Axiomatic Theories of Truth

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AXIOMATIC THEORIES OF TRUTH

At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties, and shows how the logical results impinge on the philosophical topics related to truth. For instance, he shows how the discussion of topics such as deflationism depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate and professional philosopher in theories of truth.

volker halbach is a reader in philosophy at the University of Oxford and a fellow of New College.

A X I O M AT I C T H E O R I E S OF TRUTH VOLKER HALBACH University of Oxford

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521115810
C

Volker Halbach 2011

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Contents Preface Part I

viii Foundations

1

1 Definitional and axiomatic theories of truth

3

2 Objects of truth

9

3 Tarski

15

4 Truth and set theory 4.1 Definitions and axiomatizations 4.2 Paradoxes and typing

25 25 27

5 Technical preliminaries 5.1 Peano arithmetic 5.2 Truth and satisfaction 5.3 Translations and the recursion theorem

29 29 35 36

6 Comparing axiomatic theories of truth

39

Part II

Typed truth

49

7 Disquotation

53

8 Classical compositional truth 63 8.1 The conservativity of compositional truth 68 8.2 Conservativity and model theory 82 8.3 Nonstandard models 83 8.4 Lachlan’s theorem 89 8.5 Satisfaction classes and axiomatic theories of truth 98 8.6 Compositional truth and elementary comprehension 101 8.7 Positive truth 116 9 Hierarchies 9.1 Tarski’s hierarchy axiomatized 9.2 Illfounded hierarchies

v

123 125 129

Part III

Type-free truth

135

10 Typed and type-free theories of truth

140

11 Reasons against typing

146

12 Axioms and rules

149

13 Axioms for type-free truth

152

14 Classical symmetric truth 14.1 The Friedman–Sheard theory and revision semantics 14.2 Proof theory of the Friedman–Sheard theory 14.3 The Friedman–Sheard axiomatization 14.4 Expressing necessitation via reflection 14.5 Without satisfaction

159 162 175 185 188 192

15 Kripke–Feferman 15.1 Fixed-point semantics 15.2 Completeness and consistency 15.3 Proof theory of the Kripke–Feferman system 15.4 Extensions

195 202 212 217 225

16 Axiomatizing Kripke’s theory in partial logic 16.1 Partial Kripke–Feferman 16.2 Proof-theoretic analysis of partial Kripke–Feferman

228 231 244

17 Grounded truth

257

18 Alternative evaluation schemata

263

19 Disquotation 19.1 Maximal consistent sets of disquotation sentences 19.2 Maximal conservative sets of disquotation sentences 19.3 Positive disquotation 19.4 The semantics of positive disquotation 19.5 Proof theory of positive disquotation

267 267 272 274 277 280

Part IV

Ways to the truth

287

20 Classical logic 20.1 The costs of nonclassical logic 20.2 The internal logic of the Kripke–Feferman theory 20.3 Expressive power in nonclassical logic 20.4 Containing nonclassical logic

vi

289 291 295 300 303

21 Deflationism 21.1 Disquotationalism 21.2 Conservativity

306 307 312

22 Reflection 22.1 Reflection principles 22.2 Reflective closure

322 322 326

23 Ontological reduction

330

24 Applying theories of truth 24.1 Truth in natural language 24.2 Extending schemata

333 333 335

Index of systems

343

Bibliography

345

Index

357

vii

Preface

This book has four parts. In the first part I sketch some mathematical preliminaries, fix notational conventions, and outline some motivations for studying axiomatic theories of truth. Deeper philosophical investigation, however, is postponed to the last part when the significance of the formal results is discussed. The axiomatic theories of truth and the results about them are then given in the two central parts. The first of them is devoted to typed theories, that is, to theories where the truth predicate applies provably only to sentences not containing the truth predicate. In the third part of the book I discuss type-free theories of truth and how inconsistency can be avoided without Tarski’s object and metalanguage distinction. In the fourth and final part, the philosophical implications of the formal results are evaluated. I have tried to make the book usable as a handbook of axiomatic truth theories, so that one can dip into various sections without having read all the preceding material. To this end I have also included many cross references and occasionally repeated some explanations concerning notation. It should be possible to read the final part on philosophical issues without having read the two formal parts containing the formal results. However, this last part presupposes some familiarity with the notation introduced in Chapters 5 and 6 in the first part. Of course, when discussing philosophical issues I will refer back to the formal results obtained in the two previous parts, and the reader who is interested in the last part only and skips the two formal parts will have to take my word for them. All parts of the book should be accessible to a reader who has some acquaintance with the proofs of the Gödel incompleteness theorems and therefore with the basic concepts of recursion theory and metamathematics. In sections where I have used techniques from other areas of logic like model theory, I have defined all notions I use and have made most proofs so explicit that they should be accessible to readers not used to these techniques. I also assume very little with respect to proof theory: I do not use methods from ordinal analysis as I felt unable to provide an introduction to this branch of proof theory in a short chapter. In many research papers axiomatic truth theories are analysed by relating them to well-investigated subsystems

viii

preface

ix

of second-order number theory. The truth theories can then be compared via well-known results about these subsystems. Here I have attempted, whenever possible, to relate the theories of truth to one another directly, without the detour via the second-order systems, obviating the need to appeal to welldocumented or folklore results about them. Acknowledgements. In 2009 New College granted me a term of sabbatical leave, which was extended by another term, financed by the John Fell fund. During these two terms I wrote most of this book. I am grateful for this support. I am also indebted to Eugene Ludwig for generously supporting the preparation of the final draft through the college. I owe thanks to John Burgess, Martin Fischer, Kentaro Fujimoto, Richard Heck, and Graham Leigh for making unpublished drafts of papers available to me and allowing me to use this material in the book. Work on the material in this book stretches back a long time and it is not feasible to list everybody who has helped with suggestions and criticism or by teaching me. I apologize to everyone I do not thank explicitly. I am obliged to Eduardo Barrio, Andrea Cantini, Solomon Feferman, Hartry Field, Richard Heck, Richard Kaye, Jeff Ketland, Graham Leigh, KarlGeorg Niebergall, and Albert Visser for numerous discussions, comments on papers and earlier drafts of this book. I owe much to the members of the Luxemburger Zirkel Hannes Leitgeb, Philip Welch, and especially Leon Horsten. Special thanks are due to Kentaro Fujimoto and James Studd for numerous corrections and suggestions. New College, April 2010

Part I

FOUNDATIONS

1 Definitional and axiomatic theories of truth

Philosophers have been very optimistic about the prospects of defining truth. The explicit definability of truth is presupposed in many accounts of truth: only whether truth is to be defined in terms of correspondence, utility, coherence, consensus, or still something else remains controversial, not whether truth is definable or not. The advocated definitions usually take the form of an explicit definition. Hence, if one of these proposed definitions is correct, truth can be fully eliminated as explicit definitions allow for a complete elimination of the defined notion (at least in extensional contexts). It is a quirk in the history of philosophy that many of these definitional theories, according to which truth is eliminable by an explicit definition, have come to be known as substantial theories as opposed to deflationary theories of truth, although most proponents of deflationist accounts of truth reject explicit definitions of truth and in most cases also the eliminability of truth. A common complaint against traditional definitional theories of truth is that it is far from clear that the definiens is not more in need of clarification that the definiendum, that is, the notion of truth. In the case of the correspondence theory one will not only invoke a predicate for correspondence, but one will also use facts or states of affairs as relata to which the objects that are or can be true are supposed to correspond; in the case of states of affairs one will then also have to distinguish between states of affairs that obtain and those that do not. Of course, proponents of the various varieties of the correspondence theory propounded theories of facts, states of affairs, obtaining, and correspondence in which the assumptions on which they rely in their reasoning about facts and states of affairs are made explicit. But these theories are controversial at best and most people are much clearer and firmer in their views about truth itself than in their views about facts and states of affairs. Therefore it seems sensible to make explicit the assumptions about truth rather than to take the detour via a definition in terms of notions less accessible than truth. The decision to take truth as a primitive notion that is not defined in terms of other notions need not necessarily clash with definitional approaches. To begin with, one can take truth to be a primitive notion and postulate certain

3

4

definitional and axiomatic theories of truth

principles or axioms for truth without taking a stance towards the question whether truth is definable or not. Choosing an axiomatic approach might well be compatible with the view that truth is definable; the definability of truth is just not presupposed at the outset. So an axiomatic approach might only differ from a definitional account in its methodology, and in the end both might converge to the same theory of truth. I do not think, however, that there is only a methodological motivation for an axiomatic approach to truth. In this respect the situation with truth is fundamentally different from knowledge, for instance. In the case of knowledge, epistemologists have tried for a long time to provide an adequate definition in terms of truth, belief, justification, and some further condition that allows one to handle Gettier cases. Providing an adequate definition of knowledge has proven to be very hard and some epistemologists have abandoned the enterprise of finding such: some have declared the notion of knowledge to be marginal and put justification at the centre of epistemology; still others are happy to study knowledge as a primitive notion. The main reason to view knowledge as a primitive notion and to doubt that definitional theories are feasible seems to be that nobody has been able to come forward with an generally accepted definition of knowledge. The main evidence for the undefinability of knowledge is the observation that convincing counterexamples are known against most if not all proposals for explicit definitions that are non-trivial. In the case of truth, in contrast, not only is there similar evidence that truth cannot be defined for a language within that language, but there is a theorem: Tarski’s theorem on the undefinability of truth rules out the possibility of a definition of truth under certain conditions. It states that under fairly generally applicable conditions, the assumption that there is a definition of truth within a given theory for the language of that same theory leads to a contradiction. For a sketch of Tarski’s theorem I assume that a classical first-order language L is fixed and that it contains a closed term ┌e┐ for each expression e of the language L. If a consistent theory S in the language L can prove certain basic facts about substitution of expressions in expressions and if it can describe a function taking each object to a closed term for that object, then there cannot be a formula τ(x) of L such that τ(┌ϕ┐) ↔ ϕ is provable in S for each sentence ϕ of L. But many philosophers agree with Tarski (1935) that a theory of truth for the language L should at least prove these equivalences, which are called T-sentences or disquotation sentences depending on how exactly they are formulated.

definitional and axiomatic theories of truth

5

I will not try to make Tarski’s theorem more precise, although marking out the limits of Tarski’s theorem would be worthwhile as it would illustrate just how widely applicable it is (but see almost any textbook on Gödel’s incompleteness theorems for an account of Tarski’s theorem). The amount of syntax needed is very little and can be represented in very weak arithmetical theories. If closed terms for the expressions of the language L are not available in L, then the above equivalences can be replaced with the following claims:
∀x Sentϕ (x) → (τ(x) ↔ ϕ) In these sentences the formula Sentϕ (x) expresses that x is the sentence ϕ or its code. So the notation becomes more convoluted, but Tarski’s theorem can be proved for languages like that of set theory that lack closed terms for sentences or their codes. Also Tarski’s theorem does not rely on any assumption about what a definition would look like, except that it would have to yield the disquotational or T-sentences. It is often stated for arithmetical theories and used to show that arithmetical resources do not suffice for defining the truth of arithmetical sentences. But Tarski’s theorem applies in other settings as well. For instance, the proof Tarski’s theorem does not require that truth be attributed to sentences, or their arithmetical codes. If truth is attributed to propositions and the operations on propositions corresponding to the syntactic operations mentioned above can be expressed in the theory, then Tarski’s theorem can be proved for such a setting as well. If the underlying theory S contains an axiomatic account of propositions, facts, states of affairs, or the like, Tarski’s theorem shows that truth for the propositions of the theory cannot be defined on the basis of the theory S. So Tarski’s theorem does not affect traditional definitional theories of truth any less than more mathematical theories, although the impact of Tarski’s theory may be felt less in the case of traditional theories because they are often presented in terms that are vague enough to make an application of formal results appear impossible or at least implausible. But Tarski’s theorem applies to any sufficiently precise version of the correspondence theory of truth and all the other traditional theories of truth. At any rate, Tarski’s theorem is a threat to all definitional theories whether they rely on a notion of correspondence or some other notion. I do not want to claim that any satisfactory theory of truth has to prove the equivalences T┌ϕ┐ ↔ ϕ, but if it does, Tarski’s theorem strikes and clearly truth cannot be defined in the sense just sketched, at least in a setting satisfying certain minimal conditions. If it is assumed that a theory of truth has

6

definitional and axiomatic theories of truth

to be a definition of truth, then one is excluding many theories, and in fact many of the theories that will studied below. All I am asking for is that undefinable notions of truth are not excluded from the outset as suitable accounts of truth. First one should become clear about which properties a notion of truth should have. Once one has become clear about what is expected from the notion of truth, one can investigate in a second step whether truth is definable. In semantic theories of truth – by these I mean Tarski’s theory but also, for instance, Kripke’s (1975) – truth is defined. If the metalanguage contains the object language, the equivalence T┌ϕ┐ ↔ ϕ will be provable for all sentences ϕ of the object language but not for all sentences of the metalanguage. So Tarski’s theorem is evaded by restricting the possible instances in the schema T┌ϕ┐ ↔ ϕ to sentences of a proper sublanguage of the language in which the equivalences are formulated. So it might seem that in semantic truth theories one can proceed in a different order: the definition of truth comes first, and only after truth has been defined, one explores the consequences of the definition. But in fact when one is looking at the various semantic theories of truth, they very often start from certain assumptions about truth. Philosophers often appraise semantic theories by pointing out that certain sentences or, as I would like to call them, principles are satisfied in the proposed semantics. Tarski, for instance, justifies his semantic theory by pointing out that his defined truth predicate satisfies the above mentioned equivalences for all sentences of the object language (see Chapter 3 below for a discussion of Tarski’s theory). So a certain syntactic principle stands at the beginning. Tarski’s definition of truth is then designed to show that a notion of truth satisfying the equivalences can be defined in the metalanguage, or more precisely, in a metatheory, assuming again that the metalanguage contains the object language. I propose then to focus on these principles – whether they are Tarski’s equivalences or some other principles – and to discuss them before trying to eliminate them, for instance, by providing model-theoretic semantics for a language satisfying these principles or by defining truth in terms of correspondence. One reason for pausing at this stage is that there is little agreement over which principles should be adopted. As I will show in Chapter 3, Tarski took his own equivalence to be insufficient. So even he did not fully believe in the adequacy of his principles. In the meantime many logicians and philosophers have rejected Tarski’s approach as insufficient because it excludes any application of the truth predicate to sentences that contain it. This has been

definitional and axiomatic theories of truth

7

the point of devising semantic theories of self-applicable truth; it is easy to show that Tarski’s own restrictive solution is neither plausible nor useful for many purposes. So I would like to discuss first the principles that should be satisfied by truth as there is such a wide variety of them. Moreover, once these principles have been formulated, defining the truth predicate contained in them is not the only way to eliminate the notion of truth. It is not too hard to come up with situations where truth is not definable but remains eliminable in some other way, such as being conservative over an underlying theory. In such a situation truth could be shown not to contribute anything to our knowledge outside semantics. Truth would, so to speak, supervene on the underlying base theory without contributing anything to it and truth would be in this sense eliminable without being definable. So perhaps a definition of truth is dispensable even if one aims at an eliminative theory of truth. Also, one need not provide model-theoretic semantics for analysing various properties of these principles. In some cases one will be able to prove their consistency and many other properties without appealing to model-theoretic semantics. In particular, one will be able to see what commitments are tied to these principles. If one is using a defined notion of truth it can be difficult to see which properties flow from the postulated principles of truth and which come from the particular chosen definition of truth. Furthermore, I would like not to exclude the situation where truth is added to one’s overall mathematical theory where truth is not definable. Of course, there is an opposed reductive view according to which a notion is only acceptable if it can be defined in set theory. At least I would like to consider alternatives to set-theoretic reductionism in which truth is not and cannot be defined away. After this plea for the axiomatic approach I assure the reader that I will also use model-theoretic methods throughout the book. As I mentioned above, the axiomatic approach is not opposed to definitional approaches. In fact, both approaches complement one another. For instance, one may start by formulating truth-theoretic principles like Tarski’s T-sentences for a theory like Peano arithmetic and then show that a suitable truth predicate for the language of arithmetic can be defined in set theory by defining a model for Peano arithmetic expanded by these truththeoretic principles. Hence one knows that these truth-theoretic principles are consistent, at least if set theory is to be trusted. Then one might try to formulate analogous principles for the language of Zermelo–Fraenkel set theory. There is no hope to define this truth predicate for set theory, but the

8

definitional and axiomatic theories of truth

fact that other theories such as Peano arithmetic possess a nice model when expanded by principles of this ilk supports the view that the expansion of set theory by the corresponding principles is consistent as well although Tarski’s theorem rules that this cannot be shown, unless one goes beyond Zermelo– Fraenkel set theory by introducing class quantifiers or other devices. At any rate, the model-theoretic constructions can illuminate, motivate, and to some extent support axiomatic truth theories. Model-theoretic approaches are also important for the proof theory of axiomatic truth theories. In many cases I will use formalizations of modeltheoretic constructions to provide proof-theoretic analyses of axiomatic theories of truth. So axiomatic and semantic approaches complement one another also on the technical side.

2 Objects of truth

The axioms for truth will be added to what is called the base theory. In the main part of this book I will use Peano arithmetic as the base theory, but applications to other more comprehensive base theories are intended, and the base theory may contain empirical or mathematical or still other axioms together with the appropriate vocabulary. At any rate, a base theory must contain at least a theory about the objects to which truth can be ascribed. Truth theories have been proposed where the need for objects to which truth can be ascribed and for a theory of these objects seems to be avoided. If truth were analysed in terms of special quantifiers as in the so-called prosentential theory of truth by Grover et al. (1975), for instance, it might initially appear that such objects are avoided, but it is not at all clear that the new quantifiers avoid any ontological commitment. I have no ambition to avoid ontological commitment to objects that can be true. If the axiomatic theories of truth I am going to discuss are intertranslatable with an approach without such ontological commitment, so be it. If such a translation is not possible, then I suspect that something is wrong with the approach. Here I will stick to the usual approach that takes truth to be a predicate. In almost all cases, the axioms for truth can only serve their purpose when combined with a suitable base theory. If the truth axioms, on which the theories in this book are based, are separated from the base theory, the result is a very weak theory. Since I have not introduced the theories or their axioms yet, I can only sketch some trivial results. But one might think about the theories with the disquotation sentences as axioms or with compositional axioms for truth like the one stating that a conjunction is true if and only if both conjuncts are true. Pure theories of truth without a base theory will not prove that there are more than two objects, so they will be weak in this sense. For the purely truth-theoretic axioms do not allow one to distinguish between different true objects, or different untrue objects. Therefore one can usually obtain a model for a truth theory without a base theory by taking some model of the full theory and identifying all objects in the extension of the truth predicate, on

9

10

objects of truth

the one hand, and all objects not in its extension, on the other hand. The truth predicate and its axioms will only show their potential when combined with a suitable base theory. Of course, one could strengthen the truth-theoretic axioms by building certain ontological claims and axioms about the structure of truth bearers into them. But that seems difficult if one is sticking to fairly natural axioms. However, I do not want to suggest that truth axioms do not bring any ontological commitment, as some deflationists might hope. The theory of truth cannot be completely separated from the underlying ontology of objects that can be true, as even very weak axioms for truth will imply that there are at least two different objects, if the axioms imply that something is true and something not true, as even very axioms do. This will be shown on p. 55 below. If a theory about the objects to which truth can be ascribed is required as base theory, the question arises what these objects are and objects of which kind should be described in the base theory. Philosophers have ascribed truth to propositions conceived as objects that are independent from language, or to sentence types or sentence tokens. Of course, there are also thoughts, beliefs, contents of sentences, and so on, where it is not obvious how these relate to propositions and sentences. Even once one has settled on propositions or sentences many decisions remain. Since I am not aiming at a complete (recursive) axiomatization of the ontology of the objects to which truth is ascribed – which would be impossible under even very weak assumptions – I avoid questions concerning their nature – at least to a certain extent. I do, however, presuppose that the objects possess an ontological structure analogous to that of sentences, more specifically of the sentences of the language of the base theory in the case of typed theories of truth, and of the language of the base theory expanded by the truth predicate in the case of type-free theories. In order to state an axiom to the effect that a conjunction is true if and only if both conjuncts are true, one needs to assume that the operation of conjunction is defined on the objects to which truth is ascribed. Similarly, if one wants to say that a universally quantified sentence is true if all instances are true, the operations of universal quantification and of instantiation must be defined. The axioms of the truth theory can serve their purpose only if the base theory allows one to express certain facts about the syntactic operations; otherwise they may remain void. For instance, the axiom that a conjunction is true if and only if both conjuncts are true becomes void if no information is supplied on what a conjunction is. For instance it should be provable in

objects of truth

11

the base theory that a conjunction is different from its conjuncts and different from any disjunction. Having operations such as conjunction axiomatized in the base theory does not force one to ascribe truth to sentences. Many philosophers seem happy to assume that the usual operations such as conjunction on sentences are also available as corresponding operations on propositions. If the reader shares this view, then I do not see a problem in taking the axioms about the truth bearer as axioms about propositions and the truth-theoretic axioms as axioms for propositional truth. To me the assumption that propositions happen to be structured like the sentences of a first-order language seems optimistic. Sentences are sentences of some particular language with certain atomic expressions and certain logical operations; so the structure of these sentences is specific to the language and usually different from the sentences of other languages. However, being able to ascribe the same proposition as a belief to persons who do not have a common language seems to be one of the main reasons to employ propositions. Hence the assumption that propositions share their structure with the sentences of the base language seems at least problematic. Even if one does not grant that propositions share their structure with the sentences of a certain language, and one wants to apply the truth predicate to propositions rather than sentences, axiomatic theories of truth over a base theory of objects with a sentential structure still have a sensible interpretation: the truth predicate Tx can be read as ‘x is a sentence expressing a true proposition’, or as Quine (1970, p. 10) proposed: An unsympathetic answer is that we can explain truth of sentences to the propositionalist in his own terms: sentences are true whose meanings are true propositions. Any failure of intellegibility here is already his own fault. Quine’s proposal shows only how the truth of sentences can be understood in terms of the truth of propositions and the relation of expressing. It does not necessarily show that truth of propositions is dispensable, as ‘the proposition that snow is white is true’ cannot be easily rephrased in terms of sentential truth: the phrases ‘the proposition that snow is white is true’ and The proposition expressed by the sentence ‘snow is white’ is true are not intersubstitutable salva veritate is some intensional contexts, as the proposition that snow is white is true independently of any linguistic facts. Even if ‘snow is white’ had expressed a different proposition, the proposition

12

objects of truth

that snow is white would have been true, while the proposition expressed by the sentence ‘snow is white’ could well have been the false propositional that blood is white. These modal considerations are important because they may impinge on the modal status of the axioms for truth. It has been argued that axioms for truth like the disquotation sentences such as ‘Snow is white’ is true if and only if snow is white or their propositional versions, such as The proposition expressed by the sentence ‘snow is white’ is true if and only if snow is white are only contingently true as they depend on what sentences express or how sentences are used. Hence, if such equivalences are adopted as axioms, the consequences of the truth theory need not be necessary, as its axioms are only contingent. The corresponding equivalences based on propositional truth, in contrast, are generally assumed to be necessary or to have some status of this kind. So the equivalence The proposition that snow is white is true if and only if snow is white is necessary. To my knowledge Lewy (1947) was the first to employ this argument explicitly against the necessity of the disquotation sentences, although it may be found in a less explicit form as early as 1925 or 1926 in Moore (1966).1 Under other readings, the disquotation sentences come out as more or less analytic, and a priori but not necessary. I shall not go further into the discussion of the modal status of the disquotation sentences and other truth axioms, but refer to Halbach (2001a). Here I just wanted to make the point that while in extensional contexts propositional truth may be eliminated by sentential truth, the elimination will be more difficult if intensional contexts are taken into account. Since I will study axiomatic theories in extensional settings only, the difference in the modal status of propositional and sentential formulations does not matter and I do not have to commit myself to specific assumptions on the nature of the objects to which truth is attributed. If intensional contexts are added, one may no longer be able to avoid a commitment to a specific view about the objects of truth. 1 Graham Solomon made me aware of Moore (1966).

objects of truth

13

Besides the kinds of objects that philosophers usually ascribe truth to there is a further class of objects that are often used for this purpose in formal contexts. When logicians are discussing axiomatic theories of truth, they often do not start from a theory of expressions or propositions but rather from a theory of natural numbers or even sets. Very often syntactic objects are identified with their numerical codes. This convenient approach will also be the framework employed in this book. This is in accordance with my liberal attitude towards the ontological question of which objects are the bearers of truth: as long as the underlying base theory proves that the objects have the required structural features, it is usable as a base theory. It is very well-known how the expressions of a countable formal language can be coded in the natural numbers. There are many of these codings, known as Gödel codings, and their details can make a significance difference in some contexts. Here I try to rely just on properties shared by all ‘reasonable’ codings. I will take up the issue of Gödel codings in Section 5.1. Using codes of sentences will allow me to use Peano arithmetic as base theory. The decisive advantage of using a well-known mathematical theory, and Peano arithmetic in particular, is that I do not have to develop a formal theory that can be used as the base theory and that I do not have to prove certain fundamental results about the base theory that are well known in the case of Peano arithmetic. Hence, as I want to get to the central topics of this book as quickly as possible, I will use Peano arithmetic as the base theory. The results on axiomatic truth theories with Peano arithmetic as their base theory apply in many cases to other base theories as well. Some proofs are easily transferable to other base theories; other cases are more difficult. For instance, the model-theoretic proof in Section 8.4 of the conservativity of the theory of truth based on Tarski’s clauses for the definition of truth stated as axioms relies heavily on some features specific to Peano arithmetic, while the proof-theoretic argument for the same result in Section 8.1 needs only much weaker assumptions about the base theory. In some cases, I have supplied some remarks on how proofs can be adapted to base theories other than Peano arithmetic. Proving general results for as many base theories renders the statements of results and proofs very cumbersome. Details also depend in which direction the result is to be generalized: whether, for instance, it is to be generalized to base theories weaker than Peano arithmetic, to stronger theories, to theories without closed terms for all expressions, and so on. So I have focused on theories that take Peano arithmetic as their base theory and hope that the reader can see from the

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objects of truth

proofs provided whether and how the result can be transferred and generalized to other base theories. Applying results about a truth theory over a given base theory to the same truth theory over another base theory is not always easy. Actually it is not sound to talk about the same theory over another base theory, as rewriting a truth theory for another base theory is not a trivial task and there may be more than one way to do so. What needs to be changed may depend on the language of the theory and what the base theory proves. I will give some hints in later chapters. For instance, when Zermelo–Fraenkel set theory is used as base theory, a binary satisfaction predicate may be preferred over the unary truth predicate for reasons to be explained below, depending on which axioms are to be reformulated over the new base theory. For weak base theories certain operations on the objects to which truth is ascribed may not be expressible, thereby making it impossible to apply the results obtained on the basis of a strong base theory to such a theory. So the modification required for applying a truth theory to another base theory may be substantial.

3 Tarski

For a full history of axiomatic theories of truth I would have to go back very far in history. Many topics and ideas found in what follows have been foreshadowed. For instance, even theories structurally very similar to axiomatic compositional theories of truth can be found in Ockham’s Summa Logicae, even though Ockham like many other philosophers paid lip service to the correspondence theory of truth. Relating historical to more recent accounts of truth is often difficult as it is seldom clear whether certain sentences of a particular account should be understood as definitions, descriptions, consequences of a theory, or as axioms. I think it is safe to claim that the modern discussion of formal axiomatic theories of truth began with Tarski’s The Concept of Truth in Formalized Languages (the reader might want to consult Künne (2003) on the development leading up to Tarski). Tarski proved some fundamental results about axiomatic theories although he did not adopt an axiomatic approach. Famously Tarski proposed a definition of truth for certain languages in another more comprehensive language, called the metalanguage. There were, and still are, good motives to aim at a definition rather than a mere axiomatization of truth: if one is wondering whether truth should be considered a legitimate notion at all, a definition might be useful in dispersing doubts about its legitimacy. Tarski wrote his paper when most members of Vienna Circle and the Warzaw School suspected truth to be a concept that should be avoided in good philosophy (see Wolenski ´ and Simons 1989 for a historical account). Tarski did not settle for giving a definition of truth and taking its adequacy to be self-evident. Rather he proposed an adequacy criterion for evaluating the adequacy of definitions of truth. Of course, the definition had to be a proper explicit definition, which is emphasized by his insisting on the ‘formal correctness’ of the definition. The crucial criterion is stated in his famous Convention T (Tarski 1935, pp. 187f): Convention T. A formally correct definition of the symbol ‘T r’, formulated in the metalanguage, will be called an adequate definition of truth if it has the following consequences:

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tarski (α) all sentences which are obtained from the expression ‘x ∈ T r if and only if p’ by substituting for the symbol ‘x’ a structuraldescriptive name of any sentence of the language in question and for the symbol ‘p’ the expression which forms the translation of this sentence into the metalanguage; (β) the sentence ‘for any x, if x ∈ T r then x ∈ S’ (in other words ‘T r ⊆ S’).

Condition (β) merely says that only sentences are true: S is the set of all sentences. (α) is the crucial definition. I shall first start with some terminological clarifications. In the formulation of clause (α) Tarski employs the notion of a ‘structuraldescriptive name’. When similar equivalences are stated in the modern literature, quotational names (that is, the sentence in quotation marks) or, in an arithmetical context, the numeral of the code (Gödel number) are often used. Quotational names presuppose a theory of quotation, which has its own intricacies, while Gödel numerals are confined to arithmetical contexts. In these two respects Tarski’s structural descriptive names are superior. Their main disadvantage is their awkwardness when spelled out. A structural descriptive name for ‘snow’ is the following expression (see Tarski 1943, p. 668 and Tarski 1935, p. 156n): the word consisting in the 19th, 14th, 15th and 23rd letters of the English alphabet In many respects it does not matter whether structural descriptive names, Gödel numerals, or quotational names are used. I shall not go further into the cases where it does matter: for instance, if one believes that ‘s’ happens to be the 19th letter of the alphabet only contingently, then this might impinge on the modal status of the equivalences. Tarski’s equivalences in part (α) of Convention T also differ in a further way from the equivalences that are nowadays often labelled as ‘T-sentences’: Tarski employs the notion of translation. This allows him to formulate the equivalences in cases where the language in which the equivalences are formulated does not contain the language for which T r is supposed to be a truth predicate, that is, in cases where the metalanguage does not contain the object language. Tarski does not offer much explanation concerning the notion of translation employed here and has been criticized for availing himself of so blatant a semantic notion. It is not clear whether Tarski did not have a more syntactic notion of translation in mind, such as the notion of relative

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interpretation that he later developed in Tarski et al. (1953). In his later (1943) Tarski adopted a formulation of the equivalences where the object language is a sublanguage of the metalanguage, in order to simplify the presentation of his theory. But he did not retract his earlier, more general formulation. For my present purposes I need only consider the case where the metalanguage contains the object language. Tarski’s usage of the term ‘metalanguage’ and more generally of ‘language’ differs from the modern one: today, logicians take languages to be specified by a set of well formed formulae while philosophers often take languages to be interpreted languages. In the later parts of (1935) it becomes clear that Tarski develops his theory of truth in a metalanguage that is given axiomatically: thus for him at least a set of axioms and rules belongs to a language. Tarski seems to identify languages with formal systems or theories. In the main part of (1935) Tarski shows how to provide a definition of truth satisfying Convention T. He considers certain languages and theories in which this definition can be successfully carried out such that the metalanguage, that is, in more modern terminology, the metatheory, proves the equivalences of clause (α). This proof of definability allows Tarski to establish truth as a respectable notion by his standards. Tarski (1935) also considered the option of taking his equivalences to be axioms for truth outright. This would have been an axiomatic approach, and I will consider a very closely related theory tb (Definition 7.1). Tarski proved that under certain circumstances the equivalences can be consistently added to a theory. Tarski presented this result as Theorem III in (1935, p. 256). If the class of all provable sentences of the metatheory is consistent and if we add to the metatheory the symbol ‘Tr’ as a new primitive sign, and all theorems which are described in conditions (α) and (β) of the convention T as new axioms, then the class of provable sentences in the metatheory enlarged in this way will also be consistent. Tarski gives a proof for this result which relies on certain properties of the metatheory. For instance, the metatheory must not prove for any sentence that it is identical to its negation. It is obvious that if the predicate satisfying the equivalences in (α) is definable, then adding the equivalences as axioms will yield a consistent theory, if the original theory is consistent. The strength of Theorem III lies in the fact that it can be proved under fairly general circumstances, in particular under

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conditions where Tarski’s definition of truth cannot be carried out because the metalanguage lacks the resources that are required to define truth. As pointed out above, Tarski was interested in a theory of truth that rehabilitated truth as a legitimate notion. Theorem III is clearly not enough to show that a notion of truth axiomatized by the equivalences is a safe notion: mere consistency does not establish that a theory is adequate. From Tarski’s proof of theorem III one can also extract a proof of conservativity, which would provide a better justification than mere consistency for the theory with the equivalences as axioms. Tarski mentions another reason for rejecting an axiomatization of truth relying on his equivalences (see Tarski 1935, p. 257): The value of the result obtained is considerably diminished by the fact that the axioms mentioned in Th. III have a very restricted deductive power. A theory of truth founded on them would be a highly incomplete system, which would lack the most important and most fruitful general theorems. Let us show this in more detail by a concrete example. Consider the sentential function ‘x ∈ T r or x ∈ T r’. [“∈ T r” is the truth predicate, “∈ T r” the negated truth predicate; “x” designates the negation of “x”.] If in this function we substitute for the variable ‘x’ structural-descriptive names of sentences, we obtain an infinite number of theorems, the proof of which on the basis of the axioms obtained from the convention T presents not the slightest difficulty. But the situation changes fundamentally as soon as we pass to the generalization of this sentential function, i.e. to the general principle of contradiction. From the intuitive standpoint the truth of all those theorems is itself already a proof of the general principle; this principle represents, so to speak, an ‘infinite logical product’ of those special theorems. But this does not at all mean that we can actually derive the principle of contradiction from the axioms or theorems mentioned by means of the normal modes of inference usually employed. On the contrary, by a slight modification of Th. III it can be shown that the principle of contradiction is not a consequence (at least in the existing sense of the word) of the axiom system described. Thus Tarski objects to taking the equivalences as axioms of truth because of their deductive weakness. This is remarkable because the quote shows that Tarski expects a good theory of truth to prove more than just the T-sentences;

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according to the quote, the principle of contradiction or the principle of excluded middle should also be a consequence. Tarski remarks that it is not a consequence of a theory of truth with the T-sentences as its sole truththeoretic axioms. Tarski’s complaint about the lack of deductive power of his equivalences is at odds with Convention T: if only the equivalences are used as axioms, the theory trivially has them as consequences, but the deductive weakness of the theory makes it inferior to Tarski’s own definition of truth. Tarski restricts his Convention T to definitions of truth: thus Tarski’s definition of adequacy does not apply to arbitrary axiomatizations. However, this restriction cannot resolve the tension between Tarski’s definition of adequacy and his complaint about the lack of deductive power of his equivalences: the metalanguage may contain a primitive predicate P x that satisfies the equivalences. Then one can define a truth predicate Tx by P x. Thus there are definitions of truth that are adequate in the sense of Convention T, but that are ‘highly incomplete’ (as Tarski puts it in the above quote), so they are less attractive as definitions of truth. Thus, if Convention T is supposed to characterize an appropriate notion of adequacy, it fails by Tarski’s own standards. One would not want to call a theory adequate that fails to prove theorems a ‘good’ theory of truth that is not ‘highly incomplete’ can be expected to prove. That a definition of truth has the equivalences as consequences is a necessary condition for its adequacy in the intuitive sense, but it is not a sufficient condition, because every really adequate theory should also prove the law of contradiction. Therefore Tarski’s notion of adequacy in Convention T fails to capture the intuitive notion of adequacy he is after. There are also far less trivial examples of truth definitions that are adequate according to Convention T, but do not yield the law of contradiction and similar general claims. A significant example is the definition of truth in Neumann–Bernays–Gödel set theory for the language of Zermelo–Fraenkel set theory. Neumann–Bernays–Gödel set theory is formulated in the language of first-order set theory expanded with class quantifiers. Using the class quantifiers Mostowski (1950) provided a definition τ(x) of truth in such a way that τ(┌ϕ┐) ↔ ϕ is provable in Neumann–Bernays–Gödel set theory for all sentences ϕ not containing class quantifiers, that is, for all ϕ in the language of Zermelo–Fraenkel set theory. This is not precise because the language of set theory does not feature names for codes of sentences, but these names can be eliminated using the techniques sketched on p. 5. Novak (1950) showed that Neumann–Bernays–Gödel set theory is consistent if Zermelo– Fraenkel set theory is, and Mostowski (1950) observed that Novak’s relative

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consistency proof can be formalized in Zermelo–Fraenkel set theory (and in fact in very weak arithmetical systems). Hence by Gödel’s second incompleteness theorem Neumann–Bernays–Gödel set theory does not prove the consistency of Zermelo–Fraenkel set theory. Mostowski (1950, p. 112) inferred ‘that certain properties of the notion of truth for (S) [ Zermelo–Fraenkel set theory ] cannot be established in (S0 ) [ Neumann–Bernays–Gödel set theory ]’.1 Hence the T-sentences do not allow one to show that all theorems are true and the object theory is consistent. I will show that this only becomes feasible once general principles such as the law of contradiction or of excluded middle are added. Neumann–Bernays–Gödel set theory yields the T-sentences τ(┌ϕ┐) ↔ ϕ but not generalization such as ∀x(SentZF (x) → (T¬. x ↔ ¬Tx)). Thus Neumann–Bernays–Gödel set theory yields a more interesting example of a truth definition that is adequate by the lights of Convention T, but does not have the principle of excluded middle or the principle of contradiction as a consequence, and is ‘highly incomplete’ and thus not adequate. Below on p. 99 I shall consider an analogous example within the framework of arithmetical theories: the theory aca0 stands to pa in a relation similar to the relation obtaining between Neumann–Bernays–Gödel set theory and Zermelo–Fraenkel set theory. After this excursion to results dating after Tarski’s Concept of Truth in Formalized Languages, I return to Tarski’s account of the axiomatic approach to truth. After rejecting on the grounds of deductive weakness an axiomatization of truth by the equivalences of Convention T alone, Tarski discusses the prospects of obtaining a better theory by adding further axioms. Obviously one should at least add axioms that imply the law of contradiction or of excluded middle. Tarski, however, is very pessimistic about the prospects for such an approach (see 1935, pp. 257f): We could, of course, now enlarge the above axiom system by adding to it a series of general sentences which are independent of this system. We could take as new axioms the principles of contradiction and excluded middle, as well as those sentences which assert that the consequences of true sentences are true, and also that all primitive sentences of the science investigated belong to the class of true sentences. Th. III could be extended to the axiom system enlarged in this way. [Tarski points out in a footnote that the metatheory must be assumed to be ω-consistent for 1 For a modern account of these results and sharpenings see Schindler (2002).

