Principles of Truth 9783110332728, 9783110332513

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Volker Halbach • Leon Horsten Principles of Truth

Epistemische Studien Schriften zur Erkenntnis- und Wissenschaftstheorie Herausgegeben von Michael Esfeld • Stephan Hartmann • Mike Sandbothe Band 1 / Volume 1

Volker Halbach • Leon Horsten (Editors)

Principles of Truth

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Volker Halbach and Leon Horsten

Preface The present volume is a record of the lectures given at the conference Truth, Necessity and Provability, which was held in Leuven, Belgium, from 18 to 20 November 1999. Except for the paper by Shapiro, all papers included in this volume are based on lectures given at the conference. On the one hand, the concept of truth is a major research subject in analytical philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. The aim of the conference out of which the present volume grew was to bring together prominent logicians and philosophers concerned with truth and the paradoxes. We wanted to promote a deeper interaction and collaboration between them than existed so far. The leading motivation was that recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth and that, conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. The present volume is therefore intended both for analytical philosophers working on truth and for logicians who concern themselves with the paradoxes. The contributions in this volume present an overview of recent work that has been done on the interface between these two domains. The amount of knowledge of analytical philosophy and of mathematical logic that is required for understanding the papers in this volume is of a quite modest level. Nevertheless, in order to make the contributions as accessible as possible both to analytical philosophers (graduate students and professional philosophers) and to logicians, the collection of papers is preceded by an extended introductory historical overview of the main results and tenets in philosophical and logical research on truth since the 1930s. The papers have been grouped into three parts. We have opted for one cumulative bibliography at the end of the book.

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Acknowledgements. Ingrid Lombaerts took excellent care of many of the practical aspects concerning the organisation of the conference from which this volume originates. We do not know how we would have managed without her. We are grateful to all authors we invited for their contributions to this volume and for their patience with this project. We thank the managing editor Rafael Hüntelmann and the editors of the book series Epistemische Studien, Michael Esfeld, Stephan Hartmann and Mike Sandbothe, for their support of this project. Christopher von Bülow did most of the LATEX-typesetting and proofreading. We are indebted to him for the painstaking preparation of the final version and for correcting some mistakes in the introduction. All remaining inconsistencies in the typesetting are due to the editors. Special thanks go to Ina Ratzke for her help with Word Perfect. Volker Halbach & Leon Horsten

March 2002

Contents Volker Halbach and Leon Horsten Preface

7

Volker Halbach and Leon Horsten Contemporary Methods for Investigating the Concept of Truth – An Introduction 11 John P. Burgess Is There a Problem about the Deflationary Theory of Truth?

37

Paul Horwich A Defense of Minimalism

57

Volker Halbach Modalized Disquotationalism

75

Stewart Shapiro Deflation and Conservation

103

Hannes Leitgeb Metaworlds: A Possible-Worlds Semantics for Truth

129

Vann McGee Ramsey and the Correspondence Theory

153

Michael Sheard Truth, Provability, and Naive Criteria

169

Andrea Cantini Partial Truth

183

10

Contents

Leon Horsten An Axiomatic Investigation of Provability as a Primitive Predicate

203

Bibliography

221

Index of Persons

235

Subject Index

239

Notes on the Contributors

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Volker Halbach and Leon Horsten

Contemporary Methods for Investigating the Concept of Truth An Introduction Truth as a logico-mathematical concept Truth is ubiquitous in traditional philosophy as a venerable and deep notion. Most traditional accounts of truth and modern substantialist theories of truth hold that truth is a philosophical notion that needs to be explained in philosophical terms such as correspondence, utility, coherence. It therefore comes as no surprise that truth is exactly the kind of notion that logical empiricism tried to ban from philosophy altogether. Yet today, truth is one of the important research topics in analytic philosophy. Whence the change? In the early thirties Tarski rehabilitated truth as a respectable notion in his famous work “The Concept of Truth in Formalized Languages” (1935). He gave a definition of truth for a formal language in purely logical and mathematical terms. Tarski’s method is fairly general. This suggested that truth is a logico-mathematical notion, and not an irreducible, properly philosophical notion as traditional philosophy would have it. Like all other logico-mathematical concepts, truth is most naturally investigated in formal settings, in the context of other logical and mathematical notions. Attempts should be made to analyze truth in these formal contexts, and the resulting definitions, statements and theorems ought to be regarded as logico-mathematical truths. Tarski’s contemporary heirs are the deflationists. There exists a variety of mutually incompatible theories of truth nowadays that are called deflationary. Deflationism with respect to truth therefore is a somewhat vague or at least underdetermined notion. Moreover, all such accounts differ in several

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respects from Tarski’s own views on truth. We will see, for example, that most deflationary theories of today incorporate an axiomatization of truth rather than a definition. Nevertheless deflationists side with Tarski in his insistence that truth is a logico-mathematical concept, and the axiomatic approach and Tarski’s definitional one are intimately connected, as will become obvious below. On the formal side, logicians have investigated Tarski’s theory and its variants using the methods of proof- and recursion theory. Moreover, Tarski’s work on truth in the thirties was the start of modern model theory. Some of these results are relevant also to the deflationary theories. For instance, recently certain proof- and model-theoretic results on conservativeness have proven to be important in the discussion on deflationism. Most formal work of the last quarter century focused on type-free theories. In these theories the truth predicate can be applied to sentences that contain the same truth predicate themselves. In the following we browse through some important varieties of logicomathematical theories. They include Tarski’s theory, the relatively vague family of deflationist theories, as well as formal axiomatic and semantical theories of truth.

