [(v.r)(Tr(.xj -* Sent(x)) & 1(3 f nite sequence J ) ( ( v x E s)(x is an axion, of logic v T ~ ( ~ ) V (3\.)(!: and (.v + .xj both occur before .r in s)) & r~~ = o1 E . ~ j ] .

consistent set U of sentences with E & U Sent - A, 4 is true (resp.,

false) in the classical model ('?I,U).

These evaluation schemes can be described in terms of the model-theoretic notion of definite truth. Thus, 4 is true in (YI.(E,A)) under (u,) iff

1

To get (n?) (resp., (o,)), we add the formalized version of the sentence 'The extension of "Tr" is a (maximal) consistent set of sentences'.

lf, in the proof of theorem 8.1, we set r equal to 0 and we replace the proof- theoretic by the model-theoretic notion of definite truth, we shall arrive at the least fixed point under (u2). For proving theorem 8.1, either (u , ) or (u,) would have worked just as well.

Looking back at theorem 1.5, we see that the set of sentences true at the least fixed point under evaluation scheme (n,) will satisfy conditions ( I ) , (2), and (3). Under (u,) we will get (4)b) as well, and under (4,) we will get all the conditions except (4)c).

There is a natural fourth member of our sequence of evaluation schemes, namely,

(u,) 4 is true (resp., false) in ( \ '( , (E,A)) iff, for every maximal 91-logically consistent set U of sentences with E U C Sent - A, c$ is true (resp.,

false) in the classical model (?I,U).

The set of sentences declared true at a fixed point under this evaluation scheme would satisfy all the hypotheses of theorem 1.5. But this is impossible, since , the set of sentences declared true at a fixed point would also have to be o - consistent. Consequently, there is no fixed point under this evaluation scheme. If we try to carry out the construction givm in theorem 4.1 using (u4) in place of the Kleene valuation scheme, we find that pUwEp is o-inconsistent, so that Em = A*, = Sent. But (",(Sent,Sent)) does not qualify as a partial model, since in a partial model the extension and the anti-extension have to be disjoint.

In discussing the many possible methods of evaluating truth in a partial model which we might use to get a fixed point, Kripke stresses one feature that the evaluation schemes have to have. He says,

Just about any scheme for handling truth-value gaps is usable. provided that the basic property of the monotonicity of u is preserved; that is, provided that extending the interpretation of Tr(x) never changes the truth value of any sentence of Y;,, but at most gives truth values to previously undefined cases. Given any such scheme, we can . . . construct the minimal fixed point 11975, p. 761 [Notation adjusted to fit

the notation used herc; a is the function taking (i3.A) to ({sentences true at (!)I,(E,A))}. {nonsentences) U {sentences false at (\'I.(E,A))}).]

We now see that monotonicity alone is not enough to guarantee that we will find a fixed point, for the evaluation scheme (a,) is monotone. We need some further condition to ensure that there is no stage in the inductivc construction at which the same sentence is declared both true and falsc. One such condition is the following: let us say that a partial model (?l,(E,A)) is uilequit~ocal with respect to a particular scheme for evaluating truth in partial models iff there is no sentence that is evaluated as both true and false under (?I,(E,A)). With respect to the Kleene 3-valued valuation scheme and with respect to ( a , ) , whenever E and A are disjoint sets of sentences, I!'l,(E.A)) will be unequivocal. (?I,(E,A)) will be unequivocal with respect to ( a z ) and (a,) iff E is a consistent set of sentences which does not entail any member of A . For (cr,) it is required that the union of E with the set of negations of members of A be consistent by !'I-logic. The requirement that we impose on an evaluation scheme, in addition to monotonicity, in order to guarantee that the inductive construction yield a fixed point is this:

(!)I ,(a,@)) is unequivocal. If (!)I , (E, A)) is unequivocal, so is (!)I ,u(E,A)). The union of a chain of unequivocal partial models is unequivocal.'

(a,) fails to meet the last condition. Alternatively, we could follow Woodruff 119841 in permitting truth-value

gluts. That way, monotonicity will indeed guarantee the existence of fixed points. But although we get a very rich and elegant system of fixed points when we allow truth-value gluts into the original system based on the Kleene logic, the only new fixed point we get by allowing truth-value gluts into the supervaluational systems (u , ) - (5 ) is the degenerate fixed point (Sent,Sent). For (q), (Scnt,Serzt) will be the only fixed point..

Although Kripke utilized the model-theoretic notion of definite truth, for our purposes here the proof-theoretic notion is more servieable. Hence, throughout this chapter, when I say 'definitely true', I shall mean "definitely true in the proof-theoretic sense."

Theorem 8.1 shows us that there exists a materially adequate theory of truth for the language of (!'I,T,), formulated entirely within the language of (91,r,). At first blush, i t would appear that we have succeeded in our search for a way to give a theory of truth for a language within that very language. A closer examination is less cheerful, since we have not given the theory of truth r, at all; we have only described the theory. The set of sentences T, is, according to recursion

More precisely, the last requirerrient is this: given a collection {(!)I,(E,,A,)): i s I) of unequivocal partial models, where for each i and j in I either E, C E, and A, C A, or else E, > E; and A , > A, the structure f!'[,(j;l,E,,j;jA,)) will be an unequivocal partial model.

theory, an extremely complicated object that we cannot hope to exhibit explicitly .' We can give a description of the theory I T , , formulated within an essentially richer metalanguage, by depicting a transfinite construction which produces T.. but we cannot present the theory itself. The essentially richer metalanguage is still with us.

What we would like to be able to do is explicitly to present the new axioms that we add to 1' to get the conservative extension that defines truth. To do this, we must obtain a theory that plays the role of I', by adding a recur.vive set of axioms to r. Only by doing so will we get a theory of truth that we can hope that human beings can understand and adopt. Seeing how to do this will be our next task.

THEOREM 8.3. Let "1 be an acceptable structure for a language 3'. let :f2

be a countable language which extends Y ' and which does not contain the predicate 'Tr', and let r be a set of sentences of 2; which is parameter- free inductively definable over 91. Then there is a recursive set A of sentences of a language YZ extending 2' such that (?I ,T U A) is a conservative extension of (!'L,T) and such that l' U A is materially adequate as a theory of truth for (!'I,T U A).

In fact, we shall see that, if 9' is built up from a finite vocabulary, then A will be finite. Here as elsewhere, we assume that we have fixed Godel codings for our languages so that the basic syntactic operations in the languages are represented by recursive operations on the codes.

We begin by proving a couple of lemmas that simplify our task:

DEFINITION. Let r ( x ) be an formula of a language Y that is partially interpreted by (!)l,Y) in such a way that, for any sentence 4 of we have

and

' If we assume that the language of \'I is built up from a finite vocabulary, that I' is inductively definable over \'I, and that there is at least one expansion of !'I to a model of r, we can give a precise computation: 1', will be a complete inductive set. We can see that it is inductively detinahle over !)I by formalizing the observation that I', is the smallest set of sentences that satisities the condition

( V x ) [ r E r, " ( \ E I- \ / x ha3 the form ~ r ( ' 4 ' ) for solllc sentence d~ In I', v x is an !'[-logical consequence of I',)]

To see that I', is a complete inductive set ( i . e . . that. for any inductive set R there will he a hyperelemental.y function f such that, for any .r. t s B iff,f(.r) s I-,). notice that E , l', and A ,

C {$: T$ E r,) U {nonsentences), and apply theorem 5.18.

Then r(.r) is said to be a muferially adequate truth preriirate for (\'I,Y).

The material adequacy condition stipulates that the specific formula 'Tr(x)' should - be a materially adequate truth predicate for our partially interpreted language. Our next lemma \bows that, if we have a materially adequate truth predicate for a language, we can easily rearrange things so that 'Tr(x)' becomes a materially adequate truth predicate.

LEMMA 8.4. Suppose that !)L is an acceptable structure and that (41,Y) is a partial interpretation of a countable language 2 in which the predicate 'Tr' does not occur, and suppose that the formula r (x ) is a materially adequate truth predicate for (!)l,Y). Then we can find a single sentence 5 of the language 2' got from Y by adjoining the new predicate 'Tr' such that (!'I,Y U (5)) is a conservative extension of (y1.Y) and such that 'Tr(x)' is a materially adequate truth predicate for (!'l,Y U (6)) .

PKOOF: It will not suffice to take 5 simply to be the definition (V.r)(Tr(x) ++ ~ ( x ) ) , for this would not get any sentences containing 'Tr' into the extension of 'Tr'. We need to translate 2' into 2 by substituting T for '7'r' not only where 'Tr' is used but also where 'Tr' is mentioned. Our plan is to find a function p translating formulas of 2;, into formulas of Y, , so that

p ( 4 ) = 4 for 4 an atomic formula of T,,; p(Tr(0) = ~ ( p ( D e n ( t ) ) ) ~ ( 4 v $1 = ~ ( 4 ) v P($) ~ ( 1 4 ) = ~ ~ ( 4 1 ~ ( ( 3 ~ 1 4 ) = (3v )p(4)

Then we can take our sentence 5 to be

We cannot regard this description of p as a straightforward definition, because of its impredicativity. The description shows us how to determine the translation of a compound formula once we know how to translate the simple formulas, but to know how to translate the simple formulas of the form Tr(t) we have to already know how to translate the compound formulas. We can straighten out this circularity by using the self-referential lemma.

The self-referential lemma gives us a formula 9 ( x , y ) so that

R, , t- (Vx)(Vy){Y(x,y) * [(x is an atomic formula of Y,,, & y = x) V ( X has the form Tr(r) & y = r (3y) (9 ( t , y ) &

7 ( ~ ) ) ' )

v ( 3 ~ ) ( 3 ? ) ( 3 ~ ) ( 3 v ) ( x = ( s v t ) & y = ( u v v )

Let 5 be the sentence

and let the function p: {formulas of 3;) + {formulas of 2,,) be given by

p ( 4 ) = the unique formula t,b such that 9"(r@1 ,rt,b1) is true in !"L

An induction on the complexity of sentences tells us that, for each sentence 4 of 2:,, we have

It follows that, if (91,Y) 1 p(+), then (\!I,T U {[I) 1 p ( 4 ) , and so (\!I,Y U { ( I ) 1 4 . Suppose, conversely, that (91.r U (5 ) ) 1 4 . Then, if we take an 3- logical derivation of 4 from Y U (5) and replace every occurrence of a formula of the form Tr(t) within the derivation by (3y ) (9 ( t , v ) & ~ ( y ) ) , we get an \)I- logical derivation of p ( 4 ) from Y . Hence,

(4I,Y U {tI) k 4 iff (Y( ,Y) k p(4)

To see that (41,Y U (5 ) ) is a conservative extension of (YI,Y), observe that. if 23 is an expansion of !)I to a model of Y , % can be further expanded to a model of Y U { t ) by taking the extension of 'Tr' to be {x: (3y ) (Y(x ,y ) & r ( y ) ) is true in 53).

For each sentence 4 of 2;, we have

(4LY u {tl) t 4 iff (Yl,Y) ~ ( 4 ) iff (41,Y) 1 r ( r p ( 4 ) l ) [because 7 is a materially adequate truth predicate

for (4I,Y) iff ('!r,Y) 1 (3y)(9(r+l ,y) & 70')) iff (41 , Y ) p ( ~ r ( r 4 1 ) ) iff (91,Y U { t ) ) 1 ~ r ( r 4 1 )

The proof that (Y1,Y U {t}) 1 1 4 iff (41,Y U {[}) 1 l ~ r ( r 4 1 ) is similar..

LEMMA 8.5. In proving theorem 8.3, there will be no loss of generality if we assume that 3' is built up from a finite collection of predicates and individual constants.

The fact that theorem 8.3 does not require that the language be built up from a finite vocabulary will be extremely important, if we intend to apply the theorem

to the study of an appropriately rcgimentcd fragment of English by treating the fragment like a first-order language. A first-order analysis sticks close to the surface, so that even though English is built up from a finite vocabulary, a first- order analysis must treat English as containing infinitely many predicates, for a first-order analysis must often treat compound English expressions as simple, unanalyzed primitives. PROOF: Lct US pretend that thc already have a proof of theorem 8.3 for the special case in which 2' is finitely generated, and let us see how this enables us to prove the general theorem. Our strategy is to extend 2' by adding a couple of master predicates that code up all the information expressed by all the symbols of the language, then to apply theorem 8.3 to the language containing just the master predicates.

We begin by forming a new languagc q!, from 2' by adjoining a new binary predicate 'U', a master predicate that is to encode all the information expressed by all the symbols in Y 1 . Form a list of the nonlogical symbols; we shall require the listing to be reasonably well-behaved, so that if we know the position of a symbol on the list, we can, in an effective fashion, determine the symbol's Giidel number. Produce a theory R as follows: R contains the sentence

(Vx)(Vy)(U(x,y) + ,x is a natural number)

If the nth symbol of 2' is an individual constant c, R contains

( v v ) ( ~ ( n , y ) - = C )

If the nth symbol of Y ' is an m-place predicate R , R contains

(VY)(U(;,Y) + y is an m-tuple) & (vvl ) . . . (Vv, , , ) (~( i ,<v, , . . . ,v,,,>) - R(vl, . . . , v,,,)).

If the nth symbol of 3' is an m-place function sign f , R contains

Notice that I1 is a recursive set of sentences of 3;. Let ?)I+ be the unique expansion of !)I to a model of R. Let 2! be the language

that we get from z:, by eliminating all the nonlogical symbols except for U and the symbols needed to define the arithmetical functions and to talk about finite sequences. Let !'I- be the reduction of ?I+ to 21. Then 9.1 , is the unique expansion of to a model of 0.

Now, form a new language Xi, by adding to 2', in addition to the new predicate U, a master predicate V that encodes all the information contained in thc symbols of 9' that are not symbols of 9 ' . Form a well-behaved enumeration of the nonlogical symbols in 3' that are do not occur in X1 and form a theory A describing the operation of V, as follows: A contains the sentence

(VX)(V~)(V(.Y.~) -+ n I S a natural number)

If the nth new symbol in 2' is an individual constant c , rZ contains

and so on. The construction of A is just like the construction of fl, and A, like 0, will be recursive.

Let 21 be the language whose nonlogical symbols are 'U', ' V ' , the arithmetical functions, and the terms uscd to describe finite sequences. For each sentence + of X', form an sentence $ of Y: as follows: for cach constant c, rewrite + so that c occurs only in the context of a formula c = 1%. If c is the nth symbol of Y', replace c = v by ~ ( i , v ) ; if c is the nth new symbol of 2' that is not in 2' , replace c = 1, by v ( ~ , v ) . Replace predicates and function signs in a similar fashion. For each sentence $I, we have

('?l,I1 U A) 1 ($I t. 4).

Notice that the function taking r+l to ri$l is definable in !'I, and so {ryl: r y l F

f} is parameter-free inductively definable over 91. Let r be {ryl: r y l F r} together with sentences that explicitly express the

insormation that is tacitly contained in the logical form of thesentences of yii'?. For example, if the nth symbol of Y ' is an individual constant, r will contain the sentence ( 3 x ) ( V y ) ( ~ ( n , ~ ) ++ y = x); if the nfh item on the list of symbols of 2" that are not in Y' is an individual constant, will contain ( 3 x ) ( ~ y ) ( ~ ( i , ~ ) - y = x). And so on, for each of the other symbols.

Now, r is parameter-frce inductively definable over !'I, and so r is parameter- free inductively definable over ?)(. Every set that is (parameter-free) inductively definable over ?)l will also be (parameter-free) inductively definable over 91- (though not necessarily vice versa), and so will be parameter-free inductively definable over ?I-.

Since 2' is built up from a finite vocabulary, we can apply theorem 8.3 to get a recursive set of sente~ces A such that (!'I , r U A) is a conservative extension of (9I , r ) and such that r U A is amaterially adequate theory of truth for (!'I , r U A). We want to take l- U R U A U A to be our theory of truth for (!)( ,r U f1 U A U A).

Let % be an expansion of !)I to a model of T. % can be expanded in a unique way to a model %,,,.of R U A. Let %_ be the reductic~n of v , , to 3'. % is an expansion of 9I- tp a model of and so, since (!)1-,1' U A) is a conservative extenson of (!'I , f ) , %- can be expanded to a model 6- of A . Let 6 be the unique expansion of 6- to a model of R U A. Then 6 is an expansion of % to a model of r U (1 U A U A. So (!)I,T U R U A U A) is a conservative extension of (91 ,r) .

Suppose that (91,r U R U A U A) t 4. Replacing each sentence JJ that appears in the '!(-logical derivation of i$ from r U ( 2 U A U A by $. we get an

- \!I -logical derivation of 4 from 1' U A. It follows that there is an \!L_-logical derivation of Tr(r&l) from 1- U A. Since every member of r and every atomic or negated atomic sentence true in \!I is derivable in $1-logic from r U R U A, c:)r.r u n u A u A) t ~ r ( r & l ) .

Similar arguments show that if (91.T U R U A U A) 1 ~ r ( r & ? ) then ($1.1' U R U A U A) 1 4 and that (?'I,1' U f2 U A U A) t 14 iff (!)I,T U R U iZ U

A) 1 i T r ( r & l ) . The function that takes $ to I$ can be defined in '?I, and we can take r(x) to

abbreviate the formula

getting

(U1,T u R u A U A) t 4 i f f (!)l,r u f2 U '4 U A) tr(r41) (UlJ u S1 u A U A) t 1 4 iff (91,r U f I U A u A) 1 17(r41)

Our conclusion follows by lemma 8.4.. We want to link up our current endeavor with Kripke's original investigations

using the Kleene 3-valued logic. In doing so, the following observation will be useful:

LEMMA 8.6. Where (E,,A,) is the least fixed point of the original Kripke construction applied to 2); ' as in theorem 4.1, and where KF is the Kripke- Feferman axiom system for Yl ' , we have, for each sentence 4 in E x , (71 ,KF) 1 Tr(r41) and (Y1,KF) 4.

Notice that, if we put 'k' in place of ' t ' , this lemma would be an immediate consequence of theorem 4.3. PROOF: We show by induction that, for each a , if 4 .s E,, then (\!I,KF) 1 Tr(r41). Assume that, for each P < a , if + F E,, then (?l,KF) t Tr(r$l). We show by induction on the complexity of 4 that, if 4 .s E,, then (PI,KF) /- Tr(r41). CASE I. 4 is an atomic or negated atomic sentence of Y,, . If 4 .s E,,, 4 is true in \!I, so, since KF t- ( 4 - Tr(r+l)), (\!I,KF) t ~ r ( r 4 1 ) . CASE 11. 4 has the form Tr(7). If is in Em, then, for some sentence $, 7" = r+1 and, for some p < a , $ F Ep, By inductive hypothesis, (!'I ,KF) 1 Tr(r$l). Since U1 r+l = T and KF +- (Tr(r) - Tr(rTr(r)l)), (!'L,KF) /- Tr(rTr(.r)l). CASE 111. 4 has the form ~ T r ( r ) . If 7" is not a sentence, then (91,KF) t i T r ( 7 ) because KF t- (Tr(r) + 7 is a sentence). If 7''' is a sentence, say [+I, then, if 4 is in E,, then there exists a /3 < a with i+ E ED By inductive hypothesis, (!)I,KF) t ~ r ( ~ i + l ) . Since KF + (Tr(1r) ++ T r ( r k ~ r ( r ) l ) , (H ,KF) 1 Tr(r41). CASE IV. 4 has the form (+ V 0). If 4 E Em, then either + E E, or 0 F E,. By inductive hypothesis (4 and 0 both being simpler than +), either (Y1,KF) 1 Tr(r$l) or (\!I ,KF) /- Tr(r01). Since KF + ( ~ r ( r $ v 01) - (Tr(r@) v Tr( rd ) ) , (?l,KF) /- Tr(r41).

CASE V. has the form i($ V 6). Similar argument, using the fact that KF + ( T ~ ( ~ T ( J I V 0)l) - (Tr( rk$l) & Tr(r761))). CASE VI. 4 has the form l l $ . Similar argument, using the fact that KF t-

( T r ( r i i $ l ) tt Tr(r$l)). CASE VII. 4 has the form (3v)$. Similar argument, using the fact that KF t-

(~r( r (3v)$(v) l ) ++ (3v)Tr(r$(f )I)). CASE VIII. 4 has the form i (3v)$(v) . If 4 t. E,,, then for each u in (\!I(, l $ ( n ) e E,, It follows by inductive hypothesis that, for each tr, (!'I,KF) t Tr(rl$(;)l). Using the !)l-rule, we see that (?I,KF) 1 (VV) Tr (r1$(3)1). Since KF t- (Tr( r i (3v)$(v) l ) ++ (Vv) Tr ( r ~ $ ( f ) l ) , (?I,KF) 1 Tr(4).