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this result.] But we attach little importance to this procedure. For it seems that every such enlargement of the axiom system has an accidental character, depending on rather inessential factors such, for example, as the actual state of knowledge in this field. The arbitrariness of the choice of the axioms could be removed, if a maximal theory of truth were employed. If the theory of truth fixed the extension of the truth predicate, then it could not be improved, of course; the theory would be complete for truth and no further axioms could tell us more about truth. Therefore Tarski discusses whether one should impose such a categoricity requirement on the theory of truth. Tarski presents it in the following way (see Tarski 1935, p. 258): Thus it seems natural to require that the axioms of the theory of truth, together with the original axioms of the metatheory, should constitute a categorical system. It can be shown that this postulate coincides in the present case with another postulate, according to which the axiom system of the theory of truth should unambigously determine the extension of the symbol ‘T r’ which occurs in it, and in the following sense: if we introduce into the metatheory, alongside this symbol, another primitive sign, e.g. the symbol ‘T r 0 ’ and set up analogous axioms for it, then the statement ‘T r = T r 0 ’ must be provable. But this postulate cannot be satisfied. Tarski’s claim that this categoricity requirement cannot be met is easily established: if T r = T r 0 were derivable in a theory of truth, then this theory would implicitly define Tr, that is, truth. By Beth’s Theorem a set of sentences defines a predicate implicitly if and only if it defines it explicitly (see, for instance, Chang and Keisler 1990, pp. 90f). The set of sentences may be thought of as the set of axioms for truth. Thus the theorem says that if T r = T r 0 were provable in the presence of the truth axioms for T r as well as the analogous ones for T r 0 , they would already constitute an explicit definition. But since T r is not explicitly definable by Tarski’s Theorem, it cannot be implicitly definable. Consequently, no theory satisfies the requirement of categoricity for its truth predicate and therefore the postulate of categoricity cannot be met. Tarski outlines a similar argument in order to establish his claim. Beth’s Theorem applies to all theories formulated in a first-order language and to higher-order languages, if the higher-order variables can be conceived as first-order variables of a different sort. Therefore one can evade this result only at the cost of very severe incisions into logic.

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In fact, Tarski proposes such a severe incision or rather an extension of usual first-order logic by an additional rule (Tarski 1935, p. 258): There is, however, quite a different way in which the foundations of the theory of truth may be essentially strengthened. The fact that we cannot infer from the correctness of all substitutions of such a sentential function as ‘x∈T r or x∈T r’ the correctness of the sentence which is the generalization of this function, can be regarded as a symptom of a certain imperfection in the rules of inference hitherto used in the deductive sciences. In order to make good this defect we would adopt a new rule, the so-called rule of infinite induction, which in its application to the metatheory may be formulated somewhat as follows: if a given sentential function contains the symbol ‘x’, which belongs to the same semantical category as the names of expressions, as its only free variables, and if every sentence, which arises from the given function by substituting the structural-descriptive name of any expression of the language investigated for the variable ‘x’, is a provable theorem of the metatheory, then the sentence which we obtain from the phrase ‘for every x, if x is an expression then p’ by substituting the given function for the symbol ‘p’, may also be added to the theorems of the metatheory. Tarski goes on to explain that his rule of infinite induction is only a variant of what is now called an ω-rule. This rules permits us to derive a universal sentence ‘For all natural numbers n: A(n)’ from its instances A(0), A(1), A(2),. . . The rule of infinite induction differs from the usual rules of predicate logic by requiring the availability of infinitely many premises for its application. If the rule of infinite induction, that is, the ω-rule, is added to the theory described in Tarski’s Theorem III, then a categorical system is obtained, and the extension of the truth predicate is uniquely determined, as Tarski observes. Tarski’s discussion of the rule of infinite induction is not conclusive and remains more or less an afterthought to his treatment of axiomatic theories of truth. There are various good reasons to avoid the use of infinite rules. At least if one aims at a formal system in the sense of a recursively axiomatizable system of truth, the ω-rule has to be rejected. If such a radical departure from predicate logic is not accepted and an infinite rule like the rule of infinite induction is not adopted, then an axiomatization of truth based solely on the T-sentences as truth-theoretic axioms is,

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according to Tarski, of little value, because the theory itself is too weak and does not prove desirable principles, while its extensions are always somewhat arbitrary. Taking this into account one might wonder whether this verdict does not endanger Tarski’s whole enterprise of establishing an adequacy condition: the T-sentences and Convention T do not provide one with a list of all principles that ought to be provable in a good theory of truth. Surely any good truth theory will yield the T-sentences. But this is only a necessary, or minimal, condition for a good theory, as Tarski seems to admit in the quote about the unprovability of the principle of contradiction. The derivability of the T-sentences is not a sufficient condition for the adequacy of a truth theory. Therefore the T-sentences are insufficient and should be supplemented by additional principles, but, according to Tarski, it seems hopeless to arrive at a complete list of such principles because categoricity of the truth theory is beyond our reach. In the light of incompleteness phenomena, however, one should not expect a categorical axiomatization of truth to be feasible. The impossibility of a categorical axiomatization should not keep one from studying axiomatic theories of truth, any more than the impossibility of a complete axiomatization of arithmetic or of set theory should keep one from studying axiomatic systems for integers or sets. Tarski preferred a definition of truth over an axiomatization. But from this very definition a recipe for an axiomatization of truth may be extracted: Tarski’s inductive clauses for satisfaction can be turned into axioms. So Tarski opened the way to various natural axiomatic systems of truth, which will be studied in what follows. Also in his Theorem III Tarski provided a formal analysis of typed disquotationalist theories of truth and of disquotationalism before it was born. And, as was shown above, he proved that the T-sentences lack deductive power and when taken as axioms for a truth theory are highly incomplete. His objective in The Concept of Truth in Formalized Languages were safe foundations for truth. Given a formal language, or more exactly, a formal theory, he showed how one can define a truth predicate by extending it. Tarski used second-order quantification with suitable axioms. From his perspective and presumably from the perspective of many philosophers at that time (especially in the Vienna Circle and Warsaw), these additional devices were conceived as less problematic than truth itself. While truth was viewed as suspiciously metaphysical, second-order quantifiers were accepted as mathematical or even logical tools.

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In the meantime preferences have shifted: higher-order quantifiers are no longer accepted without hesitation, while the concept of truth has become more familiar as a respectable notion. Thus I do not want to add higher-order quantification to a given theory in order to define truth, because I do not know how to justify the use of higher-order quantifiers. Rather I directly add what is minimally necessary to obtain an adequate theory of truth: namely the truth predicate itself is added to the language and the theory is extended by suitable axioms for this predicate.

4 Truth and set theory

Arguments from analogy are to be distrusted: at best they can serve as heuristics. In this chapter I am using them for exactly this purpose. By comparing the theory of truth with set theory (and theories of property instantiation, type theory, and further theories), I do not expect to arrive at any conclusive findings, but the comparison might help one to arrive at new perspectives on the theory of truth and on the question of how closely truth- and set-theoretic paradoxes are related. The theory of sets and set-theoretic membership on the one hand and the theory of truth and satisfaction on the other hand exhibit many similarities: both are paradox-ridden, allow circularities, and invite the application of hierarchical approaches. Russell’s paradox and the liar paradox are arguably the most extensively discussed paradoxes in the philosophical literature, and they seem so intractable because they are founded on very basic and clear intuitions about sets and truth. Moreover, certain remedies against the set-theoretic and the semantic paradoxes have been given the same labels; for example, ‘typing’: both kinds of paradoxes can be resolved by introducing type restrictions. While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently. 4.1 Definitions and axiomatizations There are also striking differences between the theories of truth and sets: set theory has permeated many disciplines and has become an essential part of their foundations. Many mathematicians consider set theory to be the foundations of their subject, and almost any subject with any formal rigour employs some set theory. Set theory is also used extensively in philosophy. Many philosophers seem to have hardly any problems with taking set-theoretic membership as a primitive notion, which is axiomatized rather than defined. In contrast, when epistemologists use truth in their analysis of knowledge, for instance, they hasten to add some definition of truth, usually in terms of correspondence. In the case of sets, authors of elementary logic textbooks

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usually begin their sketch of basic set theory with the obligatory quote from Cantor’s famous (1895, p. 31, my translation): By a ‘set’ we understand any collection of definite and clearly distinguished objects m of our intuition or our thought (which are called ‘elements’ of M) into a whole M. This sounds very much like a definition of set, but I strongly doubt that many philosophers would actually take Cantor’s dictum as more than a first crude approximation to a full analysis of the notions of set and membership. Even a more detailed definition or description in Cantor’s style would be considered a mere blurb, preceding the proper account in the form of an axiomatization in Zermelo–Fraenkel set theory or some similar axiomatic system. Many philosophers see a need for justifying the axioms, but it seems that hardly anyone would prefer to substitute the axiomatization of membership with a definition. This is in contrast with truth. Many philosophers prefer a definition of truth over any axiomatization. Some define truth in terms of correspondence, others opt for a definition in set theory in the style of Tarski’s definition of truth. Of course, there are exceptions: most famously Donald Davidson advanced an axiomatic approach to truth, and some proponents of deflationism rely on axioms for truth even if they prefer, bashfully, to call the entirety of axioms an implicit definition. Overall it seems that explicit definitions of truth are much more popular than explicit definitions of membership. Set-theoretic primitivism seems to be a widely accepted working hypothesis, while truththeoretic primitivism is a view that is taken to stand in need of prior defence. This is probably due to the success of set-theoretic reductions. Almost all parts of mathematics are reducible to set theory, so set theory offers a unified foundations for many of our theories. Nothing similar seems available for truth. In fact many of the results in this book show that truth-theoretic reductionism fares much worse. Not only do theories of truth need some base theory to get off the ground, but even over a base theory like Peano arithmetic they fail to reach the strength of set theory. I will investigate in this book how much is added to the base theory by the truth axioms. What can be gained by adding truth axioms depends on the specific axioms but all the consistent axiom systems in the literature add little to the base theory when compared to what set theory can afford. Hence truth theories cannot play the same role in the foundations of mathematics as set theory. Only when one focuses on predicative foundations might truth compete with other foundational systems.

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4.2 Paradoxes and typing Many authors have tried to find a source common to both the set-theoretic paradoxes and the semantic paradoxes, and, in particular, to Russell’s and the liar paradox (see, for instance, Priest 1994). Here I am not so interested in the origins of the paradoxes, as in their solutions. Obviously some solutions to Russell’s and the liar paradox share some structural features, though it is not so easy to come up with a specific and precise account of what those features are (see, for instance, Church 1976). Russell’s paradox arises as an inconsistency in systems of unrestricted comprehension or instantiation of universals or properties. The axiom schema (4.1)

∃y ∀x P xy ↔ ϕ(x)




is inconsistent in first-order predicate logic, if arbitrary formulae without y free are admitted as instances of ϕ(x). It is not so obvious that this inconsistency is directly related to the derivation of the liar paradox from the unrestricted T-sentences. In type theory, Russell’s paradox is avoided by introducing variables of different types. The first step is to distinguish between first- and second-order variables. If lower-case letters are used for first-order variables and uppercase letters for second-order variables, then a typed comprehension axiom can be stated as (4.2)

∃Y ∀x P xY ↔ ϕ(x)




The comprehension formula may be any formula not containing free occurrences of the second-order variable Y . Of course, higher-order variables can be introduced as well. As for truth, if ϕ can be any sentence, the disquotation schema T┌ϕ┐ ↔ ϕ leads to an inconsistency over any base theory that permits the formulation of a liar sentence. In the context of truth theories, typing means that the truth predicate T provably applies only to sentences not containing T.1 In the case of the disquotation sentences, typing is applied by restricting the disquotational schema to sentences ϕ without any occurrences of T. The use of the term typing may one lead to think that one solution can be applied to the semantic and the set-theoretic paradoxes. But both measures 1 For a discussion of the notion of typing see Chapter 10.

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are quite different in character. In Section 8.6 I will show that typing truth predicates corresponds to a much more severe move in the case of comprehension: typing truth corresponds to predicative typed comprehension, which can be obtained from (4.2) above by admitting only comprehension formulae ϕ that lack second-order quantifiers. Hence typing in truth corresponds more closely to ramified type theory than to simple type theory. Actually ramified type theory over Peano arithmetic as base theory, which is known as ramified analysis, is equivalent to typed compositional truth. I will elaborate on this in Chapter 9. This correspondence shows that truth theory and set – or rather type theory are very closely related and intertranslatable. Of course there are solutions to the paradoxes other than typing. This applies to the set-theoretic as well as the semantic paradoxes. I find it very difficult here to compare the solutions. On the side of set theory, the cumulative conception of set and its axiomatization have become the dominant solution. There is no corresponding solution on the truth-theoretic side. The known type-free theories of truth are not directly related to this solution of the set-theoretic paradoxes. If type-free solutions to the set-theoretic and semantic paradoxes were closely related, one might expect that the formal behaviour of the type-free systems of sets and truth is similar. But in fact type-free approaches with respect to sets and properties lead to very strong systems. Even when the restriction to predicative comprehension is dropped, truth falls behind. None of the truth theories I am aware of reach the strength of full second-order number theory, that is, of Peano arithmetic formulated in a second-order language with the comprehension schema (4.1). It would be premature to claim that there are no parallels between solutions of the set-theoretic and the semantic paradoxes – in fact, some results in this book might be taken to be such parallels – but the parallels are not so obvious as one might think at first glance: typing means different things on both sides and type-free systems on the one side do not directly correspond to type-free solutions on the other side. At any rate the work on truth theories can hardly follow the lead on set (or property or type) theory. To me it seems to be a genuinely different line of research.

5 Technical preliminaries

Before delving into the formal details and logical analysis of axiomatic truth theories, I would have preferred to discuss further philosophical issues and the motivations for the technical development. But without being able to refer to the logical machinery, I find it hard to do so. Hence I will now tackle the formal part of my project and postpone the treatment of the philosophical issues until the last part.

5.1 Peano arithmetic In discussing axiomatic systems, I will occasionally distinguish between formal systems and theories. A formal system is a collection of axioms and rules for generating theorems. Almost all the systems I am going to discuss are formulated in classical logic. In most cases it does not matter exactly which logical calculus is used. In some cases, however, it will be necessary to specify the exact logical rules, and in these cases I will use a sequent calculus, as described in many standard textbooks (Troelstra and Schwichtenberg 2000, for instance). A theory is a set of formulae closed under first-order logical consequence. Thus a theory may be generated by many different formal systems. However, in many cases, when the difference does not matter, I will not clearly distinguish between theories and the systems that generate them. All the languages I will consider have ¬, ∧, and ∨, as connectives and ∃ and ∀ as quantifier symbols. The logical axioms and rules are formulated for these connectives and quantifier symbols; they also include suitable axioms or rules for identity. An expression ϕ → ψ is understood as an abbreviation for the formula ¬ϕ ∨ ψ. Similarly, ϕ ↔ ψ abbreviates (¬ϕ ∨ ψ) ∧ (¬ψ ∨ ϕ). As the base theory, first-order Peano arithmetic pa will be used. Details of Peano arithmetic can be found in any standard textbook (Hájek and Pudlák 1993; Kaye 1991; Boolos et al. 2007). Here I restrict myself to making explicit a few assumptions that will be used in what follows.

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The system of Peano arithmetic contains the defining equations for zero, successor, addition, and multiplication as axioms but it is also convenient to assume that the language L of pa contains finitely many function symbols for certain other primitive recursive functions, together with axioms governing them. The terms of L are formed in the usual way from variables, constants, and function symbols. The only predicate symbol of L is the identity symbol; the smaller-than relation <, which will be used occasionally, is a defined predicate. The system pa features induction axioms, that is, axioms of the form ϕ(0) ∧ ∀x (ϕ(x) → ϕ(Sx)) → ∀x ϕ(x) for each formula of the language L of arithmetic. The symbol S is the function symbol for the successor function. Induction will also be used in the form of the least-number principle, which is known to be equivalent to induction (see, for instance, Kaye 1991). Here I will only prove that induction implies the least-number principle. lemma 5.1 (least-number principle). For any formula ϕ(x) of L the following is provable in Peano arithmetic:
∃x ϕ(x) → ∃x ϕ(x) ∧ ∀y < x ¬ϕ(y) The formula ϕ(x) may contain further free variables; the least-number principle should then be conceived as the universal closure of the above formula. The result of substituting the variable y for x in ϕ(x) is defined in the usual way. In particular, if ϕ(x) already contains occurrences of a quantifier ∀y with free occurrences of x in its scope, the bound variable y is substituted with the first fresh variable. In what follows renaming of bound variables will be silently assumed. proof. I prove the contraposition of the least-number principle. I assume the negation of the consequent of the least-number principle, which yields the following:
(5.1) ¬∃x ϕ(x) ∧ ∀y < x ¬ϕ(y) Next I apply induction to the formula ∀y < x ¬ϕ(y). The induction base ∀y < 0 ¬ϕ(y) is trivially provable in pa. For the induction step I assume ∀y < x ¬ϕ(y). Using (5.1) I conclude ϕ(x) and therefore also ∀y < Sx ¬ϕ(y). Hence the induction step is proved and ∀x ∀y < x ¬ϕ(y) follows by induction. Since the latter implies ¬∃x ϕ(x), this completes the proof of the least-number principle by contraposition. a

peano arithmetic

31

Adding a new unary predicate symbol T to the language L of arithmetic yields the language LT . I will use the label pat for Peano arithmetic formulated in the language LT with all instances of the induction schema in the language LT . If ax is a truth-theoretic axiom, that is an LT -sentence with the predicate T, then pa+ax is the theory obtained by closing the set containing exactly ax and all axioms of pa under logic. Typically, pa+ax will lack some instances of the induction schema with T, while pat+ax, which is defined analogously, contains all instances of the induction schema including those with the truth predicate as axioms. Similar remarks apply, if ax is a truth-theoretic rule of inference. The expressions of the language LT can be coded in the natural numbers. I will not go into the details of the coding, which can be found in standard textbooks including an account of Gödel’s incompleteness theorems (see, for instance, Shoenfield 1967 or Boolos et al. 2007). For convenience I will usually identify expressions with their Gödel codes and will not distinguish between expressions of LT and their codes. Occasionally, the coding needs to be extended to a language properly extending LT . I assume that the number 0 does not code any expression. The set of nonnegative integers is designated by ω. The numeral of a number n ∈ ω is the symbol 0 preceded by n occurrences of the successor symbol. For any number n, I will use the expression n for the numeral of n. For specific numbers I will usually omit the bar and write, for instance, 3 for SSS0 instead of 3. If ϕ is some expression of LT , ┌ϕ┐ (without a bar) is the numeral (of the code) of ϕ. In Peano arithmetic, all recursive functions can be represented. If f is a unary recursive function, there is a formula ϕ(x, y) in L satisfying the following condition for all numbers n, k ∈ ω: f(n) = k iff pa ` ∀y (ϕ(n, y) ↔ y = k) For sets there is a similar notion of representation: the formula ϕ(x) (weakly) represents a set S ⊆ ω in pa if and only if the following equivalence holds for all n ∈ ω: n ∈ S iff pa ` ϕ(n) A set S is strongly represented by ϕ(x) in pa if and only if S is weakly represented by ϕ(x) and, in addition, the following condition is met for all n ∈ ω: n∈ / S iff pa ` ¬ϕ(n) All recursive sets can be strongly represented in pa, and all recursively enumerable sets can be weakly represented in pa.

32

technical preliminaries

The representation of functions extends to functions of arbitrary finite arity; similarly the results concerning the weak and strong representation of sets extend to relations of arbitrary arity. Certain syntactic operations on expressions of the language LT are primitive recursive. For instance, the negation function that gives, when applied to a formula ϕ, its negation ¬ϕ (and 0 when applied to any other number) is primitive recursive and can be represented in Peano arithmetic. This is one of the functions for which it is convenient to have a function symbol ¬. in the language L of Peano arithmetic, so that the formula ¬. x = y represents the negation function. It follows that Peano arithmetic has ¬. ┌ϕ┐ = ┌¬ϕ┐ as a theorem for any formula ϕ. Generally, for a primitive recursive syntactic operation h, the symbol with a dot under it stands for the corresponding function symbol of L. Examples are ∧. and ∀. . So ∀. is a function symbol representing the binary function that yields, when applied to a variable and a formula, the universal generalization of that formula with respect to the variable and 0 when it is applied to other objects. Occasionally I will write x→ . y even though → is not a connective of the language L. The expression x→ . y is understood as an abbreviation for (¬. x)∨. y, matching the definition of ϕ → ψ as ¬ϕ ∨ ψ. The ternary substitution function that gives, when applied to a formula x, a term s, and a term t, the result of substituting s with t in x is also primitive recursive. I assume that the substitution is carried out in such a way that bound variables are renamed if necessary. The substitution function will be represented by the ternary function expression x(t/s) (without a dot). Closed terms are those terms that do not contain variables. The set of closed terms is recursive and strongly represented by the formula ClTerm(x). The evaluation function that gives, when applied to a closed term of L, the value of that term is primitive recursive, as L contains only finitely many function symbols. However, if there are function symbols for certain primitive recursive functions in the language (that are needed for diagonalization), there will not be a function symbol for the evaluation function itself in the language L. To keep my notation more perspicuous, however, I will use the symbol ◦ for the evaluation function although there is no symbol for the evaluation function in the language. For instance, x◦ = y expresses that the value of x is y; and s◦ = t◦ expresses that s and t coincide in their values. As the evaluation function is primitive recursive this can be expressed in L by suitable formulae. Therefore expressions such as s◦ = t◦ are understood as abbreviations of suitable non-atomic L-formulae.

peano arithmetic

33

The function that takes each number to its numeral is denoted by writing a dot above its argument. So ∀x ϕ(x) ˙ expresses that ϕ holds for all numerals. Thus x˙ represents the function that maps each number n to its numeral n. Occasionally I will use this device in order to bind variables, as in the following example. For instance, ∀x T┌ϕ(x)┐ ˙ is short for ∀x T┌ϕ┐(x/┌x┐), ˙ that is, for the claim that the result of formally substituting the numeral of x for the variable x in the formula ϕ is true. This sentence expresses that all formulae of the form ϕ(n) are true. Since the language LT also contains function symbols other than the successor symbol, the above formula does not express that ϕ(t) is true for each closed term t. The set of all sentences of L is represented by Sent(x), the set of all sentences of LT by the formula SentT (x). In order to express that x and y are sentences of L, one can write Sent(x∧. y), which is slightly shorter than Sent(x) ∧ Sent(y). I assume that the equivalence of Sent(x∧. y) and Sent(x) ∧ Sent(y) can be proved in Peano arithmetic, which will be possible under the usual coding and with natural choices for the representing expressions. As an abbreviation of ∀x ∀y(ClTerm(x) ∧ ClTerm(y) → ϕ(x, y)) I will write ∀t ∀s ϕ(s, t); so the quantifiers ∀t and ∀s range over closed terms in the intended interpretation. I will also make claims of the kind ‘there is a closed term t such that ϕ(t) is provable’ and use the letters s, t, and so on without formal quantifiers; in such cases t and s are metavariables for terms. Using the notation just introduced, one can express in L, for instance, that a sentence of the form s = t is true, where s and t are closed terms, if and
only if s and t coincide in their values. The formula ∀s ∀t T(s=. t) ↔ s◦ = t◦ serves the purpose. Translated into the metalanguage, this sentence says that for any closed terms s and t of L the identity claim s = t is true if and only if the value of s is identical to the value of t. If ϕ(x) is fixed formula, I will occasionally write ┌ϕ(t. )┐ for (┌ϕ┐(t/┌x┐)) if it is clear from the context which variable is replaced with t and if (┌ϕ┐(t/┌x┐)) is part of a formula in which t is quantified by a quantifier ∀t or ∃t in the object language. Hence ∀t T ┌ϕ(t. )┐ expresses that the result of substituting the variable x in ϕ(x) with an arbitrary closed term is true. I will assume that the formulae and function symbols just mentioned represent their respective functions and sets in a natural way. Under a natural coding and with a suitably natural representation, equivalences like the one above between Sent(x∧. y) and Sent(x) ∧ Sent(y) are provable in Peano arithmetic. Moreover, I assume that one can prove in pa that ∧. and similar functions are total and that the codes of formulae are always greater than the codes of expressions properly contained in them such as their subformulae.

34

technical preliminaries

Presumably the best way to ensure that a natural coding and natural representations are employed would be to write down the representing formulae and the defining equations for the function symbols. This assumption will allow me to demonstrate in Peano arithmetic that certain universally quantified claim are provable in Peano arithmetic. I will illustrate the point with yet another example. Normally one would assume that the claim that every numeral is a closed term is provable in Peano arithmetic: (5.2)

∀x ClTerm(x) ˙

In many places in this book I will make such assumptions without explicitly justifying them; such claims will be provable with the natural codings and natural representations, which are presupposed throughout the book. However, in pa different formulae represent the set of closed terms and (5.2) may not be provable if the set is represented in an unnatural way. To show how (5.2) might fail to be provable with the wrong representing formula, assume that the recursive relation obtaining between numbers n and k if and only if n is a proof k in pa is strongly and naturally represented in Peano arithmetic by the formula B(x, y). The proof that this relation is recursive is a standard step in the proof of Gödel’s second incompleteness theorem. Further assume that the function symbol f represents the function that assigns to every number n the nth proof in a fixed alphabetical enumeration of proofs in pa. Then I define a new formula ClTerm∗ (x) in the following way from a natural representation: ClTerm∗ (x) := ClTerm(x) ∧ ¬B(f(x◦ ), ┌0 = 1┐) The formula ClTerm∗ (x) then represents the set of closed terms. For pa ` ClTerm∗ (n) implies pa ` ClTerm(n) and therefore also that n is a closed term. Conversely, if n is a closed term, it follows that ClTerm(n) is provable in pa, but it also follows that ¬B(f(n), ┌0 = 1┐) is provable in pa because there is no proof of 0 = 1 in pa. That ClTerm∗ (x) strongly represents this set is proved in a similar way. Although the formula ClTerm∗ (x) strongly represents the set of closed terms, the general claim (5.2) cannot be proved for ClTerm∗ (x). Assuming to the contrary that ∀x ClTerm∗ (x) ˙ is provable in Peano arithmetic one would obtain pa ` ∀x ClTerm(x) ˙ ∧ ¬B(f(x˙◦ ), ┌0 = 1┐) Dropping the first conjunct and observing that ∀x x˙◦ = x is provable in pa, this would imply pa ` ∀x ¬B(f(x), ┌0 = 1┐). Hence one would have shown in pa

truth and satisfaction

35

that there is no proof of 0 = 1 in pa, that is, one would have proven the consistency of pa within pa, which is impossible by Gödel’s second incompleteness theorem, as B(x, y) was assumed to be natural and therefore satisfies Löb’s derivability conditions (see Boolos 1993). Hence even simple syntactic facts may not be provable if unnatural representations are used. Insisting on natural representations, however, is not really a problem in the present setting, as one could always write out all the representations explicitly. Then one would not have to rely on the difficult notion of naturalness. Since I always work with a fixed language, the deeper problems of intensionality studied by Feferman (1960) and others do not arise. 5.2 Truth and satisfaction If the clauses in Tarski’s (1935) definition of truth are turned into axioms, as Davidson (1984b, 1996) and others have proposed, then a primitive binary predicate symbol for satisfaction will be needed, as Tarski defined truth in terms of satisfaction. In the language LT , however, there is only a unary truth predicate, and thus, it seems, LT does not lend itself to axiomatizing Tarski’s notion of truth. Here one can exploit two convenient features of arithmetic: that there is a closed term for every object, and that one can define a function taking each object to a closed term denoting that object. The function assigning to every number its numeral satisfies this condition. Since finite sequences of objects can be coded in Peano arithmetic, one can dispense with variable assignments and satisfaction. Instead of saying that the string a1 , . . . , an satisfies the formula ϕ(x1 , . . . , xn ) one can say that the result of substituting the numeral of ai for xi in ϕ(x1 , . . . , xn ) for each i ≤ n is true. In fact, in order to state Tarski’s definition of truth, sequences are not needed if one resorts to what could be called substitutional quantification. This approach will be used throughout the book and can be found, for instance, in Definition 8.1 below. The use of substitutional quantification should be seen as a mere technical convenience that can be avoided. In a base theory like Zermelo–Fraenkel set theory, the truth-theoretic axioms would have to be restated in terms of satisfaction if the axioms involve an inductive clause for the quantifiers like Axiom ct5 on p. 65 below. Alternatively one could axiomatize a notion of truth for the language of set theory augmented with closed terms for all sets, to enable the use of the same kind of substitutional quantification as in the case of arithmetic. The

36

technical preliminaries

formulae of this expanded language would no longer form a set but rather a proper class. From the truth predicate for the extended language that contains a closed term for each objects the truth predicate for the language without these term may be definable. Of course for cardinality reasons syntactic notions such as the property of being a term or a formula of the extended language can no longer be representable in a recursively enumerable theory. The complexity of the syntax of the expanded language and various other intricacies make this approach to an axiomatization of truth for set theory less elegant. 5.3 Translations and the recursion theorem In most cases I will introduce and prove general technical results when they are needed. Here in this section I will sketch a technical trick that will be used in many places throughout the book in order to define translation functions between theories. For many systems of truth I will show that the truth predicate of that system can be defined in another truth-theory. The cases vary but for a trivial example assume that the first theory has a truth predicate T1 and an axiom ∀t (T1 T. 1 t ↔ T1 t◦ ) and that the second theory has a truth predicate T2 with an axiom ∀t (T2 T. 2 t ↔ T2 t◦ ). By the above conventions, the symbol T. 1 stands for the function that yields, applied to a term t, the formula T1 t; T. 2 is to be understood in an analogous way. It might be tempting to say that the truth predicate T1 can be easily defined in the second theory by defining T1 as T2 (assuming that this translation works also for other axioms for T1 ). This definition of T1 as T2 does not yield a truth-definition, however, as the translation of the axiom for T1 is the following sentence: ∀t (T2 T. 1 t ↔ T2 t◦ ) In this sentence the truth predicate T1 has been replaced with T2 . But the translation of the axiom for T1 did not take the axiom to the axiom for T2 , because the function expression T. 1 has not been replaced T. 2 . The translation should not only substitute all occurrences of T1 with T2 ; it should also substitute all ‘mentioned’ occurrences of T1 . So if h is the function substituting T1 with T2 , then for any term s the translation h(T1 s) should not be merely T2 s but rather T2 h. s, where h. is a function expression for h in the language of arithmetic. So it seems one would like to define the translation function h in terms of a function symbol h. for h.

translations and the recursion theorem

37

Moreover, the substitution of T1 with T2 should be carried out arbitrarily deep. So h(T1 ┌T1 ┌ϕ┐┐) should be the sentence T2 ┌T2 h. ┌ϕ┐┐. In the case of self-referential sentences there is no limit to the ‘depth’ down to which the truth predicate T1 needs to be replaced with T2 . So attempts to define h by recursion on the ‘depth’ of a formula will not work. By using the recursion theorem for primitive recursive functions, however, an appropriate recursive substitution function can be obtained. Proofs of the recursion theorem can be found in standard textbooks on recursion theory (Hinman 1978, p. 41). In the following lemma I consider the situation where the truth predicate T is replaced with a fixed formula ϑ(x) in the described way. lemma 5.2. Assume ϑ(x) is a formula. Then there is a primitive recursive function h with the following property:  n, if n is of the form s = t for terms s and t      or if n is not a sentence of LT     ¬h(ϕ), if n = (¬ϕ)     h(ϕ) ∧ h(ψ), if n = (ϕ ∧ ψ) h(n) =  h(ϕ) ∨ h(ψ), if n = (ϕ ∨ ψ)      ∀x h(ψ), if n = (∀x ϕ)     ∃x h(ψ), if n = (∃x ϕ)    ϑ(h. t), if n = (Tt) for a term t The notation suggests that there is a function symbol h. for the primitive recursive function h in the language, but I adopt here just my usual sloppy functional notation, that is, h. will be expressed by a suitable formula. proof. There is a universal binary total recursive function [e](n) that gives, applied to a code e (also known as index) of a primitive recursive function f and a number n, the value f(n), (see Hinman 1978 again). So the equation [e](n) = f(n) holds if e is an index for the primitive recursive function f. The function [e](n) is represented in Peano arithmetic by the expression [x] . (y); again there will not be a function symbol available in L, but the function can be expressed using a suitable formula. The (surface) complexity or the length of a formula is defined as the number of occurrences of quantifiers and connectives in the formula. The following primitive recursive function g can be defined by primitive recursion on the complexity of formulae so that the following holds for all numbers e and n:

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technical preliminaries

  n,          ¬g(e, ϕ),     g(e, ϕ) ∧ g(e, ψ), g(e, n) =  g(e, ϕ) ∨ g(e, ψ),     ∀x g(e, ψ),      ∃x g(e, ψ),     ϑ([e](t)), .

if n is of the form s = t for terms s and t or if n is not a sentence of LT if n = (¬ϕ) if n = (ϕ ∧ ψ) if n = (ϕ ∨ ψ) if n = (∀x ϕ) if n = (∃x ϕ) if n = (Tt) for a term t

The recursion theorem for primitive recursive functions yields an index e0 so that [e0 ](n) = g(e0 , n) holds for all numbers n ∈ ω. Now the translation function h is defined in the following way for all n ∈ ω: h(n) = g(e0 , n) The expression [e.0 ](y), where e0 is the numeral for e0 , is abbreviated by h. . The expression h. , which can be expressed using a suitable L-formula, represents the total recursive function h in Peano arithmetic. Then the lemma follows from these definitions. In particular, for the critical case Tt the desired property of h can be demonstrated in the following way: h(Tt) = g(e0 , Tt)

definition of h

= ϑ([e.0 ](t))

definition of g

= ϑ(h. (t))

definition of h.

a

Since the translation function h is primitive recursive it is provably total in Peano arithmetic. I will also invoke a variant of the lemma where the translation function replaces indexed truth predicates depending on an additional argument for the translation function. Adding a parameter in the above lemma requires only minimal modifications of the proof.

6 Comparing axiomatic theories of truth

The main technical results of this book compare axiomatic theories of truth but also compare such theories of truth with other theories like the base theory or, in some cases, second-order theories of arithmetic. These results establish that certain axiomatic theories of truth are reducible to certain other theories of truth. Philosophers and logicians have defined and discussed many different notions of reducibility, and I will employ different notions here as well. Which notion of reducibility is appropriate depends on the purpose of the comparison and one’s philosophical stance on truth. For instance, an instrumentalist about truth might want to compare truth theories on the basis of their truth-free consequences alone; the theory of truth itself is seen merely as a means to an end. However, if one is investigating whether the paradoxes are adequately resolved in certain theories of truth, then one cannot focus exclusively on truth-free consequences: one will need to compare what the different theories of truth prove about the liar sentence, for instance. To compare the conceptual strength of truth theories, one might not be so concerned about their behaviour with respect to the paradoxes, but one must still take into account the truth-theoretic consequences of the theories; one might compare the theories by examining whether one theory can define the truth predicate of the other theory. In the final part of the book I will return to the philosophical significance of various reducibility results and the notions of reducibility employed in them and look at applications such as ontological reductions, but here I first introduce various technical notions. All the theories discussed in detail in this book have Peano arithmetic as their base theory. Hence all the theories share the same arithmetical and logical vocabulary. Most of them are formulated in the language LT ; only the hierarchical theories contain many indexed truth predicates T0 , T1 ,. . . instead of the single truth predicate T. First I will introduce a very general and widely used notion of reducibility between theories: relative interpretability. This notion can be applied even in cases where the theories do not share any non-logical vocabulary. I will

39

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comparing axiomatic theories of truth

begin with this very general notion of reducibility before passing on to more specific notions with a more limited range of applicability. For the following definitions I assume that S and T are deductive systems formulated in the languages LS and LT , respectively. The first, very general notion of reducibility is the notion of relative interpretation, which plays an important role in metamathematics. It is also frequently used in philosophically motivated reductions, such as ontological reductions. In this book I will not go into the general philosophical significance of relative interpretations (but see Chapter 23). Basically a relative interpretation of S in T is a translation of LS in LT that preserves the logical structure of the formulae, possibly restricting the quantifiers, such that if S proves a formula, T proves its translation. Relative interpretations become more complex when languages containing function symbols are considered, as they are here. Since there may not be function symbols for all definable functions in T, formulae containing function symbols may need to be translated by formulae in which the functions are expressed by appropriate formulae. Following Tarski et al. (1953, pp. 20–21 and 29), I define the notion of a possible definition as follows. A possible definition of a unary predicate T in a theory T is a formula of the following form: (6.1)

∀x Tx ↔ ϕ(x)




Here ϕ(x) is a formula in the language LT of T, containing no variables other than x free. The entire sentence (6.1) is not a sentence of LT as the symbol T is assumed not belong to LT . The sentence (6.1), however, is a sentence of every extension of T containing the predicate symbol T. As I will be mainly dealing with relative interpretations between truth theories, with the truth predicate as their only non-logical predicate, the notion of a possible definition of a unary predicate will suffice. Since the identity predicate = is conceived as a logical symbol, it is always translated by itself. If other predicate symbols are present, the definition of a possible definition can be extended in the obvious way. A possible definition of an n-ary function symbol f in T is a sentence of the following form: (6.2)

∀~ x ∀y f(~ x) = y ↔ ϕ(~ x, y)




Here ϕ(~ x, y) is a formula of LT with exactly the variables y and x ~ , that is, y, x1 , . . . , xn free. Furthermore, T must prove that ϕ(~ x, y) is functional.

comparing axiomatic theories of truth

41

Therefore (6.2) is a possible definition of f only if the following sentence is provable in T:
∀~ x ∀u ∀v ϕ(~ x, u) ∧ ϕ(~ x, v) → u = v Individual constants are conceived as 0-place function symbols. Thus a possible definition of a constant c in T is a sentence of the following form:
∀y c = y ↔ ϕ(y) if ϕ(y) is in the language LT and the following sentence is probable in T:
∀u ∀v ϕ(u) ∧ ϕ(v) → u = v The relativization S P of a theory S is obtained by relativizing all quantifiers in theorems of S by a unary predicate symbol P not in the language LS of S. That is, SP is obtained from S by substituting all subformulae ∀x ϕ(x) and ∃x ψ(x) in formulae of the theory S with ∀x (P x → ϕ(x)) and ∃x (P x ∧ ψ(x)), respectively, and then taking the deductive closure. In what follows I will always assume that P is a new predicate symbol not in the language LS . A variant of a theory S is a theory obtained by replacing each nonlogical symbol of LS by one of the same kind and arity. In a relative interpretation of a theory S in T a variant of S is used if S and T have any nonlogical symbols in common. By using a variant of S one ensures that there is no such overlap. definition 6.1 (relative interpretation). A theory S is relatively interpretable in T if and only if there is a variant SvP of S P with no nonlogical symbols in common with LT , a theory R and a set D of sentences that satisfy the following conditions: (i) SvP ∪ T ⊆ R (ii) D is a recursive set of sentences, containing exactly one possible definition of P and each nonlogical symbol of LSvP in the theory T. D contains no other sentences. (iii) All elements of R are theorems of T ∪ D. A well-known example of relative interpretability is the reduction of Peano arithmetic to Zermelo–Fraenkel set theory. For this reduction a certain set, usually the set of finite von Neumann ordinals, is used to represent the nonnegative integers. That means that the predicate letter P of paP receives the possible definition ∀x (P x ↔ x ∈ ω), where x ∈ ω stands for the formula in the language of set theory expressing that x is a finite von Neumann ordinal. Then possible definitions of the

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comparing axiomatic theories of truth

nonlogical symbols of the language of arithmetic are provided in set theory. Finally it is proved that all theorems of paP can be derived in zf using the possible definitions of P and the nonlogical symbols of the language of arithmetic. If S is relatively interpretable in T, there is a translation function I from the set of formulae of LS to the set of formulae of LT . The interpretation function relativizes all quantifiers of S-theorems and replaces all nonlogical symbols of LS with their possible definitions, with function symbols expressed by predicates in the way indicated above. In many cases I will specify relative interpretations by directly specifying function symbols as translations of function symbols rather than by providing suitable formulae as required by the definition of a relative interpretation; obviously if suitable function symbols are available the required formulae can be defined in a trivial way. In particular, a relative interpretation from one truth theory to the other can be specified in many cases by saying that all arithmetical vocabulary is preserved by the relative interpretation and then specifying a possible definition for the truth predicate. Feferman (1960) established many fundamental results on relative interpretations. In the following theorem ConS is a consistency statement for S defined in the usual way from a provability predicate for S that represents S-provability in Peano arithmetic. theorem 6.2. If pa is a subtheory of T and T ` ConS obtains, then S is relatively interpretable in T. The theorem is proved by formalizing the Gödel completeness theorem in Peano arithmetic. The reader is referred to (Feferman 1960, p. 72, Theorem 6.2) for the proof and a precise formulation of the theorem. Further general properties of relative interpretability are given in Mycielski et al. (1990). Although, as Montague (1962) has shown, there are theories that are incomparable with respect to relative interpretability, all the theories in this book are comparable with respect to relative interpretability, and so they are linearly ordered by relative interpretability. A finite subtheory of S is a subtheory of S that can be generated from a finite set of axioms by closing it under first-order consequence. A theory S is locally interpretable in T if and only if every finite subtheory of S is relatively interpretable in T. A theory is reflexive if and only if it proves the consistency of each of its finite subtheories.