Tarski’s theory of truth Tarski’s theory is the point of departure for most, if not all, recent work on truth. Rather than giving a historically precise account of Tarski’s theory we elaborate those features that are most important for later developments. At some points we mention results and tools not known or available to Tarski, but which turn out to be important for Tarski’s account. We choose a setting that is different from Tarski’s original framework but which is more suitable for our present concerns. This allows us, for instance, to define truth directly, while Tarski had to take the detour via a satisfaction predicate. Avoiding this detour is only possible in certain special cases. Tarski’s truth predicate pertains to a specific formal language, the socalled object language. We choose the language LPA of arithmetic in order to illustrate Tarski’s approach. Besides the usual vocabulary of first-order predicate logic with identity, it has a constant symbol 0 for zero, a function symbol S for the successor function, and the binary function symbols + and · for the operations of addition and multiplication, respectively. We do not presuppose that the object language is interpreted in any way or that a deductive system for it is given. We only need to know what the well-formed expressions of LPA are.

Introduction

13

The truth predicate does not belong to the object language LPA . It forms part of the metalanguage LM . In the simplest case the object language is a sublanguage of the metalanguage, i. e., all formulas of the object language are also formulas of the metalanguage. If the metalanguage does not include the object language, then a translation of the object- into the metalanguage is required. Whether a translation is correct depends of course on the meaning of the sentences of the object language, whatever “meaning” is. Thus the need for a translation has caused many worries. We stick to the simple case and assume that LPA is a sublanguage of the metalanguage LM . Tarski develops his theory at first in natural language enriched by some mathematical symbols. However, Tarski is aware of the requirement for a precisely defined metalanguage that allows for strict formal proofs in it. For the metalanguage LM Tarski needs some rules governing the use of the expressions of the object language. Tarski assumed that we have axioms and rules in the language LM that allow for derivations within the metalanguage. He called the resulting theory “Metawissenschaft”, which is usually translated as “metatheory”. Tarski stated some conditions that should be satisfied by the metatheory. It is a deductive system with axioms and rules for LM . This theory contains the theory of the object language (if there is any) and allows to prove certain facts about expressions. In order to distinguish the metatheory from the pure uninterpreted and unaxiomatized metalanguage, we denote the former by MT and the latter by LM . Tarski assumed that the metalanguage LM has a name for each sentence of the object language. However, not any kind of name will do. It must be possible to read off the form of the sentence from its name. For this purpose Tarski introduced his structural-descriptive names. We do not discuss Tarski’s original approach but we present two examples of naming systems that comply with the requirement that one must be able to recover the shape of an expression from its name. In natural language the quotational name of a sentence satisfies the condition that the name has to reveal the structure of the sentence: the singular term “Snow is white” designates the sentence within the quotation marks and thus the name displays the exact shape of the sentence. For mathematical languages, codings are used. A coding (or Gödel numbering) associates with any expression e of, say, LPA a natural number peq in an effective way. The details of a typical coding are given, e.g., by Smoryński 1985. For the number peq we have a name in the language LPA : for any

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number n, the expression S(S(. . . S(0) . . .)) | {z } | {z } n times

n

is its numeral n. For instance, the numeral 3 of the number 3 is the LPA expression S(S(S(0))). Via the coding we obtain names for sentences in the language LPA : we employ the numeral pAq of the code pAq as a name for any sentence A of LPA . Since the metalanguage LM contains the object language LPA and therefore also all numerals, LM has names for all sentences of the object language LPA . Tarski aimed at a definition of truth. A (potential) definition of truth in the metalanguage LM plus the symbol T is a definition of the primitive predicate symbol T in the metalanguage LM which takes the following form:
∀x Tx ↔ TDef (x) ,

(1)

where TDef (x) is a complex formula in the metalanguage LM , containing one free variable (x). The resulting theory is a definitional extension of the theory MT. We are sloppy and call the formula TDef (x) itself a truth definition. Of course, the sentence (1) is the general pattern for introducing new unary predicates by an explicit definition. Thus, obviously, not just any choice of TDef (x) is acceptable as a definition of truth. However, there may be different acceptable choices. What are the distinguishing features of an adequate definition of truth? Tarski’s answer to this question is his material adequacy condition. Whether a definition is materially adequate depends on the metatheory MT. The metatheory must prove certain things about the predicate TDef (x) that is supposed to define truth. In the terms of our present setup involving LPA as object language and a metalanguage including LPA , Tarski’s condition can be rephrased as follows: A definition of truth is adequate if and only if the sentence TDef (pAq) ↔ A

is provable in MT for every sentence A of LPA . The sentences TDef (pAq) ↔ A are Tarski’s famous T-sentences; together they form the T-scheme. Tarski (1935) requires for the adequacy also that

Introduction

15

MT proves that only sentences (or their codes) are true; we neglect this additional restriction, which is also not essential according to Tarski. Whether there is a materially adequate truth definition patently depends on the metatheory. One obvious choice does not work: if MT is a theory in the language LPA extending the theory R of arithmetic (Robinson’s arithmetic), then it can contain no adequate definition of truth. This observation is a slight variant of Tarski’s famous theorem on the undefinability of truth for LM . Robinson’s arithmetic R is an extremely weak theory. It is much weaker than the better-known theory Peano arithmetic ( PA). Thus the theorem on undefinability tells us that the metalanguage LM must properly extend the object language LPA , if there is to be an adequate truth definition. Moreover, the metatheory MT must feature axioms and/or rules for this additional vocabulary that allow to derive all T-sentences. In order to define truth, Tarski first defined the notion of satisfaction, and from this he defined truth. As pointed out above, we can avoid this detour and define truth directly by induction on the complexity of sentences: 1. If s and t are closed terms of LPA with the same value, then s = t is true. 2. If the sentence A is not true, then ¬A is true. 3. If A and B are true, so is A ∧ B . 4. If all sentences A(n) are true, so is ∀x A(x). Here A and B are sentences of LPA . In the last clause it is assumed that bound variables are renamed in the case of a variable collision. This recursive definition of truth can be formalized in a sufficiently strong subtheory of second-order arithmetic. The theory ACA (not used by Tarski) that features a comprehension scheme for arithmetical sets and an induction scheme for all second-order formulas suffices for proving the existence of a set of formulas satisfying the above clauses. Thus one can construct a formula True(x) in the language of second-order arithmetic such that the theory ACA proves the following equivalence for all sentences of LPA : True(pAq) ↔ A.