It follows by induction that, if F E x , then ('?(,KF) 1 Tr(r41). By theorem

4.3, KF U R, ., + (Tr(r47) -+ 4) ; hence, if 4 F E,. then (?( ,KF) 1 4.. PROOF OF THEOREM 8.3: In view of lemma 8.5, we assume the vocabulary of 2' consists of a finite store of predicates and individual constants. Let 3" and 2" be, respectively, the languages obtained from and from 2' by adjoining the newhnary predicate 'Kr'. In view of lemma 8.4, it will suffice to find a formula T(X) of 3" and a recursive set A of sentences of 2'' such that (91,r U A) is a conservative extension of (?I,T) and 7(x) is a materially adequate truth predicate for (!'I ,T U A).

Let us rewrite the Kripke-Feferman axioms for %I+, using 'Kr' in place of 'Tr'; for example, if R is an n-place predicate of Y' , one of our axioms will be

We continue to use 'KF' to refer to the rewritten axiom system and '(E,,A,)' to refer to the least fixed point of the Kripke construction over ?)I.

Theorem 4.1 shows that (?I,r U KF) is a conservative extension of (!)I,T), since any expansion of \!1 to a model of T can be further expanded to a model of KF by taking the extension of 'Kr' to be Ex. We know from lemma 8.6 that E , is contained in {sentences + of X;*: (!)I,T U KF) t $), which is included in {sentences +of 3;': (\!l,r U KF) 1 $). Unless T U KF is \!(-logically inconsistent (in which case our theorem is trivially true), (4 F : (91 ,r U KF) 1 4 ) and

{nonsentences} U {+ E %,:+: ( ? I , I U KF) /- 1 4 ) will be disjoint. By theorem 7.3, both set\ will be parameter-free inductive, and so we can apply theorem 5.18 with

A = C = ( 4 F 2;': (Yi,T U KF) 1 4 )

and

B = D = {nonsentences} U ( 4 F X,:': (!)l,r U KF) 1 1 4 )

' If the symbol 'Kr' should happen to already occur in the language, introduce some other new predicate instcad. Let me let this go without saying in similar situations in the future.

to find our parameter-free formula ~ ( x ) that is a materially adequate truth predicate for (!)(,r U KF).. REMARK 8.7. DENOTATION AND SATISFACTION. AS we have focused our attention on truth, other senlantic notions such as denotation and satisfaction have receded into the shadows. We can recover these other notions by simple definitions:

Alternatively, if we prefer to have 'Den' and 'Strt' as primitive, rather than defined, terms, we can redeploy the construction used in lemma 8.4.

Thcrc arc material adequacy conditions for a theory of denotation and satisfaction, analogous to our material adequacy condition for truth. For denotation, the requirements are:

(91-Y) /- D ~ ( ~ ( L v ) ~ ~ , c I ) iff (!)l,Y) /- (Vv)(+ ct v = (I); and (Yl,Y) b -

lDetz(i(iv)+l,a) iff (91,Y) f l (b 'v)(+ H = a )

For satisfaction, the requirements are

(Yl,Y) t ~ u t ( u . ~ $ ( x ) l ) iff (91,Y) t I,!J(~); and (9 .Y) /- l ~ c r t ( a , r$(x)I) iff ( Y I ,Y) /- T$(;)

If k' is materially adequate as a theory of truth for (!'I,Y) and if denotation and satisfaction are defined as indicated above, these material adequacy conditions will be satisfied..

In chapter 5, we developed a number of useful and elegant formal features of the minimal fixed point of the Kripke construction. We now want to show that these formal properties are inherited by a partially interpreted language that contains its own truth predicate.

THEOREM 8.8. Let !)I be an acceptable structure for a first-order language built up from a finite vocabulary, Let Y' be a countable language

extending 2' , and let Y be an !)I-logically consistent set of sentences of 2; which is inductively definable over ?I. Then 3; contains a materially adequate truth predicate for (9( ,Y) just in case, for any two disjoint sets A and B that are inductive over !I(, there is a formula p(x) in 2,; such that, for any a ,

a E A iff (!)I ,Y) 1 p(a) and a E B iff (!'I,Y) t ~ ~ ( 6 )

P R O ~ F : (3) Suppose that ~ ( x ) is a materially adequate truth predicate for (Y[,Y). Let 2'' be the language got from 2' by adding the new unary predicate 'Tr' . In thc proof of lemma 8.4, we saw how to construct a formula Y(x,y) of 9' so that

the extension of !T(.i.v) in !)I is a function p: {formulas of 3;') + {formulas of $1 such that

p(+) = 4, for 4 an atornic formula of 2,;

I p(Tr(t)) = (3y)(Y(t,v) & ~ ( y ) ) ; p(+ V $1 = ~ ( $ 1 V ~ ( $ 1 ; ~ ( 1 4 ) = ~ ( 4 ) ; and ~ ( ( 3 ~ 1 4 ) = ( ~ V ) P ( + ) .

Let

C = (5entences r+l of 2:' : (91 ,Y) p(+)) and I

D = {sentences rb;l of $': (YI,Y) t li)(+)} U {nonsentences).

By theoren 7.3, C and D are inductivc over 3, and since Y is %-logically consistent, C and D are disjoint.

We want to show that (where(E,,A,) is the least fixed point of the Kripke construction for 9' ' , as described in theorem 4.1) E, C_ C and A, D. Assume, as

inductive hypothesis, that, &)reach /? < o, E,, C and A, D A straightforward

induction on the complexity of + shows that, for each sentence 4 of Y". if 4 E Ea, then $ E C, and if 9 E Am, then 4 F D Hence, by transfinite induction, E.

C and A, D, and so we can apply theorem 5.18 to get a formula v(x) of IP" such that, for any a ,

LI E A iff rv(U)l E C and a B iff rv(ii)l E D.

Let p(x) be the formula (3y)(Y(rv(i)l,y) & r(y)). We have

a E A iff rv(a)i E C iff (91 ,Y) t p(v(z)) iff (91 ,Y) /- r ( rp (v (n ) ) l~ iff (\'I,Y) k (3y) (~( rv (a ) l ,y ) & ~ ( y ) ) iff (91,Y) t pG).

Similarly, a 8 B iff (Yl,Y) k l p ( Z ) .

((=) Set A = {i@l: (YI ,Y) t $1 B = {r$l: (\'I ,Y) t - 1 4 ) U {nonscntences}

Theorem 7.3 tells us that A and B are inductive over Yl. and, because Y is 91- logically conslatent, they are disjoint. Our formula p(x) such that, for any u, a t. A iff (YI ,Y) t p(;) and a E B iff (91.Y) f ~ ~ ( 2 ) will be a materially adequate truth predicate for (\!I,Y)..

DEFINITION. A set S C (911 is weakly definable within a partial interpretation ($I,Y) iff there is a formula g(x) such that S = (a: (%,Y) t u(o)}.

COROLLARY 8.9. Let 91 be an acceptable structure for a first-order language Y 1 built up from a finite vocabulary, let Y 2 be a countable language extending 2', and let Y be an H-logically consistent set of sentences of 2; which is inductively definable over !)I. Then, if Xf contains a materially adequate truth predicate for (\!I,Y), a subset of 1911 will be inductive over ?I iff it is weakly definable within ('?I,Y). It follows that the set of true sentences of L!',,,,,,, the o(?l)th language in the Tarski-Kripke hierarchy, is weakly definable within (t'1.Y). Also, the set of true sentences of the Kripke-definable fragment of Y,,,, is weakly definable within (?)I,Y).

PROOF: TO see that every inductive set is weakly definable, apply theorem 8.8 with B = 0. TO see that every weakly definable set is inductive, use theorem 7.3. For the last two statements, use theorems 5.22 and 5.25..

DEFINITION. A set S C )!)I) IS ~fronglr, cIq'inuhle withrn a partla1 Interpretation (91,Y) iff there 1s a formula U ( X ) such that S = {cr: (91,Y) 1 a(;)} = {a: (YI,Y) v TU(;)). COROLLARY 8.10. Let !)I be an acceptable structure for a first-order language 54' built up from a finite vocabulary, let 2' be a countable extension of T', and let Y be an 91-logically consistent set of sentences of 3; which is inductively definable over YI. Then, if X: contains a materially adequate truth predicate for (91,Y), a subset of !)I will be hyperelementary over '!I iff it is strongly definable within (91,Y). It follows that the extension of any formula that occurs within the Tarski-Kripke hierarchy will be strongly definable within ('!I,Y). Also, the extension of any formula that occurs in the Kripke-definable fragment of 93,w will be strongly definable within (:)I ,Y).

PROOF: To see that, if S is hyperelementary then S is strongly definable, apply theorem 8.8 with A = Sand B = ( % I - S. Conversely, if S is strongly definable, then S and 131 - S are both weakly definable, and so both inductive. For the last two statements use corollaries 5.24 and 5.26..

One way to measure the expressive power of an interpreted language is to see what sets of individuals are definable in that language. The Tarski hierarchy shows that, by that measure, we can always increase theexpressive power ofan interpreted first-order language that is able to describe its own syntax simply by adding a new predicate to the language and interpreting the new predicate by means of a (total) implicit definition. Once we have done so, we can increase the expressive power of the language still further by adding another new predicate, giving its meaning by another implicit definition. And so on. The process of increasing the expressive power of the language by implicitly defining new predicates never closes off. How- ever much we have added on to the original language, we can add more still, and we will get a language that is strictly richer in expressive power.

The corollary below shows that the situation is quite different when we introduce new predicates via purtiul implicit definitions. Starting with an acceptable structure "I for a language 2' . we can increase the expressive power of 2' by introducing new terms and giving a theory that partially specifies their meanings. Perhaps we can increase the expressive power further by introducing new terms and a stronger theory, but eventually the process comes to an end. Once we get to a partial interpretation that includes its own truth predicate. further extensions of the theory will not increase the expressive power of the language.

COROLLARY 8.11. Let \'I be an acceptable structure for a first-order language 3' built up from a finite vocabulary, let %' be a countable extension of Y1, and let Y be an inductively definable set of sentences of %,; such that 2,; contains a materially adequate truth predicate for (?I , Y ) Let 9' be a countable extension of g2, and let R > Y be an inductively definable, 91-logically consistent set of sentences of 3:. Then a subset of /:'I/ will be weakly definable (resp., strongly definable) within (?l,Y) iff it is weakly definable (resp., strongly definable) within (%,Ll). Moreover, %,: will contain a materially adequate truth predicate for ( % , a ) .

PROOF: The proof closely resembles the argument for theorem 8.8. Let r(x) be a materially adequate truth predicate for (9I.Y). We want to show that 3,; contains a materially adequate truth predicate for (\'I,R).

As in the proof of lemma 8.4 and again in the proof of theorem 8.8, let "P(x,y) be a formula of 2' whose extension in \!I is the graph of a function p that translates sentences of into sentences of X,; so that the translation of Tr(t) is (3y)(B(t,y) & ~ ( y ) ) . Let

c = (r41: ( % , a ) 1 (3y)(B(r+l,y) & ~(y) )} and D = {r@: (91,12) 1 (3y)(?P(r+l J) & ~ r ( y ) ) } U {nonsentences}

As in the proof of theorem 8.8, we see that C and D are disjoint inductive sets with C > E , and D > A,.

Take any two disjoint sets A and B that are inductive over ?I. By theorem 5 18, we can find a formula v(x) of so that. for any element a of (?I(,

a e A iff rv(a)l E C, and a E B iff rv(a)7 E D

Let

p(x) (3y)(Y(rv(;)l, Y) & d y ) )

We have, for any a in 1'?(1,

a E A iff (91,R) t p(a). and u E B iff (\'l.fZ) 1 y ( i )

It follows by theorem 8.8 that Y" contains a materially adequate truth predicate for (!'l , f l ) .

Corollary 8.9 tells us that S is weakly definable in (Y1.Y) iff S is inductive over !)I iffS is weakly definable in (!)1,i1). Corollary 8.9 tells us that S is strongly definable in (:'l,Y) iff S is hyperelementary over $1 iff S is strongly definable in (:)r ,a)..

In the model (w,O,S), the addition operation can be implicitly defined by a recursion. but it cannot be explicitly defined. In the model (w,O,S,+), the nlultiplication operation can be implicitly defined by a recursion, but it cannot bc explicitly defined. In the model (w.O.S,+;), on the other hand, any operation that can be implicitly defined by a recursion can be explicitly defined. Thus the model (w,O,S,+ ;) is closed under the process of introducing new operations by recursion. In an analogous sense, any partial interpretation that contains a materially adequate truth predicate will be closed under the operation of defining new properties by partial implicit definition^.^

Philosophers discussing truth have naturally concerned themselves with the theoretical use of the word 'true', the use of the notion of truth as the cornerstone of semantic theory. Kripke has emphasized that there is also an everyday use of the word 'true' as a practical vehicle for conveying information. To take an example, imagine that Xochitl, an old friend whose word I regard as one hundred perccnt reliable, tells me, "Everything Wendy says is true," and that Wendy, a jazz musician with whom I have had no previous dealings, tells me, "Mogulcorp will go bankrupt." On the basis of this information, I divest myself of my holdings in Mogulcorp, happy to have avoided financial ruin. In attesting to Wendy's veracity, Xochitl has given me valuable information. Without it, I would have had no reason to believe what Wendy said, and I would have held onto Mogulcorp until the hitter end. This valuable information could not have been conveyed without using the notion of truth or some other semantic or pragmatic notion. In the sentence ' "Mogulcorp will go bankrupt" is true', the notion of truth is superfluous; one might just as well have said, "Mogulcorp will go bankrupt." In the sentence 'Everything Wendy says is true', the notion of truth is not superfluous. Xochiil can make this assertion as a judgment about Wendy's character without knowing what, specifically, Wendy is going to say. In telling me what she has told me, Xochitl has given me the information that would have been conveyed by the infinite totality of sentences of the form:

If Wendy says, "Mogulcorp will go bankrupt," then Mogulcorp will go bankrupt.

More precisely. if (!)I,Y) contains a materially adcquate truth predicate. the only way extending Y can increase the expressive power of the language will bc if our new set of axioms is monstrously complicated. I f we allow arbitrarily cornplicate_d axiom systems, wc can introduce an arbitrary set S C /?I via the axioms {R(a) : a E S) U { l R ( a ) : a 4 S } .

My reasoning has proceeded as follows: from

Everything Wendy says is true.

and

Wendy says, "Mogulcorp will go bankrupt."

I conclude:

'Mogulcorp will go bankrupt' is true

From this I infer:

Mogulcorp will go bankrupt.

Thus, I am applying the following rule of inference:

(R I ) From rT41 is true? to infer r+1. Rule (R l ) and its dual principles,

(R2) From r41 to infer rr+l is truel; (R3) From rr+l is not true1 to infer r i+ l ; and (R4) From 'I& to infer rr+l is not truel.

are central to our usage of the notion of truth as a means for conveying information. The material adequacy condition justifies rules (Rl) through (R4) by ensur- ing that these rules will be definite-truth preserving. Indeed the fact that the material adequacy condition ensures the availability of rules (Rl) through (R4) is the principal basis for the claim that the theory of truth we are developing will satisfy the ordinary usage requirement.

To illustrate the usefulness of (R l ) through (R4), let us review example 6.10. Recall that, in this example, Xochitl says:

Not everything that Yolanda tells you will be true.

Yolanda says:

Wooden matches are insoluble. Nothing Zaida tells you will be true.

Zaida tells me:

The fish in Lake Anza swim backward to keep the water out of their eyes. In Ohio, the ratio of the circumference to the diameter of a circle is 4.0. The apartments on Elm Street have rats as big as raccoons.

And 1 conclude:

The apartments on Elm Street have rats as big as raccoons.

To reach the conclusion I have reached. rules (KI) thmugh (R4) are prcci\ely what are needed. On the basis o f Xochltl's warning together with rlly own observations, I know:

Either 'Wooden matches are insoluble' is not true or 'Nothing Zaida tells you will be true' is not true.

Now, I already know:

Wooden matches are insoluble.

It is alnply confirmed by numerous observations. By (R2) 1 conclude:

'Wooden matches arc insoluble' is true.

Thus, I determine that

'Nothing Zaida tells you will be true' is not true.

Knowing this, I use (R3) to obtain:

Something Zaida tells me is true.

Knowing this and observing Zaida's behavior, 1 obtain:

Either 'The fish in Lake Anza swim backward to keep the water out of their eyes' is true or 'In Ohio, the ratio of the circumference to the diameter of a circle is 4.0' is true or 'The apartments on Elm Street have rats as big as raccoons' is true.

Now, of course,

It is not the case that the fish in Lake Anza swim backward to keep the water out of their eyes.

and so 1 get, using (R4),

'The fish in Lake Anza swim backward to keep the water out of their eyes' is not true.

Similarly, I use (R4) to get

'In Ohio, the ratio of the circumference to the diameter of a circle is 4.0' is not true.

from

It is not the case that, in Ohio, the ratio of the circumference to the diameter of a circle is 4.0.

I now know that at least one of Zaida's statements is true and that the first two are untrue; so her third statement must be true:

'The apartments on Elm Street have rats as big as raccoons' is true.

BY (Rl ) at last I obtain:

The apartments on Elm Street have rats as big as raccoons.

A great many of our everyday, nontheoretical u\es of the notion of truth can be understood as applications of (RI) through (R4). There are, however. some further principles, not following from (Rl) through (K4), which enter into our ordinary reasoning about truth in useful ways and which do not engender paradoxes. Let me list some of them:

The set of true sentences is closed under first-order consequence. The set of true sentences is consistent by first-order logic. For every sentence, either it or its negation is true.

For illustration, let us look at example 6.1 1, in which Zaida makes a fourth statement:

The residents of Elm Street are constantly menaced by wildcats.

In this example, I use (R2), (R3), and (R4) to obtain, just as before:

Either 'The apartments on Elm Street have rats as big as raccoons' is true or 'The residents of Elm Street are constantly menaced by wildcats' is true.

Using the principle that the set of truths is closed under first-order consequence, I derive:

'Either the apartments on Elm Street have rats as big as raccoons or the residents of Elm Street are constantly menaced by wildcats' is true.

(Rl) gives me my conclusion:

Either the apartments on Elm Street have rats as big as raccoons or the residents of Elm Street are constantly menaced by wildcats.

As another illustration, let us turn back to example 6.12. By three applications of (R4), 1 know that (al) , (a2), and (a4) are all untrue. Application of the principle that the set of truths is first-order consistent tells me that at most one of (a3) and (a5) is true, and so I know that at most one of A's statements is true. By (R2) I conclude that (b4) is true, and by three more applications of (R2) I determine that (b l ) , (b2), and (b3) are all true. By (K2) I conclude that (a3) is true, and by (K4) I conclude that (a5) is untrue. EXAMPLE 8.12. AS a final example, suppose that I know for a certainty that Professor Moriarty is in either England or Wales and that I am reliably informed that exactly one of Colonel Moran's three statements is true. Colonel Moran's statements are:

Moriarty is in England. Moriarty is in Wales. Either Jules's statements are both true or Jim's statements are both true.

Jules says:

Moriarty is bald. The jewels are not in the millpond.

Jim says:

Moriarty is not bald. The jewels are not buried by the lighthouse.

From this information, I can conclude that the jewels are either in the millpond or buried by the lighthouse, by the following rcasoning:

Starting from

Moriarty is in either England or Wales.

(R2) gives me:

'Moriarty is in either England or Wales' is true

From the principles that the set of truths is consistent and that, for every sentence, either the sentence or its negation is true, we see that a disjunction will be true only if at least one of its disjuncts is true. Hence,

'Moriarty is in England' is true or 'Moriarty is in Wales' is true.

Since at most one of Moran's statements is true, I get:

'Either Jules's statements are both true or Jim's statements are both true' 1s untrue.

Using (R3), from this I get:

Jules' statements are not both true and Jim's statements are not both true.

Thus, we have:

If 'Moriarty is bald' is true, then 'The jewels are not in the millpond' is not true. If 'Moriarty is not bald' is true, then 'The jewels are not buried by the lighthouse' is not true.

Now, the principle of bivalence gives us:

Either 'Moriarty is bald' is true or 'Moriarty is not bald' is true.

Whence we derive

Either 'The jewels are not in the millpond' is not true or 'The jewels are not buried by the lighthouse' is not true.

Because the set of truths is closed under consequence, we have

If 'The jewels are not either in the millpond or buried by the lighthouse' is true, then 'The jewels are not in the millpond' is true and 'The jewels are not buried by the lighthouse' is not true.