comparing axiomatic theories of truth

43

theorem 6.3 (Orey’s compactness theorem). If T is reflexive and contains pa, then S is locally interpretable in T if and only if S is relatively interpretable in T. See Feferman (1960, p. 80, Theorem 6.9) for a proof of Orey’s (1961) theorem. Any consistent extension T of Peano arithmetic with the induction schema admitting arbitrary instances in the language LT is reflexive if certain natural conditions are met. For a proof see (Hájek and Pudlák 1993, p. 189, Lemma 3.47). I will often be dealing with truth theories formulated in the language LT with pa as base theory and the full induction schema for LT . Consequently, to show that a theory S is relatively interpretable in T it will suffice to show that S is locally interpretable in T. Arithmetical vocabulary need not be preserved under a relative interpretation. In fact there will be examples of truth theories S and T with S relatively interpretable in T but where the arithmetical vocabulary has to be reinterpreted, so that + is not interpreted as addition and so on. In particular, quantifiers are translated as restricted quantifiers. Fujimoto (2010a) proposed to focus on relative interpretations that leave the arithmetical vocabulary unchanged and do not relativize the quantifiers of the source theory. This leads to a stricter notion of reduction, namely truthdefinability or relative truth-definability, as Fujimoto calls it. Truth-definability is not as generally applicable as there are restrictions on the languages in which the theories can be formulated. Truth-definability can be defined in the way I just did as a refinement of relative interpretation, or more directly as follows. definition 6.4. Assume S and T are extensions of pa and S is formulated in the language LT of arithmetic augmented with the truth predicate T. Then T defines the truth predicate of S if and only if there is a formula ϕ(x) in the language of T such that the result of uniformly substituting ϕ(x) for the truth predicate T in a theorem of S is a theorem of T. That the truth predicate of S is definable in T is also expressed by saying that T defines the truth predicate of S. In most cases considered here, both theories T and S are formulated in the language LT . In this setting, if the truth predicate of a truth theory S is definable in a theory T, the truth predicate of S is in fact explicitly definable in T in the following sense. lemma 6.5. Assume that S and T are theories formulated in the language LT , and T0 is the variant of T where the truth predicate T of LT is replaced

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with another unary predicate symbol T0 . Then the following two conditions are equivalent: (i) S is truth-definable in T. (ii) There is a formula ϕ(x) of LT0 such that S is a subtheory of T0 ∪
{∀x Tx ↔ ϕ(x) }. I will apply the notion of truth-definability also to truth theories with more than one truth predicate, for instance to truth theories with hierarchical truth predicates. The notion of truth-definability generalizes to such cases in the obvious way. When the truth predicate of S is definable in T, then all theorems of S that do not contain the truth predicate are also theorems of T. If one is not interested so much in the truth predicates as in what can be done with them in arithmetic, one can ignore the truth-theoretic content of the theories and compare truth theories by their arithmetical theorems. For any truth theory S its arithmetical content, which is also called its truth-free content, is the set of its theorems in the language L. Any two theories of truth discussed in this book either coincide in their arithmetical content, or the arithmetical content of one is a proper subset of that of the other. I turn to stricter notions of reducibility. definition 6.6 (conservativity). A truth theory T in the language LT is conservative over a theory S formulated in the language L without the truth predicate if and only if all T-theorems ϕ in the language L are also theorems of S. I will apply this definition mainly to the case where S is Peano arithmetic. Proof theorists often impose the additional restriction on a reduction that it can be carried out in the target theory, so that, for instance, to obtain a reduction of S to T one does not only have to show that T is conservative over S (or that S is relatively interpretable in T and so on) but one also has to show this in the theory T. These more refined notions are related to what is called proof-theoretic reducibility. There are various versions of proof-theoretic reducibility. Proof-theorists often do not care to specify which exact notion of reducibility is applied as the different notions of proof-theoretic reducibility coincide in many cases and most results concern specific reductions and it is clear from the proof of the specific reduction in which sense it constitutes a reduction. Feferman’s (1988) notion of proof-theoretic reducibility is presumably the best known example.

comparing axiomatic theories of truth

45

I will not give a precise definition of proof-theoretic reducibility as the exact details of the definition will not matter much in what follows. Assume that the systems S and T are formulated in recursive extensions of L. Then, very roughly, the system S is proof-theoretically reducible to T if and only if (i) every proof of a closed equation s = t in S can be effectively transformed into a proof of the same equation in T, and (ii) condition (i) is provable in the target system T. If systems are proof-theoretically reducible to each other, they are said to be proof-theoretically equivalent. A system S can be proof-theoretically reducible to a system T without being relatively interpretable in T; and S can be relatively interpretable in T without being proof-theoretically reducible to it, even when S and T are extensions of L. In many cases one can prove the proof-theoretic reducibility of S to T by providing a translation from LS to LT that preserves the logical structure and provability and then showing that this can be shown in T. Proof-theoretic reducibility is important when one tries to argue that the system S is acceptable from the standpoint of T. If one tries to show from the standpoint of T that a system S is ‘safe’ from the perspective of T, then it will hardly suffice if the existence of a translation of S into T is provable in a system going far beyond the power of T. Feferman (1988, 2000), Hofweber (2000), and Niebergall (2000) provide further discussions of proof-theoretic reducibility. In particular, Niebergall (2000) sheds some doubt on the suitability of proof-theoretic reducibility by showing that proof-theoretic reducibility is not transitive. In the reductions between truth theories I will occasionally highlight that certain reductions such as truth-definitions can be carried out in the target theory. In most cases this implies that the reduction is also a proof-theoretic reduction (see Fujimoto 2010a for results on the connection between proof-theoretic reducibility and truth-definability). All the notions of reducibility discussed so far are intended to be applied to systems in classical logic. As soon as the framework of classical logic is abandoned, several difficulties arise and the notions of reducibility have to be adapted to this generalized framework and to the particular chosen nonclassical logic(s). As I will be dealing only with theories retaining classical arithmetic, one can still compare nonclassical theories by their classical arithmetical content. Since the only nonclassical theory I consider in detail is dealt

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with in Chapter 16, I postpone the discussion of reducibility of and to nonclassical theories until this chapter. In what follows I will try to prove the strongest possible results, for instance proving that the truth predicate of a theory S is definable in T rather than showing S to be relatively interpretable in T, when possible. I leave the discussion of the significance of these reductions to the last chapter of the book, after the systems have been introduced and the formal results have been established. There are further notions of reducibility that will not be used in this book. In particular, ordinal analysis has played a prominent role in the study of axiomatic theories of truth. On this approach, using cut elimination techniques and proofs of transfinite induction, recursive ordinals are assigned to deductive systems. Famously Gentzen (1936, 1943) determined the proof-theoretic ordinal of Peano arithmetic as 0 and since then the proof-heoretic ordinals of much stronger systems have been determined. The strengths of almost all truth-theoretic systems with Peano arithmetic as base theory lie within the scope of present ordinal analysis. Despite the importance and extensive use of ordinal analysis, I have refrained from using this powerful tool here for comparing truth theories. I have several reasons for this. First, there are various different notions of the proof-theoretic ordinal of a system.1 Some of them cannot easily be accommodated to truth-theoretic systems in a suitable way, as Fujimoto (2010a) has pointed out. Second, the ordinal analyses of truth-theoretic systems are very sensitive to the base theory. Many of the results on truth-definability, for instance, can be carried over to truth systems with other base theories. For instance, one will be able to show that a typed disquotational truth predicate is definable by a typed compositional truth predicate. This is shown for Peano arithmetic as base theory in Lemma 8.4. Using the same techniques, this can be established mutatis mutandis for much stronger base theories as well. Transferring results on proof-theoretic ordinals from systems with Peano arithmetic as base theory to system with much stronger systems as base theories is at least difficult and in many cases impossible. Full second-order number theory and Zermelo–Fraenkel set theory, for instance, are beyond the reach of contemporary ordinal analysis. As I am more interested in the contribution of the truth axioms to the strength of a system and less in the absolute strength 1 For a further discussion of ordinal analysis see Girard (1987), Niebergall (1996), Pohlers (2009), and Rathjen (1999).

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of the base theory augmented with the truth axioms, ordinal analysis seems less suitable for my purposes. I would like to treat the base theory more as a changeable parameter and less as the crucial factor on which the analysis of a truth system essentially depends. Finally, ordinal analysis is a large research area and the analysis of truth theories is merely one application of many. An introduction to the techniques of ordinal analysis requires a full book and such introductions can be found elsewhere written by expert proof theorists. Having said this, the reader will find that there are points where techniques from ordinal analysis are hard to avoid. In particular, in the treatment of hierarchical systems in Chapter 9 the proof-theoretic ordinals will surface and I will refer to results that cannot be obtained, at least to my knowledge, without using cut elimination techniques, which are commonly used for determining proof-theoretic ordinals. Consequently I assume that the definitions of hierarchical systems and the results on them cannot be easily adapted to a set-theoretic setting, for instance.

Part II

TYPED TRUTH

I sort axiomatic theories of truth into two large families, namely into typed and type-free theories of truth. Roughly speaking, typed theories prohibit a truth predicate’s application to sentences with occurrences of that predicate, while type-free theories do not. I will not consider syntactically restricted theories, that is, theories in which the truth predicate cannot be combined with any term to form a sentence, but typed theories either impose no restriction on the truth of sentences with the truth predicate or they prove that all sentences with the truth predicate are not true. At any rate, in typed theories one cannot prove the truth of any sentence containing T. Type-free theories of truth are often also described as theories of self-applicable truth. Making this distinction precise is not entirely straightforward, however, and I will postpone the discussion of the distinction until Chapter 10, as only then will I have some examples of the theories to hand. Of course, axiomatic theories of truth can be classified in other ways as well. For instance, one can distinguish between compositional and non-compositional theories, or between disquotational and non-disquotational theories. By and large, I find it easier to treat typed theories together in one part as typed theories have more in common technically, than, for instance, compositional theories do. Similarly all disquotational theories, that is, theories based merely on disquotation sentences as their axioms, may initially look much alike, but as I will show, their formal properties vary wildly. The present part is devoted to typed theories. Much of the classic work on axiomatic theories belongs here. For instance, Tarski’s contribution presented in Chapter 3 falls into this category. It is remarkable that philosophers have tended to focus on typed theories when discussing the significance of prooftheoretic results for truth-theoretic deflationism, even though they do not necessarily think that typing provides the best solution of the paradoxes. For instance, many philosophers have rejected disquotational theories of truth because of their deductive weakness. In the present part, it will be shown that in the context of typed theories the charge of deductive weakness is justified, but in the next part we shall see that in a type-free context disquotational theories may be extremely strong.

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typed truth

The difference in the formal properties of typed and type-free theories of truth shows that it is very hard to discuss the merits of disquotationalism and other philosophical doctrines about truth by appealing to results on typed theories unless one underpins this appeal with good reasons for adopting a typed approach. In fact there are good reasons to favour a type-free approach, as I will show in Chapter 11. I devote an entire part of this book to typed theories for two main reasons: First, some philosophers may prefer typed theories and so the results of the present part will be directly relevant to their philosophical discussions and for certain applications typed theories are superior to typefree theories. Second, even when one finally opts for type-free theories, the results of the present part will form the basis for the technical results about type-free theories. Chapter 7 is devoted to typed disquotational theories. In the remaining chapters of this part compositional theories are considered, including iterations of typed compositional theories in hierarchical theories in Chapter 9.

7 Disquotation

The elegant simplicity of an axiomatization of truth based merely on the disquotation- or T-sentences or similar equivalences has appealed to many authors. As mentioned in Chapter 3, Tarski (1935) himself considered an axiomatization of truth by these equivalences. Interest in such an axiomatization has been revived by the recent discussion of deflationism. I call the theory that results from adding the disquotation sentences to Peano arithmetic tb for Tarski-biconditionals. definition 7.1 (tb). The theory tb comprises all axioms of pat, that is, of Peano arithmetic formulated in LT including all instances of the induction schema with the truth predicate. Moreover all sentences of the form T┌ϕ┐ ↔ ϕ are axioms of the theory where ϕ is a sentence of the language of L. Restricting the disquotation schema to sentences of L, that is, to sentences without the truth predicate makes tb a typed system. The formulae ϕ permitted as instances of the disquotation sentences of tb are sentences, that is, they are not allowed to contain free variables. In the theory utb, an acronym for uniform Tarski-biconditionals, this restriction is relaxed. To formulate the uniform disquotation sentences, I quantify over all closed terms. The uniform disquotation sentence for the formula ϕ(x) with exactly the variable x says – to speak sloppily – that for any closed term t, the sentence ϕ(t) is true if and only if ϕ(t). More formally, using the notation introduced above, this can be written as follows: (7.1)


∀t T(┌ϕ┐(t/┌x┐)) ↔ ϕ(t◦ )

As explained in Section 5.1, the expression ∀t ψ(t) is short for ∀y (ClTerm(y) → ψ(y)). Also by the conventions of this section, t◦ stands for the value of the term t; and T(┌ϕ┐(t/┌x┐)) expresses that the result of substituting all free occurrences of the variable x with the term t in ϕ(x) is true.

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The axiom ∀x (T┌ϕ(x)┐ ˙ ↔ ϕ(x)) would be weaker than (7.1), because in the scope of the truth predicate the quantifier in the latter only ranges over numerals, not over closed terms which contain function symbols other than the successor symbol. Of course, the disquotation sentences can be generalized to formulae with arbitarily many free variables. The multi-variable version of (7.1) can be formulated in a straightforward way, although the notation becomes somewhat convoluted. Hence, as explained on p. 33, I will write ┌ϕ(t. )┐ for (┌ϕ┐(t/┌x┐)) assuming that it is clear which variable t is substituted for. Using this notation for more than one free variable, the system utb is defined as follows: definition 7.2 (utb). The theory utb comprises all axioms of pat and all sentences of the form ∀t1 . . . ∀tn T┌ϕ(t. 1 , . . . , t. n )┐ ↔ ϕ(t1 ◦ , . . . , tn ◦ )




are axioms of the theory where ϕ(x1 , . . . , xn ) is a formula of the language of L with exactly x1 , . . . , xn free. definition 7.3. tb↾ and utb↾ are the theories tb and utb, respectively, without any induction axioms containing the truth predicate. The truth predicate axiomatized by the disquotation sentences is not eliminable in the sense of being definable. This is a variant of Tarski’s celebrated theorem on the undefinability of truth. theorem 7.4 (Tarski’s theorem on the undefinability of truth). Assume S is a consistent theory containing tb↾, then there is no formula τ(x) in the language L such that S ` ∀x(τ(x) ↔ Tx). proof. Assume there is such a formula τ(x). Then by diagonalization there exists a sentence γ such that S ` γ ↔ ¬T┌γ┐. Because S ⊇ tb obtains, the following sentence is a theorem of S: T┌γ┐ ↔ γ Therefore the contradiction γ ↔ ¬γ is provable in S, contradicting the consistency of S. a Nor is the truth predicate axiomatized by the disquotation sentences completely eliminable in any given proof of a T-free sentences. More precisely, the

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disquotation sentences are not conservative over logic, as shown in Halbach (2001c) because tb proves that there are at least two different objects. If ϕ is a tautology, one obtains the theorem T┌ϕ┐ from the disquotation sentence for ϕ, T┌ϕ┐ ↔ ϕ. The disquotation sentence for ¬ϕ yields ¬T┌¬ϕ┐. Thus without any base theory the disquotation sentences imply ┌ϕ┐ 6= ┌¬ϕ┐, which implies ∃x ∃y x 6= y. The latter is of course not a theorem of pure identity logic. Although the truth predicate is not eliminable in the sense of being definable or conservative over identity logic, it can be eliminated in any given tb-proof of a sentence of L that does not contain the truth predicate. In other words, tb and utb are both conservative over pa: any sentence of L provable in tb or utb is provable in pa alone. Adding the disquotation sentences in their local form, as in tb, or in their uniform version, as in utb, does not add anything to the arithmetical content of pa. The main idea for the proof of the conservativity of tb and utb can be traced back to Tarski’s (1935) proof of his Theorem III (see Chapter 3). theorem 7.5. tb and utb are conservative over pa. The idea for the proof is simple. In a proof in tb only finitely many disquotation axioms T┌ϕi┐ ↔ ϕi (i ≤ n) can occur. In the proof atomic formulae of the form Ts for some term s are then substituted with the formulae (s = ┌ϕ1┐ ∧ ϕ1 ) ∨ . . . ∨ (s = ┌ϕn┐ ∧ ϕn ), where ϕ1 to ϕn are all the sentences ϕi from the disquotation axioms occurring in the proof. The resulting structure can easily be completed into a proof. The proof for the uniform version follows this idea as well. proof. Clearly tb is a subtheory of utb, so it is sufficient to prove the claim for utb. I show how to transform any given utb-proof of an arithmetical formula into a pa-proof of the same formula. Let a proof of a formula in L in utb be given and let n be the number of uniform disquotation sentences occurring as axioms in the proof. Hence every axiom in the proof is either an axiom of pa, or an instance of the induction schema, or, for some i ≤ n, a sentence of the form:
∀t1 . . . tki T┌ϕi (t. 1 , . . . , t. ki )┐ ↔ ϕi (t1 ◦ , . . . , t◦ki ) Let the formula τ(x) be the disjunction of all formulae of the following form, for each i ≤ n: ∃t1 . . . ∃tki x = ┌ϕi (t. 1 , . . . , t. ki )┐ ∧ ϕi (t1 ◦ , . . . tki ◦ )




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Now all occurrences of the truth predicate T in the given utb-proof are replaced with τ. In particular the disquotation sentences in the proof become: (7.2)

∀t1 . . . tki τ(┌ϕi (t. 1 , . . . , t. ki )┐) ↔ ϕi (t1 ◦ , . . . , t◦ki )




These sentences are provable in pa. Proofs for all these sentences are then added to the proof, yielding a proof in pa, since the induction axioms containing occurrences of T are transformed in purely arithmetical instances. If the last sentence of the utb-proof is a formula of L, then it remains unaffected by the substitution of T with τ. a Thus in any proof in utb the truth predicate can always be replaced with a defined truth predicate for a language with only finitely many formulae. But the formula τ(x) depends on the particular proof, and Tarski’s theorem shows that there is no τ(x) that will serve this purpose for all proofs. Theorem 7.5 shows that truth as axiomatized in tb or utb is not of any use for establishing new arithmetical theorems. This accords well with deflationist views of truth according to which truth is ‘insubstantial’ and does not have any ‘real content’. Deflationists, however, do not hold that truth is completely dispensable.1 They claim that truth serves the purpose of expressing infinite conjunctions or generalizations. In sentences such as ‘All theorems of pa are true’, or ‘Everything Bush said about weapons of mass destruction in Iraq is true’, the truth predicate is not eliminable by another predicate. The following quote from Quine (1970, p. 12) is the locus classicus of this observation: We may affirm the single sentence by just uttering it, unaided by quotation or by the truth predicate; but if we want to affirm some infinite lot of sentences, then the truth predicate has its use. Quine (1990, p. 81) gives an example of how the truth predicate can be employed: The truth predicate proves invaluable when we want to generalize along a dimension that cannot be swept out by a general term. The easy sort of generalization is illustrated by generalization on the term ‘Socrates’ in ‘Socrates is mortal’; the sentence generalizes 1 For a more detailed account of the deflationist stand on the disquotation sentences and their relation to infinite conjunctions see Halbach (1999b). The following is an overview of the discussion in that paper and of some of the reactions to that paper.

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to ‘All men are mortal’. The general term ‘man’ has served to sweep out the desired dimension of generality. The harder sort of generalization is illustrated by generalization on the clause ‘time flies’ in ‘If time flies then time flies’. We want to say that this compound when the clause is supplanted by any other; and we can do no better than to say just that in so many words, including the word ‘true’. We say “All sentences of the form ‘If p then p’ are true.” It is exactly this reduction of ‘infinite lots of sentences’ to single sentences containing a truth predicate that disquotationalism takes to be the explanation for there being a truth predicate in the first place: according to disquotationalism, this is the raison d’être of truth. Later the somewhat unspecific ‘infinite lots’ were replaced by infinite conjunctions and disjunctions in various writings. One may now expect the theory of truth to allow one to prove the claims mentioned by Quine. For instance, one might expect All sentences of the form ‘If p then p’ are true to be a consequence of the theory of truth, at least when p does not contain the truth predicate. It can be shown that utb, and therefore also tb, tb↾, and utb↾, cannot prove any such ‘infinite lots’. theorem 7.6. If tb ` ∀x (ϕ(x) → Tx) and ϕ(x) is an L-formula, then there is an n such that pa proves there are at most n ϕs, or more formally, such that pa ` ∃≤n x ϕ(x). Here ∃≤n x ϕ(x) is short for the formula that expresses that there are at most n objects satisfying ϕ(x) in the language of first-order logic, with identity:
∀x1 . . . ∀xn+1 ϕ(x1 ) ∧ . . . ∧ ϕ(xn+1 ) → x1 = x2 ∨ . . . ∨ x1 = xn+1 ∨ . . . ∨ xn = xn+1 Thus every provable generalization that can be expressed by the truth predicate of tb is merely a finite generalization. Moreover, one can prove in the theory without the truth predicate that the generalization is finite. proof. The proof is based on the same technique as the proof of Theorem 7.5. Let a proof of ∀x (ϕ(x) → Tx) in tb be given. As before there are only finitely many sentences ψ such that the disquotation sentences T┌ψ┐ ↔ ψ is used in the proof. Let ψ1 , . . . , ψn be these sentences. Let τ(x) be the formula (x = ┌ψ1┐ ∧ ψ1 ) ∨ . . . ∨ (x = ┌ψn┐ ∧ ψn )

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Replace the truth predicate T everywhere in the proof by τ and, for i ≤ n, add the proofs for τ(┌ψi┐) ↔ ψi , where these sentences have replaced the disquotation sentences. The result is a proof of ∀x (ϕ(x) → τ(x)) in pa. Together with the logical theorem ∃≤n x τ(x), this yields the desired result. a In particular, if ϕ(x) expresses that x is a sentence of the form p → p, then tb does not prove ∀x(ϕ(x) → Tx). Thus the deflationist should not claim that the truth predicate serves the purpose of proving such generalizations as All sentences of the form ‘If p then p’ are true. The proof of Theorem 7.6 does not depend on pa being the base theory. Theorem 7.6 can be generalized to arbitrary base theories in place of pa. The proof for pa as the base theory is interesting as even the occurrences of the truth predicate in the induction axioms can be replaced by τ. So the argument applies even though tb contains more axioms with the truth predicate than just the disquotation sentences, namely the induction axioms with the truth predicate. So even if the deflationist extends his system to include these additional induction axioms beyond the disquotation sentences, he will not be able to prove any infinite generalizations; while for finite generalizations the truth predicate is obviously dispensable. Now one might wonder whether passing from the local disquotation sentences of tb to the uniform disquotation sentences of utb will impinge on the possibility of proving infinite generalizations. In the following theorem, lh(x) expresses the function that assigns to every formula its complexity (length).
theorem 7.7. If utb ` ∀x ϕ(x) → Tx and ϕ(x) is a formula of L, then
pa ` ∀x ϕ(x) → lh(x) ≤ n . Thus if utb proves the generalization ∀x(ϕ(x) → Tx), then pa proves that the sentences satisfying ϕ(x) do not exceed a certain fixed length n. The proof of Theorem 7.7 is very similar to the proof of Theorem 7.6. One simply uses the partial truth predicate τn instead of the defined truth predicate τ.2 From utb ` ∀x (ϕ(x) → Tx) one cannot generally infer that pa proves that only finitely many objects satisfy ϕ(x). For instance, utb proves that all sentences of the form n = n are true for n ∈ ω, where n is the numeral for n. Theorem 7.7 shows, however, that the truth predicate can be substituted by a suitable formula not containing the truth predicate. At any rate, Theorem 7.7 2 For details on partial truth predicate see Kaye (1991) or Hájek and Pudlák (1993).

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extinguishes any hope that utb might enable the deflationist to prove the generalization that all sentences of the form p → p are true by using only disquotational axioms such as the uniform disquotation sentences of utb. If the deflationist shifts his focus from Quine’s example to examples of contingent sentences such as ‘Everything Bush said about weapons of mass destruction in Iraq is true’, he will not expect his theory of truth to prove such generalizations – even if they are true. In (1999b, Proposition 2) I have suggested that the truth predicate of tb might enable one to express generalization in another sense. According to my suggestion the generalization should be equivalent to the infinite set of sentences of the form: If Bush said ‘A’ about weapons of mass destruction in Iraq, then A for all sentences A. Here again I consider only sentences A not containing the truth predicate. This suggestion is made precise in the following observation, assuming again that ϕ(x) is an L-formula: observation 7.8. Let S be the set of all sentences of the form ϕ(┌ψ┐) → ψ for all sentences ψ of L. Then S + pa and tb + ∀x(ϕ(x) → Tx) prove the same L-sentences. The observation applies also to tb↾ instead of tb. proof. From ∀x (ϕ(x) → Tx) one obtains by instantiation ϕ(┌ψ┐) → ψ for all sentences ψ of L. Therefore S + pa is a subtheory of tb + ∀x (ϕ(x) → Tx). Conversely, let a proof in tb + ∀x (ϕ(x) → Tx) of a formula χ of L be given. Very much as in the proof of Theorem 7.6 one can replace all occurrences of T by the arithmetical formula τ(x), defined as (x = ┌ψ1┐ ∧ ψ1 ) ∨ . . . ∨ (x = ┌ψn┐ ∧ ψn ), where the ψi are the sentences with T┌ψi┐ ↔ ψi in the original proof. Under this translation the axiom ∀x (ϕ(x) → Tx) becomes ∀x (ϕ(x) → τ(x)). The latter is provable from S. As before, the induction axioms containing the truth predicate become induction axioms without T. By adding some subproofs one arrives at a proof of χ in S + pa. a This result can be generalized in several ways. Clearly the proof carries over to other base theories, and it was stated in a more general form in Halbach (1999b). The present formulation shows that the addition of induction axioms in LT does not affect the equivalence of the two theories. Observation 7.8 shows that infinite generalizations understood as schemata of the form ϕ(┌ψ┐) → ψ can be expressed by a single sentence in the presence of the disquotation sentences. In a sense the infinitely many sentences

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ϕ(┌ψ┐) → ψ have been replaced by the single sentence ∀x (ϕ(x) → Tx) and the infinitely many disquotation sentences. In particular, the infinitely axiomatized theory S + pa has not been reaxiomatized by finitely many axioms (see Kemp 2005, p. 331 for a more detailed exposition of this problem). Moreover, the problem cannot be overcome by a finite reaxiomatization of tb or tb↾. theorem 7.9. None of the theories tb, tb↾, utb, and utb↾ are finitely axiomatizable over pa (where pa may or may not include all induction axioms with T). That is, there is no finite set of LT -sentences S such that pa + S yields exactly all theorems of tb, tb↾, utb, or utb↾. Moreover the set of disquotation sentences cannot be finitely axiomatized (over pure logic). I have listed here all these theories in order to show that even on the basis of an infinitely axiomatized theory such as pa or in the presence of a schema such as induction in LT , the disquotation sentences cannot be expressed finitely. proof. Assume that tb is equivalent to pa + σ. Then tb ` σ. But in any given proof in tb the truth predicate T can be replaced by a truth predicate such as in Theorem 7.6 that is definable in L. But since pa + σ proves all disquotation sentences the truth predicate in σ cannot be replaced by such a L-definable truth predicate. A similar argument applies to utb and utb↾, replacing the finite truth predicate of Theorem 7.6 with a suitable partial truth predicate such as in Theorem 7.7. a In the light of this observation it seems that tb only allows one in certain cases to replace an infinite generalization, that is an infinite sets of sentences, by another theory that cannot the finitely axiomatized. The reaxiomatization only adds an additional predicate and the problem of finite axiomatizability is shifted to this new predicate. Other theories of truth fare much better in this respect because they can be used to give finite reaxiomatizations of fairly arbitrary theories, as Craig and Vaught (1958) have shown. Nevertheless something has been gained by the addition of the truth predicate: against the background of a theory containing the disquotation sentences, infinite schemata of a certain kind can be expressed by a single sentence. A deflationist would then have to explain why the infinity of disquotation sentences is unproblematic. I think it would be coherent to claim that the disquotation sentences are in the ‘background’ in very much the same way as rules of inference (such as modus ponens) are in the background, as logic cannot be axiomatized without axiom schemata or rules with infinitely many

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instances. Given logic and the disquotation sentences, one can then quickly communicate a claim such as ‘Everything Bush said about weapons of mass destruction in Iraq is true’ by availing oneself of the truth predicate. Observation 7.8 is a strengthening of Theorem 7.5 on the conservativity of tb over pa. That tb is conservative over pa can be derived from Observation 7.8 by choosing ϕ(x) as x 6= x in the statement of Observation 7.8. Other theories of truth that will be discussed later, such as ct, are not conservative over pa. Consequently Observation 7.8 cannot be proved for them in place of tb. Thus the truth predicates of these other theories serve purposes beyond that of expressing infinite conjunctions in the sense explained. Theories such as ct actually prove infinite generalizations and thereby overcome the deductive weakness of tb illustrated by Theorem 7.6. If the deflationist thinks of Observation 7.8 as an adequate explication of his claim that truth allows for nothing more than the expression of infinite generalization or conjunctions, then he can welcome the deductive weakness of tb as support of his claim that truth does not serve any other purpose. The system tb and similar theories have been rejected as adequate axiomatizations of truth by many authors, Tarski being the first of them (see Section 3). So I shall now consider the deficiencies of tb and utb. Deflationists such as Horwich (1990) often claim that the truth predicate allows one to express infinite conjunctions and disjunctions. Observation 7.8 can be taken as a formal substantiation of the claim that with the help of the truth predicate one can express infinite conjunctions. In (1999b, p. 15) I have argued that the truth predicate can also be used to express infinite disjunctions. A theorem very similar to Observation 7.8 can be proved for infinite disjunctions. A sentence ∃x(ϕ(x) ∧ Tx) can then be taken to express an infinite disjunction, if ϕ(x) applies to infinitely many objects. Heck (2004) pointed out that the deflationist should then also expect that adding the truth-theoretic counterpart of an infinite conjunction and of an infinite disjunction to pa should prove the same T-free sentences as pa plus the respective infinite conjunctions and disjunctions. Heck (2004, p. 331) gave a counterexample to this conjecture, which will not be discussed here. By adding a sentence of the form ∀x(ϕ1 (x) → Tx) and of the form ∃x(ϕ2 (x) ∧ Tx) one can prove more sentences than with the conjunction of all sentences ϕ1 (┌ψ┐) → ψ and the disjunction of all sentences ϕ2 (┌ψ┐) ∧ ψ. So it seems, after all, that the truth predicate of tb serves a purpose beyond the mere expression of infinite conjunctions and disjunctions. The main criticism that deflationist theories based on the disquotation sentences or similar axioms have to meet was raised by Tarski: the disquotation

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sentences do not allow one to prove generalizations, as Theorems 7.6 and 7.7 show. In order to defend the disquotation sentences as the only axioms for truth, one could either simply reject the generalizations or one could argue that the truth theory itself should not be expected to prove them, maintaining instead that they are provable from other sources. I think that it is not a real option to reject such generalizations as A conjunction is true if and only if both conjuncts are true or the law of excluded middle or the principle of contradiction (see p. 20). Any semantical theory that does not yield these generalizations in some suitable form is highly incomplete. It is also unclear what should provide one with the resources to prove these generalizations other than the theory of truth: the displayed claim, for instance, is about the semantics of conjunction alone and thus it seems at least highly unlikely that any theory other than the theory of truth – together with logic, of course – could provide the means to prove such generalizations. This does not imply that any theory that relies on a disquotational conception of truth has to fall short of proving the relevant general principles. If one drops the restriction that truth only applies to sentences without the truth predicate, then one can obtain theories that are deductively very strong and prove many generalizations and perhaps all generalizations that ought to be provable. Such type-free disquotational theories will be investigated in Chapter 19. But even if one sticks to a truth predicate that only applies to sentences of L, one might try to obtain a strong theory of truth by adding proof-theoretic reflection principles to a disquotational theory. In fact, adding the uniform reflection principle for the system that is given by all uniform disquotation sentences (without any arithmetical axioms) to pa yields the ‘Tarskian’ compositional truth theory ct↾, which will be discussed in the next chapter and on p. 309.

8 Classical compositional truth

As has been shown in Section 3, Tarski (1935, p. 257) rejected an axiomatization of truth like tb based solely on the T-sentences because the resulting theory ‘would lack the most important and most fruitful general theorems’. Moreover, he did not expect that adding some of those general theorems as axioms would lead to a satisfactory theory of truth because he thought that such an axiomatization would be somewhat arbitrary (see p. 20). Ironically Tarski’s definition of truth prepared the ground for the wide acceptance of a theory – or rather, a family of theories – that go beyond purely disquotational theories but are nevertheless seen as natural, and far from arbitrary. The inductive clauses from Tarski’s definition of truth can be turned into axioms. The resulting theory is thought by many philosophers and logicians to be a theory of truth that is natural and, in a sense, complete: it proves generalizations of the kind Tarski had in mind. In particular, it proves the general principle of contradiction, the statement that not both a sentence and its negation can be true. Donald Davidson assigned an important role to this axiomatization of truth in his theory of meaning (see Davidson 1984c and Fischer 2008). He proposed to turn Tarski’s clauses for defining truth into axioms. Although significant work was done by Davidson and his disciples to extend the theory from simple formal languages to natural languages containing adverbs and other phrases not dealt with by Tarski, Davidson never specified the axioms of the theory for a simple formal language like the one considered by Tarski. There are, however, some decisions to be taken in the formulation of this theory. I will start the discussion by considering what are often called Tarski’s inductive clauses. I do not attempt to follow Tarski’s original formulation, but rather formulate the definition in a way that might now pass for a ‘Tarskian’ definition of truth for the language L of arithmetic. In particular, I will not take the detour through satisfaction to state the axioms, since a universally quantified sentence of L can be defined to be true if and only if all its instances are true. As pointed out above, sentences are identified with numbers. Hence the set of true L-sentences is a set of numbers. The identification of expressions

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and number will facilitate the formalization of the clauses of the inductive definition of truth in arithmetical theories. definition 8.1 (truth definition for L). The set of true sentences of L is the unique set S of numbers satisfying the following condition: n ∈ S if and only if (i) there are closed terms s and t such that n is s = t and the value of s is identical to the value of t; or (ii) there is an L-sentence ϕ such that n is ¬ϕ and ϕ ∈ / S; or (iii) there are L-sentences ϕ and ψ such that n is ϕ ∧ ψ and ϕ ∈ S and ψ ∈ S; or (iv) there are L-sentences ϕ and ψ such that n is ϕ ∨ ψ and ϕ ∈ S or ψ ∈ S; or (v) there is an L-sentence ∀vχ such that n is ∀vχ and for all closed terms t, χ(t/v) ∈ S; or (vi) there is an L-sentence ∃v χ such that n is ∃vχ and for some closed term t, χ(t/v) ∈ S. That there is such a unique set S satisfying conditions (i)–(vi) can be established by induction on the complexity of sentences. Instead of defining the set of true L-sentences I could have defined a truth predicate for L-sentences by conceiving ∈ S as a predicate expression, as S occurs only in formulae of the form u ∈ S in the definition (for arbitary terms u). The proof of the unique existence of the set of true L-sentences would then become a proof of the fact that all predicates satisfying (i)–(vi) are equivalent. The definition is a sentence of the following form: (8.1)

n ∈ S if and only if ζ(n, S)

So the definition shares its form with the usual explicit definitions with the exception that the definiendum S appears in the definiens ζ(n, S). This is the reason why a separate proof of the existence of a set S satisfying this equivalence is required. A slightly less cumbersome way to state the definition is obtained by stating the items (i)–(vi) in the definition as separate clauses: lemma 8.2. The set of true sentences of L is the unique set S of L-sentences satisfying the following conditions for all closed terms s, t and L-sentences ϕ, ψ, ∀vχ, and ∃vχ: (i) s = t ∈ S iff the value of s is identical to the value of t. (ii) ¬ϕ ∈ S iff ϕ ∈ / S. (iii) ϕ ∧ ψ iff (ϕ ∈ S and ψ ∈ S).

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(iv) ϕ ∨ ψ iff (ϕ ∈ S or ψ ∈ S). (v) ∀vχ ∈ S iff for all closed terms t, χ(t/v) ∈ S. (vi) ∃vχ ∈ S iff for some closed term t, χ(t/v) ∈ S. It is not hard to see that the definition of truth in the lemma is a notational variant of Definition 8.1. In order to obtain an axiomatization of this notion of truth, the definition of the set of true L-sentences can be formalized. It does not really matter whether the formulation in Definition 8.1 or in Lemma 8.2 is chosen. The formalization of Definition 8.1 yields one long sentence of ∀x (Tx ↔ . . .). Writing down such a long sentence and using it in proofs is cumbersome. Hence logicians tend to follow the formulation in the lemma in their axiomatization of the Tarskian definition of L-truth. Using the notation from Section 5.1, I will turn these clauses of the lemma into axioms, thereby formulating an axiomatization of typed compositional truth: definition 8.3 (ct↾). The system ct↾ is given by all the axioms of pa and the following axioms:
ct1 ∀s ∀t T(s=. t) ↔ s◦ = t◦
ct2 ∀x Sent(x) → (T(¬. x) ↔ ¬Tx)
ct3 ∀x ∀y Sent(x∧. y) → (T(x∧. y) ↔ T(x) ∧ T(y))
ct4 ∀x ∀y Sent(x∨. y) → (T(x∨. y) ↔ T(x) ∨ T(y))
ct5 ∀v ∀x Sent(∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v)))
ct6 ∀v ∀x Sent(∃. vx) → (T(∃. vx) ↔ ∃t T(x(t/v))) Clearly, Axioms ct1–ct6 mimic the clauses (i)–(vi) in Lemma 8.2. Axiom ct2, for instance, says that a negated sentence ¬ϕ of L is true if and only if ϕ itself is not true. Here, ¬. represents the function that yields, when applied to a sentence, its negation. The restriction Sent(x) could be replaced by Sent(¬. x) to bring ct2 into line with the other axioms but I am assuming that the negation function represented by ¬. maps a number to a sentence if and only if that number codes a sentence, and that Sent(x) and Sent(¬. x) are equivalent over Peano arithmetic. A similar assumption is also made for ∧. and ∨. . The Axioms ct1–ct6 do not stipulate anything about the truth of objects that are not sentences, whereas the set of true L-sentences contains only Lsentences. I could easily add an axiom stating that only L-sentences are true.