Therefore True(x) (or rather the result of substituting True(x) for TDef (x) in Formula 1 on the facing page) is a materially adequate definition of truth. ACA is not the weakest subsystem of second-order arithmetic that allows for a materially adequate definition of truth in Tarski’s sense. Tarski was

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aware of the fact that different metatheories can provide materially adequate theories. Later logicians like Mostowski (1950) gave a more detailed analysis of what exactly is needed in the metatheory, though the latter worked in a set-theoretic context. This rough sketch of Tarski’s theory and its later developments concludes our exposition of the (type-free) semantical theory of truth. Tarski’s paper also contained a few tentative remarks on axiomatic theories. Tarski tried to dispel the neopositivist distrust of truth by providing a definition of truth, at least for certain restricted areas of application. But in his article “The Concept of Truth in Formalized Languages” he already considered axiomatic approaches, which he finally rejected. For instance, Tarski considered the option of taking the T-sentences as axioms. One may speculate that he was sceptical of such approaches and opted for a reduction of the concept of truth by definition because it was not fashionable in neopositivism to take truth as a primitive undefined symbol. Tarski’s official arguments against axiomatic approaches, however, are different. According to Tarski, the T-sentences do not prove certain important principles. For instance, they do not prove the universal sentence saying that for any sentence the sentence itself or its negation is true. Furthermore Tarski suggests that any strengthening of the T-sentences will remain incomplete and leave out other important principles (see also Halbach 2000a). He apparently did not suspect that “nice” axiomatizations of truth would be found. This topic recurs today in the discussion of deflationism, as we shall show below.

Deflationism Unlike Tarski, most deflationists today do not attempt to define truth. Two factors have played a role here. First, Tarski’s theory diminished the distrust in the notion of truth. His work was taken to have shown that truth is a safe concept. Second, as time went by, a shift in attitude toward secondorder logic gradually took place. Until the thirties, second-order logic was regarded as relatively unproblematic, and even as a part of logic according to logicism. Second-order logic was used without reservations by Russell and Whitehead, for instance. But from the fifties onwards, second-order logic was regarded with mounting suspicion. It became increasingly clear that systems of second-order logic are quite powerful. From a philosophical point of view, Quine insisted that second-order logic is really set theory in disguise, and therefore is not without ontological commitments. Thus second-order logic and axiomatized systems of second-order logic slowly

Introduction

17

came to be regarded with apprehension. But if a definition of truth is considered unattainable because of such a definition’s entanglement with second-order logic, why not take truth to be a primitive notion? In the following we will explore the main varieties of axiomatic theories of truth. Fortunately deflationists can build on the work of logicians, in particular proof theorists, who have worked on axiomatic theories of truth. Their aim was not to provide foundations for deflationism. Rather they mostly focused on the role of truth in the foundations of mathematics, where truth turned out to be as useful as certain kinds of second-order quantification. However, their work comes in handy for the deflationist.

Typed axiomatic theories of truth By typed theories we mean deductive systems that do not prove the truth of any sentence containing the truth predicate T. Typed systems cannot prove q p T T p0 = 0q ,

for instance. They are opposed to type-free systems, which are also known as theories of self-referential truth. We shall start to sketch simple and deductively weak truth theories and then proceed to the stronger and more sophisticated theories (see Sheard 1994 and Halbach 2000b for more extensive overviews). The reader less interested in the technical elaborations may skip this section. Suppose we want to construct an axiomatic theory of truth for LPA . By employing the T-sentences as the sole axioms for truth, one obtains a minimal theory of truth, for every materially adequate theory of truth must contain the T-sentences as theorems. Thus a theory of truth based only on the T-sentences is materially adequate and contains no superfluous additional commitments. Formally, this theory is formulated in the expanded language LT , which is the language LPA of arithmetic plus an additional unary predicate symbol T. The theory is the closure under first-order logic of PA plus the infinite collection of all sentences of the form T pAq ↔ A,

with A any sentence of LPA . Let us call this theory the Disquotational Theory of Truth (DT ) for LPA . There are a couple of remaining issues to be settled here. First, one may wonder whether in instances of the induction scheme of DT, occurrences of

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the truth predicate T are allowed. Call the theory which is just like PA, except that T may occur in instances of the induction scheme, PAT. Obviously DT, which is based on PA, is weaker than the corresponding system based on PAT. One may also wonder whether in the biconditionals, A can be allowed to contain free variables. The problem, of course, is how to bind free variables in pAq. In arithmetic this can be achieved. For instance, if x is free in A then one says formally that the result of substituting the numeral x for the variable x in A is true. This way, one can quantify into the “Gödel corners”. We indicate that A may contain free variables that can be bound from outside by using the bracket notation “[A]”. Thus one may look at the universal closure of the following sentences, where A is a formula of LPA with arbitrary free variables: T[A] ↔ A.