Hence,

'The jewels are not either in the n~illpond or buried by the lighthouse' is not true.

(R3) enables us to conclude:

The jewels are either in the millpond or buried by the lighthouse..

The principle of bivalence-"For every sentence, either it or its negation is true" or, more succinctly, "Every sentence is either true or falsem-is not, strictly speaking, forced upon us by our conception of truth; that is, it is not entailed by the material adequacy condition. Nevertheless, we are inclined to regard it as requisite for a satisfactory theory of truth, on account of its usefulness in fulfilling the ordinary usage requirement. One expects some reluctance in accepting the principle, on account of the feeling that the paradoxical sentences are neither true nor false; indeed, it is sometimes thought that the moral of the paradoxes is that our initial inclination toward bivalence is misguided. This attitude, I want to argue, comes from a failure to recognize the full import of the strengthened liar. If bivalence really were the source of the problem posed by the paradoxes, then once we denied bivalence the problem ought to disappear. But the problem does not disappear; denying bivalence does not solve the problem or even bring us closer to a solution. Bivalence is a useful principle, an intuitively obvious principle, and a principle that causes no apparent harm. I propose that we keep it.

We can use brute force to modify the construction of theorem 8.1 to incorporate the principle that the set of truths is a maximal consistent set of sentences. Starting with a partial interpretation (!'l,r) with ?L acceptable, we define T,,, for each ordinal a , to be

I- U {the formalization of 'The extension of "Tr" is a maximal consistent set of sentences'} U ( ~ r ( r 4 7 ) : (?I ,oUur,) 1 $1

and we let r, = gJKTe. The proof that (?I ,K) is a conservative extension of (?I ,I-) for which r, is a materially adequate theory of truth is essentially unchanged from the proof of theorem 8.1.

We shall now see that we can likewise modify theorem 8.3 to incorporate the principle that the set of truths is a maximal consistent set.

THEOREM 8.13. Let ?I be an acceptable structure for a language 2', let 3' be a countable language extending Y ' that does not contain the symbol

'Tr ' . and let r be a set of sentences of Y' which is parameter-free inductively definable over 91. Then there is a recursive set A of sentences of a countable language Y 3 extending :A2 such that (?I ,T U A ) is a conservative extension of (?1.1'), such that I' U A is materially adcquate as a theory of truth for (!)I ,T U A) , and such that A contains the formalized version of the sentence 'The extension of "Tr" is a lnaxirnal consistent set of. sentences'. If 22 is built from a finite vocabulary, A will be finite.

PROOF: WC can use the same argument we used in lemma 8.5 to show that there is no loss in assuming that 3' is built up from a finite vocabulary of predicates and individual constants. Also, we may assume that there is at least one expansion of ?[ to a model of 1'. since otherwise we could just take A to be ('-10 = 0').

Let me present the proof by first sketching a simple strategy that does not quite work. then saying how the strategy can be improved. According to our simple strategy, A consists of three parts. First there is an inductive definition of the set of definite truths:

(Vx)(Drf(.r) - [.Y is an atomic or negated atomic sentence true in ?I V x is an element of r U A V x is an axiom of first-order logic V (3? . ) (Def(~) & Dell! --, XI) v (3y) (3v ) (x = (Vv).Y & (Vz)DLif(). "I?))])

The second part is the fact about the set of truths that we are aiming to include:

The extension of 'Tr' is a maximal consistent set of sentences.

The third part gives the connection between definite truth and truth:

(Vx)(Def(x) -+ Tr(x))

Once we have A , we intend to prove

(?t .r U A) t 4 iff ( ? l , r u A ) t ~ t f l r + l ) iff (4r.r u A) 1 ~ r ( r 4 1 )

and so

c!ll.r u A) /-- -I$ iff (?i,r u A ) t ~ r ( k 4 7 ) iff ( " , r u A) t ~ T r ( r 4 7 )

There are two difficulties with this strategy. The first is that the definition of A is circular. But we can easily use the self-referential lemma to straighten out the circularity.

The second difficulty is more serious. Definite truth in (91 ,r U A ) is the same as derivability from r U A in 3-logic. As we shall see in the next chapter, many of the familiar formal properties of derivability in first-order logic are also properties of derivability in 91-logic. In particular, there is an 91-logic analogue of Godel's second incompleteness theorem, and so an %-logically consistent

theory cannot prove its own 8-logical consistency. But U A , as we have described it, does prove its own 41-logical consistency, since it proves that the

set of definite truths is closed under 91-logical consequence, that the set of definite truths is included within the set of truths, and that the set of truths is consistent. So r U A is \'!-logically inconsistent.

To overcome this difficulty, we pay attention to the stages in which the set of dcfinite truths is constructed. Let our inductive definition of the set of definite truths of (?I,T U A) be

and let F",e the set of definite truths constructed by the olth stage, as defined in lemma 5.2. Let the consisrrnt part of the set o f definite truths be U{T',: cu is an ordinal and T i is first-order consistent). Our theory A will tell us that the set of truths is a maximal consistent set that contains the consistent part of the set of dcfinite truths. From the fact that the consistent part of the sct of definite truths is contained within a maximal consistent set, we cannot conclude that the set of definite truths is consistent, so we do not offend the second incompleteness theorem. We can give a modcl-theoretic argument that the consistent part of the set of definite truths is identical to the set of definite truths, but we cannot reproduce this argument within the formal system.

Getting down to specifics, for each open sentence q ( x ) , let A ( r T l ) be the following theory. expressed in the language 3' that is got from 9' by adding two new unary predicated predicates 'Kr' and 'Tr':

I KF, with 'Kr' in place of ' T r ' , for the language 9" got by adjoining 'Kr' to 2'

U {'The extension of "Tr" is a maximal consistent set of sentences of 2;') U { '(V sentence x of g i ) l [ ~ r ( r ~ ( ; ) l ) gi (V finite, inconsistent set R ) ( 3 r

E ~ ) ~ r ( r ~ ( r ~ ( i ) l , r ~ ( F ) l ) l ) ] -+ Tr(x)] '} . I I Here 'L(x,y)' is the formula constructed in lemma 5.12 to describe the pre-well-

ordering of E,x. The last clause of A(rq l ) tells us, for any sentence $, if there exists an ordinal

cu such that $ is inside (8: r l (rd l ) E Em} and such that any finite, inconsistent set contains some elements that are outside (8: r l (rv l ) e E,,}, then $ is in the extension of 'Tr ' . Thus, it tells us that the consistent part of {$: r l (r$ l ) e Ex)-that is, the set of all sentences $ such that, for some ordinal a , ~ ( r $ l ) is in E. and { B : 11(r81) E Em} is consistent-is contained within the extension of 'Tr ' . If {$: ?(r$1) E Ex) should happen to be consistent, then the effect of the clause will be to require that {+: ? ( r @ ) E E_} be included in the extension of 'Tr ' . But even if {$: T ( ~ J I ' ) E E x } is inconsistent, we can still consistently stipulate that the extension of 'Tr' is to be a maximal consistent set that includes the consistent part of {$: ~ ( r @ ) & E,).

For each 7 , ()!I,r U A(Trll)) will be a conservative extension of (v). To see this, let % be an expansion of \!X to a model of T. 'Zi can be further expanded

to a model 6 of r U A(r71) by setting 'Kr" equal to E, and taklng 'Tr" to bc - a maximal consistent set of sentences which includes the consistent part of {$:

,(r$I) E E x } . Let

A = {<r,l,r41>: (?r.r u ~ ( r , ? ) ) 1 4 1 B = {<rql ,r47>: (" ,r u ~ ( r , l ) ) t C = E, and D = A ,

Because we know that, for each 7. r U A(rvl) is consistent by "1-logic, we know that A and B are disjoint, so we can apply theorem 5.18 to get a formula V(X.?) such that

< r7)1, rb1> E A iff rv(rT1 ,r41)1 F C and <rT1,r+1 > E B iff rv(rrll. 11 E L)

Use the self-referential lemma to find a formula t(x) so that t(.x) is gap- equivalent (as defined in chapter 5) to v(r[l,x). Thus, for any $,

(%,I- U A(r[l)) /- $ iff r[(rJ,1)1 E E x , and (?)l ,r u A(rg1)) l$ iff r((r$7)1 t. A ,

We already know that (!)I,T U A(rt1)) is a conservative extension of (Y1,T). We want to convince ourselves that 1' U A(r[l) is a materially adequate theory of truth for (!'I,r U A(rt1)).

Suppose that (t'1.T U A(rt1)) 1 4 . Then l[(r$l)l E Ex. Now {$: r((r$l)l E E,.} = {$: (\!l,r U A(rt1)) t $1, which is consistent, and so if R is a finite, inconsistent set, R E {JJ: rt(r$l)l E E,}. Hence, there is a 0 in R such that r&rOl)l 4 E,. and so r~(1t(r41)1,r[(r01)1)1 s E,. Since whenever rX1 is in E,, (!'I,KF) t Kr(rXl), we havc

(?I ,KF) 1 ~ r ( r t ( r 4 1 ) l ) & (V finite, inconsistent R)(3 r F R) K r ( r ~ ( r t ( I 4 1 ) 1 , r[(P)l)l).

Hence,

(?l,T U A(r[I)) t ~ r ( r 4 1 ) .

For the converse, let us suppose that (!)1 ,I' U A(r[l)) 4. Let S be a maximal consistent set containing {$: (?)l,r U A(r[l)) t $ ) U { ~ 4 } . Let '?i be an expansion of !)I to a model of T, and further expand % to a model 6 of 3' by letting 'Kr'" = E, and 'Tr" = S. We want to see that Ci is a model of T U A(rt1). The only nontrivial part of showing this is to show that

(V sentence x)[[Kr(r[(i)l) & (V finite, inconsistent R ) (3 r F ~ ) ~ r ( r L ( I t ( , ? ) l ,r[(i)l)l)] -+ Tr(.r)]

is true in 6. To show this, it will certainly be enough to show that

(V sentence x ) [~r ( r [ ( i ) l ) -+ Tr(x)l

is true in 6. Suppose that Kr(r<(rX1)l) is true in 6. Then

x E {$: Kr(rt(r$l)l) is true in 6) = {$: r<(r$l)l e ' ~ r " } = {$: r<(r$l)l & E%} = {$: (\!t,r u a(ry1)) t- $1 L s = T ~ '

So ~ r ( r x 1 ) is true in 6. We now know that 6 is an expansion of :'I to a model of r U A(r{l). Since

Tr(r41) is not true in 6, it follows that ( % , r U ~ ( r f l ) ) Tr(r41) and hence that

We now know that, for any 4 ,

('1.r u A(r{l)) 1 4 iff (91,r u h(r(7)) 1 Tr(r41)

We also know that, because A(rt1) requires the extension of 'Tr' to be a maximal consistent set of sentences,

(?IJ U h(rt1)) t (V sentence x)(Tr(lx) * lTr(x))

Hence,

(%,r u a(rt1)) t 1 4 iff (9l.r u ~ ( r < l , ) 1 T r ( h $ l ) iff ('21,r U A(rt1)) t l T r ( r 4 1 ) ~

The construction given in theorem 8.13 fulfills the material adequacy condition. It is difficult to say with assurance that it also fulfills the ordinary usage requirement, because the requirement is so vague, but certainly it goes a very long way toward fulfilling it.

In meeting our first two requirements on a satisfactory theory of truth, the theory we have developed so f i r does very well indeed. However, it meets those requirements no better than the Kripke construction with supervaluation scheme (v,). What is new and interesting about the theory being developed here is that a theory of truth which satisfies the first two requirements is being explicitly presented without going outside the object language. Previous attempts to satisfy the first two requirements have not given us theories of truth; they have only given us descriptions of theories of truth. The account given here raises, for the first time, the prospect of also fulfilling the integrity of the language requirement.

The integrity of the language requirement demands that we give the semantics of our language within the language itself. Although our work so far arouses the

hope of fulfilling thiv requirement, it does not yet fulfill it. The problem is that we now have two fundamental semantical notions, truth and definite truth, and so far we have only given a theory of truth.

To give a theory of definite truth is remarkably easy. We simply give the inductive definition of the set of definite truths. Assuming that we have a way of describing the set Y and the atomic truths of 91, we get the following natural description of the definite truths of (9l , Y ) :

V x is an atomic or negated atomic sentence true in '!I V x is an axiom of logic V (3y) (Def ly) & DeAy x ) ) V ( 3 y ) ( S v ) ( x = ( v v ) y & (Vz)Drfly v / ; ) )]]

There are other natural notions of definite truth, notably the model-theoretic notion, but if we restrict our attention to the notion of definite truth as derivability from Y in ?'I-logic, this is the characterization of the definite truths which stands out as particularly simple, natural, and informative. We are drawn to this characterization by clear and distinct intuitions, and, moreover, we shall see in the next chapter that, for (!'I1 countable, we are driven to it by modal logical considerations.

We now have a theory of truth and a theory of definite truth. Our aim is to put the two together. Our material adequacy condition is expressed from an external vantage point; to express it, we require an essentially richer metalanguage. But once we have the notion of definite truth expressed within the language, we can express the material adequacy condition within the object language. The condition that is expressed within the metalanguage as

for each 4, (? i ,Y) t 4 iff (YI,Y) 1 ~ r ( r 4 1 ) I is expressed within the object language as

The other part of the material adequacy condition, I for each 4 , ('?1 , Y ) 1 1 4 iff (!'I ,Y) t iTr(rc$l ) ,

is expressed in the object language as I

Thus, if we want our theory to give a satisfactory account of truth, of definite truth, and of the connection between definite truth and truth, we shall require the following:

STRONG ADEQ~JACY CONDITION. The theory Y is strongly adequate both as a theory of truth and as a theory of definite truth for (!I , Y ) just in case the following conditions are met:

There is a formula v(x) that weakly defines Y in (\!(,Y). There is a formula ar (x ) that strongly defines the set of atomic and

negated atomic truths of \'I within (\!I,Y). The following three sentences are definitely true in ( ! ( ,Y ) :

(Vx)[Def(x) - {x is a sentence & [v(x) v 4 x 1 v x is an axiom of logic V (3y) (Def ly) & Dclf(y 7, x ) ) v ( 3 y ) ( 3 v ) ( x = (Yv)y & (Vz)Def(y "/i))l)J

(V sentence x)(Def(x) - Def(f(TTr(2)l)) (V sentence x ) ( D e f ( l x ) ++ ~ e f ( r l T r ( ; ) l ) )

Finally, for each sentence b, we have

( W , Y ) ~ e f ( r 4 1 ) iff (9I ,Y) t 4

This is a "strong" adequacy condition rather than a "material" adequacy condition, because it does not require merely that the theory pick out the set of definite truths; it requires that it pick them out the right way, via the inductive definition. Also, unlike the material adequacy condition, which merely requires that the theory pick out the right extension for 'Tr' , the strong adequacy condition requires a positive account of the connection between definite truth and truth.

The last clause,

('?I,Y) 1 ~ e f ( r 4 1 ) iff (91,Y) 1 4

is what links our strong adequacy condition for a theory of truth and definite truth with our material adequacy condition for a theory of truth. The right-to-left direction of the clause is redundant. One of the other clauses tells us that the biconditional that inductively defines definite truth is definitely true in ('21,Y), and from this we can derive that, if (!'I,Y) t 4 , then (!'l,Y) t ~ef l f (T+l) , by induction on the lengths of proofs in 91-logic. The left-to-right direction of the last clause does not follow from the other clause^,^ and it yields the following result:

"0 see this (assuming that the set of atomic and negated atomic tnrths of '?I can be defined in ?I), just take Y to be {'(V.r)(Def(x) * x is a sentence'}. Then Y will satisfy all parts of the strong adequacy condition for (?l,Y) except for the left-to-right direction of the last clause.

PROPOSITION 8.14. The strong adequacy condition entails the material adequacy condition.

PROOF: Because we have

we get

(YI,Y) t Def(r41) iff (91 ,Y) t Def( f~r ( r41) l )

Hence,

(Yl,Y) t 4 iff (41,Y) 1 ~ e f ( r 4 1 ) iff (YI,Y) /-- Def(rTr(r41)l) iff (41,Y) 1 Tr(r+1)

Similarly, (91,Y) /- 1 4 iff ('%,I") t i ~ r ( r 4 1 ) . Proposition 8.14 shows that, if the strong adequacy condition is met, then we

have a materially adequate characterization of the set of truths. We might hope to also get a materially adequate characterization of the set of definite truths, that is, a definition that meets the conditions:

(!'r,Y) Dej(r41) iff (!'I,Y) t 4 (%,Y) f- iDej(r41) iff (!'l,Y) (b

Complexity considerations show that this is too much to hope for. Whereas the first of the two conditions will be met whenever the strong adequacy requirement is fulfilled, the secondcondition will not be. Assume that !'I is an acceptable structure for a language built from a finite vocabulary, that Y is inductive over !'I, and that Y is !'I-logically consistent. Then, if both conditions were met, the set of definite truths would be strongly definable over (!'I,Y), and so, according to corollary 8.10, it would be hyperelementary over ?I. But this is impossible, since we know from theorem 8.8 together with proposition 8.14 that the set ofdefinite truths is acomplete inductive set, and so, according to corollary 5.1 1 , not hyperelementary.

Our aim is to show that the strong adequacy requirement can be met by showing that, for given (\)I,T), we can find a recursive set A so that (91,T U A) is a conservative extension of (?)L,T) and so that T U A is strongly adequate as a theory of truth and definite truth for (!'I,T U A). Let us begin by trying out the first thing that comes to mind. This will not work, but our failed attempt will show us why a more subtle approach is needed.

The first thing that comes to mind is simply to have A contain precisely those sentences that the strong adequacy condition requires A to contain. Assuming we already have our formula (YT(X) and a formula y(x) weakly defining T, we take A to consist of the following three sentences:

I (V~)[Def(x) ++ {x 1s a sentence & I y(x)

I

I v 6(x) I V "~(1)

I V x 1s an axlom of logic V (gy)(Def(y) & Def(y -r, x))

I V (3y)(3v)(x = (Vv)y & (Vz)Def(y vl;))l~l (V sentence x)(Def(,x) - Def(rTr(;)l)) (V sentence x)(Def(ix) - Dej (T i~ r ( i ) l ) )

where 6(x) is a formula, constructed by means of the self-referential lemma, that strongly represents A.

Natural though it may be, this theory does not do the job. It does not satisfy the strong adequacy condition or even the material adequacy condition. To see what the problem is, let % be an expansion of !'I to a model of T, and further expand !i? to a model 6 by letting '~ef' = Sent, 'Tr" = 0. is a model of

U A even though 'Tr(r0 = 01)' is false in 6. So while (%,I' U A) t 0 = 0, (91,I' U A) 1 Tr(r0 = 01), hence (91J U A) ~ r ( r 0 = 01).

The trouble is that the inductive definition of definite truth has a degenerate fixed point Def = Sent at which the entire construction collapses. This contrasts with the inductive definition of the Kripke fixed point, which has no trivial fixed points, as we know from theorem 5.19. Thus, if we build a Kripke fixed point into our construction, we can make sure that the construction will not collapse.

THEOREM 8.15. Let !'I be an acceptable structure for a language Z', let 2' be a countable language extending 2' which does not contain either of the predicates 'Tr' or 'Def, and let l- be a set of sentences of 3; which is parameter-free inductively definable over W . Then there is a recursive set A of sentences of a countable language 2' extending 2' such that (%,T U A) is a conservative extension of (41,I') and such that r U A is strongly adequate as a theory of truth and definite truth for (!'I,T U A). Moreover, we can arrange for A to contain the formalization of the sentence 'The extension of "Tr" is a maximal consistent set of sentences'.

As before, if 22 is built from a finite vocabulary, A will be finite. PROOF: These proofs are beginning to look like Rube Goldberg inventions. We want to take the axiom system from theorem 8.12 and add on new axioms describing the notion of definite truth and the connection between definite truth and truth. Once again, the construction from lemma 8.5 permits us to assume that 2' is built from a finite store of predicates and individual constants. Also, we may again assume that there is at least one expansion of 3 to a model of r , since otherwise our problem would be trivial.

Y" will be the language obtained from Y 2 by adjoining three new unary predicates, ' D e f , 'Tr ' , and 'Kr'.

Since 3' is finitely generated, we may take a ~ ( x ) to consist of an appropriate long finite disjunction of formulas of the form

- ( 3 v , ) . . . (3v,,,)(x = r @ G , . . . , & ~ ( I J , . . . . , v,,J). where 8 is an atomic or negated atomic formula of 2'.