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But this axiom does not make a significant difference in what follows. So I prefer to formulate ct↾ without it to minimize the number of axioms. A truth predicate satisfying Axioms ct1–ct6 and the axiom ∀x (Tx → Sent(x)) stating that only L-sentences are true is easily truth-definable in ct↾. The system ct↾ is typed as all of the quantifiers ranging over sentences in the Axioms ct2–ct6 range only over sentences of L; and Axiom ct1 concerns only L-sentences, as pure identity statements are sentences of L. If L contained more predicate symbols beyond the identity symbol, then corresponding axioms would have to be added. If, for instance, there were an additional binary predicate symbol P in the base language, the axiom
∀s ∀t T(P. st) ↔ P s◦ t◦ would have to be added. The label ct stands for compositional truth. The system ct↾ is compositional in the sense that, according to the axioms, the truth of a sentence depends on the semantic values of the constituents of that sentence: in Axiom ct1 the truth of an identity statement depends on the values of the terms involved; in ct2–ct4 the truth of a complex sentence depends on the truth or falsity of its direct subsentences. The compositionality of Axioms ct5 and ct6 might be doubted because an instance of a quantified sentence does not form part of the sentence. This problem can be avoided by formulating the axioms in a more thoroughgoingly Tarskian way, namely by using a primitive binary predicate for satisfaction. The equivalent of the last axiom using a satisfaction predicate would say that a universally quantified formula ∀v ϕ(v) is satisfied by a variable assignment a if and only if ϕ(v) is satisfied by all v-variants of a. Under this reformulation the axiom would become compositional. Such a reformulation would not make any difference to the results in the following and it may be philosophically preferable to start with satisfaction than truth; using a satisfaction predicate, however, would make the presentation much more cumbersome. First I show that ct↾ extends the typed disquotational theories tb↾ and utb↾ of truth: lemma 8.4. The theories tb↾ and utb↾ are subtheories of ct↾. proof. I derive all uniform disquotation sentences by induction in the metatheory on the length of the arithmetical formula ϕ(x1 , . . . , xn ); that is, I prove the claim
ct↾ ` ∀t1 . . . ∀tn T┌ϕ(t. 1 , . . . , t. n )┐ ↔ ϕ(t1 ◦ , . . . , tn ◦ ) . The case where ϕ(x1 , . . . , xn ) is atomic, that is of the form x1 = x2 , is covered by Axiom ct1. If it is of the form ¬ψ(x1 , . . . , xn ), one can derive the

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disquotation sentence for ϕ(x1 , . . . , xn ) in ct↾ in the following way, starting with Axiom ct2:
∀x Sent(x) → (T(¬. x) ↔ ¬Tx)
∀t1 . . . ∀tn Sent(┌ψ(t. 1 , . . . , t. n )┐) → (T(¬. ┌ψ(t. 1 , . . . , t. n )┐) ↔ ¬T┌ψ(t. 1 , . . . , t. n )┐)
∀t1 . . . ∀tn T(¬. ┌ψ(t. 1 , . . . , t. n )┐) ↔ ¬T┌ψ(t. 1 , . . . , t. n )┐ The last line holds because pa ` ∀t1 . . . ∀tn Sent(┌ψ(t. 1 , . . . , t. n )┐). Now the induction hypothesis comes into play: ∀t1 . . . ∀tn (T┌ψ(t. 1 , . . . , t. n )┐ ↔ ψ(t1 ◦ , . . . , tn ◦ )) ∀t1 . . . ∀tn (¬T┌ψ(t. 1 , . . . , t. n )┐ ↔ ¬ψ(t1 ◦ , . . . , tn ◦ )) Combining this with the last line from above gives the claim ct↾ ` ∀t1 . . . ∀tn (T┌ψ(t. 1 , . . . , t. n )┐ ↔ ψ(t1 ◦ , . . . , tn ◦ )) I skip the cases of the quantifiers and the other connectives.

a

The system ct↾ can overcome the problem of ‘restricted deductive power’ (Tarski 1935, p. 257) to some extent: in ct↾ one can prove generalizations. The law of contraction is a theorem of ct↾; actually Axiom ct2 suffices for proving ∀x (Sent(x) → (¬Tx ∨ ¬T¬. x)) in ct↾. Since ct↾ and its variants have such desirable generalizations amongst its logical consequences, they seem to have ousted purely disquotational theories such as tb, tb↾, utb, and utb↾ in the discussion on deflationism. According to Theorem 7.5, these disquotational theories are conservative over their base theory pa; so they do not have any consequences outside the theory of truth itself. Some authors have tried to understand the deflationist claim that truth is not a substantial notion as the claim that a satisfactory axiomatization of truth should be conservative over the base theory. In fact, ct↾ is conservative over pa, but the conservativity of ct↾ is harder to prove than the conservativity of the disquotational theories. I am aware of only two proofs, both of which are not completely trivial. Kotlarski et al. (1981) established a model-theoretic result that implies the conservativity of ct↾ over pa. Their model-theoretic argument, however, is relatively complicated and cannot be easily formalized in weak theories. It would be welcome to have a proof that can be carried out within the base theory, so one could see from the standpoint of the base theory that the theory of truth does not have any unwelcome consequences outside the theory of truth and, indeed, that it does not have any substantial, that is, non-semantic, consequences at all.

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Thus I will first give a cut-elimination argument for the conservativity of ct↾ over pa that can be formalized in pa and thus be used to show that ct↾ is proof-theoretically reducible to pa. I will return to the model-theory of ct↾ in a later section. 8.1 The conservativity of compositional truth In this section I sketch a proof first published in Halbach (1996). In my (1999a) I proved the result for weak base theories; the full strength of pa is not needed for the proof, although the base theory does need to prove what I called principles of unique readability; for instance, the base theory must prove that an atomic sentence cannot be a conjunction and similar syntactic claims that can be established in weak arithmetical theories. In what follows, I will highlight those parts of the proof where such assumptions about the base theory are needed. In order to carry out my cut-elimination argument, I reformulate ct↾ in a sequent calculus. Throughout the proof I will assume that the reader has a basic acquaintance with sequent calculi, as one may obtain from Troelstra and Schwichtenberg (2000) or Takeuti (1987). Sequents take the form Γ ⇒ ∆, where Γ and ∆ are finite sets of LT -formulae (rather than ordered sequences). Instead of Γ ∪ {ϕ}, I write Γ, ϕ to designate sets in sequents. Proofs have the form of trees with sequents at each node of the proof. Branches of a proof can merge. For instance, there is a rule that allows one to pass from Γ ⇒ ϕ, ∆ and Γ ⇒ ψ, ∆ to the sequent Γ ⇒ ϕ ∧ ψ, ∆. To begin with I specify which sequents are allowed to occur at the top of a proof; these sequents are called initial sequents. is1 all sequents ⇒ ϕ, where ϕ is an axiom of the base theory, that is, an axiom of pa. The formula ϕ cannot contain the truth predicate. is2 all sequents ϕ ⇒ ϕ for all atomic formulae ϕ of LT is3 the following initial sequents involving the identity symbol: ⇒ t1 = t1 s1 = t1 , . . . , sn = tn ⇒ f(s1 . . . sn ) = f(t1 . . . tn ) s1 = t1 , s2 = t2 , s1 = s2 ⇒ t1 = t2 s = t, Ts ⇒ Tt In these initial sequents s1 , . . . , sn and t1 , . . . , tn are arbitrary terms of L. These terms may contain free variables.

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In the following rules, Γ and ∆ are always arbitrary finite sets of LT formulae. I use the following structural rules: (i) weakening. Γ⇒∆ Γ, ϕ ⇒ ∆

Γ⇒∆ Γ ⇒ ϕ, ∆

(ii) cut rule. If ϕ is an element of Π and Γ, a cut in ϕ is the following rule: Γ⇒∆ Π⇒Σ Γ, Π∗ ⇒ ∆∗ , Σ The sets Π∗ and ∆∗ are obtained from Π and ∆ by deleting ϕ from them. An application of the cut rule is a t-cut, if the cut formula ϕ contains the truth predicate. The following logical rules are used: Γ ⇒ ϕ, ∆ Γ, ¬ϕ ⇒ ∆

Γ, ϕ ⇒ ∆ Γ ⇒ ¬ϕ, ∆

Γ, ϕ ⇒ ∆ Γ, ϕ ∧ ψ ⇒ ∆

Γ, ϕ ⇒ ∆ Γ, ψ ∧ ϕ ⇒ ∆

Γ ⇒ ϕ, ∆ Γ ⇒ ψ, ∆ Γ ⇒ ϕ ∧ ψ, ∆ Γ ⇒ ϕ, ∆ Γ ⇒ ϕ ∨ ψ, ∆

Γ ⇒ ϕ, ∆ Γ ⇒ ψ ∨ ϕ, ∆

Γ, ϕ ⇒ ∆ Γ, ψ ⇒ ∆ Γ, ϕ ∨ ψ ⇒ ∆ Γ, ϕ(t) ⇒ ∆ Γ, ∀y ϕ(y) ⇒ ∆

Γ ⇒ ϕ, ∆ Γ ⇒ ∀y ϕ(y), ∆

Γ ⇒ ϕ(t), ∆ Γ ⇒ ∃y ϕ(y)∆

Γ, ϕ(t) ⇒ ∆ Γ, ∃y ϕ(y) ⇒ ∆

In the rule for introducing the universal quantifier and in the last rule, which is the rule for introducing the existential quantifier, the variable x must not occur freely in the lower sequent. All other connectives are conceived as metalinguistic abbreviations. The formula ϕ → ψ, for instance, abbreviates the formula ¬ϕ ∨ ψ. The usual rules for → can then be obtained from the rules stated so far. As an example I show that the usual rule for introducing → can be derived.

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lemma 8.5. If Γ, ϕ ⇒ ψ, ∆ is derivable, Γ ⇒ ϕ → ψ, ∆ is derivable as well. proof. Writing out the definition of material implication, the claim is established by the following derivation: Γ, ϕ ⇒ ψ, ∆ Γ ⇒ ¬ϕ, ψ∆ Γ ⇒ ¬ϕ, ¬ϕ ∨ ψ, ∆ Γ ⇒ ¬ϕ ∨ ψ, ∆ The last line is obtained by passing from ¬ϕ to ¬ϕ ∨ ψ. The last sequent can be abbreviated as Γ ⇒ ϕ → ψ, ∆. a The axioms specific to truth are transformed into the following rules for the truth predicate. In these rules r, s, and t are terms of L possibly containing free variables (or even identical to variables). tr1

Γ ⇒ s◦ = t◦ , ∆ Γ, Sent(r), r = (s=. t) ⇒ Tr, ∆

tr2

Γ, s◦ = t◦ ⇒ ∆ Γ, Sent(r), r = (s=. t), Tr ⇒ ∆

tr3

Γ ⇒ ¬Tt, ∆ Γ, Sent(s), s = ¬. t ⇒ Ts, ∆

tr4

Γ, ¬Tt ⇒ ∆ Γ, Sent(s), s = ¬. t, Ts ⇒ ∆

tr5

Γ ⇒ Tt ∧ Ts, ∆ Γ, Sent(r), r = (t∧. s) ⇒ Tr, ∆

tr6

Γ, Tt, Ts ⇒ ∆ Γ, Sent(r), r = (t∧. s), Tr ⇒ ∆

tr7

Γ ⇒ Tt, Ts, ∆ Γ, Sent(r), r = (t∨. s) ⇒ Tr, ∆

tr8

Γ, Tt ∨ Ts ⇒ ∆ Γ, Sent(r), r = (t∨. s), Tr ⇒ ∆

tr9

Γ ⇒ ∀t T(s(t/v)), ∆ Γ, Sent(r), r = ∀. vs ⇒ Tr, ∆

tr10

Γ, ∀t T(s(t/v)) ⇒ ∆ Γ, Sent(r), r = ∀. vs, Tr ⇒ ∆

the conservativity of compositional truth

tr11

Γ ⇒ ∃t T(s(t/v)), ∆ Γ, Sent(r), r = ∃. vs ⇒ Tr, ∆

tr12

Γ, ∃t T(s(t/v)) ⇒ ∆ Γ, Sent(r), r = ∃. vs, Tr ⇒ ∆

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If the term s in rule tr3, for instance, contains free variables, they must not be bound in Sent(s). To avoid this bound variables must be replaced by new variables. Similar caveats apply to other rules as well. The active occurrences of formulae in a proof step are those that are newly introduced in the application of the rule. This applies equally to structural, logical and truth rules. In the truth rule tr4, for instance, the occurrences of Sent(s), s = ¬. t, and Ts in the lower sequent are all active. This formulation of the truth rules is somewhat convoluted, but useful for the proof of conservativity. More straightforward rules can be derived: lemma 8.6 (derived rules). The following rules are derived rules of the sequent system, provided that s and t are terms, possibly containing free variables: Γ ⇒ s ◦ = t◦ , ∆ Γ, Sent(s=. t) ⇒ T(s=. t), ∆

Γ, s◦ = t◦ ⇒ ∆ Γ, Sent(s=. t), T(s=. t) ⇒ ∆

Γ ⇒ ¬Tt, ∆ Γ, Sent(t) ⇒ T¬. t, ∆

Γ¬Tt ⇒ ∆ Γ, Sent(t), T¬. t ⇒ ∆

Γ ⇒ Tt ∧ Ts, ∆ Γ, Sent(t∧. s) ⇒ T(t∧. s), ∆

Γ, Tt, Ts ⇒ ∆ Γ, Sent(t∧. s), T(t∧. s) ⇒ ∆

Γ ⇒ Tt, Ts, ∆ Γ, Sent(t∨. s) ⇒ T(t∨. s), ∆

Γ, Tt ∨ Ts ⇒ ∆ Γ, Sent(t∨. s), T(t∨. s) ⇒ ∆

Γ ⇒ ∀t Ts(t/v), ∆ Γ, Sent(∀. vs) ⇒ T(∀. vs), ∆

Γ, ∀t Ts(t/v) ⇒ ∆ Γ, Sent(∀. vs), T(∀. vs) ⇒ ∆

Γ ⇒ ∃t Ts(t/v), ∆ Γ, Sent(∃. vs) ⇒ T(∃. vs), ∆

Γ, ∃t Ts(t/v) ⇒ ∆ Γ, Sent(∃. vs), T(∃. vs) ⇒ ∆

The proofs are straightforward and involve the use of the initial sequents for identity. As Sent(x) is a complicated formula, a couple of steps are required for the proof. In what follows it will be shown that this sequent system is conservative over pa. To conclude that the system ct↾ itself is conservative, the following lemma will be used:

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lemma 8.7. If ct↾ ` ϕ for a formula ϕ of LT , then the sequent ⇒ ϕ is derivable. proof. If ϕ is an axiom of pa, then ⇒ ϕ is an initial sequent and thus derivable. It remains to show that ⇒ ϕ is derivable if ϕ is one of the axioms ct1–ct6 of ct↾. As an example I consider ct2, the axiom for negation: Tx ⇒ Tx Tx ⇒ Tx Tx, ¬Tx ⇒ Tx, ¬Tx ⇒ ¬Tx ⇒ ¬Tx ¬Tx ⇒ ¬Tx tr4 tr4 Sent(y), y = (¬. x), Ty ⇒ ¬Tx ¬Tx, Sent(y), y = (¬. x) ⇒ Ty .. .. . logic . logic Sent(x), T¬. x ⇒ ¬Tx Sent(x), ¬Tx ⇒ T¬. x .. . logic Sent(x) ⇒ T¬. x ↔ ¬Tx ⇒ ∀x (Sent(x) → (T¬. x ↔ ¬Tx)) The last step employs Lemma 8.5.

a

For the proof of the conservativity of ct↾ over Peano arithmetic it remains to show that if ⇒ ϕ is derivable for a sentence ϕ of L then ϕ is a theorem of Peano arithmetic. This will be established by a cut elimination theorem: I will show that any proof can be transformed into a proof without a t-cut. Since only cuts of formulae containing the truth predicate are eliminated and these formulae are not admissible instances of the induction schema, eliminating tcuts does not require transfinite induction but only induction on the natural numbers, so the cut elimination can be carried out within Peano arithmetic and, in fact, in much weaker systems. Eliminating all t-cuts from a proof of a sequent that does not contain the truth predicate transforms the proof into a proof in pure arithmetic, as a truth predicate, once introduced, can only disappear in a t-cut. The procedure for eliminating t-cuts consists of a combination of three different kinds of transformations. Before going into the details, I give an overview of the general strategy for the proof. An application of the cut rule with a cut formula containing the truth predicate will be of the following form if Π∗ and ∆∗ are Π and ∆ without ϕ.

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left rule

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.. .. . . right rule Γ⇒∆ Π ⇒ Σ cut in ϕ Γ, Π∗ ⇒ ∆∗ , Σ

If ϕ is not an active formula in the left rule or the right rule, one moves the application of the cut in ϕ up until ϕ is introduced directly above the application of the cut. This transformation, which is carried out in Lemma 8.8, reduces what is called the depth of the cut in ϕ. The formula ϕ may be an atomic formula of the form Tt, where t is some term, or it may be a complex formula containing the truth predicate. If it is a complex formula, for instance, and ϕ has been introduced by logical rules in the left rule and the right rule, then I will show how to apply the cut rule before the logical rules, thereby reducing the logical complexity of the cut formulae in t-cuts. This will be done in Lemma 8.9. Thereafter only t-cuts in atomic formulae of the form Tt are left. In the non-trivial cases, the formula Tt will have been obtained using a truth rule, that is, one of the rules tr1–tr12. I will demonstrate how to apply the cut rule first and only then apply the truth rule, thereby reducing what I call the t-complexity of the proof. Unfortunately, the method used for reducing the t-complexity may increase the logical complexity of the cut formulae, and reducing the complexity of the cut formulae can increase the depths of the cuts. Thus a triple induction is used for the proof: the main induction is on t-complexity, the first side induction is on the complexity of the cut formulae, and the second side induction is on the depth of cuts. First, I define the t-complexity of an occurrence of a formula in a sequent within a proof. If ϕ does not contain the truth predicate, any occurrence of it has t-complexity 0. If ϕ is not an active formula in an application of a rule, the occurrence in the lower sequent has the same t-complexity as the corresponding occurrence in the preceding upper sequent. If ϕ contains the truth predicate and it is in an initial sequent or is introduced by weakening, it has t-complexity 1. The t-complexity of a formula is not affected by logical rules. For example, the occurrence of ¬ϕ in the lower sequent of the following derivation Γ ⇒ ϕ, ∆ Γ, ¬ϕ ⇒ ∆ has the same t-complexity as the occurrence of ϕ in the upper sequent. The tcomplexity of an occurrence of a formula is defined in the case of other logical rules in a similar way. In the case of the rule for introducing conjunction, I define the t-complexity of the newly introduced occurrence of ϕ ∧ ψ as the

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maximum of the respective occurrences of ϕ and ψ in the upper sequents. That is, in Γ ⇒ ϕ, ∆ Π ⇒ ψ, Σ Γ, Π ⇒ ϕ ∧ ψ, ∆, Σ the t-complexity of ϕ ∧ ψ is the maximum of the two displayed occurrences of ϕ and ψ. The rule for introducing disjunction in the antecedent, that is, in front of the sequent arrow is treated similarly. In the truth rules tr1 and tr2 Γ ⇒ s◦ = t◦ , ∆ Γ, Sent(r), r = (s=. t) ⇒ Tr, ∆

Γ, s◦ = t◦ ⇒ ∆ Γ, Sent(r), r = (s=. t), Tr ⇒ ∆

the displayed occurrences of Tr have t-complexity 1. In the rules tr3 and tr4 for negation and in the rules tr9, tr10, tr11, and tr12 for universal and existential quantification the t-complexity is increased by 1. In tr5 and tr6, that is, in Γ ⇒ Tt ∧ Ts, ∆ Γ, Sent(r), r = (t∧. s) ⇒ Tr, ∆

Γ, Tt, Ts ⇒ ∆ Γ, Sent(r), r = (t∧. s), Tr ⇒ ∆

the displayed occurrence of Tr has t-complexity n + 1, where n is the maximum of the t-complexity of Tt and the t-complexity of Ts. The t-complexity for the disjunction rules tr7 and tr8 is defined in the same way. The t-complexity of an application of the cut rule is the t-complexity of its cut formula. A thread in a proof is a sequence of occurrences of formula in a proof between an occurrence of a formula in the proof and the occurrence of an initial sequent. The initial sequent and the formula belong to the thread. So a thread is, roughly speaking, a branch in the proof above some sequent in the proof. The depth of an application of the cut rule is defined as follows. If Γ⇒∆ Π⇒Σ ∗ Γ, Π ⇒ ∆∗ , Σ is a cut in ϕ, its left depth is the maximum length of a thread immediately above Γ ⇒ ∆ such that for all sequents Ξ ⇒ Θ in this thread ϕ is in Θ. So the left depth is the number of steps one needs to go up to the inference to find the place where ϕ was first introduced after the sequent arrow ⇒. The right depth is defined in an analogous way. The depth of the cut is defined as the sum of its left and its right depth. So the minimum depth of a t-cut is 2 because ϕ has to be an element of Π and Γ.

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The rank of a formula is defined as the number of logical symbols contained in it, that is, as the surface complexity of the formula. The rank of an application of the cut rule is the rank of its cut formula. I assume that all proofs are regular in the sense that any ‘eigenvariable’ in a proof, that is, any variable with a variable condition on it, has at most one rule applied to it in the proof; and any ‘eigenvariable’ occurs freely only in sequents above the inference where it is bound. In the next lemma I show how to reduce the depth of cuts. lemma 8.8. For every proof P there is a proof P 0 of the same sequent that contains only t-cuts of depth 2. Furthermore the t-complexities and the ranks of t-cuts in P 0 are no greater than the t-complexities and the ranks of t-cuts of P . proof. Consider the first subproof of P that ends with a t-cut of a depth higher or equal to that of any other t-cut in P . In what follows I describe a procedure that reduces the depth of this last t-cut. By repeatedly reapplying the procedure to the first subproof ending with a t-cut of maximal depth, the depths of all t-cuts are reduced to 2. If the depth is greater than 2, either the left or the right depth must be greater than 1. Assume the right depth is greater than 1 (the other case is established in the same way). The last sequent on the right-hand side might have been obtained by different rules. As a first example, I consider the case where the last rule on the right side is the truth rule for negation, tr4. Hence the end of the proof looks like this: Θ, ¬Tt ⇒ Λ tr4 Γ⇒∆ Θ, Sent(s), s = ¬. t, Ts ⇒ Λ cut in ϕ Γ, Θ∗ , Sent(s), s = ¬. t, Ts ⇒ ∆∗ , Λ The sets Θ∗ and ∆∗ are just Θ and ∆ without the cut formula ϕ. The cut formula ϕ is different from Ts because the right depth was assumed to be greater than 1. Now this subproof is replaced by the following proof, where the cut in ϕ is applied before the truth rule tr4: Γ⇒∆ Θ, ¬Tt ⇒ Λ cut in ϕ ∗ Γ, Θ , ¬Tt ⇒ ∆∗ , Λ tr4 Γ, Θ∗ , Sent(s), s = ¬. t, Ts ⇒ ∆∗ , Λ The depth of the displayed application of the cut rule is thereby reduced by 1. The other rules are treated in a similar way: the cut rule is applied first and only then is the rule, like tr4 in the above example, applied.

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As a second example I consider a t-cut preceded by a cut in a formula ψ without the truth predicate: Θ1 ⇒ Λ1 † Θ1 , Θ2

Γ⇒∆ †∗

Θ2 ⇒ Λ2 †

⇒ Λ1 , Λ2 †

Γ, Θ∗1 , Θ2 ⇒ ∆∗ , Λ1 , Λ2

cut in ψ

cut in ϕ

Here, for a set M of formulae, M† is the set without the arithmetical formula ψ and M∗ is the set without the t-cut formula ϕ. As before, it is possible to apply the cut in ψ after the cut in ϕ in order to reduce the depth of the cut in ϕ:

cut in ϕ

Γ⇒∆ Θ1 ⇒ Λ1 Γ, Θ∗1 ⇒ ∆∗ , Λ1

cut in ψ

Γ⇒∆ Θ2 ⇒ Λ2 cut in ϕ Γ, Θ∗2 ⇒ ∆∗ , Λ2

∗†

†

Γ† , Θ∗1 , Θ2 ⇒ ∆∗† , Λ1 , Λ2 .. .

at most 2 weakenings †∗

†

Γ, Θ∗1 , Θ2 ⇒ ∆∗ , Λ1 , Λ2 Similarly, the other cases do not pose any problems.

a

In the next lemma it is shown how to reduce the ranks of t-cuts – the length of cut formulae in the proof – without increasing the t-complexities of their corresponding occurrences. lemma 8.9. For every proof P there is a proof P 0 of the same sequent that only contains t-cuts in atomic formulae. The t-complexities of t-cuts in P 0 can be chosen to be no greater than those in P 0 . proof. Using the previous lemma, first reduce the depths of all t-cuts in P to 2. Then consider the first subproof of P ending in a t-cut of a rank higher or equal to that of any other t-cut in P . I describe a procedure for transforming this subproof into a proof that only contains t-cuts of lower rank. This might increase the depths of t-cuts, but the depths can always be reduced to 2 by using Lemma 8.8. Assume that the cut Γ⇒∆ Π ⇒ Σ cut in ϕ Γ, Π∗ ⇒ ∆∗ , Σ

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is the last step in the proof P . As the rank of ϕ is not 0, the formula ϕ may be a universally quantified formula, a conjunction or a negated formula. In each case there are several subcases to consider. The formula ϕ is a active formula in the two preceding inferences, because I have assumed that the depth of the cut is 2. I consider the case where ϕ is a negation ¬ψ; other cases are treated in a similar way. There are two subcases: ¬ψ is introduced in ∆ by a weakening or by the logical rule for introducing ¬. In the first case, P has the following form: weakening

.. Γ ⇒ ∆∗ . ¬-intro. Γ ⇒ ¬ψ, ∆∗ Π∗ , ¬ψ ⇒ Σ cut in ¬ψ Γ, Π∗ ⇒ ∆∗ , Σ

Again, Π∗ and ∆∗ are Π and ∆, respectively, without ϕ = ¬ψ. Then the t-cut can be completely eliminated, because Γ, Π∗ ⇒ ∆∗ , Σ may be obtained from Γ ⇒ ∆∗ by some weakenings. I turn to the second case, where ¬ψ is introduced in ∆ by an application of the ¬-introduction rule. Obviously there are again two analogous subcases. If ¬ψ is introduced in Π using the ¬-introduction rule: ¬-intro

Γ, ψ ⇒ ∆∗ Π∗ ⇒ ψ, Σ ¬-intro ∗ Γ ⇒ ¬ψ, ∆ Π∗ , ¬ψ ⇒ Σ cut in ¬ψ Γ, Π∗ ⇒ ∆∗ , Σ

The end of the proof is now rewritten as follows: Π∗ ⇒ ψ, Σ Γ, ψ ⇒ ∆∗ cut in ψ Γ, Π∗ ⇒ ∆∗ , Σ The rank of the cut in P 0 is reduced by 1 compared with that of the original proof P . The other subcase is similar. a Having shown how to reduce the depths and ranks of t-cuts, I now turn to the main induction on the t-complexity of proofs. theorem 8.10. For every proof P there is a proof P 0 of the same sequent that contains no t-cuts. proof. Given a proof P , one can always reduce the depths of all t-cuts to 2 and use only atomic cut formulae in t-cuts. In a way similar to the two preceding proofs, one considers the first subproof Ps of P ending in a t-cut of maximal t-complexity in the proof and applies the modifications described in what follows. This will reduce the t-complexity of the t-cut at the end of

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Ps without increasing the t-complexity of other t-cuts in Ps , but the depth and the rank of t-cuts may be increased, so Lemma 8.8 and 8.9 may need to be reapplied. So by Lemma 8.9 the cut formula may be assumed to be atomic and hence of the form Tt for a term t. Furthermore, by Lemma 8.9, Tt may be assumed to be the active formula in the two immediately preceding inferences, so that it is introduced immediately above Γ ⇒ ∆ and Π ⇒ Σ. So the end of P looks like this: Γ⇒∆ Π⇒Σ cut in Tt Γ, Π∗ ⇒ ∆∗ , Σ There are four possible ways in which Tt could be the active formula in the inference with Γ ⇒ ∆ as lower sequent: 1. Tt is introduced in ∆ by a weakening. 2. Γ ⇒ ∆ is an initial sequent of the form Tt ⇒ Tt. 3. Γ ⇒ ∆ is an initial identity sequent s = t, Ts ⇒ Tt. 4. Tt is introduced in ∆ by one of the truth rules tr1–tr12. These four cases also cover all the possible ways Tt could be introduced in Π. So for each of the four ways Tt could be introduced in ∆, one must also consider the four ways Tt might be introduced in Π as subcases. Some of the cases are obviously symmetric. Case 1: Tt is introduced in ∆ by a weakening. If Tt is introduced by a weakening, the cut can be eliminated completely as in the proof of Lemma 8.9. Case 2: Γ ⇒ ∆ is an initial sequent of the form Tt ⇒ Tt. So the proof ends in the following way: Tt ⇒ Tt Π⇒Σ cut in Tt Π⇒Σ The cut formula Tt is not deleted from Π because it forms part of Γ, that is, the set with Tt as its only element. Obviously this cut does not change the sequent Π ⇒ Σ and it can therefore be dropped. Case 3: Γ ⇒ ∆ is an identity sequent s = t, Ts ⇒ Tt. There are four subcases depending on how the formula Tt is introduced in Π. If Π ⇒ Σ is the initial sequent Tt ⇒ Tt the cut is superfluous again. If Π ⇒ Σ is an initial sequent of the form t = r, Tt ⇒ Tr, the end of the proof looks like this: s = t, Ts ⇒ Tt t = r, Tt ⇒ Tr cut in Tt s = t, t = r, Ts ⇒ Tr

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This subproof can be replaced with a subproof without a t-cut: ⇒ t=t

t = s, t = r, t = t ⇒ s = r

cut in t = t s = t, t = r ⇒ s = r cut in s = r

s = r, Ts ⇒ Tr s = t, t = r, Ts ⇒ Tr

Finally, in the right-hand sequent Π ⇒ Σ, the cut formula Tt may be introduced using a truth rule. As an example, I consider the case in which Π ⇒ Σ is obtained using tr4; in this subcase the end of the proof will look like this: Γ, ¬Ts ⇒ ∆ tr4 r = t, Tr ⇒ Tt Γ, Sent(t), t = ¬. s, Tt ⇒ ∆ cut in Tr r = t, Tr, Γ, Sent(t), t = ¬. s ⇒ ∆ The end of the proof is then replaced by the following new ending and a proof of the sequent t = ¬. s, r = t ⇒ r = ¬. s is added: .. . logic Γ, ¬Ts ⇒ ∆ t = ¬. s, r = t ⇒ r = ¬. s Γ, r = ¬. s, Sent(r), Tr ⇒ ∆ cut in r = ¬. s Γ, t = ¬. s, r = t, Sent(r), Tr ⇒ ∆ .. . logic Γ, t = ¬. s, r = t, Sent(t), Tr ⇒ ∆ The new ending does not involve a t-cut. In particular, the skipped steps do not require an application of a t-cut. If I had not used the truth rule tr4 but rather its shorter counterpart from Lemma 8.6, it would not have been possible to substitute t with r in the way I have here without using a t-cut. Case 4: Tt is introduced in ∆ by one of the truth rules tr1–tr12. The only case that has not been covered already is the one in which Γ ⇒ ∆ and Π ⇒ Σ are obtained using truth rules. I cannot help but divide this case into two subsubcases: First, Tt is introduced in Π ⇒ Σ by the truth rule corresponding to the truth rule that is used to arrive at Γ ⇒ ∆; tr2 corresponds in this sense to tr1, tr4 to tr3, and so on. In the case of the first pair, that is, tr1 and tr2, the proof would look like this:

tr1

Θ ⇒ s1 ◦ = t1 ◦ , Λ Φ, s2 ◦ = t2 ◦ ⇒ Ψ tr2 Θ, Sent(t), t = (s1 =. t1 ) ⇒ Tt, Λ Φ, Sent(t), t = (s2 =. t2 ), Tt ⇒ Ψ Θ, Φ, Sent(t), t = (s1 =. t1 ), t = (s2 =. t2 ) ⇒ Λ, Ψ

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This subproof can be replaced by the first proof on page 81. There are no t-cuts in the new subproof. Next I consider the case in which Tt is introduced by the truth rules for the universal quantifiers tr9 and tr10: Θ ⇒ ∀t T(s1 (t/v1 )), Λ Φ, ∀t T(s2 (t/v2 )) ⇒ Ψ Θ, Sent(t), t = ∀. v1 s1 ⇒ Tt, Λ Φ, Sent(t), t = ∀. v2 s2 , Tt ⇒ Ψ Θ, Φ, Sent(t), t = ∀. v1 s1 , t = ∀. v2 s2 ⇒ Ψ, Λ This subproof is then replaced by the second proof on the next page with lower t-complexity. Other cases are treated in analogous way. Now I consider the second subsubcase: The sequents Γ ⇒ ∆ and Π ⇒ Σ are derived using non-corresponding truth rules as in the following example: tr5

Φ, ∀t T(s1 (t/v)) ⇒ Ψ Θ ⇒ Tt ∧ Ts, Λ tr10 Θ, Sent(r), r = (t∧. s) ⇒ Tr, Λ Φ, Sent(r), r = ∀. vs1 , Tr ⇒ Ψ Θ, Φ, Sent(r), r = (t∧. s), r = ∀. vs1 ⇒ Λ, Ψ

The last sequent is derivable in arithmetic without any truth axioms. For the formula Sent(r), r = (t∧. s), r = ∀. vs1 ⇒ is provable because arithmetic proves that a sentence cannot be both a conjunction and a universally quantified formula. Thus the proof will only succeed if a base theory proving some syntactic facts is presupposed. Of course, arithmetical systems much weaker than Peano arithmetic will suffice; in Halbach (1999a) I have tried to extract the required assumptions. a The point of proofs without t-cuts obtained in Theorem 8.10 is that they have the subformula property for formulae containing the truth predicate: once a formula with the truth predicate is introduced in such a proof it will occur as a subformula in all the sequents below. lemma 8.11. If P is a proof of a sequent Γ ⇒ ∆ that does not contain the truth predicate and no t-cut occurs in P , then the truth predicate does not occur in P . This lemma is established by induction on the length of proofs without any t-cut. This yields the main result of this section: theorem 8.12 (conservativity of ct↾). The system ct↾ is conservative over Peano arithmetic.

.. . arithmetic .. . logic Θ ⇒ ∀t T(s1 (t/v1 )), Λ Sent(∀. v1 s1 ), ∀t T(s1 (t/v1 )), (∀. v1 s1 ) = (∀. v2 s2 ) ⇒ ∀t T(s2 (t/v2 )) t = ∀. v1 s1 , t = ∀. v2 s2 ⇒ (∀. v1 s1 ) = (∀. v2 s2 ) Θ, Sent(∀. v1 s1 ), (∀. v1 s1 ) = (∀. v2 s2 ) ⇒ ∀t T(s2 (t/v2 )), Λ Θ, Sent(∀. v1 s1 ), t = ∀. v1 s1 , t = ∀. v2 s2 ⇒ ∀t T(s2 (t/v2 )), Λ .. . logic Θ, Sent(t), t = ∀. v1 s1 , t = ∀. v2 s2 ⇒ ∀t T(s2 (t/v2 )), Λ Φ, ∀t T(s2 (t/v2 )) ⇒ Ψ Θ, Φ, Sent(t), t = ∀. v1 s1 , t = ∀. v2 s2 ⇒ Ψ, Λ

Second proof

.. . arithmetic .. . logic Θ ⇒ s1 ◦ = t1 ◦ , Λ Sent(s1 =. t1 ), s1 ◦ = t1 ◦ , (s1 =. t1 ) = (s2 =. t2 ) ⇒ s2 ◦ = t2 ◦ t = (s1 =. t1 ), t = (s2 =. t2 ) ⇒ (s1 =. t1 ) = (s2 =. t2 ) Θ, Sent(s1 =. t1 ), (s1 =. t1 ) = (s2 =. t2 ) ⇒ s2 ◦ = t2 ◦ , Λ Θ, Sent(s1 =. t1 ), t = (s1 =. t1 ), t = (s2 =. t2 ) ⇒ s2 ◦ = t2 ◦ , Λ Θ, Sent(t), t = (s1 =. t1 ), t = (s2 =. t2 ) ⇒ s2 ◦ = t2 ◦ , Λ Φ, s2 ◦ = t2 ◦ ⇒ Ψ Θ, Φ, s2 ◦ = t2 ◦ ⇒ Λ, Ψ

First proof

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proof. Assume that ct↾ ` ϕ and that ϕ is an arithmetical formula. Then, by Lemma 8.7, there is a proof P of the sequent ⇒ ϕ in the sequent system, and hence there is also a proof P 0 of ⇒ ϕ without a t-cut by Lemma 8.10. Since ϕ does not contain the truth predicate, the truth predicate cannot occur in P 0 by the above lemma. The sequent system without the truth rules and the other rules involving the truth predicate is a standard sequent system for Peano arithmetic. Therefore ⇒ ϕ is provable in a standard sequent system for pa and therefore pa ` ϕ. a Fischer (2009) uses the result to show that ct↾ is also relatively interpretable in pa. 8.2 Conservativity and model theory As mentioned above, in the literature on deflationism the conservativity of ct↾ over pa has played a prominent role as many authors have thought that deflationism about truth implies the conservativity of the truth axioms over the base theory at least. In the discussion on deflationism and conservativity prompted by Shapiro (1998) and Ketland (1999), the main focus was on the theory ct↾ or slight variants thereof. In many papers, the conservativity of ct↾ over pa is stated without proof; Field (1999), in his reply to Shapiro, refers to Parsons (1983, p. 215), who is treating a related problem in a set-theoretic setting. If Parsons’ argument is adapted to the present setting, one would have to define the truth predicate of ct↾ in a theory of elementary comprehension such as aca0 that is known to be conservative over Peano arithmetic. Other authors have suggested proving conservativity directly by showing that any given model of Peano arithmetic can be expanded to a model of the truth theory ct↾. Both strategies are doomed: neither is the truth predicate of ct↾ definable in aca0 nor can every model of pa be expanded to a model of ct↾. That neither of these strategies can succeed follows from Lachlan’s theorem and will be proved in Theorem 8.32. This is in contrast to the systems with full induction: in the system aca with all induction axioms in the language of second-order arithmetic the truth predicate of ct can be defined; in fact ct and aca are intertranslatable, as will be shown in Section 8.6. In the discussion on deflationism, a model-theoretic notion of conservativity due to Craig and Vaught (1958) has also been mooted as a desirable feature of truth from a deflationist perspective, for instance by Shapiro (1998, p. 497) and McGee (2006, pp. 105f).