Augmenting PA by these stronger axioms yields a theory proving more sentences. But again it does not prove more arithmetical sentences than PA or DT (see Halbach 1999a). DT and its variants are attractive for deflationists: the axioms are simple, truly deflationary, and they rely on the “disquotational feature” of truth. This fits many deflationary accounts of truth where truth is described as nothing more than a device of disquotation. Furthermore, DT and the variants mentioned do not really have substantial consequences in a sense to be explained below. The deductive weakness, however, is also the Achilles’ heel of deflationary accounts of truth based on the T-sentences as the only axioms. This became clear when opponents of deflationism, e.g., Gupta (1993a), attacked deflationist accounts like Horwich’s (1998b) minimalism because they seem to rely on axioms akin to the T-sentences. Nevertheless, it is important to bear in mind that in his contribution to this volume, Horwich emphasizes that his aim was, and is, not to construct a theory of truth, in the sense of a set of fundamental theoretical postulates on the basis of which all other facts about truth can be explained. Rather, as he puts it, his approach purports to specify which of the nonsemantic facts about the word ‘true’ is responsible for its meaning what it does. And his suggestion is that our allegiance to the T-sentences is the key here. In order to illustrate the deductive weakness of DT, we employ a further notational convention for the bracket notation: if a sentence A occurs only within the brackets “[ . . . ]” within another formula, then A is treated like a free variable. For instance, in T[¬A] → ¬T[A] we may quantify over A. If we prefix a quantifier ranging over (codes of ) sentences A of LPA then we

Introduction

19

obtain the sentence saying that for all sentences A, if ¬A is true then A itself is not true. We refer the reader to the literature for an exact account. DT is deductively weak because it does not prove certain general principles that may be expected to be provable in a good theory of truth. This observation, as well as the following example, is due to Tarski (1935). DT does not prove the following sentence prefixed by a universal quantifier ranging over all sentences A of LPA : T[A] ∨ T[¬A];

(2)

DT only proves the corresponding scheme, i. e., DT only proves T pAq ∨ T p¬Aq

for all sentences A of LPA , but not the universal principle. This deductive limitation holds also for the stronger variant of DT considered above, which is based on PAT and in which the T-sentences take the form of universal closures. Already Tarski (1935) rejected theories that are based on the T-sentences as the only axioms for truth. If the T-sentences are the only acceptable truth axioms for the deflationist, then, epigrammatically speaking, Tarski refuted deflationism prior to its birth. If one wants one’s truth theory to prove such general principles like the above law of excluded third, then one has to add further axioms. Consequently deflationists tried to overcome this deficiency by opting for stronger theories. For instance, Field (1994a), Tennant (2002) and McGrath (1997) (building on Sosa 1993) have proposed theories that prove more than just the T-sentences or similar principles. These theories overcome the deductive weakness of the truth theories that have been used by deflationists previously. By Gödel’s incompleteness theorems all theories extending DT are incomplete. Thus a sound theory can never be complete in the sense that it decides all sentences formulated in LT . However, one can still hope to arrive at a complete theory in the sense that the theory proves all truth-theoretic principles we may ever think of. Such a theory would not prove certain sentences, because of the first incompleteness theorem. However, we do not expect a good theory of truth to decide those sentences. For there is no good truth-theoretic axiom for the truth predicate of LPA that settles these sentences, or so one might argue. In fact, the theory PA(S) which we are going to describe now may have such a status. It does not decide all sentences of LPA , let alone all sentences of LT . The examples of such undecided sentences fall into two categories:

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either they are Gödel sentences, consistency statements, etc., or they are strong combinatorial principles. We should not expect that these sentences can be decided by invoking a truth predicate for LPA . All known proofs of them rely on a truth predicate for more inclusive languages like LT , or on further assumptions like strong second-order axioms. We now turn to the formulation of the theory PA(S). Aside from Tarski’s material adequacy condition, another core intuition concerning truth is that the truth predicate distributes over the logical connectives and the quantifiers. That is, for instance, a sentence A∧B is true if and only if both A and B are true. This intuition lies behind Tarski’s inductive definition of truth sketched on page 15. The theory PA(S) is obtained by turning the clauses for every connective and quantifier into axioms. For simplicity we assume that LPA has only ¬, ∧ and ∀ as its logical symbols. Like DT, the theory PA(S) is again a typed theory, that is, it is a theory of truth for the language LPA . PA(S) is formulated in LT . It is the first-order closure of PA with induction in LT plus the axioms obtained by closing the following formulas with quantifiers restricted to sentences (and formulas with at most one free variable, in the last clause) of LPA : 1. T[A] ↔ A,

if A is an atomic formula of LPA ,

2. T[A∧B] ↔ T[A] ∧ T[B], 3. T[¬A] ↔ ¬T[A], 4. T[∀x A] ↔ ∀y T A(y/x) . 



The last axiom says that a universally quantified formula is true if and only if all its instances are true. If x occurs in A in the scope of a quantifier ∀y , then we rename the variable y before y is finally substituted for x. Of course, A(y/x) then is the result of this substitution.1 PA(S) is also called the Compositional Theory of Truth. The axioms reduce the truth of complex sentences to the truth of less complex sentences. Since the axioms of PA(S) prove all T-sentences, DT is a subtheory of PA(S). The theory PA(S) is much stronger than DT. For instance, 1 [This note is intended for the technically-inclined reader, and can be skipped by others without loss of continuity.] One might wonder why a new variable y is needed and why one does not employ T[∀x A] ↔ ∀x T[A] as an axiom instead. The difference may be seen by looking at nonstandard models. In Axiom 4, x may be a nonstandard variable because it occurs only within the brackets “[ . . . ]”, while y has to be standard. Consequently, Axiom 4 tells us that any sentence ∀x A is true (for any, possibly nonstandard, variable x) if and only if all its numerical instances are true. In contrast, the alternative formulation says this only for standard variables.

Introduction

21

Axiom 3 proves the principle of excluded third (Formula 2 on page 19), which is not provable in DT.