Since r is parameter-free inductive over Yl, theorem 5.10 tells us that there is a formula T(x) of x3 such that I' = {x: r-j.(i)l F E,}, where E, is the extension of 'Kr' at the least fixed point of the Kripke construction (with 'Kr' in place of 'Tr ') . Take y(x) to be K r ( r ~ ( k ) l ) .

For each open sentence q(x ) , let a(rq1) be a sentence, constructed by means of the self-referential lemma, that is R, ..-provably equivalent to

(Vx)[Def(x) - { x is a sentence &

[Y(x) v a d x ) V x is an axiom of KF (with 'Kr' in place of 'Tr ' ) v x = r ( ~ ( r q 1 ) l V x = T ( v sentence x)(Def(x) ++ Def(rTr(i)l)) l V x = r(V sentence x ) (De f ( l x ) ++ ~ e f ( r ~ ~ r ( 2 ) l ) ) l V x = the formalization of 'The extension of "Tr" is a

maximal consistent set of sentences' V x = r ( V x ) [ [ ~ r ( r r l ( i ) l ) & (V finite, inconsistent set R)

(3 r .z ~)Kr(r~(rq(i)l,rq(F)l)l)] -+ Tr(x)]l V x is an axiom of logic v (3y)(DeS(y) & &f(y 7, x ) ) V ( 3 ~ ) ( 3 v ) ( x = (Yv)y & (Vz)Def(y vlt))l)l

Let A(rq1) be the following theory:

KF (with 'Kr' in place of 'Tr ' ) d 7 7 1 ) . (V sentence x)(Def(x) ++ ~ e f ( r T r ( i ) l ) ) (V sentence x ) ( D e f ( l x ) ++ Def ( r iTr (k ) l ) ) The extension of 'Tr' is a maximal consistent set of sentences. ( V x ) [ [ ~ r ( r q ( i ) l ) & (V finite, inconsistent set R ) ( 3 r E R )

Kr(rL(rq ($1 , rq(F)l)l)] + Tr(x)]

Then rr(rq1) is R, ,-provably equivalent to the inductive definition of definite truth in (YI,T U A(rq1)).

To see that, for each q , (91,r U A(rrll)) is a conservative extension of (9l ,T), let 23 be an expansion of 91 to a model of T , and further expand % to a model (I of I' U ~ ( r q l ) by the following stipulations:

'Deft = Sent 'Tr" is a maximal consistent extension of the consistent part of {$: q ( r $ l )

I Just as in the proof of theorem 8.13, we use theorem 5.18 to find a formula

((x) SO that, for any $,

(!)l,T U A(r(1)) $ iff rt(r$l)l .z E x , and (Yl,r u h(r(1)) t i$ iff r((r$l)l .z A-

We want to see that r U A(r(1) is strongly adequate as a theory of truth and definite truth for (M,T U A(rt1)) . To do this, what we need to show is that, if (?I,r U A(r57)) t ~ < f ( r + l ) , then (Y1,T U A(r51)) +. The other clauses of the strong adequacy condition are evident by inspection.

Suppose that (Y1,T U A(rt1)) 4. Let 23 be an expansion of !)I to a model

of r, and define a further expansion (I of % as follows:

'Kr'G = E, I 'Def' = {r$ l : (91 ,r U h(r51)) t $1

'Tr" is a maximal consistent set that includes {r$1: (?)I,T U ~ ( r t l ) ) /-- $1 U (741

We want to see that is a model of T U A(r(1). Since 'Def(r+l)' is not true in 6. this will tell us that (Y1,r U A(re1-i)) Def(r+l), and so (Y1,T U h ( r t 1 ) )

~ e f ( r 4 1 ) . That Cr is a model of T U KF U (o(r(7) U {'The extension of "Tr" is a

maximal consistent set of sentences') is obvious. To see that

(t lx)[[Kr(rt(;)l) & (V finite, inconsistent set R)(3 r E R ) Kr(rL(r[(k)l ,r((F)l)l)l + TrWl

is true in 6, it will certainly be enough to show that

is true in 6. Take any X . If 'Kr(r[(rXl)1) ' is true in 6, then rt(rxl)l is in 'Kr" = E,. Hence,

So 'Tr(rX1)' is true in 6. It remains to show that

are true in 6. Looking back at the proof of theorem 8.13, we see that the proofs that, for any $,

(!)i,r u A(r(1)) t $ iff (?lJ U ~ ( r t l ) ) t ~ r ( r $ I ) , and (?[J U A(r[T)) i+ iff (?lJ U A(r(1)) t iT r ( r$ l )

are essentially undisturbed by the additions we have made to ~ ( r t l ) . The only difference is that now there is an additional new predicate in 2' to take account of, namely, 'Def. The way we see that, if (Y1,T U A(rt1)) $, then (?l,r U ~(rg)) f ~ r ( r $ l ) is to take an expansion of ?I to a model of T, then expand it further to a model of A(rt1) in which ~ r ( r $ l ) is false by letting the extension of 'Kr' be E,, the extension of 'Tr' be a maximal consistent set that includes {8: (?I,T U A(r(l)) t 0 ) U ( ~ $ 1 , and the extension of 'Dej' be the set of all sentences.

For any sentence $, we have

'~ef ( r$ l ) ' is true in Cr iff r41 8 'Dej" iff (?IJ u ~ ( r g l ) ) t- $ iff (81,r U ~ ( r g l ) ) t ~ r ( r $ l ) iff rTr(r$l)l F 'D.efc iff '~ef(rTr(r$1)1)' is true in 6.

Similarly,

'Def(ri$1)' is true in Cr iff (?X,r u ~ ( r t l ) ) 114 iff (?l,T U A([,$])) 1 i ~ r ( r $ l ) iff 'Def(ri~r(r+1)1)' is true in 6..

ILLUSTRATION 8.16. EXAMPLE 6.13 REVISITED. Let us review example 6.13, in which we have an acceptable structure 91 with 'OR'" = the set of ordinals less than K and '<'"' = the less-than relation on the ordinals < K , and in which we constructed a formula u(x) such that, for each ordinal a, cr(z) says that, for each ordinal P < a , is true. Although this is scarcely an example from everyday life, our everyday usage of 'true' gives us intuitions about the example that are unmistakable: each of the cr(G)s is true. We want to see how well these intuitions are captured by the account we are developing here.

Indeed, it is easy to see that, if Y is a materially adequate theory of truth for (91,Y), each of the o(E)s will be definitely true in (91,Y). Assume, as inductive hypothesis, that for each f i < a, (81 ,Y) t ~ ( p ) . Now, take any element a of )?I).

- - If u is not an ordinal < a, then i u < a , and so

(ar ,Y) 1 Z < (Y -+ Tr( r ~ ( Z ) l )

If a is an ordinal < a , then, by inductive hypothesis,

i (:)r .Y) (T(u) ,

and so, because Y is a materially adequate theory of truth for (\'I,Y),

(?I ,Y) t Tr(ru(n)l)

hence, - -

(?I,Y) t a < a -+ ~r( rc+(a) l )

It follows by the ?(-rule that

and so

(?( ,Y) t v(a!)

as expected. Although this bit of reasoning is intuitively satisfying, it is methodologically

unsatisfying, since the induction has to be carried out in a richer metalanguage. What I now want to show is that, if Y is strongly adequate as a theory of truth and definite truth for (!)I,Y), then, under suitable circumstances, it will be possible to cany out the argument within the language of (?l,Y), obtaining a proof of '(Vx)(OR(x) + u(x))' without resorting to a richer metalanguage.

In order to carry out the argument, we have to have a theory that we recognize as a strongly adequate theory of truth and definite truth. This means that we must accept the biconditional that forms the inductive definition of 'Def , as well as the sentences '(Vx)(Def(x) ~ef(rTr(2)l)) ' and '(Vx)(Def(lx) - ~ ~ . f ( r l ~ r ( i ) 1 ) ) ' , and we must adopt the definite-truth preserving rule of inference

From ~ e f ( r + l ) one may infer +, and vice versa.

The intuitive proof of '(Vx)(OR(x) -+ ~ ( x ) ) ' proceeds by transfinite induction. In order to formalize proofs by transfinite induction within the language of (aI,Y), we need to accept the universal closures of instances of the following transfinite induction axiom schema:

(Vx)[(OR(x) & (Vy)(y < x -+ 8(y))) -+ 8(x)I -+ (Vx)(OR(x) + B ( x ) ) .

Certainly we shall accept this schema if we acknowledge that the things that satisfy 'OR' are well ordered by '<'. Assuming we accept the schema, we can carry out transfinite inductions within the language of (?I,Y) in the ordinary way.

Assuming that we have a theory of truth and definite truth which we recognize as strongly adequate, that we accept the transfinite induction axiom schema, that we accept the universal closures of instances of the schema

with 4 an atomic or negated atomic formula of the language of $1, and that we are aware of rudimentary facts of syntax, we can reason as follows:

Take a such that OR(a) , and assume, as inductive hypothesis, that for each ! P < a , D e f ( r w ( ~ ) ~ ) . Now, take any individual a . If it is not the case that

a < a , then, because we have

and

we get

D e f ( r i a < G I )

and so

~ e f ( r ( a < a -+ ~ r ( r w ( a ) l ) l )

If, on the other hand, a < a, then, by inductive hypothesis,

Def(ru(;)l)

Because of the principle

(Vx)(Def(x) - Def(rTr ( i ) ? )

we get

Def( rTr(rcr(a)l)l)

and so, again,

~ e f ( r ( a < a -+ Tr(ru(; ) l ) l )

Thus, we have

(Vy>Def(rj < -+ ~ r ( r u ( y ' ) l ) l )

We also have the general principle

which we derive from the biconditional that inductively defines 'Def Hence,

~ e f ( r ( V y ) ( ~ < -+ Tr(rcr($)1))1)

Since we also have

we get

Def(ru((Y)l)

By transfinite induction, we conclude

Hence,

(Vx)(OR(x) -+ D ~ V O R ( ~ ) + u 6 ) 7 1)

But also,

whence

Consequently,

Because a universal generalization satisifes 'Def whenever each of its instances does,

From this we derive:

as we wished..

Theorem 8.15 shows us how, for a given partial interpretation (\!I,T) we can explicitly and concretely give a theory A such that ( % , r U A) is a conservative extension of ($I,T) which can give an account of its own semantics. A depends on r. To give A, we do not have to be able to present r explicitly, but we do need to be able to describe r well enough to get a weak definition of T. We anticipate that, if we attempt to apply the construction from theorem 8.15 to get a semantics for a suitably regimented fragment of a natural language, finding an appropriate description of r will be a task of considerable difficulty. r is supposed to contain sentences whose primary role is to establish the meanings of terms. It is easy to recognize a rough-and-ready distinction between sentences that express the meanings of terms and sentences that provide information about what the world is like; it is the familiar analyticlsynthetic distinction. But the difficulties we encounter in trying to make the distinction precise are notorious.

A related difficulty is that our construction requires us to distinguish sharply between the fully interpreted and the partially interpreted parts of our language. The obstacle here is the near invisibility of vagueness. There is no sharp break betwccn the way we use predicates with sharp boundaries and the way we use predicates with fuzzy boundaries, so we have no test to guide us in determining which predicates lie within the fully interpreted part of our language and which lie outside it.

The problems of distinguishing the fully interpreted from the partially interpreted part of our language and of distinguishing our meaning postulates from the rest of our overall theory are difficult problems. They are also unavoidable problems, if, as I have argued, giving a satisfactory semantics requircs giving an account of definite truth. Because the problems are so formidable, however, it is worth noting that, for I!'II countable, we can avoid the problems altogether, if we are willing to settle for a theory of truth without an account of definite truth. Given a language gZ, we can give a recursive theory A such that, for any partial interpretation (!'L,T) of 2', (!)l,r U A) will be a conservative extension of ('!I,T) for which r U A is a materially adequate theory of truth. A will depend upon how !)I represents finite sequences and how it encodes arithmetic, but it will not depend on any other features of '!I, and it will not depend upon where the language of !?l leaves off and the partially interpreted part of the language starts. Thus, we can give A as our theory of truth without having to make any decisions about what part of our language to count as fully interpreted or about which of our beliefs to count as meaning postulates.

THEOREM 8.17. Let 2' be a first-order language that contains just enough vocabulary to talk about finite sequences and to relatively interpret arithmetic, let ;j be a countable, acceptable structure for %'I, and let %2 be a countable language extending 2' which does not contain 'Tr' or 'Kr ' . Then we can find a recursive theory A in the language 9' obtained from 2' by adding the new predicates 'Tr' and 'Kr' such that, for any pair (?I,r), where ?I is an expansion of ;j to a model of a language 2' whose symbols are included among the symbols of Z2 and r is a theory in 2\:, ('![,I' U A) is a conservative extension of (1)I,r) and r U A is materially adequate as a theory of truth for (Y1.T U A).

As usual, if g2 is built from a finite vocabulary, A will be finite. Notice that here, unlike our previous theorems, r does not have to be inductive over '!I. PROOF: Let ?8 be an expansion of ;j to a model of 9'. Theorem 8.3 shows us how to find a recursive theory A such that (#,A) is a conservative extension of (a,@) and such that A is a materially adequate theory of truth for (%,A). Looking closely at the proof of theorem 8.3, we see that the theory A we construct does not depend upon any features of % other than the way % describes finite sequences and encodes 91, features that will be the same in any expansion of ;j to a model

of 2'. Thus, what we really get is a recursive theory A such that, for any expansion % of ;{ to a model of 9'. ( 8 , A ) is a conservative extension of ( 3 . 0 ) and A is a materially adequate theory of truth for (%,A).

Let 2' be a language that is contained in 2' and that contains %'I, let ?I be an expansion of ;j to a model of 9', and let r be a theory in 2,;. I claim that ( % , r U A) will be a conservative extension of ('!I ,I') and that 1' U A will be a materially adequate theory of truth for ( % , r U A).

Let 8 be an expansion of 91 to a model of I'. Since (%,A) is a conservative extension of (%,@), \R3 can be expanded to a model 6 of A. Thus, 6 is an expansion of % to a model of T U A. It follows that (\)I,T U A) is a conservative extension of (?I ,T).

Now suppose that (91 ,T U A) t 4. Then ('!I,T U A) 14. Let 6 be an expansion of 91 to a model of r U A, let % be the reduction of 6 to %', and let 6* be any expansion of % to a model of A. Then 6* is an expansion of !I[ to a model of r U A, and so, since ('!(,I' U A) 14, $ is true in E*. Thus, 4 is true in every expansion of % to a model of A, that is, (%,A) 1 4. It follows by theorem 7.1 that, since )%I is countable, (%,A) t +. Since A is a materially adequate theory of truth for (%,A), (%,A) Tr(r+l), and so (%,A) 1 Tr(r41). Since Ci is an expansion of % to a model of A, 'Tr(r+l)' must be true in 6. Since 6 was an arbitrary expansion of !'I to a model of r U A, we must have ('!I,T U A) 1 Tr(r$l), and so, since )\)I( is countable, ( % , r U A) 1 Tr(r$l).

The proofs that

if (YIJ U A) t Tr(r+l), then ('!l,T U A) 1 4 ; if (\!I,r U A) ti+, then (!)l,r U A) 1 l ~ r ( r 4 1 ) ; and if ( Y I J U A) f l T r ( r $ l ) , then ('!I,T U A) t 1 4

are similar..

COROLI~ARY 8.18. In theorem 8.17, we may require that A contain the formalization of the sentence 'The extension of "Tr" is a maximal consistent set of sentences'.

PROOF: The proof of corollary 8. I8 is to the proof of theorem 8.13 as the proof of theorem 8.17 is to the proof of theorem 8.3..

Definite Truth in Partially Interpreted Languages

I

In chapter 2, we investigated the connection between two notions of necessity, logical necessity-derivability in standard logic from a system of meaning postulates-and metaphysical necessity-truth in all possible worlds. In chapter 7, we developed two notions of definite truth under a partial interpretation (?l,T), definite truth in the proof-theoretic sense-derivability in !'(-logic from the system of meaning postulates r-and definite truth in the model-theoretic sense-truth in all expansions of !'I to a model of T. We may, if we like, think of the expansions of '!I to models of r as possible worlds, and if we do so we shall find a close analogy between the two notions of necessity and the two notions of definite truth. We wish to exploit the analogy by employing possible worlds semantics to investigate the logic of definite truth, in order to get a better understanding of how we may express the notion of definite truth for a language within the language itself. Our efforts will be truly successful only in the special case in which (?I( is countable, for it is in this special case that the two notions of definite truth coincide.

In chapter 8, we saw, on the basis of complexity considerations, that we should not expect our theory of definite truth to fulfill the material adequacy conditions:

(\>X,r) 1 4 iff (Y1,r) t- ~ e f ( r + l ) ;t + iff (!)l,T) /-- ~ ~ e f ( r + l ) >

Whereas the first of these conditions is unexceptionable, the second is inappropri- ate. The reason is that the second condition contravenes our intention that the set of definite truths should be an open-ended system, which we can strengthen by strengthening our system of meaning postulates.

Let r represent our system of meaning postulates at a particular time, and let Y 1 T represent a later, more refined system. If a sentence is definitely true according to T, then it should remain definitely true according to Y, for once we have settled an issue we should not be able to unsettle it by refining our linguistic usage. On the other hand, it should be possible for a sentence that is not assigned a definite truth value by (Y1,T) to be assigned a definite truth value by (?I,Y). If the material adequacy condition were fulfilled, this would not be possible, since if (!'1,r) 4 , then, by the material adequacy condition, ('!I,r) t l ~ e f ( r 4 1 ) .

Because definite truths are preserved when we strengthen our system of meaning postulates, (?I,Y) t ~ D e f ( r $ l ) , and so, by the material adequacy condition again, (?I ,'l') 4.

We are not suggesting, of course, that it never happens that we repudiate a statement that we previously regarded as definitely true. We are merely suggesting that, when this happens, we have changed our theory, not merely refined our linguistic usage. Merely adopting a stronger system of meaning postulates can adjudicate border cases, but it cannot transfer previously undisputed territory.

In the investigation of theories of truth which we undertook in the preceeding two chapters, we proceeded by first developing a criterion of adequacy for theories of truth, the material adequacy condition, then showing how the criterion could be met. We would like to find an analogous criterion of adequacy for theories of definite truth. We already have one such criterion, the strong adequacy condition from chapter 8, but that criterion appears to be excessively arbitrary, since it relies upon our particular choice of a system of ?I-logical derivation. If we had chosen a different logical calculus that acknowledged the same inferences as ?I- logically valid, we would have got a different strong adequacy condition. As we shall see in theorem 9.1 below, the strong adequacy condition is not so arbitrary as it at first appears. If we had chosen a different system of ?I-logical derivation, we would not have got a genuinely different standard of adequacy, only a different way of describing the same standard.

We would like to find a structural criterion, one that does not depend upon any particular choice of an '8-logical calculus, for testing whether a proposed theory of definite truth is adequate. The material adequacy condition for definite truth has shown itself to be unsatisfactory. Whereas the first half of the condition,

is unproblematic, the second half

(? l , r ) $- 4 iff ('21,r) t iDef(r4')

conflicts with the ideas that the system of meaning postulates should be open- ended and that the function taking the system of meaning postulates to the set of definite truths should be monotone. But the first half of the material adequacy condition is not by itself sufficient as a criterion of adequacy for a theory of definite truth, since the first half of the condition (with 'Tr' in place of 'Def) will be satisfied by any materially adequate theory of truth, and we would not suppose that an adequate theory of truth is ipso facto an adequate theory of definite truth. We need some further condition, in addition to

+ is definitely true iff ~ e f ( r 4 1 ) is definitely true

and we look to a modal analysis to give us such a condition. A possible world for (?l,T) is an expansion of ?I to a model of T. Thus a

possible world represents, not a way things might have been, but a way things might be, consistent with the totality of the empirical facts and our linguistic commitments. Since it represents the way things might be rather than the way things might have been, definite truth resembles epistemic necessity more closely than metaphysical necessity. Definite truth is by no means the same as epistemic necessity, however, since the set of definite truth extends far beyond the set of known truths.

If %? is a possible world for (Y1,T) and R is a predicate, the extension of R in ?B, R", should represent an extension R might have, consistent with the empirical facts and our linguistic conventions. R* must include the things that are definitely R and exclude the things that are definitely not R . In particular, Dej* should be a possible system of definite truths; it should represent the set of definite truths according to some system of meaning postulates that we might adopt. ~ e f ? should include the sentences that are definitely true according to ('?l,T), but it need not be restricted to those sentences, since we might in the future adopt a system of meaning postulates more potent than r .