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definition 8.13 (model-theoretic conservativity). A system S2 in LT is model-theoretically conservative over a system S1 in the language L if and only if every model of S1 can be expanded to a model of S2 . An expansion of a model 𝔐 of L to the language LT adds an extension for the truth predicate to the model 𝔐; the elements of the extension of the truth predicate must be elements of the domain |𝔐| of the model 𝔐. I write (𝔐, S) for the expansion of 𝔐 to LT that gives the extension S to the truth predicate. Applying this definition to ct↾ and pa, ct↾ is model-theoretically conservative over pa if and only if every model of pa can be expanded to a model of ct↾. The latter is often expressed by saying that every model of pa has a (full) satisfaction class. definition 8.14 (satisfaction class). Assume 𝔐 is a model of Peano arithmetic with domain |𝔐|. A set S ⊆ |𝔐| is a full satisfaction class for 𝔐 if and only if (𝔐, S) ⊨ ct↾. There are also partial satisfaction classes, but they are not considered here (though see, for instance, Kaye 1991). Thus, in what follows I will drop the qualification ‘full’: satisfaction classes are always full satisfaction classes. To prove that ct↾ is model-theoretically conservative over Peano arithmetic, one would have to show that every model of Peano arithmetic has a satisfaction class. It will be shown on the contrary that not every countable model of Peano arithmetic possesses a satisfaction class. Before taking a closer look at full satisfaction classes, I will summarize some basic facts about the models of Peano arithmetic. 8.3 Nonstandard models The standard model N of Peano arithmetic has as its domain |N| the set ω of all natural numbers; in N all function symbols are interpreted in the standard way: the constant 0 designates the natural number zero, the function symbol S is interpreted as the successor function, + as the addition function, and so on. The standard model N has exactly one satisfaction class, as there is a unique set satisfying the usual Definition 8.1 of truth-in-L. Nonstandard models of Peano arithmetic are models of pa that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem for first-order logic. A proof using the latter can be given in the following way. A single new constant c is

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added to L. Any finite subset of the set pa∪{c 6= 0, c 6= 1, c 6= 2, . . .} is satisfied by an extension of the standard model because c can be interpreted as designating a number n such c 6= n is not in the finite set. Therefore, by the soundness theorem for first-order logic, the above set is consistent and therefore it has a model 𝔐, by the completeness theorem. As the object designated by c in the model must be different from any object designated by a numeral n, the model 𝔐 cannot be isomorphic to the standard model. The object designated by c is called a nonstandard number. Generally, given a model, an object in |𝔐| is a nonstandard number if and only if it is not designated in the model by any numeral n. A nonstandard number is always successor number, that is, for nonstandard c ∈ |𝔐| one always has that 𝔐 ⊨ ∃x Sx = c, where S is the successor symbol, because otherwise c would be identical to zero by the axioms of Peano arithmetic. As there is no first nonstandard number there must be an infinitely descending sequences of nonstandard numbers: lemma 8.15. For each nonstandard number c ∈ |𝔐| the sentence Sb = c holds in 𝔐 for some nonstandard number b. Therefore each nonstandard number c has infinitely many predecessors in 𝔐. Much more is known about the structure of nonstandard numbers; the reader is referred to Kaye (1991) for further details and results. When working with a nonstandard model 𝔐 it is useful to add a constant a for each element a ∈ |𝔐|, as I did with b in the previous lemma. This is not really necessary but easier than dealing with variable assignments. The language obtained from L by adding a constant a for each a ∈ |𝔐| is designated by L𝔐 . The constant a is always interpreted by the object a. Thus every model 𝔐 can always be expanded to a model of L𝔐 in a unique way. I will not distinguish between a model and its expansion to the language L𝔐 with all the constants. For natural numbers n the expression n denotes the numeral of n as before. A basic result about nonstandard models is the following overspill lemma. I prove the following version: lemma 8.16 (overspill). Let 𝔐 be a nonstandard model of pa and b ∈ |𝔐|, and assume ϕ(x, y) is a formula with the two displayed variables. Then, if 𝔐 ⊨ ϕ(n, b) holds for every n ∈ ω, there is a nonstandard number c ∈ |𝔐| such that 𝔐 ⊨ ∀x ≤ c ϕ(x, b).

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85

The overspill lemma can be proved for more than one parameter; there may be more constants besides b in the formula. In the context of arithmetic this does not really matter as finite sequences of objects can be coded by a single number, even if the objects are nonstandard numbers. proof. By the assumption the following holds for all n ∈ ω: 𝔐 ⊨ ∀x ≤ n ϕ(x, b)

(8.2) In particular, this holds for 0: (8.3)

𝔐 ⊨ ∀y ≤ 0 ϕ(y, b)

Assume for a contradiction that (8.4)

𝔐 ⊭ ∀x ≤ c ϕ(x, b)

for all nonstandard numbers c ∈ |𝔐|. (8.2) and (8.4) yield (8.5)

∀y ≤ a ϕ(y, b) → ∀y ≤ Sa ϕ(y, b)

for each a ∈ |𝔐|; this follows from (8.2) if a is standard and from the assumption (8.4) if a is nonstandard, because the successor of a standard number is always a standard number. Combining (8.3) and (8.5) yields the following: 𝔐 ⊨ ∀y ≤ 0 ϕ(y, b) ∧ ∀x ∀y ≤ x ϕ(y, b) → ∀y ≤ Sx ϕ(y, b)




Using induction it follows that 𝔐 ⊨ ∀x ∀y ≤ x ϕ(y, b) contradicting (8.4).

a

corollary 8.17. Assume that 𝔐 is a nonstandard model of Peano arithmetic and b ∈ |𝔐|. If there are infinitely many n ∈ ω such that 𝔐 ⊨ ϕ(n, b), then there is a nonstandard number c ∈ |𝔐| with 𝔐 ⊨ ϕ(c, b). The corollary is proved by applying the overspill lemma to the L-formula ∃z > x ϕ(z, y). This corollary shows that in nonstandard models a predicate like Sent(x) also applies to nonstandard numbers: There are infinitely numbers n ∈ ω – that is, infinitely many codes of sentences – with pa ` Sent(n) and thus also 𝔐 ⊨ Sent(n). Hence, if 𝔐 is nonstandard there must be c ∈ |𝔐| such that 𝔐 ⊨ Sent(c). Such a c ∈ |𝔐| is called a nonstandard sentence (of 𝔐).

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In the same way it can also be shown that in any nonstandard model there are nonstandard variables, nonstandard terms, nonstandard formulae, and so on. A nonstandard sentence may still contain only finitely many connectives and quantifiers, if, for instance, it contains a nonstandard term. For example, if a ∈ |𝔐| is a nonstandard closed term, then there is a unique nonstandard sentence c ∈ |𝔐| with 𝔐 ⊨ c = (a=. a) which does not contain any connectives and quantifiers. Conversely, a nonstandard sentence need not contain any nonstandard terms: for instance, in a nonstandard 𝔐 there will be a nonstandard conjunction 0 ∧ . . . ∧ 0 =}0 |= 0 ∧ 0 = 0{z a many conjuncts

of length a with a nonstandard. In the Axiom ct5 of ct↾ on p. 65 concerning the universal quantifier I have stipulated that a universally quantified sentence is true if and only if all of its instances are, that is, if inserting arbitrary closed terms always yields true sentences. I justified this axiom by pointing out that the language of arithmetic features closed terms for each number. This can be proved within Peano arithmetic, that is, ∀x ∃t t◦ = x is provable in Peano arithmetic. It follows that for any given model 𝔐 of Peano arithmetic and any a in the domain of 𝔐 there is a term in the sense of 𝔐 that designates a. Before tackling the model theory of truth, I need to introduce more basic notions and results concerning models of arithmetic. For a thorough survey see Kaye’s book. Many of the notions discussed below come from general model theory (see Chang and Keisler 1990). The model theory of truth is closely connected to the notion of recursive saturation. To define recursive saturation I first give some auxiliary definitions. definition 8.18. A set p of formulae of L𝔐 with exactly the variable x free is finitely satisfied in 𝔐 if and only if for every finite subset q ⊂ p there is an a ∈ |𝔐| such that 𝔐 ⊨ ϕ(a) for all formulae ϕ(x) ∈ q. In other words, a set p is finitely satisfied if and only if, for each n, ^ 𝔐 ⊨ ∃x ϕi (x) i
for an enumeration ϕ1 (x, b), ϕ2 (x, b),. . . of the formulae in p. definition 8.19. A type over a model 𝔐 is a finitely satisfied set of formulae ϕ(x, b) that have exactly the variable x free and contain at most the

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parameter b for one fixed object b ∈ |𝔐|. A type p is recursive if and only if the set of codes of formulae ϕ(x, y) with ϕ(x, b) ∈ p is recursive. I only allow a single parameter to occur in the formulae of a type. This can be generalized to a fixed finite string of parameters, but this would make no difference since finite sequences of objects can be coded by a single object in models of Peano arithmetic. Similar remarks apply to finite strings of variables instead of the single variable x. definition 8.20. A type p over 𝔐 is (globally) realized if and only if there is an a ∈ |𝔐| such that 𝔐 ⊨ ϕ(a, b). definition 8.21 (recursive saturation). A model 𝔐 of Peano arithmetic is recursively saturated if and only if every recursive type over 𝔐 is realized. It does not really matter whether recursive types, primitive recursive types, or recursively enumerable types are used in this definition. Using Craig’s trick Kaye (1991, p. 150) shows that in all three cases the same class of models is defined. For later use I prove that any element of a recursive type is satisfied by infinitely many objects: lemma 8.22. If p is a recursive type over a nonstandard model 𝔐 that is not globally realized in 𝔐, then every finite conjunction of elements of p is satisfied by infinitely many elements of |𝔐|. Here and in what follows I will occasionally suppress the parameter in the type and write ϕ(x) instead of ϕ(x, b). proof. For each element a of |𝔐| there is a formula ψ(x) ∈ p that is not satisfied by a as otherwise the type p would be realized by a. Let ϕ(x) be a (finite) conjunction of elements of p and assume A ⊂ |𝔐| is finite. For each element a ∈ A pick a formula ψa (x) with 𝔐 ⊭ ψa (a). Then there is no c ∈ A such that ^ 𝔐⊨ ψa (c) ∧ ϕ(c). a∈A

Since p is finitely satisfied, there must be c ∈ / A such that 𝔐⊨

^

ψa (c) ∧ ϕ(c).

a∈A

Hence no finite set A can be the set of elements satisfying ϕ(x).

a

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Recursively saturated models are fairly special. The standard model N of Peano arithmetic, for instance, is not recursively saturated; and many nonstandard models are also not recursively saturated. As an example, I will now build a nonstandard model of Peano arithmetic that is not recursively saturated. Let 𝔐 be some model of Peano arithmetic and assume furthermore that D is the set of L-definable elements of 𝔐. More formally, D is the set of all a ∈ |𝔐| for which there is a formula ϕ(x) of L such that
𝔐 ⊨ ϕ(a) ∧ ∀x ϕ(x) → x = a . The model D is now defined as the submodel of 𝔐 that is generated by D. That is, D is obtained from 𝔐 by restricting the domain |𝔐| to its subset D and by restricting the interpretations of the function symbols to this subset. It can then be shown that exactly the same sentences are true in D and 𝔐. As D is a submodel of 𝔐, I only need to show that D is an elementary submodel of 𝔐, that is, I need to show that, if 𝔐 ⊨ ∃x ϕ(x) for some formula ϕ(x) possibly containing parameters from D, then there is an a ∈ D with D ⊨ ϕ(a). But if 𝔐 ⊨ ∃x ϕ(x), then by the least-number principle of Peano arithmetic: 𝔐 ⊨ ∃x ϕ(x) ∧ ∀y ϕ(y) → x ≤ y





Whence the following formula ϕ(x) ∧ ∀y < x ¬ϕ(y) defines an object a, which is thus in D. Hence I have 𝔐 ⊨ ϕ(a). So the same sentences hold in D and in 𝔐. In fact, the model D is the smallest elementary submodel of 𝔐 as it only contains the elements that have to be in the domain to ensure elementary equivalence. Such small models are not recursively saturated (see also Kaye 1991, p. 150). lemma 8.23. If Γ is a consistent extension of Peano arithmetic, there is a model of Γ that is not recursively saturated. In particular, there are nonstandard models of Peano arithmetic that are not recursively saturated. proof. First extend Γ to a complete consistent extension of Peano arithmetic (see, for instance, Shoenfield 1967). By the Gödel completeness theorem there is a model 𝔐 of this extension. From 𝔐 define D in the way outlined above. Let p be the set of all formulae (8.6)

¬ϕ(x) ∨ ¬∃y ϕ(y) ∧ ∀z (ϕ(z) → z = y)




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for all formulae ϕ(x) from L. The formula following ∨ expresses that ϕ(x) is not satisfied by exactly one object. Obviously p is recursive. I will show that it is finitely satisfied. Consider a finite conjunction of formulae of the form (8.6). If a ∈ D is not defined by a formula that occurs in this conjunction – and there will be such an a as any model of pa is infinite – then a satisfies this conjunction. For if, on the one hand, a formula ϕ(x) is satisfied by no element of D or by more than one element, then (8.6) is satisfied by all objects because of the second disjunct; if, on the other hand, ϕ(x) is satisfied by exactly one object b, then this object is defined by ϕ(x), but a was assumed not to be defined by some formula in the finite conjunction, and so a is different from b, so D ⊨ ¬ϕ(a) and (8.6) is satisfied by a again. The set of all formulae of the form (8.6) is not realized because any element a ∈ D is defined by some formula ϕ(x) and therefore a does not satisfy the corresponding formula (8.6). It follows that p is a recursive type over D but that p is not realized. Hence D is not recursively saturated. To see that there is a nonstandard model of Peano arithmetic that is not recursively saturated, start with some set Γ ⊃ pa that contains a false arithmetical statement, for instance ¬Conpa . Then 𝔐 ⊨ ¬Conpa holds and from this D ⊨ ¬Conpa follows. So D must be nonstandard. a

8.4 Lachlan’s theorem From the compositional axioms one can prove that, for any sentence, either the sentence itself or its negation is true: ct↾ ` ∀x Sent(x) → (Tx ∨ T¬. x)




Therefore, if a ∈ |𝔐| is a nonstandard sentence and S is a satisfaction class for 𝔐, either (𝔐, S) ⊨ Ta or (𝔐, S) ⊨ T¬. a must hold. Therefore, any nonstandard model must have sentences of non-standard length in its satisfaction class. As nonstandard numbers are not wellfounded, it is not possible to define a satisfaction class for a nonstandard model in the same way as for the standard model, as in Definition 8.1, because one can never reach the sentences with nonstandard length by an induction on the length of sentences. Nevertheless, for certain nonstandard models full satisfaction classes can be defined, but not for all, as will be shown below. This is not to say that there is no model-theoretic proof of the proof-theoretic conservativity of ct↾; this proof due to Kotlarski, Krajewski, and Lachlan (1981), however, is far from easy and obvious. I will say more about this

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classical compositional truth

method later. But first I will show that model-theoretic conservativity fails and that therefore a proof of conservativity, that is, of Theorem 8.12, cannot be obtained via the model-theoretic conservativity of ct↾. According to Lachlan’s theorem, only very special models of Peano arithmetic can be expanded to models of ct↾. Only if a model 𝔐 of Peano arithmetic satisfies a special condition, is there a subset S of |𝔐| such that (𝔐, S) ⊨ ct↾. It follows that ct↾ fails to be conservative over pa in the model-theoretic sense. The rest of this section is devoted to a statement and proof of Lachlan’s theorem. theorem 8.24 (Lachlan’s theorem). If a nonstandard model 𝔐 of pa has a full satisfaction class, then 𝔐 is recursively saturated. The result was first proved by Lachlan (1981); Kaye (1991, pp. 228–233) modified and generalized the proof to partial satisfaction classes, which are not discussed here. I will follow Kaye’s exposition to some extent. Another version of Lachlan’s theorem is proved by Smith (1984). For the remainder of the proof let 𝔐 be a nonstandard model that is not recursively saturated. The assumption that 𝔐 possesses a satisfaction class, that is, that there is a subset S of |𝔐| such that (𝔐, S) ⊨ ct↾, will be shown to lead to a contradiction. Since 𝔐 is not recursively saturated there is a recursive type p that is not realized. By the Enumeration theorem (see Rogers 1967) any recursively enumerable set is the range of some primitive recursive function. Hence there is a primitive recursive function that maps each i ∈ ω to ψi (x, y) so that the formulae ψi (x, b) form the type p. The formulae ψi (x, y) and the parameter b remain fixed throughout the rest of this section. This function can also be seen as a sequence of formulae, and I will write hψi (x, y)ii∈ω to refer to this sequence, that is, to the function. Since the function is primitive recursive, I can assume that there is a corresponding function symbol for it in the language L. I will write ψ i for the expression representing the function mapping . each i ∈ ω to the formula ψi (x, y). All formulae ψi (x, y) for i ∈ ω are formulae of L with the variable x and possibly y free but with no other free variables. So far I have been very sloppy when claiming that certain general claims about formulae are provable in Peano arithmetic. This sloppiness is justified because I have always assumed that the syntactic properties and operations are represented in the language of arithmetic in a natural way: these universal claims could be proved by formalizing the syntax of L explicitly. Now, however, there is no guarantee that

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the formulae ψi (x, y) are given in a natural way. Therefore I cannot assume pa ` ∀x For2 (ψ x ) . if For2 (x) naturally represents the property of being a formula of L with at most x and possibly y free. So it might not be the case that the function sending i to ψi (x, y) has only formulae of L with x and possibly y free as outputs in the sense of 𝔐. However, this can be proved numeralwise as For2 (x) strongly represents the set of such formulae: pa ` For2 (ψ n ) for each n ∈ ω . As 𝔐 is a model of Peano arithmetic, I can conclude that all these sentences hold in 𝔐: 𝔐 ⊨ For2 (ψ n ) for each n ∈ ω . By applying overspill, that is, Lemma 8.16, I conclude that there is a nonstandard number c ∈ |𝔐| with the following property: 𝔐 ⊨ ∀x ≤ c For2 (ψ x ) . From the primitive recursive function, denoted by hψi ii∈ω , mapping each number i ∈ ω to the formula ψi (x, y), define a further recursive function mapping each i to the L-formula ϕi (x, y). The formulae ϕi (x, y) are obtained by a primitive recursive (syntactical) operation sending each formula ψi (x, y) to ϕi (x, y). Hence it is provable in Peano arithmetic that ϕi (x, y) is a formula . with the appropriate free variables from the assumption that ψ (x, y) is such . a formula. So, this function also has the following property, where ϕx is a . function expression representing the function mapping hϕi ii∈ω : (8.7)

𝔐 ⊨ ∀i ≤ c For2 (ϕi ) .

Similar remarks will also apply to other similar recursive functions, defined from hψi ii∈ω . lemma 8.25. Assume 𝔐 is not recursively saturated. Then there is a primitive recursive function hϕi (x, y)ii∈ω satisfying the following conditions for all i ∈ ω: (i) 𝔐 ⊨ ∀x ϕ0 (x, b) (ii) 𝔐 ⊨ ∀x (ϕi+1 (x, b) → ϕi (x, b)) (iii) 𝔐 ⊨ ∃x (ϕi (x, b) ∧ ¬ϕi+1 (x, b)) (iv) There is no a ∈ |𝔐| with 𝔐 ⊨ ϕi (a, b) for each i ∈ ω.

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classical compositional truth

According to the items (i)–(iii) in the lemma, the formulae ϕ0 (x, b), ϕ1 (x, b), ϕ2 (x, b), define a strictly decreasing sequence of subsets S0 , S1 , S2 , . . . of the domain |𝔐| of 𝔐 with |𝔐| as the first set in the sequence. For any n there is an element of |𝔐| that is in all sets S0 , S1 , S2 , . . . , Sn but by (iv) there is no object that is in all sets of the sequence. So for each object in the domain of the model there is an i ∈ ω such that it is in Si but not in Si+1 . proof. Let {ψi (x, b) : i ∈ ω} be the unrealizable recursive type from above. From the recursive function {ψi (x, y) : i ∈ ω} one defines a further recursive function of formulae by a syntactic operation as sketched above, suppressing the parameter b and variable y: ϕ0 (x) := (x = x) ϕi+1 (x) := ψi (x) ∧ ϕi (x) ∧ ∃z < x ϕi (z) It is plain that the formulae ϕi (x, y) satisfy (i) and (ii). Item (iii) is established as follows: Setting Si = {a : 𝔐 ⊨ ϕi (a, b)}, S0 is the entire domain of 𝔐. The set Si+1 is then
Si \ min(Si ) ∩ {a : 𝔐 ⊨ ψi (a, b)}. So Si+1 contains those elements of Si that satisfy ψi (x, b) and are not the smallest element of Si . By Lemma 8.22 each conjunction of formulae ψi (x, b) is satisfied by infinitely many objects. Each Si is such a finite intersection with at most finitely many objects missing, namely the smallest elements of each Sj for j < i (the least-number principle Lemma 5.1 is used to establish this). The third conjunct in the definition of ϕi+1 (x, y) ensures that at least one element of Si is missing from Si+1 . This concludes the proof of (iii). Since any a in the domain |𝔐| of 𝔐 that satisfies ϕi (x, b) for each i ∈ ω would also satisfy ψi (x, b) for each i ∈ ω, contradicting the assumption that 𝔐 is not recursively saturated, item (iv) holds as well. a Using the recursive function hϕi (x, y)ii∈ω from the previous lemma, I inductively define a further function hϑi (x, y)ii∈ω with L-formulae as outputs. As before, Peano arithmetic will prove that ϑi (x, y) is a formula with at most x and possibly y free if ϕi (x, y) is. The formulae ϑi (x, b) define the sets Pi = {a : 𝔐 ⊨ ϑi (a, b)} in such a way that P0 = Ø and then Pn+1 = Sn \ Sn+1 . This can be achieved by defining the formulae ϑi (x, y) in the following way: ϑ0 (x) := ¬(x = x) ϑi+1 (x, y) := ϕi (x, y) ∧ ¬ϕi+1 (x, y)

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In the next lemma it is shown that the domain |𝔐| of the model is partitioned by the sets Pi with i ≥ 1: lemma 8.26. (i) For each a ∈ |𝔐| there is exactly one i ∈ ω with 𝔐 ⊨ ϑi (a, b). (ii) 𝔐 ⊨ ∃x ϑi (x, b) holds for each non-zero i ∈ ω. proof. For each a ∈ |𝔐| there is an i ∈ ω such that a ∈ Si but a ∈ / Si+1 by Lemma 8.25 (i) and (iv). Uniqueness follows from Lemma 8.25 (ii): there is a unique i ∈ ω with a ∈ Si \ Si+1 = Pi . Item (ii) follows from Lemma 8.25 (iii). a From the preceding lemmata only the last one showing that the formulae ϑi (x, y) with i ≥ 1 yield a partition {Pi : 1 ≤ i ∈ ω} of the domain of the model will be needed. Before continuing the proof, I will outline the strategy for its remaining part. A new function hγi (x, y)ii∈ω will be defined from the function hϑi (x, y)ii∈ω . The formulae γi (x, y) define sets Ci = {a : 𝔐 ⊨ γi (a, b)}. The definition of the γi (x, y) is guided by the following idea. The set C0 = P0 is empty. The set Ci+1 is defined as follows:    P1 = S0 \ S1 = |𝔐| \ S1 if Ci = Ø Ci+1 = Pk+1 if k is the least k with Pk ∩ Ci 6= Ø   Ø otherwise By induction on i one can show that Pi = Ci , so the Ci s determine the same partition of the domain of 𝔐 as the Pi s, but by the lights of the model (𝔐, S), these sets will come apart, and this will be used to deduce a contradiction. The function hγi (x, y)ii∈ω is represented in the language of arithmetic by γ i , . so in accordance with the remark (8.7) we can show that even for nonstandard numbers i ≤ c we have 𝔐 ⊨ For2 (γ i ). Consequently, for i ≤ c nonstandard, γ i . . is still a formula in the sense of the model. Of course, nonstandard formulae cannot be used; they are not formulae of L (not even of L with additional parameters). But, if there is a satisfaction class S for 𝔐, one can express in the model (𝔐, S) that γi (for i ≤ c) applies to a number a ∈ |𝔐| and to b by claiming (𝔐, S) ⊨ Tγ i (a, b). .

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So in a sense one has extended the sequences of sets Ci from above to a sequence up to the nonstandard number c using the truth predicate, via the assumption that there is satisfaction class S for 𝔐. The assumption that there is such a satisfaction class will be shown to be inconsistent with the assumption that 𝔐 is not recursively saturated. Using the satisfaction class, and reasoning in the model, one can show that if Ci with i ≤ c is empty then Ci = P1 . This will be shown formally in Lemma 8.27. Next a function f is defined on the set of all i ≤ c by stipulating  the smallest j with P ∩ C 6= Ø j i f(i) := 0, if there is no such j As the sets Pj with j ∈ ω partition the domain of 𝔐, the range of the function f contains only standard numbers. From the definitions of f and of the Ci s one can conclude the following: (8.8)

If Ci 6= Ø, then Ci+1 = Pf(i)+1

This will be shown formally in Lemma 8.28. Now Ci−1 = Ø implies Ci = P1 6= Ø, and thus also Ci 6= Ø. Hence, by contraposition, for any given i ≤ c, Ci = Ø implies Ci−1 6= Ø. So by (8.8), Ci = Pf(i−1)+1 . But Pf(i−1)+1 is not empty because Pf(i−1)+1 is a partition, so Ci cannot be empty and by 8.8 again generally (8.9)

Ci = Pf(i) for all i ≤ c.

This will be shown formally in Lemma 8.29. Now one obtains a contradiction by continuing as follows: Since Pf(i+1) = Ci+1 by (8.9) and Ci+1 = Pf(i)+1 by (8.8), one has f(i + 1) = f(i) + 1. But a nonstandard number i has infinitely many predecessors by Lemma 8.15. So there would also be an infinitely descending sequence . . . < f(i − 2) < f(i − 1) < f(i) of natural numbers. This contradiction shows that 𝔐 cannot have a satisfaction class, if it is not recursively saturated. After this outline I now return to the proper proof by defining the sentences γi (x, y). For technical reasons I define two primitive recursive functions hγi (x, y)ii∈ω and hδi,k (y)ii,k∈ω by simultaneous recursion.

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I will work informally inside the model 𝔐. In particular, I will form disjunctions of the nonstandard length c. To be more technically precise, I would have to add a further parameter j below and define sequences hγi,j (x, y)ii,j∈ω and hδi,k,j (y)ii,k,j∈ω , then go to the model 𝔐, set j = c, and argue that all objects hγi,c (x, y)ii∈ω are formulae with x and possibly y free in the sense of 𝔐. Instead I will simply use c as a parameter in the definition of hγi (x, y)ii∈ω and hδi,k (y)ii,k∈ω . First for each j ∈ ω set γ0,j as follows: γ0,j (x) := (x 6= x) Assuming that the formulae δi,k (y) have been defined for all k ≤ c, I define γi+1 (x, y) in the following way:
γi+1 (x, y) := δi,1 (y) ∧ ϑ1 (x, y) ∨  ¬δi,1 (y) ∧ (δi,2 (y) ∧ ϑ2 (x, y))∨ (¬δi,2 (y) ∧ (δi,3 (y) ∧ ϑ3 (x, y))∨ (¬δi,3 (y) ∧ (δi,4 (y) ∧ ϑ4 (x, y))∨ .. . ¬δi,c−1 (y) ∧ ((δi,c (y) ∧ ϑc (x, y))∨
 (¬δi,c (y) ∧ ¬x = x) |{z} ... ) brackets

Assuming that γi (x) has been defined for i, the formulae δi,k (y) are defined in the following way: δi,1 (y) := ¬∃z γi (z, y) δi,k+1 (y) := ∃z (ϑk (z, y) ∧ γi (z, y)) The function sending ϑi (x, y) (and the above mentioned parameter j in the official version) to γi (x, y) is primitive recursive and so is the function mapping ψi (x, y), k (and j) to δi,k (y). Moreover, these primitive recursive functions yield, for i, k ≤ c, formulae with the appropriate free variables as output. Hence, as Lemma 8.7 ensures that all ψi and thus ϕi are formulae in the sense of 𝔐 for i ≤ c, I infer the same for the formulae ϑi , more formally: 𝔐 ⊨ ∀x ≤ c For2 (ϑ. x )

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I conclude that similar claims also hold for γi (x, y) and δi,k (y); the following two conditions obtain: 𝔐 ⊨ ∀i ≤ c For2 (γ i ) . 𝔐 ⊨ ∀i, k ≤ c For2 (δ. i,k ) Since utb is a subtheory of ct↾ by Lemma 8.4, I can use the uniform Tsentences in the following. So, for instance, the following obtains for all numbers i ∈ ω: ◦
(𝔐, S) ⊨ ∀t Tϑ. i (t, b) ↔ ϑi (t◦ , b ) This will be used freely in what follows. Next I will prove the formal counterpart to the claim that if Ci = Ø, then Ci+1 = P1 .
lemma 8.27. (𝔐, S) ⊨ ¬∃t Tγ i (t) → ∀t Tγ i+1 (t) ↔ ϑ1 (t◦ ) for each i < c. . . proof. Using the axioms of ct↾, I reason in the model as follows:

(8.10)

(𝔐, S) ⊨ ¬∃t Tγ i (t) → T(¬. ∃. xγ i (x)) . . → Tδ. i,1

Moreover, from the definition of γi (x) using the axioms of ct↾ one again obtains the following for all i ≤ c: 
 (𝔐, S) ⊨ ∀t Tγ i+1 (t) ↔ (Tδ. i,1 ∧ Tϑ. 1 (t)) ∨ (¬Tδ. i,1 ∧ Ts) . Here s is a term designating the remaining part of γi+1 (x). The claim now follows from (8.10). a As promised in the outline above, I now define the function f(i) from the set {i : i ≤𝔐 c} into the set of all natural numbers:  the unique j with 𝔐 ⊨ ∃x (ϑ (x) ∧ Tγ (x)) ˙ if 𝔐 ⊨ ∃t (Tγ i (t◦ )) j .i . f(i) := 0 otherwise Lemma 8.26 (i) guarantees that in the first case there is always a suitable j ∈ ω. In the next lemma I formally establish the implication (8.8) above. lemma 8.28. For all i <𝔐 c the following holds: (𝔐, S) ⊨ ∃t Tγ i (t◦ ) → ∀t ϑf(i)+1 (t◦ ) ↔ Tγ i+1,c (t) . .




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proof. I suppress the index c here and in the following proofs. Assume (𝔐, S) ⊨ ∃x Tγ i (x). ˙ In this case (𝔐, S) ⊨ T¬. δ. i,1 obtains. In virtue of the defini. tion of f the following obtains:
(𝔐, S) ⊨ ∀k < f(i) ¬∃t Tϑ. k (t) ∧ Tγ i (t) . By definition of δi,k (x) this implies the following: (𝔐, S) ⊨ ∀k < f(i) T¬. δ. i,k+1 But also Tδ. i,f(i)+1 holds in (𝔐, S) by assumption and by the definition of f(x). Using the compositional axioms of ct↾, one can then push the truth predicate into the formula γi+1 (x) to obtain the following equivalence, where s is a term designating the remaining part of γi+1 (x): 
(𝔐, S) ⊨ ∀t Tγi+1 (t) ↔ Tδ. i,1 ∧ Tϑ. 1 (t) ∨  T¬. δ. i,1 ∧ (Tδ. i,2 ∧ Tϑ. 2 (t))∨ (T¬. δ. i,2 ∧ (Tδ. i,3 ∧ Tϑ. 3 (t))∨ (T¬. δ. i,3 ∧ (Tδ. i,4 ∧ Tϑ. 4 (t)) ∨ . . . .. . T¬. δ. i,f(i) ∧ ((Tδ. i,f(i)+1 ∧ Tϑ. f(i)+1 (t)∨ 
 (T¬. δ. i,f(i)+1 ∧ Ts) . . .)
It follows that ∀t ϑf(i)+1 (t◦ ) ↔ Tγ i+1 (t) holds in (𝔐, S). .

a

lemma 8.29. The following claim obtains for all i such that 1 < i ≤ c: (i) (𝔐, S) ⊨ ∃t Tγ i (t) .
(ii) (𝔐, S) ⊨ ∀t ϑf(i) (t◦ ) ↔ Tγ i (t) . proof. I reason in the following way for 0 < i ≤ c:
(𝔐, S) ⊨ ¬∃t Tγ i−1 (t) → ∀t Tγ i (t) ↔ Tϑ. 1 (t) . . → ∃t Tγ i (t) .

Lemma 8.27 Lemma 8.26 (ii)

Contraposing, we obtain: (𝔐, S) ⊨ ¬∃t Tγ i (t) → ∃t Tγ i−1 (t) . .
→ ∀t ϑf(i−1)+1 (t◦ ) ↔ Tγ i (t) . → ∃t Tγ i (t) .

Lemma 8.28 Lemma 8.26 (ii)

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Thus I conclude (𝔐, S) ⊨ ∃t Tγ i (t) for all 0 < i ≤ c; this is claim (i). Claim (ii) . follows then from Lemma 8.28. a lemma 8.30. f(i + 1) = f(i) + 1 for all 1 < i ≤ c. proof. I continue reasoning in the model:
(𝔐, S) ⊨ ∀t ϑf(i+1) (t◦ ) ↔ Tγ i+1 (t) .
↔ ϑf(i)+1 (t◦ )

Lemma 8.29 (ii) Lemmata 8.28 and 8.29

From this one can deduce the claim using Lemma 8.26 (i).

a

Finally, Lachlan’s theorem, that is, Theorem 8.24, can be proved: Proof of Lachlan’s theorem. From the previous lemma and Lemma 8.15 I conclude that, for some nonstandard number i ≤ c, the following inequations hold: f(i) > f(i − 1) > f(i − 2) > . . . Since f(i) ∈ ω (which is established by exploiting the assumption that 𝔐 is not recursively saturated), one has produced an infinite descending chain of natural numbers from the assumption that 𝔐 has a satisfaction class and the assumption that 𝔐 is not recursively saturated. Hence, if a nonstandard model has a satisfaction class, it is recursively saturated. a

8.5 Satisfaction classes and axiomatic theories of truth Lachlan’s theorem shows that ct↾ is not model-theoretically conservative over Peano arithmetic. theorem 8.31. There is a model of Peano arithmetic that cannot be expanded to a model of ct↾. That is, there is a model 𝔐 ⊨ pa for which there is no set S such that (𝔐, S) ⊨ ct↾. proof. By Lemma 8.23 there is a nonstandard model of Peano arithmetic that is not recursively saturated. By Lachlan’s theorem the model does not have a satisfaction class. a As pointed out on p. 82, some authors have suspected that the schema of elementary comprehension suffices for defining a truth predicate. Lachlan’s theorem shows that this is not correct. The theory aca0 of arithmetical comprehension is formulated in the language for second-order arithmetic and

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given by the same axioms as aca (Definition 8.41), with the exception that the induction axioms are replaced by the following single axiom: 0 ∈ X ∧ ∀y (y ∈ X → Sy ∈ X) → ∀y y ∈ X Otherwise aca0 has the comprehension axioms of aca, that is, all axioms of the form
∃X ∀y y ∈ X ↔ ϕ(y) , where ϕ(y) is a formula of L2 in which neither X nor any second-order quantifiers occurs. Free second-order variables are allowed in ϕ(y). All the induction axioms of Peano arithmetic are theorems of aca0 . So theory aca0 is basically aca with induction restricted to sets. theorem 8.32. The theory aca0 cannot define a truth predicate satisfying the ct↾ axioms. This result can be established for any extension S of Peano arithmetic so that any model of Peano arithmetic can be extended to a model of S. proof. First I show that any model of Peano arithmetic can be expanded to a model of aca0 . Let a model 𝔐 ⊨ pa be given. Take the second-order quantifiers to range over the class of all L𝔐 -definable sets, that is, the sets {a : 𝔐 ⊨ ϕ(a)}, where ϕ(x) is some formula of L𝔐 which the language L with additional constants for all elements of 𝔐 (these constants are used to verify instances of the comprehension schema with first-order parameters). It can then be shown that the resulting expansion of 𝔐 is a model of aca0 . So every model of Peano arithmetic can be expanded to a model of aca0 . By Lemma 8.23 there is a model 𝔐 of pa that is not recursively saturated. Expand 𝔐 to a model 𝔐2 of aca0 . If there were a formula ψ(x) in the language of aca0 such that the axioms of ct↾ with ψ(x) in the place of the truth predicate were satisfied in 𝔐2 , then 𝔐 would be recursively saturated by Lachlan’s theorem, contradicting the assumption. a The proof technique used to prove Theorem 8.32 can be adapted to obtain further undefinability results like the following. The Kripke–Feferman theory kf of truth, which will be defined in Chapter 15, is a strong type-free truth system, which will be defined in Definition 15.2. kf is much stronger that the typed theory ct. Both theories, kf and ct, include all induction axioms in the language LT with the truth predicate. If induction is restricted to the

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language L in both theories, then at least with respect to truth-definability the typed theory becomes stronger than the type-free theory. For every model of Peano arithmetic can be expanded to a model of the type-free theory kf↾, as Cantini (1989) showed. Hence the theory kf↾ cannot define the truth predicate of ct↾ (in the sense of Definition 6.4). After these negative results I sketch some positive results without proofs. As has been mentioned already, Kotlarski et al. (1981) established a result that yields a model-theoretic proof of the conservativity on ct↾ over pa: theorem 8.33 (Kotlarski et al. 1981). Every countable recursively saturated model of Peano arithmetic has a satisfaction class. More detailed proofs of this result and some of its variants are given by Engström (2002), Kaye (1991), and Smith (1984). Kaye’s and Engström’s proofs are closer to my setting here as they prove the result for a language with function symbols, in contrast to Kotlarski et al. (1981). The restriction of the theorem to countable models is necessary, as Smith (1984, 1989) has shown. Of course, this theorem in itself does not suffice for proving the conservativity of ct↾. The following lemma, a weakening of a result due to Barwise and Schlipf (1976), can be used to obtain the conservativity result: theorem 8.34. Every countable model 𝔐 of Peano arithmetic has a countable recursively saturated extension 𝔑 such that the same sentences hold in 𝔐 and 𝔑. By saying that 𝔑 is an extension of 𝔐 I mean that the domain of 𝔐 is a subset of the domain of 𝔑 and that the interpretations of all nonlogical symbols in both models agree on |𝔐|. Theorems 8.33 and 8.34 together yield an alternative proof of Theorem 8.12, which states that ct↾ is conservative over pa. Alternative proof of Theorem 8.12. Assume ϕ is an L-sentence with pa ⊬ ϕ. By the completeness theorem there is a countable model 𝔐 so that 𝔐 ⊨ ¬ϕ, and by Theorem 8.34 there is a countable recursively saturated model 𝔑 of pa with 𝔑 ⊨ ¬ϕ. The existence of a satisfaction class S for 𝔑 with (𝔑, S) ⊨ ct↾ +¬ϕ follows then from Theorem 8.33; this implies ct↾ ⊬ ϕ by soundness. a Compared to the cut-elimination proof on p. 80, the model-theoretic proof involving Theorem 8.33 has the disadvantage that it cannot easily be established within Peano arithmetic. As McGee (2006) has argued, the deflationist will want to argue that truth is conservative over the base theory without going beyond the base theory, that is, in the present case Peano arithmetic.

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The model-theoretic proof has the advantage that it can be adapted to prove results that are stronger than the plain conservativity result for ct↾. For instance, using a result mentioned by Kotlarski et al. (1981), proved via their methods, one can show that ct↾ cannot prove that all sentences of the form 0 = 0 ∧ . . . ∧ 0 = 0 are true. In general, for any nonstandard model of Peano arithmetic there will be a wide variety of satisfaction classes that can be used to establish that ct↾ does not prove certain claims. There has also been some work on full inductive satisfaction classes. A set S ⊂ |𝔐| is a full inductive satisfaction class for 𝔐 just if (𝔐, S) ⊨ ct, that is, if (𝔐, S) also validates the induction axioms of pat containing the truth predicate. Kotlarski (1991) summarizes the main results on these satisfaction classes. They can be used to gain some proof-theoretic results about ct. Here I will not go into these results, but turn now to the analysis of ct, from a proof-theoretic perspective. 8.6 Compositional truth and elementary comprehension The system ct↾ does not feature instances of the induction schema as axioms: ϕ(0) ∧ ∀x (ϕ(x) → ϕ(Sx)) → ∀x ϕ(x) is an axiom of ct↾ only if the truth predicate does not occur in ϕ(x). If one accepts the induction principle of Peano arithmetic, however, it is very natural to continue to apply the induction principle to formulae when new symbols are added to the language of arithmetic. Induction is a general principle that does not rely on the expressive limitations of the language of Peano arithmetic, L. As Kreisel (1967) emphasized, the motivation behind the induction principle is independent of the symbols of the language and supports the extension of the induction principle to the full language when new symbols are added. It might be illustrative to understand Peano arithmetic as a schematic theory: one might take Peano arithmetic as a system that contains an additional schematic unary predicate symbol P with the single induction schema P (0) ∧ ∀x (P (x) → P (x+1)) → ∀x P (x) and a rule that allows one to substitute arbitrary formulae for P . If more formulae become available, for instance, by the addition of a truth predicate or second-order quantifiers, the newly added formulae can be used as well. Under this conception induction looks more like an open-ended principle that can be applied to whatever condition we might be able to formulate.