Conservativeness Variants of DT, and, to a lesser extent, PA(S), are the theories of truth of choice for deflationists, at least if one sticks to typed theories. We have already mentioned some obvious proof-theoretic properties of DT, its variants, and PA(S). In the recent discussion on deflationism, further proof-theoretic observations have been used. Some deflationists even claim that truth is a purely logical device. Deflationists generally do not use mathematical tools for setting up their theory. They axiomatize truth, and thus they do not need mathematics for defining it. But this alone does not establish that their concept of truth is purely logical. If truth actually is purely logical, then it should be “neutral”. It should not decide “substantial” nonlogical matters; the theory should not yield consequences underivable by pure logic (or perhaps a little bit more). In technical terms, these philosophers claim that their theories of truth are conservative. Clearly, the weaker the theory, the more likely it is to be conservative. Thus the T-sentences are the deflationist’s best bet with respect to conservativeness. However, the T-sentences themselves already prove “something about the world”: they prove that there are at least two different objects, because the codes of a tautology and of a contradiction must be different (see Halbach 2001b). In other words, the T-sentences are nonconservative over the underlying first-order logic. In the light of this trivial observation it is highly implausible to say that truth is a logical concept. For even disquotational truth is not ontologically neutral, as one might expect from a logical notion. The ontological commitment of the T-sentences is very small: they do not prove the existence of further objects, i. e., the T-sentences do not exclude that all true sentences are identical (and the same for all false sentences). Now, even first-order predicate logic is not completely neutral; for it already proves that at least one thing exists.2 In the light of this, we should not be concerned too much because our theory of truth proves the existence of a further object. This becomes plausible if we agree with Tarski that a theory of truth without an underlying theory of the objects which may 2 For this reason many philosophers claim that free logic, which does not prove the existence of anything, is more adequate as pure logic.

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be true does not make much sense. Tarski wanted such a theory to form part of his metatheory. Thus a deflationist could still try to save a kind of conservativeness by claiming that a notion of truth should be conservative over the underlying arithmetical theory, Peano arithmetic. Somehow, the minimal ontological commitment of the T-sentences has to be tolerated by all deflationists. However, as the contributions to this volume bear out, there is disagreement in the deflationist camp whether an acceptable theory of truth ought to be conservative over the arithmetical theory to which it is added. If it were conservative then truth would be neutral at least in the context of a certain presupposed theory. At least in the context of PA truth would be like a logical notion because it would not allow for new mathematical insights. In this respect truth would be free of mathematical commitments. The theory DT is successful in this respect: DT is conservative over PA, that is, DT does not prove any sentence that is not already a consequence of PA. As we have seen, however, DT is hardly sufficient as a theory of truth because it is deductively too weak. Thus we turn to PA(S). This theory proves new arithmetical truths that are not provable in PA, whence it is not arithmetically conservative over PA. In fact PA(S) is proof-theoretically strong: it does not only prove the consistency of PA, but many more and stronger arithmetical truths (see, e.g., Feferman 1991). In this sense, it appears that truth yields new “substantial” insights. Ketland (1999) and Shapiro (1998) have attacked deflationism on these grounds: How can a theory of truth be “deflationary” if it proves mathematically substantial theorems? The commitments of the theory PA(S) are the same as those of the second-order theory ACA mentioned earlier (see again Feferman 1991). Hence the truth predicate of PA(S) is surely not deflationary with respect to its mathematical consequences. In response, Field (1999) maintains that deflationism implies only that PA(S) without the induction axioms involving truth has to be conservative over PA. However, it must be kept in mind here that although this theory is indeed conservative over PA, this fact is far from obvious. Kotlarski, Krajewski, and Lachlan (1981) employed an ingenious model-theoretic argument in order to prove that PA(S) is conservative over PA. That the proof cannot be trivial follows from a deep theorem proved by Lachlan (1981), according to which only very special models of PA can be expanded to models of PA(S). Recently, proof-theoretic arguments for the conservativeness of PA(S) over PA have also become available (Halbach 1999a). Some deflationists, however, explicitly distance themselves from the com-

Introduction

23

mitment that an acceptable theory of truth must be arithmetically conservative. It seems that in this context, the distinction between considering truth to be a logical notion and considering truth to be a logico-mathematical notion really makes a difference. Even the truth predicate of PA(S) is deflationary in the sense that it is mathematical. I. e., the theory is independent of the physical world and does not depend on any contingent features.

Semantic theories of type-free truth Tarski’s inductive definition of truth is a semantic theory. The axiomatic counterpart is the theory PA(S) because PA(S) has the inductive clauses as axioms. The recursion-theoretic properties of the semantic construction and the proof-theoretic properties of the corresponding typed theories are fairly well understood. Logicians sought new challenges by studying type-free theories of truth. Many different theories have been developed and in this introduction we cannot even attempt to sketch all of the most important ones. The reader may turn to the monographs Brendel 1992, McGee 1991, Gupta and Belnap 1993, Cantini 1996, and Halbach 1996. Here, we restrict ourselves to a particularly successful semantical theory and its axiomatizations, namely Kripke’s theory of truth. Several logicians experimented with partial logic, supervaluations and transfinite inductive definitions in order to set up type-free theories of truth. Kripke (1975) took up these developments, unified them in a very general setting, and found phenomena that had gone unnoticed. Moreover, he availed himself of the theory of positive inductive definitions that had been studied in detail and in generality shortly before by Moschovakis (1974) and others. Kripke exploited the fact that Tarski’s theorem on the undefinability of truth does not apply in an unqualified manner to partially interpreted languages. He constructed partial models for LPA of which it can be said in a somewhat weakened sense that they make the unrestricted Tarskibiconditional sentences true. His models are formed in stages, indexed by ordinals. The simplest of Kripke’s models, the so-called least-fixed-point model of the strong Kleene scheme, is constructed along the following lines. One wants to build a model for the language LT . The arithmetical vocabulary is interpreted throughout as in the standard model N; the truth predicate T will be the only partially interpreted symbol: it will receive, at each ordinal stage, an extension E and an anti-extension A. The union E ∪ A does not exhaust the domain; otherwise T would be a total predicate. The

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extension E of T is the collection of (codes of ) sentences which are (at the given stage) determinately true; the anti-extension A of T is the collection of (codes of ) sentences which are (at the given stage) determinately false. A partial model M for LT can then be identified with the ordered pair (E, A). In general, the union of the extension and the anti-extension will not exhaust the collection of all sentences: some sentences will at each ordinal stage retain their indeterminate status. An example of an eternally indeterminate sentence is the Liar sentence L such that PA ` L ↔ ¬T pLq.