D e P should consist of all the sentences that are true in all the worlds that would be possible if we committed ourselves to the definite truth of Def". That is, we should have

~ e f " = {sentences true in all possible worlds for (!) i ,r U Def") ) = {r+l: (? l , r u D ~ F ) 14)

If we say that a world is accessible from % iff every sentence in DeF is true in 6 , we can restate this condition as

for every 4 , ~ e f ( r 4 1 ) is true in % iff 4 is true in every world accessible from %?.

This requirement gives us a further condition, in addition to the unproblematic half of the material adequacy condition, that we shall require if a proposed theory of definite truth is to be regarded as adequate.

DEFINITION. Given a partial interpretation (!'1,17), with Y1 acceptable, the canonicalframe for (?I ,T) is the pair < W,R>, where W (the set of "worlds") is the set of all expansions of ?I to models of r and R C W x W (the "accessibility relation" on W) is given by:

%?RK iff, for every 4 , if Def(r41) is true in 8, 4 is true in 6

DEFINITION. The theory r is modally adequate as a theory of definite truth for (?I,T) iff

( M A I) (!'I ,T) 1 (Vx)(Def(x) + Sent(x)) ( M A 2 ) for each 4 , (Y1,F) 14 iff (91,r) 1 Def(r41) and ( M A 3 ) for any 4 , for any world % in the canonical frame for (9[,1'),

~ e f ( r 4 1 ) is true in % iff 4 is true in every world accessible from %.

Here 'De f can be taken to be either a primitive predicate or else an abbreviation for a complex open sentence, like the formula 'Bew' we talked about in chapter 2.

The left-to-right direction of ( M A 3 ) follows immediately from the way the accessibility relation was defined. The force of ( M A 3 ) is in the right-to-left direction.

The left-to-right direction of ( M A 2 ) is a consequence of ( M A 3 ) . To see this, suppose that (!)I,r) 1 4, and let % be an expansion of '?I to a model of r. 4 is true in every world in the canonical frame for ('?'L,T), and so, in particular, 4 is true in every world accessible from %. It follows by ( M A 3 ) that Def(r41) is true in %?. Since % was arbitrary, ( H , T ) 1 ~ e f ( r 6 1 ) .

The right-to-left direction of ( M A 2 ) is nontrivial, since ( M A 3 ) leaves open the possibility that the definite truths as described by r are actually the definite truths of some theory much stronger than T . As an extreme case, consider the fact that the theory { '(Vx)(Def(x) ++ Sent(x))') satisfies conditions ( M A I ) and ( M A 3 ) .

In chapter 8 we described what it was for a semantic theory to be what we called "strongly adequate" as a theory of both truth and definite truth under a partial interpretation. Looking back at the definition, we see that there were four clauses, two of which described the notion of definite truth and two of which described the connection between definite truth and truth. Let us now focus our attention on the "definite truth" part of the strong adequacy condition:

DEFINITION: Suppose that we have a partial interpretation (? I , r ) , with ?I acceptable, and open sentences y(x) and ~ ( x ) such that, under (9I ,T), y (x) strongly defines r and ar(x) strongly defines the set of true atomic and negated atomic sentences of ?I. r is strongly adequate as a theory of definite truth for (?I , r ) iff

( S A I ) (')l,r) t (b'x)[Def(x) ++ {x is a sentence & [Y(x) v ~ T ( x )

V x is an axiom of logic V ( j v ) ( D e f ( y ) & Defly -, x ) ) v (3y ) (3v ) ( x = (Vv)y &

(Vz)Def(y b'!ii))l)l and

( S A 2 ) for each 4 , (?I,T) 1 4 iff (91,r) 1 Def(r41)

Here we require that y(x) strongly define T rather than merely weakly define T , because we want to avoid worrying about worlds in which there are nonstandard

members of r, that is, worlds % in the canonical frame for (?I ,T) in which there are objects a such that y(a) is true in % even though a 4 F.

The strong adequacy condition we presented in chapter 8 was motivated solely by its intuitive appeal, but now we see in terms of modal logic that the condition we gave was precisely the correct one:

THEOREM 9.1. Given !'I countable and acceptable and a partial interpretation ( ! ) l , r ) in which the formulas y(x) and nr (x ) strongly define. respectively, T and the set of true atomic and negated atomic sentences of Y1. Then r is modally adequate as a theory of definite truth for (\)I,T) iff it is strongly adequate.

We break up the proof into two lemmas:

LEMMA 9.2. Even without the countability assumption, if r is modally adequate we shall have

( S A I (? l , r ) 1 (Vx)[Def(x) - {x is a sentence &

Iy(x) v ~ T ( x )

V x is an axiom of logic

v (3y) (Def (y) & Def(y 1, x ) ) v (3y ) (3v ) ( x = (tlv)y '9

(Vz)Def(y "/?))I1 I and

(SA2)1= for each 4 , (YI,T) 14 iff (?I,T) 1 ~ e f ( r 4 1 )

( S A I),, and (SA2)1s are just the conditions that define strong adequacy, with '1' replaced by '1 ' . If 91 is countable, replacing '1' by '1' will make no difference.

PROOF: TO get the left-to-right direction of (SAI ) , , let 23 be an expansion of 91 to a model of T, and suppose that Def(Z) is true in %. Because of ( M A l ) , a must be a sentence. Because of ( M A 2 ) , Def(a 1, a) must be true in %, and so (3y) (Def (y) & Def(y i. a)) is true in 93; hence, a satisfies the right-hand side of (SA l ) ,= .

To get the right-to-left direction of (SAI) , , , let 93 be an expansion of ?I to a model of T , and let be a sentence such that

lr('47) v f f ~ ( r 4 1 ) V r+l is an axiom of logic V (3r>(Def(y> & Def(y 1, r49) v ( 3 ~ ) ( 3 v ) ( r 4 ' = (Vv)y & (Vz)~ef (r+lv lz - ) ) l

is true in 23. We want to see that ~ e f ( r 4 1 ) is true in 93. If 4 is a member of r , an atomic or negated atomic sentence true in 91, or an

axiom of logic, then (9I ,T) 1 4 , and so, according to ( M A 2 ) , (?I,T) 1 ~ e f ( r + 7 ~ Hence, ~ e f ( r 4 . 1 ) is true in %.

Suppose that, for some sentence $, ~ e f ( r $ I ) and ~ e f ( r ( $ + +)I) are both true in 9. So, if 6 is any world accessible from $, $ and ($ + 4) are both true in 6, and hence 4 is true in 6. Thus, 4 is true in every world accessible from % and so, according to ( M A 3 ) , Dtlflr41) is true in %.

Finally, if 4 has the form (Vv)$(v) and, for each ' I , ~ ~ f ( r $ ( i ) l ) is true in %, then for each object a and each world 6 accessible from %, $(a) is true in 6, and so (Vv)$(v) is true in every world accessible from %. Thus, according to ( M A 3 ) , Def(r(Vv)$(v)l) is true in %.

This proves (SAI ) ,= . (SA2),- is the same as (MA2)..

LEMMA 9.3. Assuming [?I( is countable, if 1' is strongly adequate as a theory of definite truth for (?I ,T), then it is modally adequate.

PROOF: ( M A I ) is obvious. ( M A 2 ) follows from (SA2) , since the countability of (?I\ implies the equivalence of 1 and 1. The left-to-right direction of ( M A 3 ) is immediate, so what remains is to show the right-to-left direction of ( M A 3 ) .

Let 93 be a world in the canonical frame for (\!I,r). We need to show that, if 4 is true in every world accessible from 9, then Def(r41) is true in %. A world accessible from 93 is just an expansion of \!I to a model o f r U Def". Thus, what we need to show is that, if (91,r U ~ e f ? \ ) 1 4 , then ~ e f ( r + l ) is true in %. In view of the equivalence of /- and 1, this is the same as showing that, if (9X.T U D r y ) t 4 , then ~ ~ f ( f ( T 4 1 ) is true in %, which we prove by induct~on on the lengths of proofs in 91-logic.

That Def(rc,bl) is true in % whenever C#J is in De? is immediate. That Def(r41) will be true in % whenever 4 is an element of r, an atomic or negated atomic sentence true in ?)I, or an axiom of logic follows from the fact that

( V x ) [ [ y ( x ) // CYT(X) V x is an axiom of logic] -+ Deflx)]

is a consequence of ( S A I ) . That ( 4 : Def(r+I) is true in %) is closed under modus ponens and the %-rule follows from the fact that

and

are consequences of (SAl) . . . We now want to see how, if we are given an arbitrary partial interpretation

(!?I,r), with ]!'I( countable, we can extend r to a theory I?"" such that r"" is a modally adequate theory of definite truth for (\!l,TD").

DEFINITION. Given a theory I', let I""' be the smallest set of sentences which

contains r contains '(V.'.r)(Deflx-) --+ Se~zt(.r))' is closed under 91-logical consequence contains each instance of the schema:

contains each instance of the schema:

contains Def(Lf41) whenever it contains d, and contains 4 whenever it contains Dcf(:f4l).

PROPOSITION 9.4. Suppose we are given a partla1 ~nterpretation (91,T) of a countable language Y , with !)l countable and acceptable. Then T1'" is the weakest theory extending r such that r"" is modally adequate as a theory of definite truth for (91,TD"). That is, I'"" 1s modally adequate as a theory of definite truth for (!'I,T""), and ~f fl > r is modally adequate as a theory of definite truth for (91,R) then (!)l,R) 1 r"".

PROOF: In showing that r""' is modally adequate, (MAI), (MA2), and the left- to-right direction of (MA3) are immediate. Just as in the proof of lemma 9.3, to prove the right-to-left direction of (MA3) it is enough to show that, for any world B in the canonical frame for (!)I,rD"), if (Yl,TD" U Dt:f 4 , then Def(r41) is true in 8. Proving this by induction on the lengths of proofs in 91-logic is routine.

T""' is defined as the smallest set that meets seven closure conditions. The argument that, if fl > r is modally adequate for (!)1,R), then ( % , a ) 1 To" is a straightforward verification that (4: (Y1,fI) 1 4 ) satisfies all seven conditions; this argument does not depend upon the countability of 1411..

In strengthening our theory from r to r"", we diminish the class of possible worlds. Unless is already an adequate theory of definite truth for (!)I,r), there will be some expansions of ?It that T regards as possible worlds but that T"" does not regard as genuinely possible. Let me now give a more model-theoretic version of proposition 9.4 which works directly with the Kripke frames, rather than get at Kripke frames indirectly by way of theories, and which does not depend upon the countability of 1911.

DEFINITION. Let <W,,,R,,> be the canonical frame for (!'I,@). An !)llframe is a pair <W,R>, where W C W,, and R = R,, f l (W x W). A sentence is valid in <W,R> iff it is true in every world in W. An ?(-frame <W,R> is an czdequate frame iff it meets the following three conditions:

(Fl ) For every world % E W, Def" Sent.

(F2) For any sentence 4, 4 is valid in <W,R> iff ~ e f ( r 4 1 ) is valid in <W,R>.

(F3) For any % c W. for any 4 , ~e f ' ( r+ l ) is true in $3 ~ f f (V6)(%R6 -+ (4 is true in 6 ) ) .

Clearly, a theory r is a modally adequate theory of definite truth for (!)I ,T) iff the canonical frame for (?l,T) is an adequate frame.

THEOREM 9.5. For any r , there is a largest adequate frame <W, ,R,> that is contained within the canonical frame for ( ? m . <W, ,R,> is contained within the canonical frame for (!)l,rD''). If \\!(I is countable, <W, ,R,> is identical to the canonical frame for (?(,To"').

PROOF: For each ordinal a , let W, be the set of worlds % in the canonical frame for (!)I ,r) that meet the following three conditions:

~ e f " is a set of sentences. For each sentence 4 , if Def(r41) is valid in < ,nctW,, ,n,R,>, then 4 is true in %. For any sentence 4 , if 4 is true in every world in pn,Wp that is accessible from %, then Def(r41) is true in %.

Let R, be R,, n (W, x W,,). It is easy to see that the sequence of W,s is nonstrictly decreasing, that is, if

a. 5 y , then W, > W,, Let

We need to show that <W, ,R,> has the desired properties. Notice that the left-to-right direction of condition (F3) is automatic, and that the

left-to-right direction of (F2) follows from (F3). Thus, to show that <W, ,R,> is an adequate frame, we need to verify (Fl ) and the right-to-left directions of (F2) and (F3). (Fl ) follows from the first condition defining W,. It follows from the second of the defining conditions for W, that, if W, = W,, , , then W, satisfies the right-to-left direction of (F2). It follows from the third of the defining conditions for W, that, if W, = W,,,, then W, satisfies the right-to-left direction of (F3).

Now suppose that < W,R> is an adequate frame contained within the canonical frame for (!)i,r). We want to show by induction that, for each a, W C We. Thus, suppose that, for each P < a , W C W,, and take 8 E W. Because <W,R> satisfies (Fl ) , ~ e p is a set of sentences. 1f Def(r41) is valid in < ,?,,W,, ,Q,R,>, ~ e f ( r + l ) is valid in <W,R>, and so, by (F2), 4 is valid in <W,R> and so true in 8. If 4 is true in every world in fin,Wo that is accessible from 8, then 4 is true in every world in W that is accessible from %, and so, by (F3), Def(r+l) is true in B. It follows that 3 is in W,.

It is easy to verify that every member of 1'"" is valid in every adequate frame that is included in the canonical frame for (!)l,r). It follows that every such frame in included in the canonical frame for (?!1,TD").

We now know that the <W, ,R,> is included in the canonical frame for (91 ,rD"). But if I?)ll is countable, then, according to proposition 9.4, the canonical frame for (?(,roe') is an adequate frame and so included in <W, ,R,>. Thus < W, .R,> = the canonical frame for (!)I,T""').m

Up to now, in this chapter, we have been looking at the problem of how to recognize when we already have an open sentence in our language that adequately represents the set of definite truths. We now want to look at how we might represent the set of definite truths by introducing a new predicate. Let us suppose that (?!I,r), with ?!I countable and acceptable, is a partial interpretation of a countable language 2 that does not contain the predicate 'Def; we suppose further that 2 contains formulas y(x) and a ~ ( x ) that, under (YI,T), strongly define r and the set of atomic and negated atomic truths of ?!I. We form an enlarged language 2" from 2 by adding the unary predicate 'Def; the models of TU will take the form (%,U) , where 8 is a model of 2 and U C is the extension of 'D'. We want to see how to produce within 2" a conservative extension of (91 ,T) which is modally adequate as a theory of definite truth. A highly nonconstructive way to do this will be to replace r by TD". This procedure does not actually give us the new axioms that we need to add to to get our conservative extension.' It merely describes the new set of axioms within an essentially richer metalanguage.

If we want to write down explicitly the new axioms that we need to add to r , we can do the following: use the self-referential lemma to find a sentence cr that is R,, .-provably equivalent to

(V,x)[Defi.x) - {X is a sentence &

[Y(x) v x = ru1 v V x is an axiom of logic v (3y)(De.y) & @fly -? x)) V (3~) (3v) (x = i'fv) & (Vz)Def((y vi;))l}l

Then (91,T U {u)) will be a conservative extension of (?!I,r), and F U {u) will be strongly adequate as a theory of definite truth for ('!I,r U {cr}). To see that (YlJ U {u)) is a conservative extension of (!)l,T), notice that, if !23 is an expansion of !)I to a model of T, (B,Senr) is an expansion of 8 to a model of r U {u). To verify the right-to-left direction of (SA2), notice that, if (!'I,r U {u}) 4 , then,

I The simplest way to see that (\'i.TD") is a conservative extension of (!1,1') is to note that the theory (91,r U {v}) constructed in the next paragraph is a conservative extension of (9i.T) and that, according to proposition 9.4, (\'I,T U (cr}) Tn".

if %? is an expansion of 91 to a model of T, (%, {?'1-logical consequences of T U {a}}) will be an expansion of ?I to a model of r U {a} in which ~ e f ( r 4 1 ) is false.

As a matter of fact, these two constructions produce the same result; one

- can show by using corollary 9.7 below that rU" and r U {a ) are ?I-logically equivalent.'

If we think of u as a first-order positive inductive definition over \!X, we see that the possible worlds in the canonical frame for (YI,T U {cr)) will be models (%,U), where % is an expansion of 91 to a model of r and U is a fixed point of a. Thus, the possible extensions of 'Def will be the fixed points of cr. Among

I i these fixed points is the degenerate fixed point Def = Sent So there are possible worlds in which every sentence is definitely true.

This is surprising. To intuitions guided by the ordinary English usage of 'definitely', it seems nonsensical to suggest that, although '2 + 2 = 5' could not I possibly be true, it might nonetheless be definitely true. in ordinary usage.

1 'definitely true' implies 'true'. According to everyday intuitions, the schema,

I

If r41 is definitely true, then 4

is so obvious that we are inclined to regard it as part of the meaning of 'definitely true', so that we are inclined to accept the schema

I It is definitely true that, if r$Jl is definitely true, then $J

But, in fact, if Harry has a full head of hair, a world in which every sentence is definitely true will be a world in which

I I If 'Harry is bald' is definitely true, then Harry is bald

is false. One might suspect that all this result shows us is that, even though r U {cr)

is an adequate theory of definite truth, in terms of our formal criteria of adequacy, U {u) is an unwise choice for a theory of definite truth, and that a more

sophisticated theory of definite truth would eliminate the possible worlds in which every sentence is definitely true. The next theorem shows that, on the contrary, such worlds are unavoidable. As long as our theory of definite truth is ?I-logically consistent, that is, as long as our theory admits any possible worlds at all, the theory will inevitably admit possible worlds in which every sentence is definitely true. The reason is familiar from Godel's second incompleteness theorem: any YI-logically consistent theory that can give a decent account of %-logical provability will be unable to prove its own ?!I-logical consistency.

That (!)l,r U {v}) rt'" is an immediate consequence of proposition 9.4. T o see that (\'1,To") t v, first show that (\'l,TD") t (&f0v1) -* v), then apply corollary 0.7 below.

THEOREM 9.6. Let !'I be an acceptable structure, and let (9I.f) be a partial interptetation in which the formula y(x) weakly defines I' and the formula ar(x) strongly defines the set of atomic and negated atomic sentences true in !'I. If 1' is an !)(-logically consistent, strongly adequate theory of definite truth for (!'[,T), then ' i ~ e f l r 0 = 11)' will not be definitely true in <4I,T>.

Notice that this theorem does not require that 1411 be countable. The proof actually uses only clause (SAI) of the strong adequacy condition, not (SA2). PROOF: Substituting 'Defcx)' for ' ~ ( x ) ' in theorem 1.5, we see that (SAI) guarantees that (4 : (41,r) 4) satisfies all the conditions except for (4)b). If ' i ~ r f ( r 0 = 11)' were an ?(-logical consequence of T, (4: (\)I,T) f 4 ) would satisfy (4)b) as well, and so it would be w-inconsistent. But an w-inconsistent set is !)I-logically inconsistent..

According to our ordinary way of using the phrase 'definitely true', it seems preposterous to propose that 'Not every sentence is definitely true' should not be definitely true. But our usage here is not entirely ordinary. We are proposing a coordinated change in the language we use to talk about semantics, so that as we exchange our ordinary notion of truth for a scientifically reconstructed notion of truth, we simultaneously replace our ordinary notion of definite truth with a scientifically reconstructed notion.

If one thinks of definite truth in the way we have been thinking of it here- to be definitely true is to follow, in an appropriate sense of 'follow', from a system of meaning postulates-theorem 9.6 is not so surprising. As Carnap taught us, laying down meaning postulates is a risky business. If nature is uncooperative, the meaning postulates

If a body is placed into water, then it is soluble iff it dissolves.

and

Two bodies of the same substance are either both soluble or both insoluble.

will enable us the judge the same body both soluble and insoluble. Because laying down meaning postulates is a risky business, there is nothing we can do now to foreclose the possibility that, sometime in the future, we should adopt a system of meaning postulates that is 91-logically inconsistent.

Theorem 9.6 brings to our attention the striking differences between the structural properties of the set of truths and the set of definite truths. If we substitute 'Tr' for '7' in theorem 1.5, the set of sentences that are definitely true under a materially adequate theory of truth will satisfy conditions ( I ) , ( 2 ) , and (3). Theorem 8.13 shows us that, with sufficient effort, we can arrange things so conditions (4)a) and (4)b) are also satisfied. Material adequacy guarantees that rules (RI) through (R4), which we discussed in chapter 8, will be definite-truth preserving.