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If induction is formulated with a schematic predicate letter, it becomes more plausible to think of the induction axioms involving the truth predicate as axioms one has already accepted implicitly with the system of Peano arithmetic; once one accepts this schematic version of induction, one is prepared to accept all its instances independently of which particular conceptual resources are available. The truth predicate is no exception: once it is available it can be used in formulae replacing the schematic letter P in the induction schema. Feferman (1991) offers a more thorough discussion of the role of schemata. Having made this point, it is technically more straightforward to return to the usual formulation of induction as a set of axioms. The alternative formulation based on a schematic letter would not make a significant difference to what follows. At any rate, it seems very natural to add all instances of the induction schema to ct↾ including those that contain the truth predicate. definition 8.35 (ct). The system ct is obtained from ct↾ by adding all induction axioms in the language of LT . In other words, ct is the theory ct↾ without the unnatural restriction of induction to formulae in L. As Kotlarski, Krajewski, and Lachlan (1981) have shown, the system ct↾ with arithmetical induction alone does not even prove that all sentences of the form 0 = 0 ∧ . . . ∧ 0 = 0 are true. This is easily established in ct, however, by induction on the number on conjuncts. Thus there are generalizations that are not provable in ct↾, which are provable in ct. In what follows I shall also make use of some further generalizations that are provable in ct but not in ct↾. Another generalization not provable in ct↾ is that substitution of identicals preserves truth (compare Cantini 1989, p. 102). In ct, however, this is provable: lemma 8.36 (regularity). The following sentence is provable in ct:
ct ` ∀x ∀s ∀t Sent(∀. vx) ∧ s◦ = t◦ → (Tx(s/v) ↔ Tx(t/v)) sketch of proof. Using Axiom ct1 one can prove in ct that the claim holds for atomic formulae x. Then one proves the claim formally by induction on the length of x. a Next I prove another generalization that is not provable in ct↾. It is actually a strengthened version of axiom ct5: in ct the truth predicate commutes not only with single quantifiers but with entire blocks of quantifiers of arbitrary

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length. Informally speaking, this means that all instances ϕ(t1 , . . . , tn ) of a formula are true for all closed terms t1 , . . . , tn if and only if ∀x1 . . . ∀xn ϕ is true. The crucial problem is that the number n may be arbitrarily large and only in ct can induction on n be employed for the proof. In order to formulate this generalization of ct5, I introduce some notation. The expression ∀~ v ranges over (finite) sequences of objects; it will be used to quantify over strings of variables. The expression ∀. v ~ x is the result of formally prefixing x with the expression ∀(v)1 . . . ∀(v)n where (v)i is the ith member of the sequence v ~ . Thus, one can prove in ct that all members in v ~ are variables and that x is a formula of L with only the variables in v ~ free if and only if Sent(∀. v ~ x) obtains. Similarly, ∀~t is used for quantifying over strings of closed terms of L. Furthermore, x(~t ) stands for the result of substituting the ith variable with the ith element of the sequence ~t in x for each i if the length of ~t is not smaller than i, or with 0 if ~t does not have an ith element.
lemma 8.37. ct ` ∀x ∀~ v Sent(∀. v ~ x) → (∀~t Tx(~t ) ↔ T∀. v ~ x) proof. The claim is proved in ct by an induction on the number of free variables in x. When the number of free variables in x is 0 one proves the claim, using Axiom ct5, by a side induction on the length of v ~ . This is fairly ~ trivial because in this case x(t ) = x. For the induction step, Axiom ct5 is employed again. I skip the details. a The universal closure of a formula is the formula preceded by universal quantifiers binding all free variables of the formula. For the sake of definiteness, one may assume that the variables have some alphabetical order and that the prefixed quantifiers are added in this order. If ucl(x) is a function expression representing the function that gives, when applied to a formula, its universal closure, then the following is a corollary of the preceding lemma:
corollary 8.38. ct ` ∀x ∀~ v Sent(∀. v ~ x) → (∀~t Tx(~t ) ↔ Tucl(x)) Here I assume, as in the case of ∀. , that the function represented by ucl(x) yields a sentence only if x is a formula and that this can be proved in Peano arithmetic. We have seen several instances in which ct can prove generalizations not provable in ct↾. By proving a certain generalization, namely the so-called global reflection principle1 , I will show that, unlike ct↾, the system ct is not 1 The term global reflection principle is taken from Kreisel and Lévy (1968).

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conservative over Peano arithmetic. The global reflection principle expresses the soundness of Peano arithmetic in ct: one proves that all theorems of Peano arithmetic are true, as long they do not contain free variables. Thus, one reasons in ct, if 0 = 1 were provable in Peano arithmetic 0 = 1 would be true. But since 0 = 1 is ct-provable false, Peano arithmetic is consistent. As Gödel’s second incompleteness theorem prevents Peano arithmetic from proving its own consistency, ct cannot be conservative over Peano arithmetic. In what follows Bewpa (x) expresses that x is a formula provable in Peano arithmetic formulated in the language L without the truth predicate. theorem 8.39. The theory ct proves the global reflection principle for Peano arithmetic, that is,
ct ` ∀x Sent(x) ∧ Bewpa (x) → Tx and therefore ct ` ¬Bewpa (┌0 = 1┐). proof. I assume that the axioms of Peano arithmetic are sentences, not open formulae. There are only finitely many axioms of pa beyond the induction axioms. If ϕ is an axiom of pa, it follows from Theorem 8.4 that ct ` T┌ϕ┐. Next I will prove in ct that all induction axioms are true. The following is an instance of the induction schema of ct:
˙ Tx(┌0┐/v) ∧ ∀y Tx(y/v) ˙ → Tx(Sy/v) → ∀y Tx(y/v) ˙ As explained in Section 5, x(y/v) ˙ stands for the result of substituting v with the numeral of y in x, and ˙ x(Sy/v) for the result of substituting v with the numeral of the successor of y in x. ˙ Since (x(Sy/v)) = (x(S. y/v)) ˙ is a theorem of Peano arithmetic, where S. represents the function that prefixes the successor symbol S to each term, the theory ct proves the following:
Tx(┌0┐/v) ∧ ∀y Tx(y/v) ˙ → Tx(S. y/v) ˙ → ∀y Tx(y/v) ˙ By applying the Regularity lemma 8.36, I obtain the following theorem in ct:
Tx(┌0┐/v)∧ → ∀t Tx(t/v) → Tx(S. t/v) → ∀t Tx(t/v) By the quantifier Axiom ct5 and the axioms ct2 and ct4 for negation and disjunction, taking the symbol → as an abbreviation, I infer the following in ct:

Sent(∀. vx) → Tx(┌0┐/v) ∧ T ∀. v (x→ . x(S. v/v)) → T∀. vx

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Now the truth predicate is put in front: Sent(∀. vx) → T x(┌0┐/v)∧. ∀. v (x→ . x(S. v/v))→ . ∀. vx




Hence it can be proved in ct that all closed instances of the induction schema are true. Closed instances of the induction schema are sufficient as axioms of Peano arithmetic. If instances with parameters were allowed, one could, using the quantifier Axiom ct5, also prove that the universal closures of all instances of the induction schema are true. Now that I have proved in ct that all the axioms of Peano arithmetic are true, it remains to prove in ct that (the universal closures of) the logical axioms are true as well. For the present proof it is convenient to assume that Peano arithmetic is formulated in an axiomatic calculus.2 Here I do not specify the exact axioms and rules, but it should be clear how to prove in ct, using the axioms ct1–ct5, that the universal closures of all (reasonably chosen) logical axioms are true. To prove the claim that the universal closure of each theorem of Peano arithmetic is true, one can prove by induction on the length of proofs that if a formula is provable then the universal closure of the formula is true. To prove this, let Prv(x, y) be a formula expressing that y is provable with a proof of length at most x, and, as before, ucl(x) represents the function that assigns to every formula its universal closure. To show that all universal closure of pa-theorems are true, I appeal to the following induction axiom:
(8.11) ∀x Prv(0, y) → Tucl(y) ∧
∀x ∀y (Prv(x, y) → Tucl(y)) → ∀y (Prv(Sx, y) → Tucl(y)) → ∀x Prv(x, y) → Tucl(y)




The result that all axioms of Peano arithmetic are ct-provably true covers the induction base, that is, the first line in (8.11). To establish the formula in the second line, I need to show that the rules of inference preserve truth. This is made more difficult because formulae with free variables may occur as steps in a proof. I did not specify a specific logical calculus, but one possible rule might be a conjunction introduction rule that licenses the step from two premisses ϕ and ψ to the conclusion ϕ ∧ ψ, where ϕ and ψ are formula of L. So to prove this case of the induction step, I start from 2 Takeuti (1987, p. 186) sketches a similar proof for the sequent calculus.

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Axiom ct3 and derive, in ct, the following special case: ∀x ∀y Sent(ucl(x))∧ Sent(ucl(y)) →
(Tucl(x) ∧ Tucl(y) → ∀~t Tx(~t ) ∧ ∀~t Ty(~t ))
→ ∀~t T(x(~t )∧. y(~t ))
→ ∀~t T((x∧. y)(~t )))
→ Tucl(x∧. y))
Therefore, Sent(ucl(x∧. y)) → Tucl(x) ∧ Tucl(y) → Tucl(x∧. y) is a theorem of ct and one case of the induction step of (8.11) is proven. The other rules of inference of the chosen calculus are treated in a similar way; the case of the introduction rule for the universal quantifier from ϕ to ∀x ϕ is especially easy because the universal closure of ϕ is provable by induction hypothesis. a Now the proof can be completed in the way sketched above. The sentence Sent(┌0 = 1┐) ∧ Bewpa (┌0 = 1┐) → T┌0 = 1┐ is an instance of the global reflection principle. Since pa ` 0 6= 1, Theorem 8.4 implies that ct ` ¬T┌0 = 1┐. This yields ct ` ¬Bewpa (┌0 = 1┐), that is, ct proves the consistency of Peano arithmetic, which is not provable in Peano arithmetic by Gödel’s second incompleteness theorem. This immediately implies the following: corollary 8.40. The theory ct is not conservative over Peano arithmetic. Refuting the conservativity of ct by instantiating the global reflection principle of Theorem 8.39 with 0 = 1 requires Gödel’s second incompleteness theorem. The usual proof of the second incompleteness theorem relies on the Löb derivability conditions and, in general, on a natural formalization of the property of being a theorem of pa. If this property is formalized by what is conceived as a less natural formula, then the consistency of pa may well become provable. For instance, if provability is expressed in L by the socalled Rosser provability predicate, then the consistency statement becomes provable.3 If one aims to avoid this problem with intensionality, instantiating the global reflection principle with the Gödel sentence may yield a better proof: If Bew(x) (weakly) represents provability in pa – that is, pa ` Bew(┌ϕ┐) if
and only if pa ` ϕ – and ct proves ∀x Sent(x) ∧ Bew(x) → Tx , then ct 3 The Rosser provability predicate was introduced by Rosser (1936) to dispense with the assumption of ω-inconsistency in the proof of Gödel’s first incompleteness theorem. For more information on these intensionality phenomena see Feferman (1960).

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proves any Gödel sentence γ. To see this, assume γ is a Gödel sentence, that is, assume pa ` γ ↔ ¬Bew(┌γ┐), and proceed as follows: ct `Bew(┌γ┐) → T┌γ┐

global reflection

Bew(┌γ┐) → γ

Lemma 8.4

Bew(┌γ┐) → ¬γ

choice of γ

¬Bew(┌γ┐)

two preceding lines

γ

choice of γ

Hence the Gödel sentence for pa is provable in ct, though not in pa by Gödel’s first incompleteness theorem. The observation that ct or a very similar system decides the Gödel sentence from the first incompleteness theorem is due to Tarski (1935, p. 274). This attempt to free the proof of non-conservativity from intensionality problems by appealing to the first rather than the second incompleteness theorem has its limitations, though. For the proof of Theorem 8.39 relies on certain properties of the provability predicate; it can be proved for the Rosser provability predicate, because Rosser provability implies standard provability and this implication can be established in pa. But other provability predicates may call for different proofs of analogues of Theorem 8.39. At any rate, Theorem 8.40 and other Gödel phenomena do not provide the strongest considerations against conceptions of truth which take the truth axioms to be conservative over the base theory. I think that a comparison with second-order quantification is more revealing: quantification over arithmetically definable sets of natural numbers is as deflationary as the truth predicate of ct, for this kind of quantification and ct-truth are interdefinable. The result is also more informative than Corollary 8.40 because the secondorder system aca to which ct will be related is well understood and much stronger than the mere consistency statement for Peano arithmetic. Thus, it is the comparison with aca rather than Corollary 8.40 that reveals the full strength of the compositional truth predicate of ct. definition 8.41 (aca). The system aca is formulated in the language of second-order arithmetic L2 , that is, in the language of L extended by secondorder variables and a binary predicate symbol ∈. The expression t ∈ X is an atomic formula of L2 if t is a term of L and X is a second-order variable. The system aca is given by the axioms of pa, the set of all induction axioms in the language L2 and all axioms of the form
(8.12) ∃X ∀y y ∈ X ↔ ϕ(y) ,

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where ϕ(y) is a formula of L2 in which neither X nor any second-order quantifiers occurs. Free second-order variables are allowed in ϕ(y). The language L2 can be considered as a two-sorted first-order language. The quantifiers are governed by the usual logical rules for first-order quantifiers. Schema 8.12 is called the schema of arithmetical comprehension; it expresses that if a set is definable in the arithmetical language (possibly with secondorder parameters), then it exists. Further information on aca can be found in Buchholz et al. (1981), Takeuti (1987), and Simpson (1998). The system aca is predicative insofar as it only postulates the existence of sets that can be defined without quantifying over other sets. The secondorder parameters that may occur in the comprehension formula are usually not seen as a problem for the predicativity of aca as for formulae with such parameters the comprehension axiom only implies the existence of sets relative to that parameter. As I will show below, however, allowing such parameters is crucial. The comprehension schema without second-order parameters is much weaker that the actual version. Here I cannot go into the voluminous discussion on predicativity. For an overview see Feferman (2005). The systems aca and ct are very closely related and intertranslatable: theorem 8.42. The systems ct and aca are proof-theoretically equivalent. More precisely, the truth predicate of ct can be defined in aca and there is a relative interpretation of aca in ct that does not reinterpret arithmetical expressions (with the exception of renaming bound variables). The theorem belongs to the proof-theoretic folklore. Takeuti (1987) provides a definition of the truth predicate of ct in aca. I am not aware of a detailed proof of the converse direction in print; but Feferman (1987, pp. 20f) sketches a proof in an unpublished draft. I prove the theorem by splitting up its proof into smaller lemmata. The strategy for interpreting aca in ct is straightforward: sets of numbers are definable in aca by arithmetical formula. The claim that n is an element of the set defined by ϕ(y) may thus be replaced by the claim that ϕ(y) is true of n. Quantification over sets is then replaced by quantification over formulae. There is a quirk that makes the interpretation less perspicuous: the language L2 has two sorts of variables, first-order variables and second-order variables. Second-order quantification is interpreted as quantification over formulae, but the target language LT does not have a special sort of variables ranging over formulae; and I need to keep the variables replacing the second-order variables and the variables replacing the first-order variables

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of L2 separate. This is done by substituting the nth second-order variable with the 2n + 1st variable of LT and by replacing the nth first-order variable with the 2n + 2nd variable of LT . The noughth variable x0 is reserved for a special purpose: when saying that a formula with one free variable is true for a number, x0 will always be this variable. The interpretation function ∗ is defined on all terms and formulae of the language L2 . Let an enumeration x0 , x1 , x2 , x3 , . . . of all first-order variables and an enumeration X0 , X1 , X2 , X3 , . . . of all second-order variables be given. The formula For(y, ┌x0┐) expresses that y is a formula of L with exactly x0 free. There is a function h that gives, when applied to a number n and a formula ϕ(x0 ) with exactly the noughth variable x0 free, the sentence ϕ(n) that is obtained from ϕ(x0 ) by substituting all free occurrences of x0 with the numeral of n. If k is not an L-formula with exactly x0 free, then h(n, k) is the sentence n = n. The function h is naturally represented by the function symbol h. (x, y). The expression Sat(x, y) abbreviates the formula Th. (x, y). So Sat(x, y) expresses that x satisfies the formula y if y is a formula with exactly the noughth variable free. The interpretation function ∗ is then defined as follows: (i) xn∗ = x2n+2 (ii) If t is a term of L, then t∗ is the result of substituting each variable x in t with x∗ . (iii) Xn∗ = x2n+1 (iv) (t ∈ X) = Sat(t∗ , X ∗ ) (v) (¬ϕ)∗ = (¬ϕ∗ ) (vi) (ϕ ∧ ψ)∗ = (ϕ∗ ∧ ψ ∗ ) (vii) (ϕ ∨ ψ)∗ = (ϕ∗ ∨ ψ ∗ ) (viii) (∀x ϕ)∗ = (∀x∗ ϕ∗ ) (ix) (∀X ϕ)∗ = ∀X ∗ (For(X ∗ , ┌x0┐) → ϕ∗ )




(x) (∃x ϕ)∗ = (∃x∗ ϕ∗ ) (xi) (∃X ϕ)∗ = ∃X ∗ (For(X ∗ , ┌x0┐) → ϕ∗ )




According to clause (iv), an atomic formula t ∈ X is translated by the claim that X ∗ is true of t∗ . Sat(t∗ , X ∗ ) contains exactly the variables of t∗ and the variable X ∗ . Of course, the variable X ∗ needs to range over formulae with exactly x0 free. For this reason the first-order quantifiers replacing the second-

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order quantifiers of L2 are restricted to such formulae. In clauses (ix) and (xi), For(y, ┌x0┐) expresses that y is a formula with exactly the variable x0 free. lemma 8.43. If ϕ is a formula of L2 and aca ` ϕ, then ct ` ϕ∗ . proof. Obviously, logical axioms and induction axioms of aca are translated into logical axioms and induction axioms of ct; logical rules are also trivially preserved. The only non-trivial part of the proof is to show that the translations of the comprehension axioms are provable in ct. Typically, a comprehension axiom takes the following form: (8.13)

∃X ∀y y ∈ X ↔ ϕ(y, z, t ∈ Z) | {z }




comprehension formula

The variable z here is a parameter, that is, a variable that occurs freely in the comprehension formula. As the comprehension formula may also contain second-order parameters, that is, free second-order variables, it can contain subformulae of the form t ∈ Z where t is some term. The variables z and Z serve only as examples: the comprehension formula may of course contain first-order parameters besides z and occurrences of second-order parameters besides the displayed occurrence in t ∈ Z; these can be dealt with in the same way as z and Z. I now prove the translation of the comprehension Axiom 8.13, reasoning in ct. The following is a theorem of logic: ∀x0 ϕ∗ (x0∗ , z∗ , Sat(t∗ , Z∗ )) ↔ ϕ∗ (x0∗ , z∗ , Sat(t∗ , Z∗ ))




Here ϕ∗ indicates that all bound variables in ϕ are renamed in accordance with the definition of the translation function. On the left-hand side of the equivalence, I move the truth predicate in front of the formula using the axioms ct1–ct6, the Regularity lemma 8.36 and the definition of Sat(x, y) as Th. (x, y): (8.14)


∀x0 Tϕ∗ (x˙0 , z˙∗ , h. (t∗ , Z∗ )) ↔ ϕ(x0 , z∗ , Sat(t∗ , Z∗ )) .

Here ϕ∗ (y, z, v) represents the function that yields, when applied to y, z, . and v, the sentence that is obtained by substituting x0 with y, z with the numeral of z and the subformula t ∈ Z with v. Applying existential weakening and renaming the bound variable x0 yields the following formula: (8.15)


∃X ∗ For(X ∗ , ┌x0┐) ∧ ∀y Sat(y, X ∗ ) ↔ ϕ(y, z∗ , Sat(t∗ , Z∗ ))

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This is just the translation of the instance of the comprehension schema of aca:

∗ ct ` ∃X ∀y y ∈ X ↔ ϕ(y, z, t ∈ Z) a In the embedding of aca in ct, the compositional axioms, as opposed to the disquotational axioms of utb, are only needed to handle second-order parameters in the comprehension formula in the derivation of (8.14). If no second-order parameters are admitted, then the resulting system can already be interpreted in utb with the same translation function as above. Thus aca without second-order parameters in the second-order formulae is conservative over Peano arithmetic just as utb is (see also Halbach 1999a). To formulate this claim more precisely, let acapf be like aca of Definition 8.41 but with the comprehension formula ϕ(y) in the comprehension axioms
∃X ∀y y ∈ X ↔ ϕ(y) restricted to formulae of L. corollary 8.44. If ϕ is a formula of L2 and acapf ` ϕ, then utb ` ϕ∗ . Hence acapf is conservative over pa by Theorem 7.5. proof. The Corollary can be proved in the same way as the above lemma. To move the truth predicate in front of the formula only the uniform disquotation sentences are needed. a Lemma 8.43 provides a reduction of the second-order theory aca to the compositional theory of truth ct. To complete the proof of Theorem 8.42 stating the equivalence of aca and ct, I show how to define the truth predicate of ct in aca. Of course, this task is very akin to Tarski’s original problem of providing a definition of truth in a higher-order language. Here I am working in a setting that differs from Tarski’s because I am using an arithmetical language and because aca provides only minimal resources for carrying out the definition of truth. In fact, the comparison between aca and ct can be seen as an attempt to pinpoint the minimal resources required for defining truth. As mentioned above, the truth definition in aca can be found in the literature, for instance in Takeuti (1987, pp. 183ff). I will follow Takeuti’s proof to some extent. The plan for the proof is as follows: Using an instance of the schema of arithmetical comprehension, I prove that there is a set that contains exactly the true atomic arithmetical sentences, that is, all sentences s = t where s and t are closed terms coinciding in their values. Then I show that, using the set

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of all true arithmetical sentences of at most length n as a parameter, one can define the set of all true arithmetical sentences of maximal length n + 1 by an arithmetical definition. So, given that the set of all true sentences of length n exists, the set of all true sentences of length n+1 also exists according to the schema of arithmetical comprehension. The formula Tset(X, n), expressing that X is the set of true sentences of length at most n, is defined in the following way: ∀x (x ∈ X → Sent(x) ∧ lh(x)≤n)∧
∀x ∀s ∀t x = (s=. t) → (x ∈ X ↔ s◦ = t ◦ ) ∧
∀x ∀y x = ¬. y ∧ Sent(x) ∧ lh(x)≤n → (x ∈ X ↔ y ∈ / X) ∧
∀x ∀y ∀z x = (y∧. z) ∧ Sent(x) ∧ lh(x)≤n → (x ∈ X ↔ y ∈ X ∧ z ∈ X) ∧
∀x ∀y ∀z x = (y∨. z) ∧ Sent(x) ∧ lh(x)≤n → (x ∈ X ↔ y ∈ X ∨ z ∈ X) ∧
∀x ∀v ∀y x = ∀. vy ∧ Sent(x) ∧ lh(x)≤n → (x ∈ X ↔ ∀t (y(t/v)) ∈ X) ∧
∀x ∀v ∀y x = ∃. vy ∧ Sent(x) ∧ lh(x)≤n → (x ∈ X ↔ ∃t (y(t/v)) ∈ X) Here n is an ordinary first-order variable. The function expression lh(x) represents the function that yields, when applied to a formula, the number of logical symbols, that is, the number of quantifier symbols and connectives, in the formula. In the next lemma I prove that the truth sets, that is, the sets satisfying the formula Tset(X, n) for some n do not disagree on sentences of maximal length n. lemma 8.45. The following is a theorem of aca: Tset(X, n) ∧ Tset(Y, k) ∧ n≤k → ∀x lh(x)≤n → (x ∈ X ↔ y ∈ Y )




proof. I argue informally in aca. Let n and k be fixed and assume n≤k and lh(x)≤n. Then the claim can be proved by induction on lh(x). If x is atomic, then, by definition of Tset(X, y), the sentence x is in X if and only if it is a true atomic sentences; the same applies to Y . Therefore X and Y contain exactly the same arithmetical sentences of length 0. The induction step is proved by distinguishing five cases: x can be a negated sentence, a conjunction or disjunction, or a universally or existentially quantified sentence. If, for instance, x ∈ X is the negated sentence ¬y, then, by definition of Tset(X, y), y is not in X, and, by induction hypothesis, not in y; whence x ∈ Y . The proof that x ∈ X if x ∈ Y is symmetric. The other cases can be proved in a similar fashion. a

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In the special case n = k, it follows from this lemma that for every n ∈ ω the truth set for n is unique up to extensionality. Adding the extensionality axiom to aca would allow me to prove full uniqueness, but I prefer the formulation of aca without an axiom of extensionality to keep its axiomatizations as simple as possible. corollary 8.46. The following sentence is a theorem of aca: ∀n ∀x ∀X ∀Y Tset(X, n) ∧ Tset(Y, n) → (x ∈ X ↔ x ∈ Y )




As mentioned above, I will prove the existence of truth sets, that is, of the sets satisfying Tset(X, n) for every n inductively. The case n = 0 is covered by the following lemma: lemma 8.47. aca ` ∃X Tset(X, 0) proof. The following formula is an instance of the arithmetical comprehension schema of aca: (8.16)

∃X ∀x x ∈ X ↔ ∃s ∃t (x = (s=. t) ∧ s◦ = t◦ )




In Peano arithmetic one can prove that atomic arithmetical sentences are identity statements: pa ` Sent(x) ∧ lh(x) ≤ 0 ↔ ∃s ∃t x = (s=. t) This yields the claim together with (8.16).

a

Now I prove the induction step: I show that if there is a truth set for sentences of maximal length n, then there is a truth set for sentences of maximal length n+1. For the induction step an instance of the comprehension schema of aca with a free second-order parameter is required. That the use of such an instance of the comprehension schema is indispensable for the proof of the lemma below follows from the observation that the system acapf , that is aca without second-order parameters, is conservative over pa (Corollary 8.44), because ct is not conservative over pa by Corollary 8.40, and the proof of Theorem 8.50 below does not require further instances of the comprehension schema with second-order parameters. lemma 8.48. aca ` ∃X Tset(X, n) → ∃X Tset(X, Sn) In the lemma S is the successor symbol.

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proof. The following is an instance of the arithmetical comprehension axiom. The right-hand side of the equivalence contains free occurrences of the second-order parameter Y .  ∃X ∀x x ∈ X ↔  x ∈ Y ∨ Sent(x) ∧ lh(x) = Sn ∧ ∃y (x = ¬. y ∧ Sent(y) ∧ y ∈ / Y )∨ (8.17)

∃y ∃z (x = (y∧. z) ∧ (y ∈ Y ∧ y ∈ Y ))∨ ∃y ∃z (x = (y∨. z) ∧ (y ∈ Y ∨ y ∈ Y ))∨ ∃y∃v(x = ∀. vy ∧ ∀t y(t/v) ∈ Y )∨ 
 ∃y∃v(x = ∃. vy ∧ ∃t y(t/v) ∈ Y )

From the assumption Tset(Y, n) that Y is a truth set for sentences of length smaller than n+1, I conclude, using (8.17), that there is a truth set for sentences of length at most n + 1: aca ` ∃X Tset(X, n) → ∃X Tset(X, Sn)

a

Combining Lemmata 8.47 and 8.48 and applying an instance of the induction schema of aca, one can prove the following main lemma: lemma 8.49. aca ` ∀n ∃X Tset(X, n) theorem 8.50. The truth predicate Tx of ct is defined in aca by the formula
∃Y Tset(Y, lh(x)) ∧ x ∈ Y . proof. It will be shown that the definition satisfies the axioms of ct. To prove Axiom ct1
∀s ∀t T(s=. t) ↔ s◦ = t◦ with the truth predicate substituted by its definition in aca, I observe that the following claim follows directly from the definition of Tset(X, n): aca ` ∀x ∀X ∀n Tset(X, n) → ∀s ∀t (x = (s=. t) → (x ∈ X ↔ s◦ = t ◦ ))




From this, using Lemma 8.47, one can easily obtain the desired translation of ct1: aca ` ∃X (Tset(X, lh(s=. t)) ∧ x ∈ X) ↔ s◦ = t◦

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In the case of the negation Axiom ct2 I reason in aca in the following way:  ∀x Sent(x) →
 ∃X(Tset(X, lh(¬. x)) ∧ ¬. x ∈ X) ↔ ∃X(Tset(X, lh(¬. x)) ∧ x ∈ / X)
 ↔ ∃X(Tset(X, lh(x)) ∧ x ∈ / X)
 ↔ ¬ ∃X(Tset(X, lh(x)) ∧ x ∈ X) The first equivalence follows from the definition of Tset(X, n), the second from Lemmata 8.49 and 8.45, and the last from Corollary 8.46. The last line is Axiom ct2 with the truth predicate of ct replaced with the truth predicate that is defined in aca: 
 ∀x Sent(x) → ∃X (Tset(X, lh(¬. x))∧¬. x ∈ X) ↔ ¬∃X (Tset(X, lh(x))∧x ∈ X) From the remaining axioms I consider only Axiom ct5. For this axiom I reason in aca in the following fashion:  ∀x ∀v Sent(∀. vx) → ∃X(Tset(X, lh(∀. vx)) ∧ ∀. vx ∈ X) ↔ ∃X (Tset(X, lh(∀. vx)) ∧ ∀t x(t/v) ∈ X)




↔ ∃X(Tset(X, lh(x(0/v))) ∧ ∀t x(t/v) ∈ X)




↔ ∀t ∃X(Tset(X, lh(x(t/v))) ∧ x(t/v) ∈ X)




As before, Lemmata 8.49 and 8.45 and Corollary 8.46 are used to show these equivalences. The last line is Axiom ct5 with the primitive truth predicate replaced by its definition in aca. a Further and more detailed results are known about systems of compositional truth between ct↾ and ct, that is subsystems of ct that properly contain ct↾. In particular, Kotlarski and Ratajczyk (1990a,b) have investigated versions of ct with induction restricted to formulae of restricted complexity (but with the truth predicate) and obtained detailed results on their strength. Fischer (2009) considers also other restrictions of the induction schema of ct to define further systems between ct↾ and ct and analyses their strength. He also proves further results about connections between subsystems of ct and subsystems of aca.

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classical compositional truth 8.7 Positive truth

The truth-theoretic axioms of the systems ct↾ and ct can be seen as the result of turning the clauses in the ‘Tarskian’ definition of truth in Lemma 8.2 into axioms. There is another way of defining the set of true L sentences that is slightly different from the usual definition, that is, Definition 8.1 or Lemma 8.2. The set can also be defined by a positive inductive definition. A positive inductive definition of a set S takes the form (8.18)

n ∈ S if and only if ζ(n, S),

where ζ(n, S) is a formula in which the predicate expression ∈ S occurs only positively, that is, only in the scope of an even number of negation symbols (assuming that conjunction and disjunction are the only other connectives used).4 In most cases of positive inductive definitions in this book, the formula ζ(n, S) will be in the language L of arithmetic with the exception of the unary predicate expression ∈ S. So ζ(n, S) will be in the language L of arithmetic augmented with the additional predicate expression ∈ S, in which all atomic subformulae of the form u ∈ S only occur in the scope of an even number of (occurrences of) negation symbols. This extension of L is considered to be a sublanguage of our informal language. So I can write ζ(n, S) to state that ζ(n, S) holds (for n and S) rather than saying that ζ(n, S) holds in the standard model of arithmetic under the usual interpretations of n and S). As an example of a positive inductive definition, I state the definition of L-truth as a positive inductive definition: definition 8.51. The set of true sentence of L is the smallest set S ⊆ ω satisfying the following condition: n ∈ S if and only if (i) there are closed terms s and t such that n is s = t and the value of s is identical to the value of t; or 4 Positive inductive definitions have nice and well understood properties (see Moschovakis 1974) and definitions of truth have been used extensively in the theory of positive inductive definitions. Moreover, positive inductive definitions have fed into the theory of type-free truth, in particular into Kripke’s (1975) theory of truth (see also McGee 1991). The reader is referred to Moschovakis’ book for more details, as I will present only some basic notation in this section.

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(ii) there are closed terms s and t such that n is ¬s = t and the value of s is different from the value of t; or (iii) there is an L-sentence ϕ such that n is ¬¬ϕ and ϕ ∈ S; or (iv) there are L-sentences ϕ and ψ such that n is ϕ ∧ ψ and ϕ ∈ S and ψ ∈ S; or (v) there are L-sentences ϕ and ψ such that n is ¬(ϕ ∧ χ) and (¬ϕ ∈ S or ¬ψ ∈ S); or (vi) there are L-sentences ϕ and ψ such that n is ϕ ∨ ψ and (ϕ ∈ S or ψ ∈ S); or (vii) there are L-sentences ϕ and ψ such that n is ¬(ϕ ∨ ψ) and ¬ϕ ∈ S and ¬ψ ∈ S; or (viii) there is an L-sentence ∀vχ such that n is ∀v χ and for all closed terms t, χ(t/v) ∈ S; or (ix) there is an L-sentence ∀vχ such that n is ¬∀vχ and for some closed term t, ¬χ(t/v) ∈ S; or (x) there is an L-sentence ∃vχ such that n is ∃vχ and for some closed term t, χ(t/v) ∈ S; or (xi) there is an L-sentence ∃vχ such that n is ¬∃vχ and for all closed terms t, ¬χ(t/v) ∈ S. The usual ‘Tarskian’ definition, that is, Definition 8.1, is not positive because of the clause for negation, that is, clause (ii): a sentence ¬ϕ is defined to be in S if and only if ϕ itself is not an element of S. This means that ∈ S has a negative occurrence in the definiens. In the case of the Tarskian definition, the existence of a set S satisfying the defining equivalence was shown by induction on the complexity of sentences. In the case of the positive inductive definition the existence of a set S satisfying the definitional equivalence of the form 8.18 follows from general abstract considerations. Assume again that ζ(x, S) is a formula in the language of arithmetic possibly containing in addition positive occurrences of atomic formulae u ∈ S (where u is some arbitrary term of L). Then the following implication is easily established for all n ∈ ω: (8.19)

If S1 ⊆ S2 , then ζ(n, S1 ) implies ζ(n, S2 )

Here it is crucial that the predicate expression ∈ S occurs only positively in ζ(n, S). Now an operator Γζ on sets of numbers is defined for all S ⊆ ω: Γζ (S) := {n : ζ(n, S)}

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Using (8.19), it is easily shown that Γζ is monotone: If S1 ⊆ S2 , then Γζ (S1 ) ⊆ Γζ (S2 )

(monotonicity) β

The sets Iζ are defined by transfinite recursion in the following way: β

Iζ := Γζ

[

Iζα




α<β

In particular, Iζ0 = Γ(Ø) holds. It follows from the monotonicity of Γζ that β

Iζα ⊆ Iζ if α ≤ β. Hence, for cardinality reasons, there must be a smallest ordinal γ γ+1 Iζ = Iζ . γ

For this ordinal, the set Iζ is identical to Iζ :=

[

Iζα ,

α∈On

where On is the class of all ordinals. As I am considering only sets of natural numbers, the set of countable ordinals will suffice. It follows that Iζ is a fixed point of Γζ in the sense that Γζ (Iζ ) = Iζ and that Iζ is the smallest fixed point (see Moschovakis 1974 for full proofs). It is also a fixed point of the formula ζ(n, S), in the sense that the following equivalence obtains for all n ∈ ω: n ∈ Iζ iff ζ(n, Iζ ) These general considerations can be applied to the positive inductive definition of L-truth (Definition 8.51). It follows that there is a set S satisfying the conditions (i)–(xi) in Definition 8.51. To show that the positive inductive definition, that is, Definition 8.51, and the Tarskian definition, that is, Definition 8.1, define the same set, induction on the complexity of sentences is needed. That induction involving truth is needed can be demonstrated by formalizing both definitions in axiomatic systems with arithmetical induction only and then showing that the two systems are not equivalent. The systems ct↾ and pt↾ are the axiomatic counterparts of Definition 8.1 and 8.51, respectively, and both lack the induction axiom with the truth predicate. Therefore the metatheoretic proof of the equivalence of Definitions 8.1 and 8.51 cannot be formalized in them. In particular, as will be shown below, the axioms of ct↾ cannot be proved in pt↾.