The intuition that the Liar argument tells us that the Liar sentence L cannot have a determinate truth-value is the basic motivation for constructing a theory of truth in which T is treated as a partial predicate. At stage 0, one lets the extension E0 and the anti-extension A0 be empty. This yields a partial model, M0 = (E0 , A0 ) = (∅, ∅). Next, one uses a popular evaluation scheme for partial logic, the so-called strong Kleene scheme. The strong Kleene evaluation scheme sk is defined as follows: 1. For any atomic formula F x1 . . . xn : M sk F x1 . . . xn (resp., M sk ¬F x1 . . . xn ) iff the n-tuple (o1 , . . . , on ), where oi is assigned to xi for i = 1, . . . , n, belongs to the extension (anti-extension) of F . 2. For all formulas A, B : • M sk A ∧ B iff M sk A and M sk B ; • M sk ¬(A ∧ B) iff either M sk ¬A or M sk ¬B (or both); • M sk ∀x A iff for all n, M sk A(n/x); • M sk ¬∀x A iff for at least one n, M sk ¬A(n/x).

One now uses this strong Kleene scheme to determine the collection E1 of sentences of LT that are made true by M0 , and the collection A1 of sentences of LT the negation of which is made true by M0 . Thus a new partial model M1 = (E1 , A1 ) is obtained. Using M1 , a new extension E2 and a new anti-extension A2 are then determined on the basis of M1 , and so on. In general, for any ordinal α, Eα+1

≡



A ∈ LT Mα sk A

Aα+1

≡



A ∈ LT Mα sk ¬A .

and

25

Introduction

For limit stages λ, we set Eλ

≡

[

Eκ

κ(s)    > ϕ , g(e, ϕ) := ¬g(e, ψ),     g(e, ψ1 ) → g(e, ψ2 ),    ∀x g(e, ψ),

if ϕ is t, if ϕ is T(t, s), if ϕ is an atomic formula of LB , if ϕ is ¬ψ, if ϕ is ψ1 → ψ2 , if ϕ is ∀x ψ.

In the first case, [e. ] is the function symbol for the primitive recursive function with index e. The assumption that function symbols for all primitive recursive functions are available in the language of arithmetic renders the notation ˙ stands for somewhat simpler; it can easily be avoided. The expression t(s) the result of replacing all free variables in t by the numeral of s. The values of g where e is not the index of a primitive recursive function and ϕ no formula of L T do not matter. Now apply the recursion theorem to obtain the index e of a primitive recursive function [e] satisfying the following equation: [e](ϕ) = g(e, ϕ).

I write Im for the function [e] with index e; it has the following properties: 
BewACA−
I. m (t) ,     ˙ , Tr t>(s)    > ϕ , Im (ϕ) := ¬Im (ψ),     I (ψ ) → Im (ψ2 ),    m 1 ∀x Im (ψ),

if ϕ is t, if ϕ is T(t, s), if ϕ is an atomic formula of LB , if ϕ is ¬ψ, if ϕ is ψ1 → ψ2 , if ϕ is ∀x ψ.

I. m is here the function symbol in the language of arithmetic for the function Im .

Assume n < m and that In+1 has been already defined, and put Sn :=



In+1 (ζ) ζ is at level n+1 ∪ ACA−.

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Sn is a finite extension of ACA , so there is a canonical proof predicate BewSn for the theory Sn . Then the function In is defined as follows: 
BewSn I.
n+1 (t) , if ϕ is t,     ˙ , if ϕ is T(t, s), Tr t>(s)    > if ϕ is an atomic formula of LB , ϕ , In (ϕ) := ¬I (ψ), if ϕ is ¬ψ,  n    I (ψ ) → I (ψ ), if ϕ is ψ1 → ψ2 ,  n 2   n 1 ∀x In (ψ), if ϕ is ∀x ψ. I(Π) is the tree structure which results from replacing for any level n all formulas ϕ at level n with In (ϕ). Obviously, it is sufficient to show that I(Π) can be complemented into a proof in ACA in order to establish the theorem. The following is a known general result on ACA.

Lemma 3 ACA is uniformly reflexive, that is: Let S be any finite subtheory of ACA and let BewS (x) be the canonical provability predicate for S . Then ACA proves uniform reflection for S , i. e., 
 ACA ` ∀x BewS pϕ(x) ˙ q → ϕ(x) .

Proof: This is established in a similar way as the reflexivity of PA is proved. First, cut-elimination for S is shown in ACA, then partial truth predicates for the second-order language are used to prove uniform reflection by induction on the length of cut-free proofs. Note that the axiom of choice is not available in ACA, which renders the definition of truth predicates somewhat clumsy. Details are too messy to be included in this paper. In particular, the preceding theorem applies to ACA− and all sets Sn defined above. Now it is shown that I(Π) can be supplemented to a proof in the theory ACA. Proof of Theorem 3: It has to be checked that all axioms and applications of rules of MDPA translate into provable ACA-formulas and ACA-admissible rules, respectively. In the proof Π axioms may appear at any level n ≤ m, where m is again the maximum level of any occurrence of a formula in Π. So ACA ` In (ϕ)

must be established for any axiom ϕ of MDPA and any n ≤ m. I skip the logical axioms and the axioms of PA including the induction axioms, because

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they are translated by any In into the corresponding axioms of MDPA . So consider the translation of Axiom MT under In , if n < m:

  ∀ϕ ∈ LB BewSnp∀x Tr pϕ>(x) ˙ q ↔ I. n+1 ϕ(x) q.