If we substitute 'Def for '7' in theorem 1.5, we find that a strongly adequate theory of definite truth will give us (11, (2), (3), (4)a), and (4)c), but not (4)b). If we substitute 'definitely true' for 'true' in rules (R1) through (R4), only the analogues of (RI) and (R2),

From rr41 is definitely true1 to infer r41

and

From r$i to infer rrqbl is definitely true1

will be both valid (definite-truth preserving) and usable. The analogue of (R3),

From rr$l is not definitely true1 to infer Ti41

is vacuously valid; it never fails us because we never get to use it. The analogue of (R41,

From r ~ 4 1 to infer rrqb1 is not definitely true?

always fails. We can derive an analogue of Lob's theorem from theorem 9.6 in just the way

Lob's Theorem is derived from Godel's second incompleteness theorem:

COROLLARY 9.7. Let 91 be an acceptable structure, and let (91,r) be a partial interpretation in which the formula y(x) weakly defines r and the formula a ~ ( x ) strongly defines the set of atomic and negated atomic sentences true in '!I. If T is a strongly adequate theory of definite truth for ('!I,T), then, for any sentence 8 , (\)X,T) b (~ef( re1) + 8) iff (%,T) 1 8.

Once again, the result really requires only that (91,T) satisfy (SAI).) PROOF: The right-to-left direction is immediate. For the left-to-right direction, substitute ' ~ e j l r - 1 8 1 x)' for 'T(x)' in theorem 1.5. We see that {qb: (91,r) t ( i d + 4)) satisfies conditions ( l ) , (2), (3), (4)a), and (4)c) If (3.T) k (~ef( r01) + 8), then (4: (91,r) k ( 1 8 -, 4 ) ) also satisfies (4)b). Hence, (4: (%,TI /-- ( 7 6 + 4) is w-inconsistent and so 91-logically inconsistent. Hence, (%, r ) 1 8..

Reflecting upon our ordinary usage of the phrase 'definitely true', we find ourselves in a familiar position, trapped between three conflicting theses:

111 No legitimate notion of definite truth can fail to acknowledge that all instances of the schema


are definitely true. [2] The notion of definite truth described in the strong adequacy condition

is a legitimate notion of definite truth. 131 Except in cases of inconsistency, the notion of definite truth described

in the strong adequacy condition will not acknowledge that all instances of the schema


are definitely true.

With regard to [2], English speakers use the phrase 'definitely true' in a variety of ways, some of which have scarcely any connection with our concerns here; for example, one says that a statement is definitely true as a way of attesting that one has good evidence for it. Another way of using the phrase 'definitely true' is to distinguish clear from dubious applications of vague terms; this usage is intimately connected with our concerns here. Our partially interpreted first-order languages are intended to provide a model of how clear applications of vague terms are distinguished from dubious applications; that is, on one way of using the phrase 'definitely true', they are intended to provide a model of definite truth. They might not provide a very good model-it is highly idealized and greatly oversimplified-but, even so, the notion that was referred to as "definite truth" in the strong adequacy condition is, quite unmistakably, a notion of definite truth. In view of corollary 9.7, this implies, contrary to thesis [ I ] , that there are recognizable and intelligible notions of definite truth for which the principle

If r41 is definitely true then 4

is invalid. My response to this trilemma is the same as my response to the corresponding

problem in chapter 2. Although principle [ I ] is powerfully supported by deep intuitions, we must repudiate it, allowing that there are some legitimate uses of 'definitely true' that render the principle

If r+l is definitely true then q5

invalid. In developing a scientific worldview, one looks for the best overall fit between theory and experience. There are inevitably conflicts, in which unex- pected experimental or theoretical results, such as corollary 9.7, come into conflict with deeply entrenched intuitive beliefs. In the interest of global theoretical harmony, even the most firmly held pretheoretical intuitions, such as the intuitions that support thesis [ I I , must sometimes be cast aside.

Toward a Semantics of Natural Language

Looking over the semantics we have developed for partially interpreted first-order languages with an eye to seeing what, if anything, will be useful in developing semantics for natural languages, what strikes us as most interesting and unex- pected is that our intuitive notion of truth has broken down into two notions, truth and definite truth. In retrospect, this does not seem so surprising, for we put the intuitive notion of truth to two very different kinds of use, one practical and the other theoretical, and we had no very good reason to suppose that the same notion would serve both purposes.

The notion of truth is practically useful because it enables us to, in effect, assert or deny the conjunction or disjunction of a nameable set of sentences without being required to list the set, sentence by sentence. If we tried to achieve the same effect without using any semantic notion, utilizing instead some pragmatic notion such as warranted assertibility or warranted acceptability, we would not succeed. When Xothitl tells me

Not everything Yolanda says is true.

she has, in effect, asserted the disjunction of the denials of all Yolanda's statements. To have told me

You will not be warranted in asserting each thing Yolanda says.

would have been much too weak, whereas

You will be warranted in denying some of the things Yolanda says.

would have been too strong. Semantics, according to Tarski (1936, p. 4011, is concerned with the

connection between expressions of a language and the objects and states of affairs to which the expressions refer. So understood, there is nothing inherently semantical about the use of the notion of truth as a means of expressing infinite conjunctions and disjunctions. Were it not for the paradoxes, we could take the notion of truth simply to be implictly defined by schema (T), without saying anything about the connections between expressions and objects or states of affairs. Even when we take the paradoxes into account, we see that the principles we need to ensure the usability of the notion of truth to simulate

infinite Boolean operators-rules (R I ) through (R4), the principle of bivalence, and so on-do not inherently depend on any connection between expressions and the things the expressions are about.

In contrast to our practical use of the notion of truth, its theoretical use in enabling us to understand human verbal behavior depends essentially upon its semantical character. It will not do, as an account of language acquisition, simply to say that we are taught rules that permit us to utter a sentence on occasions at which the utterance is appropiate; for such an account fails to explain how we manage to learn how to use a sentence quite different from any sentence we have heard before under circumstances quite dissimilar from the circumstances in which we were trained. An alternative account has us learning a language by learning the truth conditions for its sentences. For very simple sentences like 'Theaetetus sits', the account is already contained in the Sophist 12631. We learn that the referent of the name 'Theaetetus' is the boy Theaetetus and the referent of the verb 'sits' is the action of sitting. We get a sentence by "weaving together" names and verbs. 'Theaetetus sits' is true iff the agent denoted by the name performs the action referred to by the verb. This gives us our truth conditions for the very simplest sentences. Truth conditions for compound sentences are got from the truth conditions for simple sentences by means of a Tarski-style compositional semantics.

It is not so clear how this story is to continue beyond the realm of the predicate calculus. For modal statements, for instance, we can give truth conditions in terms of possible worlds, but such truth conditions are no help in understanding how we acquire the ability to use modal language; for we do not teach our children to verify a sentence by inspecting other possible worlds. There is, in fact, a continual tension between the desire to give simple, graceful truth conditions and the desire to develop a semantics that has explanatory value in understanding the acquisition and use of language.

We see this tension even with respect to the simplest sentences. It is natural to give the truth condition for 'Theaetetus is brave' in terms of the satisfaction relation:

'Theaetetus is brave' is true iff the individual named by 'Theaetetus' satisfies the general term 'brave'.

On the other hand, it is not plausible to suppose that we learn to use and understand a sentence like 'Theaetetus is brave' by learning which individuals satisfy 'brave'. The satisfaction relation provides an exhaustive classification. Every individual satisfies either 'brave' or 'not brave';

(Vx)(x satisfies 'brave' V x satisfies 'not brave')

follows logically from

(Vx)(x satisfies 'brave' * x is brave)

I and

1 (Vx)(x satisfies 'not brave' ++ x is not brave)

Yet it is not plausible to suppose that, when we learn to use the word 'brave', we are taught an exhaustive classification. We are taught how to classify some individuals as brave, others as not brave, but inevitably there are borderline cases. There are persons whose bravery is lukewarm and persons who are in some ways brave and in other ways cowardly. These are cases that are left unresolved by the criteria we are taught when we learn to use the word 'brave'. Nor is this merely a defect in the linguistic training of a particular individual. There are cases that the totality of conventions and practices that constitute the English-speaking community's usage of 'brave' fails to adjudicate.

The satisfaction relation provides an exhaustive classification into 'brave' I l

and 'not brave'. Our linguistic conventions do not determine an exhaustive classification into 'brave' and 'not brave'. Therefore, the satisfaction relation is not determined by our linguistic conventions.

This is not so surprising as it first appears. We learn to use the word 'satisfies' after we learn most of the rest of the language by learning the disquotational satisfaction condition,

(Vy)(y satisfies r+(x)7

[Here 1 am setting the paradoxes aside, pretending there were no self-reference.] This implicit definition does not say anything about linguistic conventions, and it gives us no reason to suppose that our linguistic conventions uniquely determine what pairs of things satisfy 'satisfy'. Our linguistic conventions leave the applicability of 'satisfies' underdetermined, just as they leave the applicability of 'brave' underdetermined.

It is initially plausible to suppose that general terms get their meanings in virtue of linguistic conventions that determine their satisfaction conditions. But we now see that our linguistic conventions do not fix the satisfaction relation, and that, if we want an account of linguistic usage that has causal explanatory power, we need to employ subtler notions than satisfaction, notions that are more firmly grounded in linguistic practices.

In the same way, it is initially plausible to suppose that sentences get their meanings in virtue of linguistic conventions that determine their truth conditions. But if Jones is a borderline case for 'brave',' our linguistic conventions, together

' We change our example from 'Theaetetus' to 'Jones' because the historical Theaetetus was distinguished by his courage.

with the empirical facts, will not assign a truth value, either true or false, to 'Jones is brave'. Yet 'Jones is brave' is either true or false.

('Jones is bravc' is true V 'Jones is brave' is false)

follows logically from

('Jones is brave' is true t* Jones is brave)

and

('Jones is brave' is false ++ Jones is not brave).

Truth is bivalent, but our linguistic conventions produce only a trivalent partition. There are sentences that our linguistic conventions, together with the empirical facts, determine to be true; sentences that our linguistic conventions, together with the empirical facts, determine to be false; and sentences that are left unsettled.

The notion of definite truth we have been developing is an attempt to give a model, albeit a highly idealized and excessively formalized model, of how our linguistic conventions, together with the empirical facts, give a sentence a determinate truth value. The notion of truth we have been developing here is not the naive notion that is implicitly defined by the disquotational truth condition,

rq57 is true ++ c$

but it is still a bivalent notion. We employ a bivalent notion of truth because bivalence has proved to be practically useful when we utilize the notion of truth to simulate infinitary conjunctions and disjunctions.

We use partial interpetations in giving our formal models of definite truth to reflect the fact that there is more to the acquisition of general terms than learning what Gupta calls rules of application. The two components of a partial interpretation are intended to represent, in a crude and oversimplified way, the fact that there are two components to our acquisition of general terms: learning how to apply the term in particular cases, and learning the position of the term within a theoretical network of concepts. To learn the meaning of 'acid', for instance, it is not enough to learn how to classify liquids as "acid" or "nonacid" using litmus paper; one must also learn the position of the notion of acidity within our constellation of chemical concepts.

Partial interpretations give us a highly idealized model of how we acquire general terms. In practice, we do not expect the part of the language learned by acquiring a rule of application to be neatly separated from the part of the language learned by acquiring a theory; instead, the two components of language learning are thoroughly intertwined. To get a theory of definite truth that accurately reflects how language

is learned would require adeep investigation into the psychology of language acquisition; what we have here is a logician's highly simplified model.'

The insight that we use the notion of truth in two fundamentally different ways is due to Hartry Field [1986], who distinguishes "disquotational truth" from "correspondence truth." The former conception uses the notion of truth, which it takes to be implicitly defined by schema (T), solely to effect infinite conjunctions and disjunctions, whereas the latter conception attempts to give a causal explanation of the connection between sentences and the objects and states of affairs to which they refer, in order to justify the use of semantic notions in explanations of human behavior. Field draws the distinction by pointing to the counterfactual

If we had used the word 'white' to mean green. 'Snow is white' would have been true.

which is, he says, true according to the disquotational conception, which regards the counterfactual as logically equivalent to

If we had used the word 'white' to mean green, snow would have been white.

and false according to the correspondence conception. This way of presenting the distinction is, I think, unfortunate. As we noted in remark 0.1, what we really seem to have in this example is one meaning of 'true' and two readings of the scope of a modal operator; we have

If we had used 'white' to mean green, 'Snow is white' would have been a true sentence of English as we actually speak it.

versus

If we had used 'white' to mean green, 'Snow is white' would have been a true sentence of English as we would have been speaking it.

One particularly sticky point is that the fully interpreted part of our language is required to contain an isomorphic copy of the natural number system. Our reason for requiring this is simply that we need a copy of the natural numbers in order to be able to talk about sentence types, and unless we can talk about sentence types we cannot get our project started. To make our notion of definite truth plausible as a model of what speakers learn when they acquire the meanings of the terms of their language, we would need to provide an account of how numerical terms make their way into the fully interpreted part of the language. The picture I have in mind, derived from the early work of putnam and the recent work of Chihara, is that the universe of !'I, which is our system of urelements as described in digression 7.4, should consist of physical objects and constructible physical objects. "Existential" quantifiers ranging over the urelements should be read "it is possible to construct." "It is possible to construct" can be either taken as primitive or explicated in terms of modal operators. I do not intend to attempt to develop this picture here. Here the problem is simply being swept under the rug.

Even though it is not successfully marked by the counterfactual, Field's distinction between the two fundamentally different ways we use the notion of truth is an important one.

Following Gupta, let us imagine that we have singled out a fragment of English that can reasonably be regarded as having a first-order structure. The fragment will consist entirely of sentences used to make assertions, and it will be free of indexicals, modalities, propositional attitude constructions, and so on. We also assume that no semantical terms occur in the fragment. We intend to add the adjective 'true' to the fragment. As noted in remark 8.6, we can simultaneously add 'denotes' and 'satisfies' without extra effort. Some of the other semantical and quasi-semantical notions-knowledge, for instance-would require quite a lot of extra effort.

There are two main obstacles that stand in the way of applying the methods of chapter 8 to obtain a semantics for our fragment of English. The first is that we have no criterion for distinguishing the fully interpreted part of the language from the rest of the language, nor for recognizing which of our basic beliefs are to count as meaning postulates. This is one of many difficult problems that we are choosing to ignore. To obtain a satisfactory system of meaning postulates would require a deep and detailed investigation into the psychology of language acquisition. Such an investigation, while undoubtedly valuable, would likely have little effect on the answers to the central problems we have been working on here. With the aim of simplifying our task as much as possible by setting aside all problems that are not immediately relevant to the questions at hand, I propose to pretend that we have a way of picking out a set of meaning postulates in the same way that we pretend that we have a way of distinguishing semantical from nonsemantical terms and we pretend that we have a way of distinguishing context- free from context-dependent sentences.

The second obstacle, by contrast, is at the very center of our concerns here. We pretend that we have divided our fragment of English-call it Z2-into a fully interpreted part 3' and a partially interpreted part and we pretend that we have a system of meaning postulates T. The problem is that 2' cannot be regarded as a partially interpreted language because its universe of discourse is not a set.

Although Z2 is not a partially interpreted language, we can treat it as if it were a partially interpreted language by applying to it the same formal methods we would apply if the quantifiers were restricted to a set. Let me be more specific. If we have an acceptable structure for X', theorem 8.15 shows us how to produce a theory A so that r U A is strongly adequate as a theory of truth and definite truth for (%,T U A). Looking over the proof of theorem 8.15, we see that the theory A we construct does not really depend upon the structure 91. What we really have is a theory A such that, for any model !)I for %' that interprets the arithmetical symbols and the symbols for finite sequences standardly, r U A will be a strongly adequate theory of truth and definite truth for (!'I,r U A) . Thus,

any way we choose to restrict the intended interpretation of Y ' to a set-sized subuniverse, we shall find that r U A is a strongly adequate theory of truth and definite truth for the resulting partial interpretation. Every single time we try out r U A on a set-sized scale model, we get precisely the results we wanted. This gives us good reason to believe that a full-scale application of r U A will give satisfactory results.

1 propose that we take our semantic theory for Y 2 to be T U A. I propose further that we adopt the rules of inference ( R l ) through (R4). together with two

1 intuitively clear rules for definite truth:

From r+l to infer rr$l is definitely truel, and From rr+1 is definitely true? to infer r+7

and one clearly counterintuitive rule, taken from corollary 9.7,

From rlf r+l is definitely true, then 41, to infer r$l.

1 Now, it makes no sense to ask whether the resulting theory of truth and definite

1 truth is strongly adequate, for we can only formulate the strong adequacy condition from the vantage point of an essentially richer metalanguage. What the proof of theorem 8.15 gives us is good evidence-though not a formal proofi-that the resulting theory gives results consistent with the empirical facts.

The procedure proposed gives us a semantic theory for a sizable fragment of English. Extending the theory beyond the fragment will require further work, some of it straightforward, some of it quite difficult. If we have a theory that

I gives us the truth conditions for assertions, we can easily extend the theory to one that gives the conditions of satisfaction for the propositional contents of other speech acts. The fulfillment conditions for the directive, 'Please bring me a beer from the kitchen', coincide with the truth conditions for the assertion, 'You will

1 bring me a beer from the kitchen7.1 By contrast, giving the truth conditions for sentences containing indexicals is likely to be an enormously difficult task. Yet we may find that including indexicals will actually simplify some aspects of our semantic theory. The places where the methods developed here appear to be particularly clumsy and complicated are places where we have to produce convo- luted self-referential constructions. Having indexicals in our language might substantially simplify these constructions, since we can sinlply say 'this sentence' rather than appeal to the self-referential lemma.

The program of developing a semantics for English by directly applying theorem 8.15 to a fragment of English that sufficiently resembles the language of the predicate calculus, then extending the theory to encompass larger and

In view of theorem 9.6 , no formal proof will be forthcoming. ' See Searle [1969]. One of the great merits of Searle's theory is that it enables us to advance almost

effolilessly from a semantics for assertive5 to a general semantics for speech acts.

larger fragments. is not intended to bc taken seriously. Thc methodology is all backward. Rather than let the properties of the language under study guide the development of the logical theory, we are using our powerful logical machinery brutally to force the language into a preconccivcd logical mold. While I earnestly hope that the methods developed in chapter 8 will eventually be useful in the study of natural language, I anticipate a much less heavy-handed application.

What is impressive about theorem 8.15 is not that i t actually will be used or ought to be used to give a semantics for English. What is impressive is the fact that we can cvcn coherently imagine using it to give a semantics for English, for this shows us that it is possible to develop a semantics for a language within the language itself.

A semantics for natural language will be a long time coming. In the meantime, it is worthwhile to set down some interim results that appear likely to emerge from our eventual semantic theory:

(Vx)(Tr(x) + Sent(x)) (VA)(X is an axlom of log~c + Tr(x) ) (Vx)(Vg)(Tr(x ? g) -+ (Tr(x) + Tr(y))) 1 ( 3 x ) ( T r ( x ) & T r ( 1 x ) ) (Vx)(Sent(x) -+ (Tr(x) V T r ( l x ) ) ) (Vx)(Dej(x) -+ Sent(x)) (Vx)(x is an axiom of logic --+ DLif(x)) ( V x ) ( V ~ ) ( D e f ( x ? Y ) -+ (Def(x) -+ Def(y) ) ) (Vy)(V variable v)[(Vz)Dej(y vls) -+ Def((vv)y)] (Vx)(Def(.r) ++ L)ef( r T r ( i ) l ) ) ( V x ) ( D e f ( l x ) - D e f ( r i T r ( k ) l ) )

( R l ) From rTr(r+1)1 to infer r41. ( R 2 ) From to infer rTr(rq51)1. ( R 3 ) From r i ~ r ( r $ 1 ) 1 to infer r i 4 1 . (R4) From T i 4 1 to infer r l T r ( r 4 l ) l .

From r ~ e f ( r 4 1 ) l to infer r+l. From r41 to infer r ~ e f ( r+1 ) I . From r(Def(r+l) --, 4)l to infer r$l

We have proposed to solve the liar antinomy by treating the adjective 'true' as a vague predicate. That 'true' is a vague predicate should come as no surprise. Intuitively, when we assert or deny that 'Harry is bald' is true, we are saying the same thing as when we assert or deny that Hany is bald. If that is so, then, if the linguistic conventions that govern the use of the vague term 'bald' leave it

unsettled whether or not Harry is bald, the linguistic convcntions that govern the use of the term 'true' likewise leave it unsettled whether or not 'Harry is bald' is true. The biconditional

'Harry is bald' is true iff Harry is bald.

is intuitively obvious, and, since it involves no self-reference, the intuitions that support the biconditional are not seriously undermined by the paradoxes. But if we firmly accept the biconditional but we regard the righthand side as unsettled, we must regard the lefthand side as unsettled. Thus, 'true' inherits the vagueness of all the vague nonsemantical predicates of our language.