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By substituting the expression ∈ S with the truth predicate in the equivalence in Definition 8.51, one could turn the positive inductive definition of truth into a single axiom. To avoid an ungainly long axiom, I reformulate Definition 8.51 in the same way I reformulated Definition 8.1 in Lemma 8.2. This allows me to give an axiomatization in which one long axiom is broken down into eleven axioms. The result of formalizing Definition 8.51 directly yields an equivalent theory. lemma 8.52. The set of true L-sentences is the smallest set of satisfying the following conditions, assuming that s and t are closed terms and that ϕ, ψ, and ∀vχ are sentences of L: (i) s = t ∈ S iff s and t coincide in their values. (ii) s 6= t ∈ S iff s and t differ in their values. (iii) ¬¬ϕ ∈ S iff ϕ ∈ S. (iv) ϕ ∧ ψ ∈ S iff ϕ ∈ S and ψ ∈ S. (v) ¬(ϕ ∧ ψ) ∈ S iff ¬ϕ ∈ S or ¬ψ ∈ S. (vi) ϕ ∨ ψ ∈ S iff ϕ ∈ S or ψ ∈ S. (vii) ¬(ϕ ∨ ψ) ∈ S iff ¬ϕ ∈ S and ¬ψ ∈ S. (viii) ∀x χ ∈ S iff for all n, χ(n) ∈ S. (ix) ¬∀x χ ∈ S iff there is an n with ¬χ(n) ∈ S. (x) ∃x χ ∈ S iff there is an n with χ(n) ∈ S. (xi) ¬∃x χ ∈ S iff for there is no n with ¬χ(n) ∈ S. (xii) S contains only L-sentences. It should be obvious that the lemma is an inessential reformulation of the original positive definition. In the theory pt↾ of positive truth the clauses of the lemma are taken as axioms. definition 8.53 (pt↾). The system pt↾ is given by all axioms of pa and the following axioms:
pt1 ∀x ∀y Sent(x=. y) → (T(x=. y) ↔ x◦ = y◦ )
pt2 ∀x ∀y Sent(x=. y) → (T(¬. x=. y) ↔ x◦ 6= y◦ )
pt3 ∀x Sent(x) → (T(¬. ¬. x) ↔ Tx)
pt4 ∀x ∀y Sent(x∧. y) → (T(x∧. y) ↔ T(x) ∧ T(y))
pt5 ∀x ∀y Sent(x∧. y) → (T¬. (x∧. y) ↔ T(¬. x) ∨ T(¬. y))
pt6 ∀x ∀y Sent(x∨. y) → (T(x∨. y) ↔ T(x) ∨ T(y))
pt7 ∀x ∀y Sent(x∨. y) → (T¬. (x∨. y) ↔ T(¬. x) ∧ T(¬. y))

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pt8 ∀v ∀x Sent(∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v)))




pt9 ∀v ∀x Sent(∀. vx) → (T(¬. ∀. vx) ↔ ∃t T(¬. x(t/v)))
pt10 ∀v ∀x Sent(∃. vx) → (T(∃. vx) ↔ ∃t T(x(t/v)))




pt11 ∀v ∀x Sent(∀. vx) → (T(¬. ∃. vx) ↔ ∀t T(¬. x(t/v)))




As in ct I do not include an axiom stating that only sentences of L are true. By adding all instances of the induction schema in the language LT with the truth predicate, the system pt is obtained. In the presence of full induction, one can formalize the proof of equivalence of the two semantic Definitions 8.1 and 8.51: lemma 8.54. The systems ct and pt have the same theorems. I will not give the proof here. The truth-theoretic axioms of pt are easily derived in ct. To prove the negation axiom of ct, that is Axiom ct2, in pt, induction on the complexity of sentences is used. Without using induction for an LT -formula, Axiom ct2 cannot be derived. This will be shown in the rest of this section. When the systems with just arithmetical induction are considered, however, the picture changes: ct↾ and pt↾ do not have the same theorems. lemma 8.55. Every model of Peano arithmetic can be expanded to a model of pt↾. proof. Let 𝔐 be some model of Peano arithmetic. I will now define a subset S of the domain of 𝔐 as a suitable extension of the truth predicate such that (𝔐, S) ⊨ pt↾ by adapting Definition 8.51 to the (possibly nonstandard) model 𝔐. The binary function expression ∧. is interpreted in 𝔐 by some binary function on the domain of 𝔐; I write ∧𝔐 for this function and similarly for other function expressions of the language of arithmetic. Similarly, Sent𝔐 (a) says that a is in the extension of the predicate expression Sent(x) in 𝔐, and a ∈ ClTerm says that a is in the extension of the predicate expression ClTerm(x). So S may then be defined as follows:  An element a in the domain of 𝔐 is in S if and only if a ∈ Sent𝔐 and there are b, c such that a = (b = 𝔐 c) and b◦𝔐 = c◦𝔐 or there are b, c such that a = (¬. 𝔐 b = 𝔐 c) and b◦𝔐 6= c◦𝔐 or there is a b such that a = (¬𝔐 ¬𝔐 b) and b ∈ S or there are b, c such that a = (b ∧𝔐 c) and b, c ∈ S or there are b, c such that a = ¬𝔐 (b ∧𝔐 c) and (¬𝔐 b ∈ S or ¬𝔐 c ∈ S)

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or there are b, c such that a = (b ∨𝔐 c) and (b ∈ S or c ∈ S) or there are b, c such that a = ¬𝔐 (b ∨𝔐 c) and ¬𝔐 b ∈ S and ¬𝔐 c ∈ S or there are b, c such that a = ∀𝔐 bc and for all d with d ∈ ClTerm𝔐 , c(d/b)𝔐 ∈ S or there are b, c such that a = ¬𝔐 ∀𝔐 bc and for some d with d ∈ ClTerm𝔐 , ¬𝔐 c(t/b)𝔐 ∈ S or there are b, c such that a = ∃𝔐 bc and for some d with d ∈ ClTerm𝔐 , c(d/b)𝔐 ∈ S or there are b, c such that a = ¬𝔐 ∃𝔐 bc and for no d with
 𝔐 𝔐 𝔐 d ∈ ClTerm , ¬ c(t/b) ∈ S This is a positive inductive definition: the expression ∈ S occurs only positively on the right hand side of the equivalence. The only difference to the case considered above is that this definition is applied to a nonstandard model rather than the standard model. The existence of a fixed point S of the definition can be proved as in the standard cases. The model (𝔐, S), where S is some fixed point of the above definition, is a model of pt↾, as can be easily seen by comparing the truth-theoretic axioms of pt↾ and the clauses of the above definition. a A fixed point S of the definition in the proof of Lemma 8.55 may fail to satisfy the following clause for negation: for all a in the domain of 𝔐 either a ∈ S or (¬𝔐 a) ∈ S This follows from Lachlan’s theorem, of course, as Lemma 8.55 cannot be proved for ct↾ in place of pt↾: not every model of pa can be expanded to a model of ct↾. In particular, in the minimal fixed point of the definition in the proof of Lemma 8.55 there will be no sentence of nonstandard length, that is, with a nonstandard number of connectives or quantifiers. If the model is nonstandard, the negation axiom of ct↾, that is, ∀x Sent(x) → (T(¬. x) ↔ ¬Tx)




will fail in (𝔐, S) for the minimal fixed point S of that definition. corollary 8.56. pt↾ is a proper subtheory of ct↾.

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proof. Deriving the axioms of pt↾ in ct↾ is not difficult. ct↾ and pt↾ are not identical theories because every model of pa can be expanded to a model of pt↾ but not to a model of ct↾. a The proof of Lemma 8.55 basically is an application of the positive inductive definition of truth to a nonstandard model 𝔐. This yields an extension of 𝔐 that satisfies the axioms of pt↾. The Tarskian Definition 8.1 of truth, in contrast, cannot be applied to a nonstandard model, at least not without the elaborate tricks employed by Kotlarski et al. (1981). The proof that Definition 8.1 has a fixed point relied on an induction on the complexity of sentences, but for a ∈ Sent𝔐 the complexity might be a nonstandard number, and as nonstandard numbers are not wellfounded the induction cannot be performed.

9 Hierarchies

The typed theories of the ilk of tb, ct↾, or ct have been formulated with the language L as their base language. I have hinted at how one might formulate analogous theories for other base theories. In particular, if the base language is L with an additional predicate symbol, one can easily revise the typed theories to obtain truth theories for the expanded language. In the case of disquotational theories one can simply allow new instances of the disquotation or T-schema. In the case of the compositional theories one will have to add a new axiom analogous to the Axiom ct1 for atomic sentences of the form s = t where s and t are closed terms. If the base language contains an additional unary predicate symbol P , the following axiom can be used (see p. 66): ∀t TP. t ↔ P t◦




The axiom is a particular uniform disquotation sentence. Now one can consider the truth predicate T of the language LT as such an additional predicate and formulate typed theories with LT as their base language. If the truth predicate of LT is employed as the truth predicate for formulating the new theory of truth and the sentence ∀t TT. t ↔ Tt◦




is used as an axiom together with the other compositional truth axioms of ct↾ generalized to the language LT , the resulting system is inconsistent (see Lemma 14.3). However, one may instead use a new truth predicate T1 and add the axiom (9.1)


∀t T1 T. t ↔ Tt◦ .

The old truth predicate T is treated here merely as an arbitrary predicate symbol of the base language LT . One can then add compositional axioms for both truth predicates with the proviso that the base language for T is only L, so that T is not applied to sentences containing the higher-level truth predicate T1 , while T1 is axiomatized as the truth predicate for the language LT .

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For the typed theories tb, ct, and so on, a coding of all sentences and other expressions of L in the natural numbers was required. For the new theory that treats LT as the base language a natural coding of LT is required. If the formation of a new theory of truth is to be iterated, a new truth predicate is required at every step. The truth predicates can be indexed by natural numbers T0 , T1 , T2 ,. . . For the language containing all the truth predicates indexed with natural numbers one can again state a theory of truth with still another truth predicate Tω . Continuing the addition of truth predicates, a further truth predicate Tω+1 is added, and so on for further ordinal numbers. All the languages and their truth predicates with ordinal numbers as indices must be coded in the natural numbers. Otherwise one would not be able to formulate the corresponding theories of truth. There are obvious limits to this coding: there are uncountably many ordinals but only countably many natural numbers to encode the truth predicates. Countable sets of ordinal numbers can be put into a one-to-one correspondence with sets of natural numbers, but not any such correspondence can be used as a coding of the ordinals, and for certain countable segments of the ordinals there are no suitable codings at all. In particular, the set of natural numbers coding a certain initial segment of the ordinals in some particular way might fail to be definable in the language of arithmetic;1 the ordering relation on the codes of the ordinals might not be describable in the language of arithmetic; and even if the ordering can be defined in the language of arithmetic, the deductive power of Peano arithmetic might not suffice for proving facts about this ordering relation on the codes. The set of all ordinals for which the ordering relation can be expressed by a recursive or arithmetical ordering on their codes is a natural halting point and leads to a hierarchy of truth predicates with certain mathematically important properties. In Halbach (1995) and (1997) I considered models of such hierarchies of truth predicates and their relation to hierarchies in recursion theory and to Kripke’s (1975) theory of truth. Here I will concentrate on axiomatic approaches. For them much smaller ordinal notation systems seem appropriate as one will want to prove certain facts about the ordering of ordinal numbers in Peano arithmetic or the truth theories. If notation systems for very large initial segments of the ordinal numbers are employed, the strength of the truth-theoretic systems might arise from the ordinal notation systems and not the truth-theoretic content. 1 See, for instance, Rogers (1967) for more information on ordinal notation systems

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Even though I do not think that hierarchical solutions provide an attractive approach to the analysis of truth, they prove valuable in the technical analysis of various theories of truth. In particular, one can measure the strength of type-free theories of truth by investigating how they compare to hierarchical theories of truth. For instance, one can ask whether a type-free theory of truth allows one to define hierarchical truth predicates up to some ordinal level such that all the truth predicates satisfy the compositional axioms. Instead of indexing the truth predicates with ordinal numbers or their codes one can also use other orderings. The truth predicates need not be wellordered. Perhaps these illfounded sequences of languages and their truth predicates do not deserve the tag hierarchy anymore, but they can yield some insights into certain phenomena related to the paradoxes. In Section 9.2 I will consider an illfounded hierarchy indexed by the natural numbers under their reverse-ordering, that is, the finite ordinals in reverse order. But first I will start with the more obvious approach: a hierarchy along an initial segment of the ordinal numbers. 9.1 Tarski’s hierarchy axiomatized I have tried to avoid the use of ordinal analysis and proof-theoretic ordinals so far. Some reasons for this are given at the end of Chapter 6. As pointed out in this chapter, proof-theoretic ordinals are highly sensitive to the base theory and I am trying to obtain results that can be generalized to base theories other than Peano arithmetic. When it comes to transfinite hierarchies of truth theories I do not see any way of avoiding these ordinals. Consequently most of the results on transfinite hierarchies of truth predicates depend very much on the use of Peano arithmetic as base theory. In a set-theoretic framework, for instance, transfinite hierarchies of truth theories might be dealt with in a completely different way. In what follows, I will assume some familiarity with the ordinal numbers. Usually ordinal numbers, or ordinals for short, are introduced within a settheoretic framework as certain sets. Operations like addition and exponentiation are then defined on these sets in set-theoretic terms (see, for instance, Lévy 1979). More information on ordinals used in predicative proof theory can be found in various textbooks on proof theory, such as Schütte (1977), Pohlers (1998, 2009). The class of ordinals is denoted by On. In particular, I presuppose the notion of ordinal exponentiation. The binary Veblen function ϕ is defined by transfinite induction. For each α ∈ On, ϕα is a function from On into

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On, which are defined in the following way for all ordinals α, β and limit ordinals λ: ϕ0 β := ωβ ϕα+1 β := the βth ordinal ξ with ϕα ξ = ξ ϕλ β := the βth ordinal ξ such that ϕα ξ = ξ for all α < λ The ordinal β is defined as ϕ1 β. Hence 0 is ϕ1 0 and 0 is the limit of all the ordinals ω ωω 1, ω, ωω, ωω, ωω, . . . The Feferman–Schütte ordinal Γ0 is the least ordinal γ greater than 0 such that α, β < γ implies ϕα β < Γ0 . Here I do not prove that the Veblen function and the above mentioned ordinals are well defined; details can be found in the texts mentioned above. Every ordinal smaller than Γ0 can be designated by a unique term. The terms, in turn, can be coded in the natural numbers and I take the codes of these terms as the codes of the ordinals themselves. The set of all (codes of ordinals) less than Γ0 can be represented in Peano arithmetic as well as their ordering. I will write ∀α (also with other Greek letters) to express quantification over all codes of ordinals in Peano arithmetic. Moreover α is the numeral of the code of the ordinal α. Truth predicates indexed with ordinals smaller than Γ0 can then be understood as ordered pairs with the symbol T as first component and the code for the ordinal as second component. In this way the language of arithmetic expanded by all truth predicates indexed with ordinals smaller than Γ0 can be coded in the natural numbers and the usual syntactical operations and properties can be appropriately expressed in Peano arithmetic. I define the languages of the hierarchy and the associated theories up to any ordinal level smaller than Γ0 . For the moment being, I do not want to suggest that Γ0 actually is the or a natural halting point for the hierarchy. But it follows from considerations below that the natural halting points will occur at level Γ0 or below. So defining all levels of the Tarskian hierarchy will suffice for the purposes of this book. Now the languages of the hierarchy are defined in the following way: definition 9.1. For an ordinal γ < Γ0 the language L<γ is the language of Peano arithmetic expanded by all truth predicates Tβ for β < γ; Lγ is the language L<γ+1 , so it also contains in addition to all the symbols of L<γ the symbol Tγ .

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The set of sentences of L<γ is naturally represented in Peano arithmetic by the formula Sent<γ (x). Similarly, Sentγ (x) represents the set of all Lγ sentences. In some contexts, in particular, in Axiom rt8β below, I will treat γ in the formula Sent<γ (x) as a bindable variable. So Sent<γ (x) should be seen as a formula with two free variables representing the relation obtaining between an index of an ordinal γ and a formula if and only if the formula is a formula of the language L<γ . Hence for fixed γ, I should write Sent<γ (x) to indicate that the numeral of the code of the ordinal is substituted for the free variable standing for the level index. To keep the notation tidier I will not overline the level index as this should not cause confusion. Even when the variable is bound from outside, as in Axiom rt8β , I will not use the usual overdotting. When it comes to the axioms for the truth predicates, one has a choice between the different typed theories of truth; one could iterate the theories tb↾, tb, utb↾, utb, pt↾, ct↾, or ct. It is plausible to assume (although the proofs may be somewhat tricky) that iterating any of the conservative theories – that is, any of the theories other than ct – again yields a conservative extension of Peano arithmetic, even though their truth predicates are not definable in Peano arithmetic and some subtle issues arise about the definability of the truth predicates. I will focus on the only theory that yields a non-conservative extension of Peano arithmetic, namely ct. Iterations of this theory may be called theories of ramified truth and are defined in the following way: definition 9.2. For γ ≤ Γ0 the theory rt<γ is given by all the axioms of Peano arithmetic including all induction axioms in the language L<γ and the following axioms for all α < β < γ: rt1β ∀s ∀t (Tβ (s=. t) ↔ s◦ = t◦ ) rt2β ∀x Sent<β (x) → (Tβ (¬. x) ↔ ¬Tβ x)




rt3β ∀x ∀y Sent<β (x∧. y) → (Tβ (x∧. y) ↔ Tβ (x) ∧ Tβ (y))





rt4β ∀x ∀y Sent<β (x∨. y) → (Tβ (x∨. y) ↔ Tβ (x) ∨ Tβ (y))
rt5β ∀v ∀x Sent<β (∀. vx) → (Tβ (∀. vx) ↔ ∀t Tβ (x(t/v)))
rt6β ∀v ∀x Sent<β (∃. vx) → (Tβ (∃. vx) ↔ ∃t Tβ (x(t/v)))
rt7β ∀t Sent<α (t◦ ) → (Tβ (T. α t) ↔ Tα t◦ )
rt8β ∀t ∀δ ≺ β Sent<δ (t◦ ) → (Tβ (T. δ t) ↔ Tβ t◦ ) In Axiom rt8β the expression ≺ represents the relation on natural numbers that obtains between n and k if and only if n is the code for an ordinal α, k

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the code for β, and α < β. The system rtβ is rt<(β+1) , that is, the system with all axioms of Peano arithmetic in the language Lβ containing all the truth predicates up to and including Tβ and the axioms rt1β –rt8β above. The first six Axioms rt1β –rt6β are straightforward generalizations of the axioms of ct to the expanded language. Also Axiom rt7β can be seen as a generalization of the ct-axiom ct1 that governs the truth of atomic sentences. Axiom rt7β deals with sentences with the additional unary predicate symbols in L<β – the truth predicates Tα , for α < β – in the manner proposed in Axiom (9.1), that is, the axiom
∀t T1 T. t ↔ Tt◦ . In contrast to (9.1), the universal quantifier ranging over term is restricted to values that are sentences in the language L<α in Axiom rt7β . This is in line with the policy of not stipulating anything about the truth and falsity of sentences that are not in the appropriate language. In Axiom rt7β the code α cannot be quantified over as it is only used as an index as in the formula Tα t◦ , not as an argument to which the truth predicate is applied. The last Axiom rt8β does not have an analogue in the uniterated version of ct: it expresses the cumulativity of the truth predicates: the truth predicates Tα and Tβ agree on sentences of the language Lα if α < β. If β is a limit ordinal, then Axiom rt8β expresses that Tβ is the limit of all truth predicates of lower levels: the set of true sentences of Lβ is the union of all true sentences of lower levels. Following and slightly modifying proposals by Feferman (1991), Fujimoto (2010a) mentioned a hierarchical theory of truth with one binary truth predicate that applies to codes of ordinals and sentences. On this approach Tx y has the two free variables x and y and quantification over levels becomes possible. So, for instance, one can form the formula ∃x Tx y expressing that y is true at some level. In forthcoming publications Fujimoto proves various results about this axiomatization of hierarchical truth. For any ordinal γ smaller or equal to Γ0 a theory rt<γ is defined. But this does not imply that all theories up to and including rt<γ should be accepted. So the question arises whether there is a natural halting point in the hierarchy of the theories rt<γ . In Section 22.2 I will discuss which iterations are acceptable. For instance, rt<0 looks like a natural candidate, as for any ordinal smaller than 0 transfinite induction is provable in Peano arithmetic but not for higher ordinals. I postpone the discussion of which levels of the hierarchy should be accepted until Section 22.2, when more concepts and results that are useful for the discussion will have been introduced.

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In Section 8.6 it was shown that the truth predicate of the theory ct is definable in the theory aca of arithmetical comprehension, and that there is an interpretation of aca in ct. This result can be extended to iterations of aca and ct. Iterations of the theory of elementary comprehension were studied by Feferman (1964) and Schütte (1977). The resulting system of ramified analysis up to γ is designated by ra<γ . The match between elementary comprehension and compositional truth can now be extended to their iterations: proposition 9.3. The truth predicate of rt<γ is definably in ra<γ . Both theories are mutually interpretable. Feferman (1991) hints at a proof of this result, though for a system of iterated truth that is different from the formulation chosen here. As I am here not so much interested in the connection between truth theories and subsystems of second-order arithmetic, I will not give a proof of Proposition 9.3. The systems rt<γ of ramified truth will be used extensively in what follows. In particular, I will use them to measure the strength of type-free theories of truth. This will be done by examining how many levels of ramified truth Tβ can be defined in the type-free theories. 9.2 Illfounded hierarchies Kripke (1975, p. 697) mentioned the possibility of illfounded hierarchies of languages. Subsequently McCarthy (1988) and Visser (1989) investigated such hierarchies. I will consider a theory formulated in the language L<ω , which has truth predicates indexed by finite ordinals. That is, L<ω has, in addition to the expressions of the language of Peano arithmetic all truth predicates Tn for all n ∈ ω. In contrast to the standard wellfounded hierarchal theories the order of the languages is now reversed: the truth predicate T0 is now the topmost truth predicate; it applies to all sentences of L<ω containing truth predicates with index n > 0. Obviously there is no first language with a truth predicate that only applies to arithmetical sentences, as any truth predicate Tn applies to all sentences containing only truth predicates Ti where i > k. definition 9.4. The language L>n is the language L expanded by all truth predicates Ti for finite ordinals i > n. In contrast to the usual wellfounded hierarchies, it is not possible to define a standard model by recursion over the language level as one cannot carry out such a definition along an illfounded ordering. In fact it will be shown

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in what follows that there are no nice models for the illfounded hierarchy of languages. It is, however, possible to iterate the theory tb↾ of Tarski biconditionals – or any other hitherto discussed theory of typed truth – along an illfounded ordering. definition 9.5. The theory itb↾ is given by the axioms of Peano arithmetic and all axioms Tn ┌ϕ┐ ↔ ϕ, where n is a finite ordinal and ϕ is a sentence of L>n . The consistency of theories very similar to itb↾ was proved by McCarthy (1988). In fact, one can establish the consistency of the following stronger compositional theory. In what follows Sent>n (x) expresses that x is a sentence of the language L>n . The axioms of the system ict and its subsystem ict↾ with induction restricted to arithmetical formulae are modelled along the lines of the axioms of the system rt<ω of ramified truth. The system ict only differs from rt<ω by having its truth predicates in reverse order. definition 9.6. The theory ict↾ is given by all axioms of Peano arithmetic and the following axioms: ict1 ∀s ∀t (Tn (s=. t) ↔ s◦ = t◦ ) ict2 ∀x Sent>n (x) → (Tn (¬. x) ↔ ¬Tn x)




ict3 ∀x ∀y Sent>n (x∧. y) → (Tn (x∧. y) ↔ Tn (x) ∧ Tn (y))





ict4 ∀x ∀y Sent>n (x∨. y) → (Tn (x∨. y) ↔ Tn (x) ∨ Tn (y))
ict5 ∀v ∀x Sent>n (∀. vx) → (Tn (∀. vx) ↔ ∀t Tn (x(t/v)))
ict6 ∀v ∀x Sent>n (∃. vx) → (Tn (∃. vx) ↔ ∃t Tn (x(t/v)))
ict7 ∀t Sent>i (t◦ ) → (Tn (T. i t) ↔ Ti t◦ ) for i > n
ict8 ∀t ∀i > n Sent>i (t◦ ) → (Tn (T. i t) ↔ Tn t◦ ) That itb↾ is a subtheory of ict↾ can be established by generalizing the proof of the result that tb↾ is a subtheory of ct↾. proposition 9.7. The theory itb↾ is a subtheory of ict↾. The theory ict, which is ict↾ plus all induction axioms containing arbitrary truth predicates from L<ω , is consistent. Visser (1989, p. 637) sketches a proof

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of the consistency of a subtheory of ict. McCarthy (1988) observes that a theory akin to itb↾ is consistent. The proofs are applications of the compactness theorem. Axiom ict8, which is not contained in Visser’s theory, adds a complication to the proof. theorem 9.8. The system ict is consistent. proof. Let S be a finite subtheory of ict. Then S contains only finitely many truth predicates. Let Tn1 , Tn2 ,. . . ,Tnk be those truth predicates with their indices in descending order, that is, for any i, j < k with i < j the finite ordinal ni is larger than nj . I define a standard model (N, S0 , S1 , . . .) for the language L<ω . The structure N is the standard model of arithmetic; each set Sn is the extension of the truth predicate Tn in this model. For all i > n1 , that is, for all indices greater than all indices of truth predicates in S, the truth predicate Ti does not occur in S and I set Si = Ø. Now assume that all Si with i > n have been defined. If i−1 is not one of the indices n1 , n2 ,. . . ,nk set Si−1 = Si . If i is one of these indices, take Si−1 to be the set of all those sentences of L>i−1 that are true if the truth predicates Tj with j ≥ i are interpreted by the already defined extensions Sj , that is, define Si−1 in the following way: Si−1 = {ϕ ∈ L>i−1 : (N, Si , Si+1 , . . . ) ⊨ ϕ} These sets yield a model (N, S0 , S1 , . . .) for the language L<ω satisfying the theory S. I do not go through all the axioms, but Axiom ict8 might be worth mentioning: to verify the left-to-right direction of the axiom one exploits the fact that at most sentences with lower indices are added at each step of the recursive definition of the extensions; the right-to-left direction can be proved by observing that the extensions Sn1 , Sn2 ,. . . are increasing. As all finite subtheories of ict have a model, by compactness, ict also has a model, and therefore ict is consistent. a Of course, the theorem also implies that the subsystems itb↾, itb, and ict↾ are consistent. None of the models used in the consistency proof of ict provides a correct interpretation of all truth predicates of L<ω . In fact, there are no standard models of ict or ict↾ whatsoever. A model for a language extending the language of Peano arithmetic is said to be an ω-model if and only if the model restricted to the language L of arithmetic is the standard model N of arithmetic. The following proof essentially is due to Visser (1989):

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theorem 9.9. The theory itb↾ does not have an ω-model. proof. Using the parametrized version of Lemma 5.2, I obtain a recursive function h:  k, if k is of the form s = t for terms s and t      or if k is not a sentence of L<ω     ¬h(ϕ), if k = (¬ϕ)     h(ϕ) ∧ h(ψ), if k = (ϕ ∧ ψ) h(n, k) =  h(ϕ) ∨ h(ψ), if k = (ϕ ∨ ψ)      ∀x h(ϕ), if k = (∀x ϕ)     ∃x h(ϕ), if k = (∃x ϕ)    Ti+n h. (n, t), if k = (Ti t) for a term t The representation h. of the function h can be expressed by a suitable formula in the language L. Thus the function h(n, k) increases the index of all truth predicates in the formula k by n, even when they are only mentioned. Therefore, for any formula ϕ of L<ω , h(n+1, ϕ) is a formula in L>n . Therefore the following equivalences are theorems of itb↾ for all n > j: (9.2)

h(n, ϕ) ↔ Tj h. (n, ┌ϕ┐)

Moreover, the formalization of the following claim can be proved in Peano arithmetic for all n, l, k ∈ ω: (9.3)

h(n, h(l, k)) = h(n+l, k)

If a LT -formula ϕ is a theorem of Peano arithmetic, replacing subformulae of the form Tt in ϕ will yield another pa-theorem. Hence the following implications obtains: (9.4)

If pat ` ϕ, then pat ` h(n, ϕ)

Using the Diagonal lemma, one obtains a sentence γ such that the following equivalence is provable in Peano arithmetic: (9.5)

γ ↔ ¬ ∃x > 0 T0 h. (x, ┌γ┐)

Applying (9.4) to this equivalence and using the fact that h commutes with all connectives by definition, I conclude that the following is a theorem of Peano arithmetic: (9.6)

h(n, γ) ↔ h(n, ¬ ∃x > 0 T0 h. (x, ┌γ┐))

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~ of itb↾ where S ~ is a sequence of I assume that there is an ω-model (N, S) sets of numbers providing interpretations for the truth predicates L<ω . This ~ one can assumption will be shown to be contradictory. Reasoning in (N, S) proceed in the following way: ~ ⊨ h(n, γ) → h(n, ¬ ∃x > 0 T0 h. (x, ┌γ┐)) (N, S) → ¬ ∃x > 0 Tn h. (x+n, ┌γ┐)

(9.6) (9.3) and def. of h

→ ¬ Tn h. (1+n, ┌γ┐) → ¬ Tn ┌¬ ∃x > 0 Tn+1 h. (1+n+x, ┌γ┐)┐ (9.3) and def. of h → ∃x > 0 Tn+1 h. (1+n+x, ┌γ┐)

itb↾

→ Tn+1 h. (1+n+i, ┌γ┐)

for some i > 0 ~ is standard as (N, S)

→ h(1+n+i, γ)

(9.2)

→ Tn h. (1+n+i, ┌γ┐)

(9.2)

Thus, the following is established for some number i > 0: (9.7)

~ ⊨ h(n, γ) → Tn h. (1+n+i, ┌γ┐) (N, S)

Starting in the same way as in the above proof, I continue as follows: ~ ⊨ h(n, γ) → ¬ ∃x > 0 Tn h. (x+n, ┌γ┐) (N, S) → ¬Tn h. (1+n+i, ┌γ┐)

for i from (9.7)

Thus the following is esatablished for the i from (9.7): (9.8)

~ ⊨ h(n, γ) → ¬Tn h. (1+n+i, ┌γ┐) (N, S)

Combining (9.7) and (9.8), I obtain a refutation of h(n, γ): (9.9)

~ ⊨ ¬h(n, γ) for all n (N, S)

But one can also show that h(n, γ) must hold for some n, that is, one can show the following: (9.10)

~ ⊨ h(n, γ) for some n (N, S)

To prove (9.10) I conclude the following from (9.9): ~ ⊨ ¬h(0, γ) (N, S)

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~ ⊨ ¬γ. Now (9.10) can be As h(0, γ) and γ are equivalent, this implies (N, S) shown in the following way: ~ ⊨ ∃x > 0 T0 h. (x, ┌γ┐) (N, S)

from (9.5)

T0 h. (n, ┌γ┐)

for some n > 0 ~ is standard as (N, S)

h(n, γ)

(9.2)

From (9.9) and (9.10) one can see that there cannot be such a model and any models of itb↾, which exist by Theorem 9.8, must be nonstandard. a The result can be stated in more proof-theoretic terms. definition 9.10. A theory S is ω-inconsistent if and only if there is a formula ϕ(x) such that S ` ϕ(n) for every n ∈ ω and S ` ¬∀x ϕ(x). According to a standard result, a theory without ω-models is ω-inconsistent (Chang and Keisler 1990, p. 82, Proposition 2.2.13). Applying it to itb and itb↾ yields the following result: corollary 9.11. The theories itb and itb↾ are ω-inconsistent. The theory itb↾ of these illfounded or ungrounded hierarchies provides the first example of a consistent theory of truth that does not have an ω-model. McGee’s (1985) and Yablo’s (1993) paradoxes also belong to this category of results. An overview of the various ω-inconsistency results in this direction is given by Barrio (2010) and Leitgeb (2001).

Part III

TYPE-FREE TRUTH

Tarski’s resolution of the liar paradox by distinguishing an object language from a metalanguage underlies much of modern logic. All the theories in the previous part rely on this approach, although both languages are very similar. In Tarski’s terminology a language is more like what one would call a deductive system or a theory in modern terminology. The axioms and rules of the object language do not matter for Tarski’s purpose; what are crucial are the axioms and rules of the metalanguage. For instance, in typed theories like tb or ct the object language is the language L of arithmetic with pa as the associated system and the metalanguage is the same language expanded merely by an additional predicate symbol T for truth. In each case the object theory is a subtheory of the metatheory and the metatheory is formulated in a language – here LT – properly extending the object language. In the hierarchical theories the theory rtγ is a metalanguage for rt<γ . Despite the success of Tarski’s resolution of the paradoxes in this way, many philosophers have been discontent with a Tarskian approach and so have developed alternative proposals, trying to minimize or close the gap between object and metalanguage. Kripke’s (1975), Herzberger’s (1982), Gupta’s (1982), Barwise and Etchemendy’s (1987), and Gaifman’s (1992) proposals are perhaps the best known among philosophers in this direction. On each of them the truth predicate can be sensibly applied to sentences containing the truth predicate, so that object and metalanguage are no longer clearly distinguished. All of the accounts just mentioned are semantic theories. When discussing aspects of truth like deflationism that are not directly related to the paradoxes, philosophers tend to rely on Tarski’s solution of the paradoxes and to focus on typed theories like the ones analysed in the previous part, even if they would not seriously endorse Tarski’s view that the paradoxes should be resolved by distinguishing between an object and a metalanguage. In the discussion of deflationism, the explanatory power of the disquotation sentences and some other topics, proof-theoretic aspects such as

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the conservativity of truth over the base theory or the ability of the truth theory to prove reflection principles are often considered. Semantic theories of truth are obviously useless for investigating such proof-theoretic properties of truth. Moreover, some prominent proponents of deflationism like Horwich (1990) have advanced use-theoretic accounts of meaning, which mesh with axiomatic approaches of truth much better than with semantic theories. I presume that many authors working on deflationism and some other topics have resorted to typed axiomatic theories because they are well known, well understood, and more familiar than most of the type-free axiomatic theories. However, if they prefer type-free semantic theories to typed ones and they reject the distinction between object and metalanguage as a solution to the semantic paradoxes, it seems inevitable that proof-theoretic results about type-free axiomatic theories of truth are required in order to come to definitive conclusions about deflationism, the explanatory power of the disquotation sentences and similar topics. Hence, in recent years, more philosophers have increasingly begun to base their investigations on type-free axiomatic theories and consequently more has been published on them not only by proof-theorists but by philosophers with an interest in the less technical aspects of truth. In this part of the book I will introduce the most important type-free axiomatic theories of truth and investigate their formal properties. While I discussed disquotational theories first in the previous chapter on typed theories, this part begins with theories that can be seen as type-free generalizations of compositional theories in the ilk of ct and pt. Only later will I consider type-free disquotational theories. Treating the theories in this order has a technical reason: perhaps somewhat surprisingly my technical analysis of a certain type-free disquotational theory presupposes the previous analysis of a well-known compositional theory. The relationship between type-free disquotational and compositional theories is of a completely different nature to the relationship between typed disquotational and compositional theories. While in the typed case disquotational theories tend to be much weaker than compositional theories, typefree disquotational theories can be very strong. This is one point that should worry philosophers who have based their philosophical views on deflationism and disquotationalism purely on the technical results about typed theories such as tb or ct↾. If one does not accept Tarski’s solution to the liar paradox, the philosophical value of technical results about typed theories is very restricted as these results cannot easily be transferred to type-free theories. Hence I expect that many of the results in this part about type-free theories

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will have high philosophical significance and, in fact, greater significance than those in the previous part about typed theories. Before coming to the type-free theories themselves, I will first take a closer look at the distinction between type-free and typed theories, which has so far only been made informally. I will then look at the reasons why one might prefer type-free theories to the typed theories of the previous part.

10 Typed and type-free theories of truth

Having used the distinction between typed and type-free axiomatic theories of truth before, I shall now try to be more precise about what it means for a theory of truth to be typed. Typically, a system of truth will be classified as typed if it is based in some way on an object and metalanguage – or languagelevel – distinction. Type-free theories of truth are often also called theories of self-referential or self-applicable truth. The terminology typed and type-free comes from and has been originally applied to theories about sets, concepts, universals, and the like, of course. Typing could be applied to theories of truth by imposing syntactic restrictions. The terms of the language would have to be classified by their types and the truth predicate restricted to terms of appropriate type. In particular, a truth predicate would only be applied to formulae containing variables if they only ranged over formulae not containing that truth predicate. Accordingly, variables of different types would have to be used. Such an approach seems incompatible with my approach here, as the language L of the base language features only one sort of quantifier that ranges over natural numbers. One could achieve a restriction, however, by choosing a coding of formulae in the natural numbers that only codes sentences of the language L without the truth predicate. On a semantic approach, the truth predicate would be interpreted or axiomatized in such a way that it does not apply to sentences with the truth predicate or any restrictions are imposed on the truth of such sentences, without imposing any syntactical restrictions. The distinction between typed and type-free theories applies to axiomatic as well as semantic theories of truth, and the distinction can be made in both cases in a syntactic and a semantic way. So, for instance, one can focus on a language with typed variables and restrictions on the formation of formulae and give either a semantics or formulate axioms for this language. In the former case a semantic theory of truth with syntactic typing is obtained, the latter case yields an axiomatic system of truth with syntactic typing again. Conversely, one can consider a language such as LT without any syntactic type restrictions and provide a semantics for it or formulate an axiomatic

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system in it that renders it a typed system. I would describe the latter case as a semantically typed axiomatic truth theory. Here I do not pursue the syntactic approach to typing, mainly because it would be very awkward and cause notational difficulties. It would also diminish the possibility of applying the formal theories to the analysis of informal discourse, which does not exhibit type restrictions. Although certain axiomatic theories or semantic theories are called type-free without much hesitation, the demarcation of typed and type-free accounts of truth is not completely straightforward. In the case of semantic theories of truth, such as Tarski’s or Kripke’s, the distinction between typed and type-free approaches might be less problematic than for axiomatic approaches. If a model for the language LT with the truth predicate is provided, it can be called a typed model if no code of a sentence containing the truth predicate is in the extension of the truth predicate; otherwise it is type-free. A semantic approach is then called typed if the proposed model or all such models are typed. Under this characterization of typing, the Tarskian approach sketched in Definition 8.1 comes out as typed while, for instance, Kripke’s (1975) comes out, as expected, as type-free. In order to distinguish typed from type-free axiomatic theories of truth, one could resort to an intended model of the theory and define a theory of truth as typed if the intended model is typed. For reasons provided earlier, I do not want to assume that there is a single intended model for any axiomatic theory of truth. Rather I would like to classify theories or systems of truth as typed or type-free without reference to models and solely on the basis of the theorems of the theory, or, perhaps, on the basis of the chosen axioms and rules of the system. When considering proposals for delineating typed from type-free systems, it would be useful to specify first what counts as an axiomatic theory of truth. I will focus exclusively on extensions of Peano arithmetic in the language LT with the truth predicate; but not any extension of Peano arithmetic in the language LT ought to be classified as a truth theory. The axioms of the theory could characterize the predicate T as the set of all primes, for instance. If the criterion for distinguishing typed from type-free theories does not work on such a theory, this can hardly be used as an argument against the criterion. Generally, in the discussion of possible criteria, one would have to specify the space of theories that are classified. Here I will not attempt to explain what counts as a truth theory, but I trust that the examples I use will be seen as truth theories by the reader. In more thorough discussion it would have to be explained what truth theories are.

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A first proposal for the distinction between typed and type-free axiomatic theories of truth might be to call a theory of truth type-free if and only if there is a sentence ϕ containing the truth predicate such that T┌ϕ┐ is provable in S. A truth theory, however, might well prove the existence of true sentences without proving that a specific sentence is true. The Completeness axiom, which is a theorem of several of the theories to be discussed, states that for any sentence of LT either the sentence itself or its negation is true.

(comp)

∀x SentT (x) → (Tx ∨ T¬. x)




In comp the formula SentT (x) naturally represents the set of LT -formulae. The Completeness Axiom comp implies that there are true sentences as it implies the following disjunction for any sentence ϕ: T┌ϕ┐ ∨ T¬. ┌ϕ┐ Hence the claim that there are true sentences with the truth predicate, that is,
∃x Tx ∧ SentT (x) ∧ ¬ Sent(x) is provable from comp. But one will not be able to prove in pa + comp for any specific sentence ϕ containing the truth predicate that it is true. Theories like pa + comp, however, ought to be classified as type-free as they say something about the truth of sentences with the truth predicate. Hence the condition that a theory has to prove T┌ϕ┐ for some sentence ϕ with the truth predicate is not a necessary condition for being type-free. In the light of the example, one could improve on the previous criterion and propose to define a theory as type-free if it proves that there are true sentences containing the truth predicate, that is, if it proves the following existential claim:
∃x Tx ∧ SentT (x) ∧ ¬ Sent(x) There might be counterexamples to this proposal as well, although they would need to be more pathological than the previous one. A truth theory might impose certain restrictions on the truth of sentences with the truth predicate without actually proving that any of these sentences are true. For instance, axioms like the following

(10.1)

∀x ∀y SentT (x∧. y) → (T(x∧. y) ↔ T(x) ∧ T(y))




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do not imply that there is a true sentence containing the truth predicate. It is compatible with such axioms that all sentences with the truth predicate are neither true nor false. Nevertheless, I would tend to classify systems with axioms of this kind as type-free as they prove theorems about the truth of sentences with the truth predicate. Hence I propose to call a theory typed if it is compatible with any claim about the truth of sentences with the truth predicate, unless the theory proves that all these sentences are not true. This yields a dual criterion: An axiomatic theory of truth is typed if and only if one of the following two conditions is satisfied: (i) The theory proves that only L-sentences are true (or, perhaps, that sentences with the truth predicate are not true). (ii) The theory does not impose any restriction on the truth of sentences with the truth predicate. The first close is easily expressed in more formal terms; the second clause, in contrast, is much harder to state in formal terms. At any rate, the following proposal should be seen as preliminary: definition 10.1. A theory S of truth extending Peano arithmetic is typed if and only if one of the following two conditions is satisfied: (i) S ` ∀x (Tx → Sent(x)) (ii) There is no formula ϕ(x) of L satisfying the following condition: S ` ¬∀x SentT (x) ∧ ¬ Sent(x) → (ϕ(x) ↔ Tx)




A theory of the above kind is type-free if and only if it is not typed. Clause (ii) says that the theory does not refute the truth predicate’s being coextensive with any L-formula on sentences containing the truth predicate; semantically speaking the theory is compatible with the set of true sentences that contain the truth predicate’s being any L-definable set. Perhaps one may want to extend condition (ii) to ensure that a typed theory should permit the set of true sentences that contain the truth predicate to be any set whatsoever, whether or not this set can be defined by an L-formula. However, I can see no easy way to express this and I am not aware of a natural example where the restriction to an L-definable set matters. So I stick to the definition above. If Definition 10.1 is employed, then it can be shown by considering standard models, that the theories tb, utb, and ct come out as typed because of

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clause (ii). Sloppily speaking, these theories do not say anything about the truth of sentences containing the truth predicate. Any system with ∀x (Tx → Sent(x)) as a theorem is also typed. In part β of his Convention T, Tarski (1935) demanded that any adequate definition of truth should prove that only sentences of the object language are true. Hence any definition (or rather theory) of truth that is adequate by Tarski’s definition is typed in the sense of Definition 10.1. Any theory with the Completeness Axiom comp as a theorem is type-free, according to Definition 10.1. To see this, let S be some consistent system proving comp. Choose the formula x 6= x as ϕ(x) in clause (ii) of the definition. As S proves ∃x (SentT (x) ∧ Tx ∧ ¬ Sent(x)), condition (ii) is not satisfied as
S ` ¬∀x SentT (x) ∧ ¬ Sent(x) → (Tx → x 6= x) obtains. For a theory without comp but with (10.1) as its axiom one chooses the formula x = ┌T┌0 = 0┐┐ ∧ x 6= ┌T┌0 = 0┐ ∧ T┌0 = 0┐┐ as ϕ(x) in Condition (ii). It is not hard to check that Condition (ii) is not satisfied for this ϕ(x) as a theory with (10.1) and some syntax axioms will prove the following: T┌T┌0 = 0┐ ∧ T┌0 = 0┐┐ ↔ (T┌T┌0 = 0┐┐ ∧ T┌T┌0 = 0┐┐) Hence such a theory, under the obvious provisos, will not satisfy Condition (ii), and it will thus be classified as type-free, as it should. Inconsistent theories are classified as typed by Definition 10.1 because the formula ∀x (Tx → Sent(x)) is provable in an inconsistent theory. Of course, one could add an additional clause in the definition requiring that typed theories must be consistent. Since I do not have a clear view about the classification of inconsistent systems, I shall not modify the definition. There may be alternative ways of defining the property of being typed. Perhaps one should not try to classify theories, considered as deductively closed sets of formulae, but rather deductive systems with exactly specified axioms and rules of inference. One could and perhaps should classify single axioms and axiom schemata as typed or type-free. This may be useful as very often single axioms or schemata are rejected because they seem to incorporate some type or language-level distinction. But then a theory would be needed about how single typed or type-free axioms, axiom schemata, or rules of inference render a deductive system typed or type-free.