(7)

For ϕ ∈ LB both formulas ϕ> and In+1 (ϕ) are identical and this is provable in PA. So in order to prove (7), the following sentence has to be shown in ACA:
  ∀ϕ ∈ LB BewSnp∀x Tr pϕ>(x) ˙ q ↔ ϕ>(x) q. (8) This is established by observing that the uniform disquotation sentences are provable in ACA− and that this fact itself can be proved in PA. Formula (7) follows now from (8) and the fact that the translations are arithmetical formulas: ∀ϕ ∈ LB ∀n ≤ m I. n (ϕ) ∈ LPA .

(9)

It is also easy to show that the translation of Axiom MT under the function Im is provable in ACA. The translations of Axioms N1 and N2 follow easily from the general properties of provability predicates. The reflection axiom N3 is treated as follows. For any n < m, the translation In (N3) is the following expression: 

  ∀x BewSn I. n+1 pϕ(x) ˙ q → In ϕ(x) ,

(10)

where ϕ is a formula of the language of LT , that is, without . If a formula ϕ does not contain , then In (ϕ) and Ik (ϕ) are provably in PA identical formulas. Therefore (10) is a theorem of ACA by virtue of Lemma 3. The case n = m is handled in a similar way. Finally I turn to necessitation. For necessitation, induction over the length of the proof is used. To this end I assume that the lemma holds for the proof Π up to the occurrence of the necessitation rule in concern. An instance ϕ pϕq

of the necessitation rule receives in I(Π) the following shape: In+1 (ϕ) In (pϕq )

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The conclusion of the rule, however, is BewSn I. n+1 (pϕq) by definition of In . By assumption, the set Sn contains axioms that are sufficient to prove In+1 (ϕ). Therefore S ` In+1 (ϕ) holds and from this I conclude:



PA ` BewS I. n+1 ( pϕq ) .

This shows that the applications of the necessitation rule in Π become admissible rules in I(Π). By a similar argument MDPA proves the same formulas of LT as the theory T(PA), that is, PA plus the clauses of Theorem 2, that is, PA + “there is a full inductive satisfaction class” (see Halbach 1996 for a definition of T(PA)). Thus the truth predicate of modalized disquotationalism (in the sense of MDPA ) coincides by this observation and Theorem 2 with the usual ‘inductive’ truth predicate. This is noteworthy as modalized disquotationalism leads to the same theory of truth which is obtained by turning the Tarskian definition of truth into an axiom system, as was proposed, e.g., by Davidson.

5 Further perspectives The remark at the beginning of the paper that the theory of truth and the theory of necessity are both conservative over the base theory, whereas the combination is not, might have given a somewhat startling impression, because Robinson’s consistency theorem states that the union of a pair of conservative theories is again conservative if they agree on the base theory in the language that is shared by both. What is meant by my remark should be obvious now: The account of truth advanced in this paper is disquotationalist, the corresponding formal theory is the theory of the uniform Tarskian equivalences. This system is easily seen to be conservative over the base theory (see Halbach 1999a), as is the theory of necessity, i.e., N1–N4, by Proposition 1, even if PA is used as base theory and full induction is allowed in the axiomatization of both notions. With this notion of necessity at hand the disquotationalist account of truth can be refined and strengthened by claiming that the equivalences are necessary. The single axiom for truth, which incidentally does not even contain the truth predicate, is now all that is left from the infinite number of disquotation sentences of strict disquotationalism. Still this axiom is conservative. Only if brought together with the axioms for necessity that do contain T, it develops a strength going beyond the base theory. Rather than contradicting the Robinson consistency

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theorem, Theorem 3 shows that Robinson’s theorem must not be seen as a result providing a permit to treat notions like truth and necessity separately. In the main axiom MT of modalized disquotationalism the necessity predicate is used as a device that allows to quantify over sentences. Are there other means to do this that do not require the use of a necessity predicate? Several options might come to mind that deserve to be explored. Here I will discuss shortly only one alternative proposal, namely substitutional quantification where substitutional variables can stand in the place of formulas and the substitution class is formed by a set of sentences. This allows to express the disquotation sentences by the following single formula with a universal substitutional quantifier Πp: Πp (Tppq ↔ p).

(11)

Moreover, my arguments for a necessity predicate instead of an operator of sentences fail in such a setup. In particular, it can be expressed that all disquotation sentences are necessary with an operator of sentences only: Πp (Tppq ↔ p).

As was observed by Field (1994a), (11) proves with a suitable simple base theory that, e. g., the disjunction of two sentences is true if and only if at least one disjunct is true. This can be generalized to establish that all local versions of the ‘Tarskian’ clauses can be obtained from (11) with exception of the quantifier clause (see Theorem 2). I have several reservations against this account. In the first place, substitutional quantification is mainly a method to provide semantics for quantifiers. Roughly speaking, according to it, a universally quantified sentence is true if and only if all its substitution instances are true. If Πp were really explained in this way, (11) should not be considered to be disquotationalist, because an account of truth in LB (plus the operator  of sentences) were required in order to explain substitutional quantification, which is then in turn used to characterize truth in LB . Thus the notion of truth which is going to be explained was used (indirectly) in its explanation. Are there ways to introduce substitutional quantification without reference to a theory of truth? The distinctive feature of Πp that matters for the purposes here is after all not its semantics, but the possibility to put ‘substitutional’ variables in place of formulas. Since, as far as I know, no precise information is supplied by the defenders of this proposal as to which conditions substitutional quantification has to meet, there is no unmistakable