It is sometimes supposed that vagueness is an altogether bad thing, to be eliminated wherever possible, but a little reflection will reveal that the vagueness of scientific discourse is often quite useful. Imagine, if you will, a little creature Jocko right on the border between protozoa and plants. We normally identify protozoa by a cluster of properties we typically find all together. But Jocko is an exception. Jocko satisfies too few of our normal criteria for protozoa for us to say with conviction that Jocko is a protozoon, yet Jocko satisfies too many of our normal criteria to say with conviction that Jocko is not a protozoon. We could, if we chose, settle the matter by stipulation, picking an arbitrary sublist of the usual list of criteria and stipulating that an organism is to count as a protozoon if and only if it satisfies all the items on the sublist. Although such a stipulation would help stamp out vagueness, it would be bad scientific practice. For it might turn out that there is some micromolecular property, unsuspected by us now and certainly not a part of our present-day criteria for using the word 'protozoon', that underlies our present distinction between animals and plants and explains why some organisms exhibit the characteristic qualities of animals and others the characteristic qualities of plants. Thus, our arbitrary stipulation would have produced a sharp, unnatural boundary where, unknown to us, nature provides a sharp, natural boundary. Because there would be some things that nature classifies as animals but we classify as plants, our scientific theory would develop into an ugly, cumbersome thing, cluttered with exceptions. Because of our squeamish- ness about vagueness, we would have produced a degenerate science that studies artificial categories rather than natural kinds.

The notion of truth inherits the vagueness of vague nonsemantical terms; if it is undetermined whether Jocko is a protozoon, it is likewise undetermined whether 'Jocko is a protozoon' is true. Thus, whatever reasons we have for wanting to keep the edges of the concept protozoon a bit fuzzy are also reasons for wanting to keep the edges of the concept of truth a bit fuzzy. There is a further reason, special to the way we use the notion of truth, for not wanting the border between truth and untruth to be conlpletely sharp. Essential to our everyday usage of the notion of truth is the application of rules ( R I ) through (R4). According to the theory we have been developing, this usage is justified by the fact that (Rl)

through (R4) are definite-truth preserving. If the notion of truth had sharp boundaries, (RI) through (R4) would not be definite-truth preserving. To see this, let us suppose, on the contrary, that the prcdicate 'Tr' has a sharp boundary, so that, for each sentence 4 , either Tr(r41) is definitely true or i T r ( r 4 1 ) is definitely true, and, moreover, that (Rl) through (R4) are definite-truth preserving. If Tr(r47) is definitely true, then, by ( R l ) , 4 will be definitely true. If i T r ( r 4 1 ) is definitely true, then, by (R3), 14 will be definitely true. In either case, (Tr(r41) - 4) will be definitely true. But, taking 4 to be the liar sentence A, we see that i (Tr ( rh1) - A) is definitely true, so that the set of definite truths is contradictory.

If our notion of truth, either our naive notion or our scientifically reconstructed notion, were precise, rules (R l ) through (R4) would not be valid. Thus the vagueness of the notion of truth is essential to its unique logical usefulness.

If asked whether Jocko is a protozoon, I shall have to reply "I do not know," without intending to intimate thereby that there is any fact of the matter there to be known. Likewise, if I am asked whether 'Jocko is a protozoon' is true or whether the liar sentence is true or whether the truthteller is true, I shall be forced to reply "I do not know," without intending to intimate that there is any fact of the matter there to be known.

Often one needs a detailed diagnosis of an illness before one can design an effective plan of therapy. Sometimes, however, one is only able to diagnose a disease by seeing how the disease responds to treatment. We began our investigation with the crudest possible diagnosis of the liar antinomy; the antinomy arises, we said, because our naive theory of truth is inconsistent. Now, retrospectively, we are in a position to offer a more detailed diagnosis. The liar antinomy arises from an improper application of rules (RI) through (R4).

Let us look at the reasoning by which thinking about the starred sentence from chapter 0 leads us to a contradiction. We start with the observation:

( 1 ) The starred sentence = 'The starred sentence is not true'.

Assume, for conditional proof,

(2) The starred sentence is true.

Substituting, we get

(3) 'The starred sentencc is not true' is true.

From this we derive, using (R 1 ),

(4) The starred sentence is not true.

We assumed ( 2 ) and derived (4). By the rule of conditional proof, we get the following conclusion without the premiss (2):

(5) The starrcd sentence is true -+ the starred sentence is not true.

Hence,

(6) The starred sentence is not true

(Alternatively, we could have gone directly from (4) to (6) by the rule of reductio ud absurdurn; this amounts to the same thing.) By (R2) we get

(7) 'The starred sentence is not true' is true.

Substituting again, we obtain

(8) The starred sentence is true

Contradiction. The fallacious step in this argument occurs at line (4). (RI ) is a legitimate rule

because the inference

4 is true. Therefore 4 .

is definite-truth preserving. (RI) cannot be legitimately used within conditional proofs because such a use amounts to applying the much stronger rule

($ + (r41 is true)). Therefore (+ 4 4 ) .

There is no reason to suppose that this stronger rule is definite-truth preserving, and, indeed, the liar antinomy shows that the rule is not definite-truth preserving.

(RI) is a legitimate rule of proof that cannot be legitimately applied within conditional proofs. In this respect, (R l ) is like the rule of necessitation in modal logic. Necessitation is a legitimate rule because it preserves modal validity. Necessitation cannot, however, be applied within conditional proofs, for such an application would enable us to derive the obviously invalid schema

as follows: Assume, for conditional proof,

Derive

by necessitation. Conditionalize to get

We have seen that we can get a contradiction from the application of (RI ) within conditional proofs. Simple variants of the argument get a contradiction i from the application of (R2). (R3), or (R4) within conditional proofs. Schema I

(T) is derived by applying rules (RI) and (R2) within conditional proofs, as follows: Assume

(9) ~I#J: is true

( R l ) gives

Conditionalizing, we get

Now assume

Use (R2) to get

( 13) r+l is true

Conditionalizing,

(14) (4 + is true))

Combining (1 1) and ( l4) ,

(T) ((r$l is true) ++ 4)

We get the paradoxes by applying rules (R l ) through (R4) within conditional proofs. We have correctly seen that (Rl) through (R4) are valid rules of inference, and we have incorrectly supposed that (RI) through (R4) could be correctly applied even within conditional proofs.

In chapter 0, trying to motivate the naive theory of truth, I suggested that the instances of the schema

(T) rq!11 is true iff 4

have to be true because the sentences on either side of the 'iff' say the sarne thing. We now see that we have to be more circumspect. There is a sense in which, when we say that r41 is true, we are saying the same thing as when we say 4 , for the two sentences ir41 is truel and have precisely the same assertion and denial conditions. Nevertheless, rr41 is truel and i41 do not I express the same proposition, for compound sentences containing ir41 is truel are

I not always equi-assertible with the corresponding compound sentences containing I

I

.' The correct observation that r&l and rrd-1 is true? are equi-assertible and equi-deniable is what legitimates the valid rules of inference ( R l ) through (R4). The mistaken further assumption that and rr41 is true1 actually mean the same thing, so that they can be substituted for one another everywhere, is what provokes the fallacious application of ( R l ) through (R4) within conditional proofs.

We give a similar diagnosis of the definite liar paradox. A typical derivation of the definite liar paradox will involve the sentence marked with a spade below:

4 The sentence marked with a spade is not definitely true.

Our careless everyday intuitions would sanction the following line of argument: Observe

( 15) The sentence marked with a spade = 'The sentence marked with a spade is not definitely true'.

Assume, for conditional proof,

(16) The sentence marked with a spade is definitely true.

Substituting, we get

(17) 'The sentence marked with a spade is not definitely true' is definitely true.

Applying the rule of inference that permits us to infer r41 from rr41 is definitely truel , we obtain

(18) The sentence marked with a spade is not definitely true.

Conditionalizing, we get

(19) The sentence marked with a spade is definitely true + the sentence marked with a spade is not definitely true.

whence

(20) The sentence marked with a spade is not definitely true.

We have rigorously proven (20), and whatever we can prove rigorously is definitely true. Thus, we have the rule of inference that enables us to infer r.41 is definitely truel from r41, a rule which permits us to derive

(21) 'The sentence marked with a spade is not definitely true' is definitely true.

Substituting again, we get

Cf. Dumrnett 11958. pp. 153ffl

(22) The sentence marked with a spade is definitely true.

Contradiction. The fallacy occurs at line ( 18). The rule that permits us to infer r41 from rr+l

is definitely true1 is a legitimate rule of inference, but it cannot be legitimately applied within conditional proofs.

An alternative derivation of the definite liar paradox gets the conditional

'The sentence marked with a spade is not definitely true' is definitely true -+ the sentence marked with a spade is not definitely true.

by direct appeal to the schema

r41 is definitely true -+ 4. This schema is invalid. One explanation for its appearance of validity is that there are other uses of the phrase 'definitely true', different from the use we have been employing here, for which, presumably, the schema is valid. Another explanation is that we have allowed ourselves an illegitimate ascent from a rule of inference to an axiom schema. W e have correctly seen that the rule of inference

From r r 4 7 is definitely true1 to infer

is valid, and we have fallaciously supposed that the axiom schema

r47 is definitely true -+ 4 is valid as well.

Perhaps someday, secure in the possession of a satisfactory theory of truth, we shall regard the reasoning in the liar antinomy as a simple fallacy, an amusing intellectual sleight-of-hand, similar to the paradox from the Euthydemus [298]:

This dog is a father. This dog is yours. Therefore this dog is your father

At present, however, we d o not possess a satisfactory theory of truth, and the liar antinomy is invaluable as a guide toward getting one.

Bibliography

Artemov. S. N . 1985 "Nonarithmeticity of Truth Predicate Logics of Provability." Doklady Akademii

Nauk SSSR 284: 270-71. English translation in Soviet Mathematics-Doklady 32 (1986): 403-5.

Barwise, K . Jon. 1975 Admissible Sets and Structures. Berlin: Springer-Verlag

Belnap, Nuel D. , Jr. 1982 "Gupta's Rule of Revision Theory of Truth." Jourrzal oj Philo.sophica1 Logic- I I:

103-16. Boolos. George. 1979 The Unprovability ofConsistenky: An Essay in Modal Logic. Cambridge: Cambridge

University Press.

Boolos, George, and Vann McGee. 1987 "The Degree of the Set of Sentences of Predicate Provability Logic That Are True

Under Every Interpretation." Journal c?f'S~~mbolic Logic 52: 165-7 1.

Burali-Forti, Cesare. 1897 "Una qucstione usi numeri transfinite." Rendiconti del Circolo matematico di Pal-

ermo 8: 154-64. English translation by Jean van Heijenoort, "A Question on Transfinite Numbers," in van Heijenoort 1 1967, pp. 104-121.

Cantor, Gcorg. 1899 Letter to Richard Dedekind. English translation by Jean van Heijenoort and Stefan

Baure-Mengelberg in van Heijenoort [ 1967, pp. 113-171.

Carnap, Rudolf. 1936 '"Testability and Meaning." Philosophy of Scienc.e 3 (1936): 419-71 and 4 (1937):

2-40. 1947 Meaning and Necessit?;: A Study in Semantics and M o h l Logic.. Chicago: University

of Chicago Press.

Chang, C. C. , and H. Jerome Keisler. 1973 Model Theory. Amsterdam: North Holland.

Chihara, Charles S. 1973 Ontology and the Vicious-Circle Principle. Ithaca, N.Y .: Cornell University Press. 1979 "The Semantic Paradoxes: A Diagnosis." Philoscjphical Review 88: 590-618. 1984 "A Simple Type Theory without Platonic Domains." Journal of Philosophical Logic

13: 249-83.

Craig, William. 1953 "On Axiomatizability within a System." Journ~zl oJ'Symholic- Logic. 18: 30-32.

Davidson. Donald. 1967 "Truth and Meaning." Synthese 17: 304-23. Reprinted in Davidson 11984, pp. 17-

361. i

1977 "Reality without Rcfcrcncc." Dialectics 3 1: 247-53. Rcprintcd in Davidson 11984, ! pp. 2 15-25]. Page references are to the reprint.

1979 "Moods and Perfomlances." In Meaning und Use, edited by A. Margalit. Dordrecht, Holland: D. Reidel. Reprinted in Davidson [1984, pp. 109-211.

1980 "The Logical Form of Action Sentences." In Davidson's Essays on Actions and 1

Evetzts, pp. 105-22. Oxford: Oxford University Press. 1984 Inquiries into Truth urzd Intc~rpretution, Oxford: Clarendon Press.

Davidson, Donald, and Jaakko Hintikka, eds. 1969 Words und 0bjec.tions: ESSLIJS 011 the Work of W . V . Quirze. Dordrecht. Holland:

D. Reidcl.

Dummett, Michael A. E. 1958 "Truth." Proc~eedings of the Aristotlean Sociey 59 (1958-59): 141-62.

Feferman, Solomon. 1960 "Arithmetization of Mctamathcmatics in a Gcncral Setting." Fu~idamenta Mathema- I

tic-ae 49: 35-92. 1984 "Toward Useful Type-Free Theories, I." Journal of Symbolic Logic- 49: 75-1 1 1 .

Rcprinted in Martin 11984, pp. 237-871.

Field, Hartry. 1972 "Tarski's Theory of Truth." Journal of Philosophy 69: 347-75. 1980 Science without Numbers. Princeton, N.J.: Princeton University Press. 1986 "The Deflationary Conception of Truth." In Graham MacDonald and Crispin Wright,

i eds., Fucr, Science, and Moruli~y, pp. 55- 1 17. Oxford: Blackwell.

Frege, Gottlob. 1879 Begriffssc,hrqt. English translation by Stefan Bauer-Mengelberg in van Heijenoort

[1967, pp. 1-82]. 1x91 "Function and Concept." English translation by Peter Gcach and Max Black in their

Trun.slatic~nsji-om the Philosophical Writings ofGottlob E'rege, pp. 2 1-41 . Oxford: Blackwell (1966).

1915 "My Basic Logical Insights." English translation by Peter Long and Roger White in Frege's Po.sthumou.s Writings, pp. 251-52. Chicago: University of Chicago Press (1979).

I

Gaifn~an, Haim 1987 "Solution to the Semantic Paradoxes." Address to the Pacific Division of the Ameri-

can Philosophical Association.

Gadel, Kurt. 1930 "Die Vollstandigkeit der Axiome des logischen Funktionenkalkiils," Munutshefte

,fur M~zrhematik und Physik 37: 349-60. English translation by Stefan Bauer- Mengelberg, "The Cnrnpleteness of the Axioms of the Functional Calculus of Logic," in van Heijenoort 11967, pp. 582-911.

1931 " ~ b e r formal unentscheidbare Satze der Print-ipicr Mrithem~lricw und verwandter Systeme I ." Monatshefie fur Mathernutik und Physik 38: 173-98. English translation by Jcan van Heijenoort, "On Formally Undecidable Propositions of Principin Muthe- matica and Related Systems I." in van Heijenoort (1967, pp. 596-6161.

Grelling. Kurt, and L. Nelson. 1908 "Bemerkungen zu den Paradoxien von Kusscll und Burali-Forti." Ahhandlungerr der

Fries'sc.hen Schule neue Folge 2: 301-34.

Grice, H. P.. and P. F. Strawson. 1956 "In Defense of a Dogma." Philosopl~ical Review 65: 141-58.

Grover, Dorothy, J . L. Camp, Jr., and Nuel D. Belnap, Jr. 1975 "A Prosententittl Theory of Truth." Phrlo.sophicc~l Studie.\ 27: 73-1 25.

Gupta, Anil. 1982 "Truth and Paradox." Jourtiul cf'Philosophical Logic I 1 : 1-60. Reprinted in Martin

[1984, pp. 175-2351, Page rcferences are to the reprint.

Hempel, Carl G . 1958 "The Theoretician's Dilemma: A Study in the Logic of Theory Construction." In

Herbert Feigl. Michael Scrivcn, and Grovcr Maxwell, eds., Minrlesoru Studies in the Philosophy of Science, v. 2. Minneapolis: University of Minnesota Press. Reprinted in Hempcl [ 1965, pp. 173-2261,

1965 Aspects of Sc,ienrlfic E.xplanc~tic~n. New York: The Free Prcss.

Herzberger, Hans G. 1982 "Naive Semantics and the Liar Paradox." Journal of Philosophy 79: 479-97. 1982a "Notcs on Naive Semantics." Journal of Philos(1phica1 Logic 1 1 : 1-60. Reprinted

in Martin 11984, pp. 175-2351, Page references are to the reprint.

Howe, William H. 1975 Butterflies cf North America. Gardcn City, N .Y: Doubleday.

Jech, Thomas. 1978 Ser Theory. Ncw York: Academic Press.

Kaplan, David. 1969 "Quantifying In." In Davidson and Hintikka 11969, pp. 206-421.

Karp, Carol. 1964 Languages wirh Expressions of Infinite Length. Amsterdam: North Holland.

Kleene, Stephen Cole. 1952 Introduction to Metamathematics. Ncw York: American Elsevier.

Kneale, William Calvert. 1962 .'Modality De Dicto and De Re." In Logic. Methodology, anclPhilo.sophy ofScience,

edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, pp. 622-33. Stanford, California: Stanford University Press.

Kneale, William Calvcrt. and Martha Kneale. 1962 The Development of Logic-. Oxford: Oxford University Press.

Konig, Julius. 1905 "iJber die Grundlagen der Mengenlehre und das Kontinuumproblem (Zweite Mittei-

lung)." Mathemarisehe Annalen 61: 156-60. English translation by Stefan Bauer-

Mcngelberg, "On the Foundations of Set Theory and the Continuum Problem," in van Heijenoort [I 967, pp. 145- 1491.

Kremer, Michael 1988 "Kripke and the Logic of Truth." J n ~ ~ r n n l ~fPlrilo.vor~lric~r~l Logic. 17: 225-78.

Kripke. Saul A. I

1959 "A Completeness Theorem in Modal Logic." Jourrr~l of Symbolic Logic 24: 1-14. i I

1963 "Semantical Considerations on Modal Logic." Ac.tc~ Philosophica Ferrnicrr 16: 83- 94.

1972 "Naming and Necessity." In Semcrntics of Natural Langucrge, edited by Donald i Davidson and Gilbert Harman, pp. 253-355 and 763-69. Dordrecht, Holland: D. Reidel.

1975 "Outline of a Theory of Truth." Journal of'Philosophj 72: 690-716. Reprinted in Martin 11984, pp. 53-81]. Page rcferenccs are to the reprint.

Leibniz, GottSried Wilhelm von. 17 14 The Monodology. English translation by George Martin Duncan in Thp Philo.sophic~a1

Works of Leibniz, pp. 308-23. New York: Tuttlc, Morehead. and Taylor ( 1908).

Lemrnon, E. J. i 1977 An Introduction to Mode11 Logic. Americun Phil~sophic~~il Quarterly Monograph

Series, no. 11. 1

Lewis, Clarence Irving. 1918 A Survey of S~~rnbolic Logic,. Berkeley. California: University of California Press.

Lewis, David K. 1968 "Counterpart Theory and Quantified Modal Logic." Journal ofPhilosophy 65: 1 13-

26. Reprinted in Michael J. Loux, ed., The Actual and the Possible, pp. 110-28. Ithaca, N.Y.: Cornell University Press (1979).

Liib. M. H. I

1955 "Solution to a Problem of Leon Henkin." Journal of Symbolic. Logic. 20: 115-18.

Lukasiewicz, Jan 1920 "0 Logice trojwartoiciowej." Ruth Filozc$c.zny 5: 169-71. English translation by

H. Hiz, "On Three-Valued Logic." in McCall, [1967, pp. 16-18].

McAloon, Kenneth. 1982 "On the Complexity of Models of Arithmetic." Journal o f Symbolic Logic 47: 403-

15.

McCall, Storrs, ed. 1967 Polish Logic,: 1920-1939. Oxford: Clarendon Press.

i McCee, Vann. 1985 "How Truthlike Can a Predicate Be'? A Negative Kesult." Journal o f Philosophical

Logic 14: 399-410. I

Martin, Robert L. , ed. 1984 Recent Essays on Truth and the Liar Paradox. Oxford: Oxford University Press.