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answer to this question.21 But I am suspicious that a suitable kind of substitutional quantification can be set up without introducing at the same time at least implicitly a notion of truth. Very often a language embracing substitutional quantification can be considered as a notational variant of the language with the truth predicate satisfying the Tarskian clauses for truth, if a suitable base theory is presupposed. The quantifiers Πp can be superseded by ∀ϕ ∈ LB , the used occurrences of p are replaced by τ (pϕq), and mentioned occurrences by inserting a ‘referential’ variable for the corresponding substitutional variable.22 Thus replacing the disquotation scheme by the principle (11) means begging the question, because it is unclear how to explain the quantifier Πp without presupposing a truth predicate or an equivalent device. In this respect modalized disquotationalism can do better. Here a necessity predicate is required and the proof-theoretic investigations of this paper show that the notion of necessity required is much weaker than a full truth predicate satisfying the Tarskian clauses. The theory of necessity N1–N4 is even conservative over the base theory (and interpretable in it) and in this sense reducible to the base theory. Thus modalized disquotationalism does not rely on a device equivalent to truth (or as strong as truth) in order to explain truth. Now I will turn to some modification of the formal systems of modalized disquotationalism studied so far. Obviously, Theorems 2 and 3 constitute only a first step to a comprehensive investigation in formal theories of modalized disquotationalism, as many other interesting questions are still open. I will try to give at least some hints to problems which might be worthwhile of being pursued. For example, it might be asked whether MDPA is conservative over Peano arithmetic if the local versions of Axioms MT and N3 are used instead of the uniform ones. This question is especially interesting because pure strict disquotationalism does not employ the uniform Tarskian equivalences. Moreover, uniformity may be dropped either in Axiom MT or N4 but retained in the other, and the two resulting systems may be investigated for their conservativity. Still one could go farther afield and modify the basic principles of necessity in Section 2. As pointed out there, necessitation and full reflection pϕq → ϕ for any sentence ϕ cannot be consistently combined under usual 21 Field (1994a) seems to have an axiomatic approach in mind, because he is using axioms in his treatment of the connectives. 22 Consequently, substitutional quantification of the mentioned kind is threatened by the same paradoxes as truth, and the same solution can be applied. See, for instance, Kripke 1975, footnote 31.

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conditions. I have explained why I prefer necessitation over reflection and have therefore admitted only a restricted reflection principle; but according to different tastes or different philosophical conceptions of necessity, one might adopt full reflection and restrict necessitation or take a middle course and restrict both principles in such a way that inconsistency is avoided. The remaining principles of necessity I advocated in Section 2 are hardly objectionable for the mentioned kinds of necessity. I do not know of any reasonable reading of the box rendering these axioms false. So only the pair of necessitation and reflection is up for discussion. Furthermore, there are additional principles for necessity one might consider. There are readings of  where the Barcan formula seems adequate. In the present paper the instances of the disquotation scheme were restricted to those sentences that contain neither the truth- nor the necessity predicate. This is, however, a coarse way to avoid paradox. Thus one might try to relax this restriction of the allowed instances of the (uniform) disquotation scheme and adopt more liberal versions of Axiom MT, that means, the quantifier ∀ϕ ∈ LB in Axiom MT might be replaced by a quantifier ranging over the formulas of a language properly extending LB . In the first place, it might be asked whether LB can be replaced by L . The axiom MT modified in this way claims that the uniform disquotation scheme is necessary for all formulas not containing T, but possibly the necessity predicate. This strengthening of Axiom MT, however, is inconsistent. Proposition 5 Assume the base theory proves the diagonal lemma. Then MD is inconsistent with ∀ϕ ∈ L pTpϕq ↔ ϕq.

(12)

Proof: Using the diagonal lemma pick a “Liar sentence” γ ∈ L and reason in MDPA + (12) as follows: (1) (2) (3) (4) (5) (6) (7) (8) (9)

γ ↔ ¬pγq pγ ↔ Tpγqq pγq → pTpγqq pγq → Tpγq ppγq → Tpγqq ppγq → γq pγq → ¬γ ppγq → ¬γq p¬pγqq

diagonal lemma Formula (12) from (2) by N2 from (3) by N3 from (4) by necessitation from (5) by (2) from (1) from (7) by necessitation from (6) and (8)

100 (10) (11) (12) (13) (14) (15)

Volker Halbach

pγ ↔ ¬pγqq pγq ¬γ p¬γq p⊥q ⊥

from (1) by necessitation from (9) and (10) from (11) by (1) from (12) by necessitation from (11) and (13) from (14) by N3

By the proposition it is not possible to allow all instances of formulas of L in the disquotation scheme. But it still leaves the possibility to allow instances in which the necessity predicate occurs only in a way that does not yield inconsistency. Whereas it requires some reasoning in order to show that quantifying over all formulas of L in Axiom MT renders the system inconsistent, allowing all formulas of LT as instances in the disquotation scheme obviously leads to contradiction because of the Liar paradox. Hence the quantifier ∀ϕ ∈ LB in Axiom MT cannot be consistently replaced by ∀ϕ ∈ LT . For the sake of simplicity, I have not allowed any instance of the disquotation scheme containing T. There are, however, more sophisticated strategies that might be considered as partial approximations of Horwich’s proposal to admit only the “unproblematic” instances. It is an obvious suggestion to admit only those formulas as instances in which T occurs only positively.23 To make the definition smoother, assume that ¬, ∧, ∨, ∃ and ∀ are the logical symbols of LB . Then T occurs only positively in ϕ ∈ LT if and only if all occurrences of T in ϕ are within the scope of an even number of negation symbols. Let L+ T be the set of all formulas of LT where T occurs only positively; then strengthening the main axiom MT of modalized disquotationalism to 
 p q p ˙ q ∀ϕ(v) ∈ L+ T  ∀x T ϕ(v) , x ↔ ϕ(x)

(13)

yields a consistent system. The simple uniform disquotation scheme in its modalized form MT produces already a system of considerable strength, as was shown in Section 4. Now how strong is the enhanced form (13) of Axiom MT? Theorem 4 The system MDPA +(13) proves the same arithmetical formulas as RA