Martin, Robert L. , and Peter W. Woodruff. 1975 "On Representing 'True-in-L' in L." Philosophra 5: 213-17. Reprinted in Martin

11984, pp. 47-51].

Mates, Benson. 1965 Elernenfury Logrc. New York: Oxford Un~versity Press Montague. Richard. 1963 "Syntactic Treatments of Modality, with Corollaries on Reflexion Principles and

Finite Axiomatizability." Acra Philosophica Fennic.~ 16: 153-67. Reprinted in Montague 1 1974, pp. 286-3021,

1974 Formal Philosophv. New Haven: Yale University Press. Montague. Richard, and David Kaplan. 1960 "A Paradox Regained." Notre Dome Journal ofFormal Logic I : 79-90. Reprinted

in Monague [1974, pp. 271-851. Mosehovakis, Yiannis N . 1974 Elementary Induction on Abstract Struc,tures. Amsterdam: North Holland. Pollock, John L. 1982 Language and Thought. Princeton: Princeton University Press. Putnam, Hilary. 1963 "The Analytic and the Synthetic." In Herbert Feigl and Grover Maxwell, eds.,

Scienrijic. Explanation, Space, onel Time. Minnesotcl Studies in the Philosophy of' Science, v. 3. Minncapolis: University of Minnesota Press. Reprinted in Rosenberg and Travis [I97 1, pp. 94-1261.

1968 "The Logic of Quantum Mechanics." In Putnam [1975a, pp. 176971. 1975 "The Meaning of 'Meaning. "' In K. Gunderson, ed., Language, Mind, and Knowl-

edge. Minnesota Studies in the Philosophy ofscience. v.7. Minneapolis: University of Minnesota Press. Reprinted in Putnam 1 197%. pp. 2 15-27 I]. Page references are to the reprint.

1975a Mathemutics. Matter, and Method. Philosophiccrl Papers, v. 1 . Cambridge: Cam- bridge University Press.

1975b Mind, Langucrge, and Realit?. Plzilosophic~ul Papers, v. 2. Cambridge: Cambridge University Press.

1980 "Models and Reality." Journal of Symbolic Logic 45: 464-82. Quine, Willard Van Orman. 1950 Methods c?f' Logic. New York: Holt, Reinhardt, and Winston. 195 1 "Two Dogmas of Empiricism." Pkilosophicul Review 45. Also in Quine 11953,

pp. 20-461 and in Rosenberg and Travis [1971, pp. 63-80]. 195 l a Mathemutical Logic,, 2nd ed. New York: Harper and Row. 1953 Frorn n Logical Point of' View,. Cambridge, Mass.: Harvard University Press. 1953a "Three Grades of Modal Involvment." Proc,eedings ofthe Xlth Irrternational Con-

gress of Philosophy, v. 14. Amsterdam: North Holland. Reprinted in Quine I 19661. 1960 Word and Object. Cambridge, Mass.: M.I.T. Press. 1961 '.Keply to Professor Marcus." Synthese 20. Reprinted in Quine 119661. 1966 The Ways of Parad0.x clnd Other Essays. Cambridge. Mass.: Harvard University

Press. 1968 "Ontological Relativity ." Journcrl cf Philosophy 65: 185-2 12. 1970 Philosophy of' Logic. Englewood Cliffs. N. J . : Prentice-Hall. Ramsey, Frank Plumpton. 1929 "General Propositions and Causality ." In Ramsey 1 1 950. pp. 237-551.

1929a "Theories" In Ramsey L1Y50, pp. 712-361. 1950 The Foundation.~ c!f'Mtrtlzrmutic~~ urzd O ~ h e r E.v.sa\s. New York: Humanities Press. Reinhardt, William N . 1986 "Somc Remarks on Extending and Interpreting Theories with a Partial Predicate for

Truth." Journal of Philosophic,cil Logic 15: 2 19-5 1

Richard, Jules. 1905 "Les principcs dcs mathkmatiques et le problkme des ensembles." Revue gknkrale

de sc.ienc.e.spurrs et appliyukes 16: 541. English translation by Jean van Heijenoort, "The Principles of Mathematics and the Problem of Scts," in van Hcijcnoort [1967, pp. 142-441.

Rogers, Hanley. Jr. 1967 Theory rf Recursive Func.tions and Efiecti~,e Computabiliy. New York: McGraw-

Hill.

Rosenberg, Jay F.. and Charles Travis. 1971 Readings in the Philosophy c?fLongungc>. Englewood Cliffs. N.J.: Prentice-Hall. Rosser, John Barclay, and Atwell R. Turquette. 1952 Mar~~~-Val i~ed Logics. Amsterdam: North Holland. Russell, Bertrand. 1902 Letter to Gottlob Frege. English translation by Bcvcrly Woodward in van Heijenoort

[1967. pp. 124-251. 1905 "On Denoting." Mind. new series, 14: 479-493. Ryll-Nardzewski, C. 1952 "The Role of the Axiorn of Induction in Elementary Arithmctic." Funtlnrnenta

Matharnaricue 39: 239-63. Schiffer, Stephen. 1987 Remni7nt.s of Meaning. Cambridge: Cambridge University Press. Searle, John R. 1969 Speech Acts. Cambridge: Cambridge University Press. 1978 "Literal Meaning." Erkenrlmis 13: 207-24. Reprinted in Searle [1979, pp. 1 17-

361. 1979 Expression iind Meaning. Cambridge: Cambridge University Press. Sierpinskii, Wacbaw 1958 Cardinal and Ordinal Numbers. Warsaw: Panstwowe wydawnnictwo naukowe.

Smorynski. Carl. 1985 Selj'-Reference and Modal Logic. New York: Springer-Verlag. Smullyan, Raymond M. 1961 Theon r,f'Formul Systems. Princeton, N . J . : Princeton University Prcss.

Solovay, Robert. 1976 "Provability Interpretations of Modal Logic." Israel Journal of'Muthemutics 25:

287-304. Tarski, Alfred. 1935 "Der Warheitsbegriff in den Formalisierten Sprachen." Studiu Log i~u 1: 261405 .

English translation by J. H. Woodger. "The Concept of Truth in Formalized Lan- guages," in Tarski 11956, pp. 152-2781. Page references are to the translation.

1936 "0 ugruntowania naukiowej scmantyki." Pr:e,ylnrl File,-c!/ic,:n! 39: 50-57. English translation by J . H. Woodger, "The Establishment of Scientific Semantics," in Tarski 11956, pp. 401-81. Page refcrcnces are to the translation.

1944 "The Semantic Conception of Truth and the Foundations of Semantics." Philo.sc~plzy and Phenomenologicc~l Research 4: 341-76.

1956 Logic., Semantic-s. Metamcithematic.~. Edited and translated by J. H. Woodger. Oxford: Oxford University Press.

1969 "Truth and Proof." Scientific American 220, no. 6: 63-77.

Tarski. Alfrcd. Andrzcj Mostowski, and Kaphacl M. Robinson. 1953 Unrlrc~idablr Theories. Amsterdam: North Holland.

Tenncnbaum, Stanley. 1959 "Non-archimedian Models of Arithmetic." Notices of the American Mathematical

Society 6: 270.

Van Fraassen. Bas. 1966 "Singular Terms, Truth Valuc Gaps, and Free Logic." Journal of Philosophy 63:

464-95.

Van Heijenoort, Jean. 1967 From Frege to Gijdel. Cambridge, Mass.: Harvard University Press.

Vaught. Robert L. 1967 "Axiomatizability by a Schema." Jourrlal of Symbolic Logic 32: 473-79

Vardanyan, V. A. 1985 "0 Prcdikatnoi Logike Dokazuemosti." Preprint. Moscow: Nauchnyi Sovet po

Kompleksnoi Probleme "Kiberneika," Akademiya Nauk SSSR.

Whitehead, Alfrcd North. and Bertrand Russell. 19 10 Principia Mathemarica. 3 volumes. Cambridge: Cambridge University Press. Page

references are to the second edition, published in 1927.

Wittgenstein, Ludwig. 1953 Philosophic~al Investigcrtions. German text with English translation by G . E. M.

Anscombe. New York: Macmillan. Page references are to the third edition, published in 1968.

Woodruff, Peter W. 1984 "Paradox, Truth, and Logic." Journal of Philo.sophica1 Logic 13: 21 3-32.

Zcrmelo, Ernst. 1904 "Proof That Every Set Can Be Well-ordered." English translation by Stephan Bauer-

Mengelberg in van Heijenoort [1967. pp. 139-411.

Index

/a/ , 113 /!'L/. 20 A,,, 89 A,, 89 '?I-consistency. 50-52 '?[-frame, 202 ?[-logic, 150 YI-logical derivation, 150 ? ) I - ~ l e , 150 ?l-validity, 50-52, 55. See also '3i-validity Abelard, Peter, 41 Acceptable structure, 38 Accessibility relation, 52-53, 5&58. 198 Adams, Ernest W. , ix Adequacy conditions for a detinition of truth,

10, 156-57, 158-59. 185. S e r r t l ~ o materially adequate definition.

Adequacy conditions for a theory of definite truth, 185-86, 198-99

Adequacy conditions for a theory of denotation, 170

Adequacy conditions for a theory of satisfaction, 170

Adequate frame. 202-203 Admissible fragment, 126n Agatha, 105 Agathymus a rwnu , 83-85 Agme rhrysuntha. 83-84 A g a ~ v palmeri, 83-84 h',, 3411 as (x ) , 204

Analyticity. See logical necebsity; analytic- synthetic distinction.

Analytic-synthet~c d~stinction, 41-42. 44, 193 Anti-extension. 87. 120n Application procedure. 127. 139-40, 2 12 Aristotle, 3911% 41. 42. 4313, 54. 101 Arithmetic, language of. 19-20. See nlso

second-order arithmetic. Artemov. S.N , 5 1 n, 59-63 Asher, Nicholas, x Avron, Arnon. 51n Axioms of logic. 2 1

Barcan, Ruth. See Marcus, Ruth Barcan. Barcan formula, 28, 58 Barw~se, K . Jon, 17. 7611, 10711 Redrimr ,for Bonzo, 54 Belnap, Nucl D., Jr., x , 17, 127-38. SEC ~ l s o

rule-of-revision semantics. Belnnp scquence, 130 Bemay\. Paul. 76 Berry. G G . 3 1-32 Beth. Evert, 7 1 Beth'\ theorem. 71 Rr!z, 46-47 B~ldterdl reduction sentence. 148 B~valence, 104. 177-79. 21 1-12, 216 Boolos. George, x. 48. 50, 51n. 59-63. 6311 Bounded formula. 20 Bounded quantifier. 20 Boyd, Richard, 77n Huchanan. James. 49 Building-block method, 13 Built-in coding scheme. 3X Burali-Forti, Cesare, 32n Burge, Tyler, 4n

Camp. J.L.. Jr . 138 Canonical frame, 198

Cantor. Georg. 3211. 139 Carnap. Rudolf, 7. 43. 148-49. 152 -53, 155.

206 Categorical Implicit delinition, 73 Chang, C.C., 5811, 73. 136n, 150, 15 In. 152n Chihara, Charles S . . ix, 2. 1911, 39n. 40n. 138,

14011. 21311 Church, Alonzo. See Church's thesis. Church's thesis. 17. 25. 2511. 45. 60 C'lassical model, 87 Coding schemc, 38 'Coextensive'. 77 Coinduct~ve set, 108 Compactness. 17.50-5 l,98-99, 138. See al.so

countable compactness; recursive compactness.

Complete lattice, l 1 In Completeness theorems, 17, 21, 23, 97-98,

151 Compositional semantics, 12- 13 Conditional proof, rule of. 2 19-22 Conservative extension. 159 Consistent part. 18 1 Context dependence, In, 8-9, 146-47. 214,

215 Convention T, 10 Converse Barcan formula, 58 Correspondence truth. 209- 14 Countable compactness, 50-52, 65-66 Counterpart relation, 57 Covering lemma, 120-21 Craig, William. See Craig's theorem. Craig's theorem, 45-46, 52, 90-100

David, Marian, x Davidson, Donald, 13 decode, 125-26 De dicto necessity, 41--42, 54 Definite liar, 7, 221-22 Dcfinitely true, 150, 153, 162 Definitely truc in the model-theoretic sense,

150 Definitely true in the proof-theoretic sense, 150 Delinitely untrue in the model-theoretic sense,

1 50 Definitely untrue in the proof-theoretic sense,

1 50 A:' relation, 20 Den, 69, 82

Denotat~on. 16, 31-37, 170-71. 214. See ulso Den.

De re belief attributions, 55-56 Dc re necessity. 41 -42. 54 I)lagnostic problcm, 2, 2n, 138, 218--22 Disposition terms. 148-49 Disquotational truth. 209-14 Dollar sign, sentence marked with a, 4 Dummett, Michael A.E.. 22111

E,r. 89 E-. 89 61m Street. 141-42, 175-77 Epimenides. 3, 10, 25 Equi-assertible. 220-2 1 Equi-deniable, 220-2 1 Essentially richer metalanguage, 67-68 Eurhydrmus, 221 Excluded middle, law of the. 104 Expansion, 150 Explicit definition. 68 Extended omitting types theorem. 58,5811. See

cllso omitting types theorem. Extension of a model, 150 Extension of a predicate. 87, 12011 (FI )-(F3). conditions, 202-3 False, 5n Feferman, Solomon, 1811, 9611, 100, 1 12. See

also Kripkc-Feferman axiom system. Field. Hartry, 15-16, 7611, 82-86, 102, 213-

14 Fine, Kit, 7 , 155 Fineline, Ms., 40n Finite sequence, code of a, 38 Finite set, code of a, 69 First-order positive inductive definition, 107 Fixed point. 89, 107 4-valued logic, 1 1 0 - 1 1 Fraenkel, Abraham, 17, 36, 124 Frege, Gottlob, 12, 73-74. 80 Fully defined, 120 Fully interpreted part of a language, 12, 149 Functionally represents, 22

G. modal axiom system, 47-48, 58 G' , modal logic system, 50, 58 Gaifman, Haim. 15n l'i, I07 I-,, 107 r D". 201 -2

I'-cons~\tency, 50-52. 65-66 ['-validity. 47-48. 55. 63-65 Gap-equivalent. 1 12 Gaps, truth-value. Srr 3-valued log~c. Gluts, truth-value. SPC 4-valueij logic. GBdcl. Kurt. 17. 18, 21. 28n. 38. 38n. 43.46.

48. 49. 58. 73-74, 76. See nlso completeness theorems; Godel codes: incompleteness theorems; self-referential lemma.

Giidel codcs. 18-19. 23. 26. 29. 35. 37-38. 42, 4511, 59, 122. 12211. 125, 163. 166

Goldberg, Reuben. I87 Grelling, Kurt. 31n. 32n Grice. H.P.. 44 Grounded sentence. 9 1. 91n Grover. Dorothy, 138 Gupta, Anil. x, Ion. 17. 127-41. 142, 145-

47. 212. 2 14. See also rule-of-rcv~sion semantics.

Gupta sequence. 1 28

Hackett. Frances. x Harrington, 1,eo. 12 In Hayes-Bautista, Roberta, x Heart, sentence marked with a. 14 Hempel, Carl G. . l00n Henkin, Leon, 98, 151 Her~berger, Hans, 17, 127-39. See a l ~ o rule-

of-revision semantics. Her~berger sequence, 120 Hierarchy of languages. See Tarski hierarchy,

Tarskl-Kripke hierarchy. Holism. 47-49. 66, 145. 208. See c~lvo holistic

method, unity of science. Holistic method, 13 Homophonic translation, 13. 1311 Howe, William, 83-84 Hullett, James, x HYP,, , 12611 Hyperarithmetical set, 108n Hyperelementary set, 108

Idiolect. 10-1 1 . 13-16 Implication, material, 42-43 Implication. strict. 42-43 Implicit definition, 72-73. 172 Incompleteness theorems, 17, 28. 46-47. 58.

180-8 1 , 205, 207 Indeterminacy of translation. 10. 13- 15 Indexicals. See context dependence.

Inducr~on axiorn hchcrna. 24 Induct~ve detjnition. 107-8 Induct~ve 4et. 108 l n ~ t ~ a l segment. 57n Inwrutability of reference, 11-15 111.\oluhiiicl. 66. 66n Integr~ty of the language requirement. 159.

183-84 lnterpretat~on o f a tirst-order language. 20 Interpretation of a modal language, 47. 54

Jech. Thoma\. 124n J~rn. 178 Jocko. 101-2. 217-18 .lone{. 21 1-12 Jule\, 178

Kaplan. David. 54. 5511, paradox of knowledge, 16, 27. 27n. 39

Karp. Carol. 80n Keisler, H. Jerome, SXn, 73. 13611. 150, 15 In.

152 Kelly. Michael. x KF. See Kripkc-Feferman axiom system Kirklln. Daniel, x Kleene. Stephen Cole. 87. 108, 10811. See t~ lso

.?-valued logic. Kneale, Martha. 66n Kncale. William Calven. 41, 66n Knowledge, paradox of. See Kaplan, David:

paradox of knowledge. Kiinig. Julius. 3 1 , 3211 Kr. 169 Kremer, Michael. 96n Kripkc, Saul A., 4711; theory of necessity, 28.

56, 196; theory of truth. 1 6 1 7 , 87-126, 148, 159. 160-62, 168. 170-71, 174, 18 1 , 183, 187, 209. See also Kripke semantics for modal logic; Kripke-definable fragment; Kripke-Feferman axiom system; Tarski-Kripke hierarchy.

Kripke-definable fragment, 125-26. 172 Kripke-Feferrnan axiom system, 92-93. 99.

100. 104-6, 168-170 Kripke semantics for modal loglc, 52-53, 56-

58. 196-204

(L). schema, 47 Y,,, 80. 80n, 124-26 ~ ( ; , 6 ) , 113-15

Statements, 9 States of affalrs, 39-40 Strawson, Peter F., 44 Strengthened liar response. 5 Strengthened 11ar sentence, 4 Strong adequacy cond~tlons, 185-86, 199-200 Strongly dehnable. 172 Strongly repre9ents. 22 Structure. 20 Sturgeon. Scott. x SUB. 126 Submodel. 150 Substitutional quant~ticatcon, 3711, 40n Supcrvaluations. See van Fraassen. Has

T, convention, 10 (T). schema, vii, 6 , 8 Tarski, Alfred, vii, 9 , 10, 16, 21, 22, 25, 2511,

26, 27, 3911. 67 -86, 104, 147, 209. See also Tarski hierarchy; Tarski-Kripke hierarchy.

Tarski hierarchy. XO&XI. 122. 172. See ulso Tarski-Kripke hierarchy.

Tarski-Kripke hierarchy, 122-23, 172. See also Tarski hierarchy.

Tennenbaum, Stanley, 62, 6211, 64 Theaetetus, 210, 21 In Theoretical entities, 154-55, 21311 Therapeutic problem, 2 3-valued conscquencc, 96-98 .?-valued derivation, 96-98 3-valued interpretation. 96 3-valued logic. 87-106, 148,155, 161-62, 168 Toby's cat, 6 Tokens v.,. types, 19 Translation, 10-16 Truthteller sentence. 8 Truth undcr an interpretation. 2 1 Truth-value gaps. See 3-valued logic. Truth-value gluts. See 4-valued logic. Turing. Alan. See Turing machine. Turing machine, 59. 61, 63

Twin Earth argument, 10 Types vs. tokens, 19

Unequivocal. 162 Unity of science, ix, 79, 81. Set, ulso holism. Universal gcncralization, rulc of. 58 Universality of natural languages. 70 Untruth, 5n Urelements. 7611, 154-55, 21 3n

V,, 3411 Vagueness. viii. 7-8. 4511. 155-57, 210--12,

2 1 6 1 8 Valence detinition. 82 Van Fraassen. Bas, 7. 77n. 94, 15.5, 16662 .

183 Vardanyan, V.A., 63-65 Variable assignment, 70n Variables, 20 Vaught, Robert. 65 Vcrmazcn. Bruce, ix Vissar, Albert, x. 51n

Weakly detinable, 17 1 Weakly represents, 22 Weakly btable, 129 Wendy, 174-75 Whitehead, Alfred North, 3211, 42, 4211, 44 Wittgenstein, Ludwig, vii, ix, 4 , 6 , 15, 25-26 Woodruff, Peter, x. 89-90, 1 I & I I . 1 19-20,

I62

Xochitl, 141-42, 174-77, 209

Yablo, Stephen, ix Yolanda, 141-42, 175-77, 209 Yuan dynasty, vii

Zaida, 141-42, 175-77 Zerrnelo. Emst, 17, 36, 124, 153 Zorn, Max, 90 Zorn's lemma, 90

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

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