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For systems with more than one truth predicate, like the systems rt<γ of ramified truth, Definition 10.1 can be adapted by generalizing from the single truth predicate T to all truth predicates. The type-free systems discussed in the following are all ruled to be typefree by the criterion whose necessity has been doubted but that is evidently sufficient for a system’s being type-free: all these systems prove the truth of a specific sentence containing the truth predicate.

11 Reasons against typing

Tarski’s (1935) solution to the liar paradox was highly successful and has become the standard in formal semantics. But philosophers have doubted the adequacy of this solution for various reasons. Of course, Tarski’s distinction between object and metalanguage forms the basis for model theory and formal semantics, but for other purposes Tarski’s solution is less adequate in the eyes of many philosophers. Much of the work on semantic theories of truth like Kripke’s (1975), Herberger’s (1982), and Gupta’s (1982) and many further accounts building on them was prompted by the desire to devise a semantics for a language with a type-free or self-applicable truth predicate. These semantic theories then sparked and motivated the research on axiomatic type-free theories. When it comes to axiomatic theories, the doubts about Tarski’s solution and the arguments in favour of the semantic theories of self-applicable truth translate into doubts about the adequacy of typed axiomatic theories of truth and into arguments in favour of type-free systems of truth. In this section I will briefly consider some of the arguments that motivate the investigation of type-free systems of truth. First of all, it is often emphasized that typing is not natural. Although I agree that typing is not found in natural language, I do feel that it would hardly be appropriate to employ this argument in the present setting: I am working with Peano arithmetic as the base theory, expressions of the language are coded in the natural numbers, context dependent features of natural language are completely ignored: it would be preposterous to reject typing as not natural given that I am working in a highly artificial framework. Only if this formal framework were seen as the starting point for the development of a semantics for natural language and if approaches that assign levels of truth to truth predicates in natural language were rejected would type-free theories be better candidates for theories that can finally form part of a comprehensive reconstruction of natural language in a formal theory. But this is not my project. However, there are also theoretical contexts where a type-free truth predicate is more usable than a typed one. When saying that all tautologies are

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true, one would like to be able to make this claim for all tautologies, including those that contain the truth predicate. In fact, in the Friedman–Sheard theory, which will be discussed in the following, this claim is provable. On a typed account the claim that all tautologies are true cannot be provable, as is easily seen from Definition 10.1. The same applies to claims from epistemology and other areas in which truth is used: in these claims the truth predicate is not typed. Hence, if a typed theory of truth is employed, the truth predicate can hardly serve the purpose it is assigned in many philosophical contexts, at least not without severe incisions into these areas. Some philosophers, and proponents of deflationism in particular, think that the purpose of the truth predicate is to facilitate the expressions of generalizations. Typed theories already serve this purpose to some extent, but typefree theories might serve it better in proving more generalizations. Feferman (1991) proposed the type-free theory kf and gave a proof-theoretic analysis in terms of infinite conjunctions that makes it obvious that a typed theory such as ct can only prove a small fraction of the generalizations provable in kf. This should be a point in favour of the type-free approach not only for deflationists, who see the expression of generalizations as the main or only purpose of truth, but also for other philosophers who believe that the expression of generalizations is an important purpose of the truth predicate. There is also a more specific observation that should support the type-free approach from the perspective of deflationists. As I will show in Section 19.3, certain type-free disquotational theories enable one to prove more generalizations than the strongest typed compositional theories like ct. If truth is, as some deflationists have it, ‘at bottom disquotational’ (Field 1994b), and if it is the chief purpose of truth to enable one to express and prove generalizations, then typing causes a problem: typed disquotational theories like tb and utb are notoriously weak at proving generalizations: compositional theories like ct are much better at it, as the proof-theoretic analysis of these theories in the previous part has shown. Thus one doctrine of (certain varieties of) deflationism, namely that truth is a device of disquotation, seems to be in conflict with another of its doctrines about the purpose of truth: truth as a device of disquotation axiomatized by the disquotation sentences serves the alleged purpose of truth much worse than compositional theories – if only typed theories of truth are considered. When the restriction to typed disquotational theories is relaxed, the two doctrines need not clash anymore. In fact, type-free disquotational theories serve the purpose of truth of expressing generalizations at least as well as most type-free compositional theories. Of course, these claims need to be

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made more precise and substantiated. Here I only emphasize that there are prospects for resolving tensions between deflationist doctrines by passing from typed to type-free theories of truth. The higher expressive and deductive power of type-free theories of truth also makes them more usable for reductions. In the theory ct certain amounts of second-order quantification can be mimicked as the proof-theoretic analysis in Section 8.6 has shown. In type-free theories more second-order quantification can be mimicked. Hence if one considers truth theories as tools for eliminating second-order quantification, type-free theories look more promising than typed theories. Historically the dominance of typed semantic and axiomatic theories of truth can be traced back to Tarski’s distinction between object and metalanguage. But if then Tarski’s argument for the distinction is the main argument for typing, then it seems strangely weak: the set of all disquotation sentences T┌ϕ┐ ↔ ϕ for sentences of LT is inconsistent with weak base theories. But it does not directly follow that typing should be used as a remedy for this inconsistency. One might not be content with the set of all unrestricted disquotation sentences for various reasons and typing is by no means the only solution. I surmise that the dominance of typed theories has another source: typing facilitates reductive definitional theories of truth. Tarski’s own definition of truth relies on a metalanguage (or rather metatheory) that is ‘essentially’ stronger than the object language. I do not attempt to make this precise, but it is plain that the distinction between object and metalanguage is required for the definability of truth. Without this distinction there is no way to define truth in Tarski’s style. If, however, the project of defining truth is abandoned and an axiomatization of truth becomes the goal, then the distinction between object and metalanguage is no longer required for the viability of this project. The adherence to typed theories is borne, I suspect, from our habituation to Tarski’s definition of truth. Once this reductive approach is abandoned and replaced by an axiomatic approach, typing no longer seems natural. All this does not imply that type-free theories are better than typed ones. I have not yet introduced any type-free theories of truth and it still has to be investigated whether the advantages of this approach still outweigh its disadvantages.

12 Axioms and rules

When browsing through various papers on axiomatic theories of truth, one may get the impression that the goal is to find a combination of attractive axioms and rules of inference that can be combined without rendering the system inconsistent, that is, at least when the systems are formulated in classical logic, without rendering the system inconsistent. The axioms and rules can only be evaluated against the background of a base theory and base logic, and only in combination with the other proposed axioms and rules. I think a holistic approach to judging axiomatic systems for truth is preferable to a piecemeal treatment where one tries to evaluate axioms and rules separately and then tries to combine as many desirable rules and axioms as possible, or where one tries to optimize some kind of degree of desirability where one starts by scoring each proposed truth-theoretic axiom and rule only against a background logic or the background of a certain base theory without other axioms for truth. Only in connection with a specific logic, a specific base theory, and a set of other axioms and rules for truth, can a candidate axiom for truth be evaluated. Many discussions seem to hinge on the typographic shape of axioms and rules. It does not make much sense to discuss, for instance, a disquotational axiom T┌ϕ┐ ↔ ϕ on its own. First, the use of a double arrow may be very suggestive, but if it is not governed by the usual introduction and elimination rules for the classical biconditional, the axiom may have very little to do with classical disquotationalism. Only against the background of a fixed logic can we start discussing a disquotational principle. There is of course the possibility of fixing certain minimal requirements weaker than classical logic that fix a symbol like ↔ as a biconditional, and many logicians have argued that their ↔ is a biconditional because it satisfies certain properties. Intuitionistic logic shares many rules with classical logic; perhaps ↔ deserves to be called a biconditional if it is governed by intuitionistic logic, but once Strong Kleene logic is adopted the common basis seems to become quite meagre. If I am granted the word means, I would say that T┌ϕ┐ ↔ ϕ does not mean the same when ↔ is understood classically and when it is understood according to Strong Kleene rules. Hence to compare

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truth-theoretic axioms and rules in the indicated way, at least the underlying logic has to be fixed. In particular, it is highly misleading to say that one has succeeded in preserving certain truth-theoretic principles by sacrificing classical logic: one might have preserved theorems or axioms that share the same graphical shape with certain truth-theoretic principles of classical systems. But the mere coincidence in their typography does not seem to warrant the claim that one has preserved the same truth-theoretic principle. Generally, I do not want to assume that logical symbols like ↔, ¬, or ∀ come with a fixed meaning that can be made explicit, so that one can disagree about the adequacy of an analysis of ↔, ¬, or ∀ based on classical or some other logic. Presupposing a fixed meaning of these symbols might be feasible if one sees them as formal counterparts of certain expressions in English or some other natural language and takes the meaning of the logical symbols to be fixed by the meaning of the correlated English expressions. But this is not the route I am taking here. I do not want to pretend that I am using ∧ as a symbol whose meaning is fixed by the English word and, and I certainly do not want to claim that → corresponds to if then. For me, although the enterprise of philosophical analysis is driven by natural language, its goal is not a linguistic analysis of English but rather an expressively strong framework that may at best be seen as a revision of English. That said, I do not want to claim that systems based on classical logic and systems based on nonclassical logic are incommensurable. But the observation that they both share theorems that happen to coincide in shape is not a very useful criterion for their comparison. Comparisons of their prooftheoretic ordinals, for instance, might be more telling. That way one can at least compare the strength of the systems in some way. When comparing two deductive systems that both have a certain sentence (as a string of symbols) as a theorem, one has to take the base theory, as well as the underlying logic, into account. For instance, an axiom like
∀x Sent(x) → (T¬. x ↔ ¬Tx) is only compositional against the background of a base theory that contains certain restrictions on the expressions Sent(x) and ¬. . Many formal results, for instance about conservativity and the correspondence between compositional truth and comprehension, rely on the base theory. My worries about evaluating the merits of truth-theoretic axioms and rules separately from the base theory might prompt a wider suspicion: to obtain a suitable background against which to evaluate truth theories, perhaps it is not enough to stop with the inclusion of the base theory; instead we should need

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to evaluate systems of truth in the context of our total theory of the world. When considering truth-theoretic systems with a base theory like Peano arithmetic one might miss out on some bigger problems beyond the confines of the usual metamathematical playground. Actually some of these problems may be found nearby. For instance, if the theory of truth is not just engrafted on a base theory like Peano arithmetic but instead on a system that already contains a theory of necessity, then the merits of truth-theoretic axioms and rules might appear in a completely different light. In Section 24.2 I will consider an example where the combination of a truththeoretic system with a system for necessity yields an inconsistency (at least if ‘combination’ is understood in a certain way). Hence it may be advisable not to try to evaluate a theory of truth only on the basis of a fixed logic and base theory but also in the context of other theories that interact with truth. Generally I think that worries about only investigating axiomatic theories of truth in the context of a fixed logic and base theory are justified: when we try to integrate truth theories with a theory of necessity or a theory of knowledge, we may need to go back and revise or sharpen our truth theory. So the worries stated at the beginning of this chapter run even deeper: the axioms for truth need to be evaluated not only against the background of a minimal base theory of the truth bearers but of a very comprehensive base theory. For the time being, however, I cannot do any better than to consider the truth theories on the background of Peano arithmetic as base theory. At the end of the book I will give some hints on how to apply truth theories to more comprehensive base theories.

13 Axioms for type-free truth

There are many limitative results on type-free approaches to truth. In this section I will list some of the results that are needed later on. They also have some independent interest as they show that certain approaches are not viable. The most notorious limitative result is surely the liar paradox. In particular, as the liar paradox shows, the full type-free disquotation sentences T┌ϕ┐ ↔ ϕ for each ϕ of LT are inconsistent over a base theory that allows for diagonalization. I will now look at what happens when the two directions T┌ϕ┐ → ϕ and ϕ → T┌ϕ┐ of the disquotation schema are replaced with corresponding rules. The rule corresponding to ϕ → T┌ϕ┐ is often called the rule of necessitation, as it is reminiscent of its modal analogue. The rule will be written in the following way and be labelled nec: nec

ϕ T┌ϕ┐

If the rule is formulated in the sequent calculus, then this rule means that one can pass from the sequent ⇒ ϕ to the sequent ⇒ T┌ϕ┐. No other formulae, that is, no side formulae are allowed in the sequents. Analogously, if nec were applied in a system of Natural Deduction, all assumptions would have to be discharged before one could pass from a sentence ϕ to T┌ϕ┐. If side formulae were allowed, one could go from the initial sequent ϕ ⇒ ϕ to ϕ ⇒ T┌ϕ┐ and then to ⇒ ϕ → T┌ϕ┐ and thereby derive the corresponding axiom. The rule, however, is weaker than the axiom schema ϕ → T┌ϕ┐. Peano arithmetic can consistently be closed under the rule nec of necessitation and its converse T┌ϕ┐ conec . ϕ The following strengthening of the liar paradox, which is called Montague’s paradox and due to Montague (1963), shows that if the direction ϕ → T┌ϕ┐ of the disquotation schema is replaced with the rule of necessitation, the resulting system is still inconsistent:

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lemma 13.1 (Montague’s theorem). Any system that is closed under the rule nec for all sentences of LT and contains pa and the schema T┌ϕ┐ → ϕ for all sentences ϕ of LT is inconsistent. proof. By the Diagonal lemma there is a sentence λ, the liar sentence, with the following property: S ` λ ↔ ¬T┌λ┐ Now the liar paradox may be derived as follows in any system S containing the schema T┌ϕ┐ → ϕ and closed under nec: S `T┌λ┐ → λ

instance of T┌ϕ┐ → ϕ

T┌λ┐ → ¬λ

def. of λ

¬T┌λ┐

two preceding lines

λ

def. of λ

T┌λ┐

nec

The last and third lines are contradictory.

a

Montague’s paradox is usually applied to theories of necessity rather than theories of truth. The dual of Montague’s paradox is not relevant for necessity, but it imposes further limits on theories of truth. lemma 13.2 (dual of Montague’s paradox). Any system that contains pa and the schema ϕ → T┌ϕ┐ for all sentences ϕ of LT is inconsistent with the rule conec

T┌ϕ┐ . ϕ

The proof is analogous to the proof of Montague’s theorem. However, nec and conec can be consistently combined with one another, and also with many further axioms. The Friedman–Sheard system fs, which will be studied below, provides an example of a consistent system containing both rules. In a system S closed under nec and conec the internal logic, that is, the set of all sentences ϕ with S ` T┌ϕ┐ and the external logic, that is, the set of all sentences ϕ with S ` ϕ coincide. Such systems prove exactly those sentence true that are provable in the system. I call such systems symmetric. definition 13.3 (symmetry). A theory of truth is symmetric if and only if it is closed under nec and conec. No such system decides a liar sentence:

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lemma 13.4. In any consistent symmetric theory S a liar sentence is neither provable nor refutable, that is, S ⊬ λ and S ⊬ ¬λ, if λ is a sentence with λ ↔ ¬T┌λ┐ provable in the base theory. proof. Assume that S is symmetric. The assumption that S proves λ leads to an inconsistency: S`λ S ` T┌λ┐

nec

S ` ¬T┌λ┐

def. λ and first line

The assumption that λ is refutable also leads to an inconsistency: S ` ¬λ S ` T┌λ┐

def. λ

S`λ

conec

a

An inspection of the proof of Montague’s theorem shows that the schema T┌ϕ┐ → ϕ implies the liar sentence. This observation can be used to show that the schema is also inconsistent with the so-called Completeness axiom that states that every sentence is either true or false: ∀x SentT (x) → (Tx ∨ T¬. x)

(comp)




Since this axiom rules out truth-value gaps, it will be incompatible with approaches which take the internal logic of truth to be a partial logic . lemma 13.5. The schema T┌ϕ┐ → ϕ for all sentences of LT is inconsistent with comp over Peano arithmetic. proof. Assuming that S proves comp and the above schema, I reason as follows using the liar sentence λ from above: S ` T┌λ┐ → λ

instance of T┌ϕ┐ → ϕ

T┌λ┐ → ¬λ

def. of λ

¬T┌λ┐

two preceding lines

λ

def. of λ

T┌¬λ┐

from third line and comp

¬λ

instance of T┌ϕ┐ → ϕ

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Again there is a dual of this observation with an obvious proof: the schema ϕ → T┌ϕ┐ is inconsistent with the Consistency axiom; it says that no sentence is true and false: (cons)

∀x Sent(x) → ¬(Tx ∧ T¬. x)




Thus cons rejects truth-value gluts. On dialethic approaches to truth paradoxical sentences like the liar sentence have both classical truth-values. The axiom cons explicitly denies that any sentence has two truth-values and therefore the consistency axiom will be incompatible with such approaches to truth. For later reference I introduce alternative formulations of the Completeness and the Consistency axiom. lemma 13.6. The Consistency axiom cons is logically equivalent to
∀x Sent(x) → (T¬. x → ¬Tx) , and the Completeness axiom comp is logically equivalent to
∀x Sent(x) → (¬Tx → T¬. x) . Therefore the axiom (fs2)

∀x Sent(x) → (T(¬. x) ↔ ¬Tx)




is logically equivalent to the conjunction of cons and comp. As I mentioned above, nec and conec can be combined consistently. Next I shall prove a result due to Friedman and Sheard (1987) that shows that they cannot be combined with two other attractive axioms. I first prove a version of Löb’s theorem (or Curry’s paradox), which is a result about provability. The proof that Löb’s theorem follows from the socalled Löb derivability conditions also applies to the truth predicate as long as it satisfies the analogues of the derivability conditions. For more information on Löb’s theorem see, for instance, Boolos (1993). lemma 13.7 (Löb’s theorem). Assume that S is a system that contains pa and has the following properties: (i) S is closed under nec. (ii) S ` T┌ϕ → ψ┐ → (T┌ϕ┐ → T┌ψ┐) (iii) S ` T┌ϕ┐ → T┌T┌ϕ┐┐ Then S proves the sentence T┌T┌ϕ┐ → ϕ┐ → T┌ϕ┐.

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axioms for type-free truth

proof. Using the Diagonal lemma I obtain a sentence γ satisfying the following condition: γ ↔ (T┌γ┐ → ϕ)

(13.1)

The result can then be established by applying nec to (13.1) and reasoning in the following way: S `T┌γ ↔ (T┌γ┐ → ϕ)┐ T┌γ┐ → T┌T┌γ┐ → ϕ┐

(ii) and modus ponens

T┌γ┐ → (T┌T┌γ┐┐ → T┌ϕ┐)

(ii) and modus ponens

T┌γ┐ → T┌ϕ┐

(iii)

(T┌ϕ┐ → ϕ) → (T┌γ┐ → ϕ)

logic

(T┌ϕ┐ → ϕ) → γ

(13.1)

T┌(T┌ϕ┐ → ϕ) → γ┐

nec

T┌T┌ϕ┐ → ϕ┐ → T┌γ┐

(ii) and modus ponens

T┌T┌ϕ┐ → ϕ┐ → T┌ϕ┐

fourth line

a

Halbach et al. (2003) have assigned Löb’s theorem a crucial role in the theory of paradoxes. They have reduced a large number of inconsistency results, such as McGee’s ω-inconsistency result Theorem 13.9 below, to Löb’s theorem, even including theories not satisfying Axiom (iii). I can now prove the result due to Friedman and Sheard (1987, p. 16) mentioned above. theorem 13.8. Any theory S containing Peano arithmetic and closed under the rules nec and conec with the following two sentences as theorems is inconsistent:
(i) ∀x ∀y Sent(x→ . y) → (T(x→ . y) → (Tx → Ty)) (ii) ∀t (TT. t ↔ Tt◦ ) The material conditional → is not a primitive symbol of the language LT ; by the conventions adopted in Chapter 5 the formula ϕ → ψ abbreviates ¬ϕ ∨ ψ. The function symbol → . stands for the function taking formulae ϕ and ψ to the formula ¬ϕ ∨ ψ. Assumption (i) is needed, as is shown in Theorem 19.21. The consistency of the systems without either nec, conec, or one direction of (ii) is established by Friedman and Sheard (1987).

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157

proof. It will be shown that any sentence ϕ is provable in S. I begin by instantiating (ii) and then I continue in the following way: S ` T┌T┌ϕ┐┐ ↔ T┌ϕ┐ T┌T┌T┌ϕ┐┐ → T┌ϕ┐┐

nec

T┌T┌T┌ϕ┐┐┐ → T┌T┌ϕ┐┐

(i) and modus ponens

Clearly, any system satisfying the premisses of the theorem also satisfies the premisses of Löb’s theorem, that is of Lemma 13.7. Applying Löb’s theorem to the last line, I proceed as follows: S ` T┌T┌ϕ┐┐ T┌ϕ┐

(ii)

ϕ

conec

As ϕ is arbitrary, S is inconsistent.

a

Finally I prove a negative result due to McGee (1985). It is not a proof of an outright inconsistency but of another property that seems almost as bad as inconsistency to many logicians. The following slightly adapts McGee’s result to the current setting. theorem 13.9 (McGee’s theorem). Any theory S containing all the axioms of pa that is closed under nec and proves the following sentences is ωinconsistent:1
(i) ∀x Sent(x) → (T¬. x → ¬Tx)
(ii) ∀x ∀y SentT (x∨. y) → (T(x∨. y) → T(x) ∨ T(y))
(iii) ∀v ∀x SentT (∀. vx) → (∀t T(x(t/v)) → T(∀. vx)) proof. To formulate the diagonal sentence used in his proof, McGee employs a binary primitive recursive function f that yields, when applied to a number n and a sentence ϕ, the sentence ϕ preceded by n+1 truth predicates: f(n, ϕ) := TT. . . . T. ┌ϕ┐ | {z } n occurrences

The value of f(0, ϕ) is the sentence ϕ again. 1 For the definition of ω-inconsistency see Definition 9.10 above.

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axioms for type-free truth

This primitive recursive function is representable in Peano arithmetic. I will avail myself to a function symbol f . If L does not have a function symbol for f . it can be expressed using a suitable formula. I now reason in S as follows, beginning with an application of the Diagonal lemma: (13.2)

(13.3)

γ ↔ ¬∀x Tf (x, ┌γ┐) . ¬γ ∨ ¬∀x Tf (x, ┌γ┐) . T┌¬γ ∨ ¬∀x Tf (x, ┌γ┐)┐ . T┌¬γ┐ ∨ T┌¬∀x Tf (x, ┌γ┐)┐ . ¬T┌γ┐ ∨ T┌¬∀x Tf (x, ┌γ┐)┐ . T┌γ┐ → T┌¬∀x Tf (x, ┌γ┐)┐ . T┌γ┐ → ¬T┌∀x Tf (x, ┌γ┐)┐ . T┌γ┐ → ¬∀x Tf (x+1, ┌γ┐) . T┌γ┐ → ¬∀x Tf (x, ┌γ┐) . T┌γ┐ → γ

prop. logic nec (ii) (i) def. → (i) (iii) and def. of f weakening from (13.2)

Line (13.2) also implies the first of the following lines: ¬γ → ∀x Tf (x, ┌γ┐) . ¬γ → Tf (0, ┌γ┐) . ¬γ → T┌γ┐

univ. inst. def. of f

The last line and (13.3) imply γ and therefore also by (13.2) the following: (13.4)

¬∀x Tf (x, ┌γ┐) .

By iterated applications of nec to γ, one obtains the following: T┌γ┐ T┌Tf (0, ┌γ┐)┐ . T┌Tf (1, ┌γ┐)┐ . .. .

and so Tf (0, ┌γ┐) . and so Tf (1, ┌γ┐) . and so Tf (2, ┌γ┐) . .. .

Hence one has S ` Tf (n, ┌γ┐) for all n ∈ ω, which yields an ω-inconsistency . together with (13.4). a

14 Classical symmetric truth

An obvious way to generalize the compositional theory ct is to lift the restriction to L-sentences in the compositional Axioms ct2–ct6. It can be postulated that truth commutes with connectives and quantifiers not just for arithmetical sentences but also for sentences with the truth predicate. This yields the following type-free system: definition 14.1 (fsn). The system fsn is given by all the axioms of pat (including induction axioms containing the truth predicate) and the following axioms:
fs1 ∀s ∀t T(s=. t) ↔ s◦ = t◦
fs2 ∀x SentT (x) → (T¬. x ↔ ¬Tx)
fs3 ∀x ∀y SentT (x∧. y) → (T(x∧. y) ↔ (Tx ∧ Ty))
fs4 ∀x ∀y SentT (x∨. y) → (T(x∨. y) ↔ (Tx ∨ Ty))
fs5 ∀v ∀x SentT (∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v)))
fs6 ∀v ∀x SentT (∃. vx) → (T(∃. vx) ↔ ∃t T(x(t/v))) The first axiom of ct, that is, Axiom ct1, governing the truth of atomic sentences of the form s = t cannot be generalized easily, so it is left as it is and relabelled as fs1. Clearly fs proves all the axioms of ct. By Lemma 8.4 the typed disquotation sentences are also provable in fsn: lemma 14.2. The theories utb and ct are subtheories of fsn. The system fsn does not contain an axiom specifically about the truth of atomic sentences of the form Tt for t a closed term. For instance, as will be established by the model theory of fsn, the sentence T┌T┌0 = 0┐┐ is not a theorem of fsn. Nevertheless, certain sentences containing the truth predicate can be proved to be true in fsn. For instance, T┌Tt ∨ ¬Tt┐ is a theorem of fsn by Axioms fs4 and fs2 for each closed term t. But fsn does not prove the truth of any atomic sentences with the truth predicate. I will now look at possibilities for adding an axiom about the truth of such

159

160

classical symmetric truth

sentences. To motivate a further axiom, I will consider again how one would treat other predicate symbols in the language of the base theory if it was expanded with further predicate symbols. If one were to generalize fsn to contain a further unary predicate symbol P , one would proceed as in the case of ct (see p. 66) and add the following axiom, which is akin to fs1:
∀t TP. t ↔ P t◦ In order to obtain an axiom about the truth of atomic sentences with the truth predicate, one might follow this pattern and formulate the following axiom: t-sym

∀t (TT. t ↔ Tt◦ )

Friedman and Sheard (1987) have implicitly shown that the resulting theory is inconsistent (see also Halbach 1994). I prove a sharpening of this inconsistency result by showing that only Axiom fs2 is needed to arrive at an inconsistency with t-sym. lemma 14.3. The theory fs2+t-sym is inconsistent with Peano arithmetic. By Lemma 13.6 the axiom t-sym is equivalent to the the conjunction of cons and comp. Therefore the lemma can be restated as the claim that the system with all axioms of Peano arithmetic and the axioms t-sym, cons, and comp is inconsistent. proof. The usual liar sentence γ is of the form ¬T t for a closed term t such that (14.1)

pa ` t = ┌γ┐.

From this identity the derivability of γ ↔ ¬T┌γ┐ follows. This term t can be used to arrive at an inconsistency in the following way: (14.2)

pat ` γ ↔ ¬Tt

tautology

(14.3)

pat+t-sym ` γ ↔ ¬T┌Tt┐

(14.4)

pat+fs2+t-sym ` γ ↔ T┌¬Tt┐

fs2

(14.5)

pat+fs2+t-sym ` γ ↔ T┌γ┐

def. of γ

t-sym

Since γ ↔ ¬T┌γ┐ is theorem of pa, pa+fs2+t-sym is inconsistent.

a

The lemma shows that fsn, which has fs2 as an axiom, is inconsistent with Axiom t-sym.

classical symmetric truth

161

One could opt to add only one direction of t-sym. For instance, one could ∀t (Tt◦ → TT. t)

t-rep

as an axiom. But fsn together with this axiom proves the liar sentence and is therefore incompatible with the rule that licenses the inference from a sentence ϕ to T┌ϕ┐. Friedman and Sheard (1987) provided models for a system containing fsn+t-rep and for a system containing fsn and the converse of t-rep under the labels It is true that everything is true and It is true that everything is false, which describe the models constructions very well. Friedman and Sheard’s constructions show that both theories have fairly trivial models. Hence I think it is more promising to aim at the theory that allows one to prove the truth of atomic sentences with the truth predicate in another way. The rule nec allows one to prove the truth of atomic sentences that contain the truth predicate. So I will focus on extensions of fsn closed under this rule. A system containing fsn that is symmetric in the sense of Definition 13.3 of truth can be obtained simply by stipulating symmetry. This is achieved by adding the following two rules, which have been introduced above, for all sentences ϕ in the language LT : nec

ϕ T┌ϕ┐

T┌ϕ┐ ϕ

conec

definition 14.4 (fs). The system fs is obtained from fsn by closing it under the rules nec and conec. The acronym fs is short for Friedman–Sheard. Friedman and Sheard (1987) introduced a slightly different version of the system under a very different axiomatization. Since ct is a subtheory of fsn, fsn proves the global reflection principle for Peano arithmetic in the language L by Theorem 8.39. In fact, as the truth axiom of fs are type-free, fs proves the global reflection principle for pat, that is, for Peano arithmetic formulated in the language with the truth predicate and induction in the language LT . In the following lemma the formula Bewpat (x) expresses provability in pat in a natural way.
lemma 14.5. fs ` ∀x SentT (x) ∧ Bewpat (x) → Tx proof. The proof resembles the proof of Theorem 8.39, that is, of the global reflection principle for Peano arithmetic formulated in the language L in

162

classical symmetric truth

the system ct; a soundness proof by induction on the length of proofs is formalized in fsn. I will only highlight the parts of the proof of Theorem 8.39 that need modifying for the present lemma. I assume again that pat is formulated in a Hilbert-style calculus. In contrast to pa the axiom schemata of pat also apply to sentences with the truth predicate. Using axioms fs2–fs6 one can prove that the universal closures all logical axioms of pat are true. The claim that all instances of the induction schema in the language LT are true follows from a single instance of the induction schema. The proof is similar to the proof of the claim that all L-induction axioms are true in Theorem 8.39. Some of the identity axioms also contain the truth predicate. For instance, the formula x = y ∧ Tx → Ty might be an axiom of the chosen calculus. But by using nec
T┌∀x ∀y x = y ∧ Tx → Ty ┐ is easily established. For the induction step, that is, for the proof that pat is closed under the logical rules of the chosen calculus the axioms fs2–fs6 are employed as before. a The following corollary contains the rule version of Lemma 14.5: corollary 14.6. If ϕ is a sentence of LT such that pat ` ϕ, then fs ` T┌ϕ┐ obtains. proof. Assume that ϕ is a sentence of LT with pat ` ϕ. Then by the usual properties of the provability predicate, since Bewpat (x) weakly represents provability in pat, one can conclude pat ` Bewpat (┌ϕ┐). Using the reflection principle from the above lemma, I conclude fs ` T┌ϕ┐. a 14.1 The Friedman–Sheard theory and revision semantics The Friedman–Sheard system fs satisfies the premisses of McGee’s ω-inconsistency theorem, that is, Theorem 13.9. Hence fs is ω-inconsistent. Nevertheless it has an elegant semantics. Natural models for subsystems of fs with a limited number of applications of nec and conec are obtained via revision semantics, which was introduced by Gupta (1982) and Herzberger (1982) and

revision semantics

163

defended and further investigated by Belnap and Gupta (1993). Revision semantics is arguably the main competitor of Kripke’s theory of truth among semantic truth theories. I will begin with an exposition of revision semantics for finite levels of the revision process as infinite levels are not needed for the semantics of fs and related systems. The idea of revision semantics for finite levels is simple: one starts with a classical standard model (N, S) for the language LT : the arithmetical vocabulary is interpreted in the standard way, and the truth predicate receives some set S of natural numbers as its extension. The extension S can be any set; it need not be a natural interpretation of the truth predicate; it may be the empty set, for instance. The model (N, S) makes some sentences true, others false. The set of all sentences that are true in (N, S) (in the usual classical sense), that is, the set S0 of all sentences ϕ satisfying (N, S) ⊨ ϕ can now be used as a new extension of the truth predicate, and a new model (N, S0 ) is obtained. The step from (N, S) to (N, S0 ) is then repeated. Through this revision process one may hope to arrive at better and better models. Whether the models and extensions actually become better in some sense depends on the extension that one starts with. The revision procedure will now be presented in more formal terms. First I define the revision operator Γ: definition 14.7. For S ⊆ ω the revision operator Γ is defined in the following way: Γ(S) := {ϕ : ϕ is a sentence of LT and (N, S) ⊨ ϕ} The following is an obvious consequence of this definition: lemma 14.8. For all sentences ϕ of LT the following equivalence obtains: (N, Γ(S)) ⊨ T┌ϕ┐ iff (N, S) ⊨ ϕ Iterations of the revision operator are defined inductively as follows: Γ0 (S) := S Γn+1 (S) := Γ(Γn (S)) For the following lemma I define an abbreviation for the n-fold application of the truth predicate to a term t: T1 t abbreviates Tt Tn+1 abbreviates T┌Tn t┐

164

classical symmetric truth

Thus, using the function f from Theorem 13.9, f(n, ϕ) is equivalent to Tn ┌ϕ┐ over Peano arithmetic for all n ≥ 1. lemma 14.9. (i) The operation Γ is injective: Γ(S1 ) = Γ(S2 ) implies S1 = S2 for arbitrary sets S1 , S2 ⊆ ω. (ii) For all sentences ϕ of LT the following equivalence holds for any number n ≥ 1: (N, Γn (S)) ⊨ Tn ┌ϕ┐ iff (N, S) ⊨ ϕ (iii) Γn (S) 6= S for all n ≥ 1 and S ⊆ ω proof. To prove (i) assume that S1 6= S2 . Hence there is a sentence ϕ with (N, S1 ) ⊨ T┌ϕ┐ and (N, S2 ) ⊭ T┌ϕ┐. Therefore one has T┌ϕ┐ ∈ Γ(S1 ) but not T┌ϕ┐ ∈ Γ(S2 ). Item (ii) is obtained by iterating Lemma 14.8. For the proof of (iii) I invoke Gödel’s diagonal lemma to obtain a sentence γ with the following property: pa ` γ ↔ Tn ┌¬γ┐ Applying item (ii) to this, I conclude the following equivalence: (N, Γn (S)) ⊨ ¬γ iff (N, S) ⊨ γ Hence the sets Γn (S) and S must be different.

a

The term revision process suggests that by applying the operator Γ better and better extensions for the truth predicate and thus better models for the language LT are obtained. But neither is it clear what better could mean here nor is it plausible to assume that the revision process leads to better models as one may start with a good model and then apply the revision process to this model. It is better to view the revision process as a method for sorting out unsuitable extensions of the truth predicate. One begins with the class P(ω) of all sets of natural numbers as possible extensions for the truth predicate. Each set S ⊆ ω can be seen as an arbitrary guess at the extension of the truth predicate. The operator Γ is then applied to each set in P(ω), that is, to each set S ⊆ ω. Some sets are re-obtained by applying Γ to some set S ⊆ ω, others cannot be obtained by applying Γ to an element of P(ω). Therefore the

revision semantics

165

sets obtained by applying Γ to some element of P(ω) form a proper subset Γ[P(ω)] := {Γ(S) : S ⊆ ω} of P(ω). The really inadequate elements have been excluded, while other sets remain in Γ[P(ω)]. Now Γ is applied again to Γ[P(ω)]. This yields a new set of possible extensions for the truth predicate, which is in turn a proper subset of Γ[P(ω)]. Hence the choice of possible extensions for the truth predicate has been slimmed down even further. This process is then iterated and more and more sets will be ruled out as possible extensions of the truth predicate. I will describe the revision process of excluding more and more extensions of the truth predicate in more technical terms. First I will introduce some notation for applying Γ to all elements of a set M ⊆ P(ω) of sets of natural numbers simultaneously: Γ[M] := {Γ(S) : S ∈ M} In other words, Γ[M] is the image of M under Γ. As in the case of single sets, I will write Γn [M] for the result of applying this operation n-times to the class M. lemma 14.10. Let P(ω) be the power set of the set ω of natural numbers. Then the following implication holds for all natural numbers n, k ∈ ω: If n ≤ k, then Γk [P(ω)] ⊆ Γn [P(ω)] proof. Since Γ0 [P(ω)] = P(ω), Γ1 [P(ω)] ⊆ Γ0 [P(ω)] trivially holds. The proof can be completed by showing inductively by showing Γn+2 [P(ω)] ⊆ Γn+1 [P(ω)]. To see this assume S2 ∈ Γn+2 [P(ω)]. Under this assumption there must be a set S1 ∈ Γn+1 [P(ω)] that satisfies Γ(S1 ) = S2 . By induction hypothesis, S1 is in Γn [P(ω)] and therefore S2 , that is, Γ(S1 ) must be in Γn+1 [P(ω)]. a As already pointed out, the set P(ω) is the class of all possible extensions of the truth predicate if the arithmetic vocabulary is interpreted in the standard way, that is, by N. By iterated application of Γ, as in the lemma, more and more possible extensions are eliminated. The empty set, for instance, is eliminated in the first step: Ø ∈ / Γ1 [P(ω)] because, for instance, (0 = 0) ∈ Γ(S) for any S ⊆ ω. All extensions in Γ1 [P(ω)] already satisfy the axioms of the compositional theory ct. More formally, one can verify the following for all S ⊆ ω: If S ∈ Γ[P(ω)] then (N, S) ⊨ ct

166

classical symmetric truth

So all sets of numbers that do not yield an interpretation of the truth predicate that behaves appropriately on arithmetical sentences are eliminated in the first step. In the second step, all sets S with (N, S) ⊨ T┌T┌0 = 1┐┐, for instance, are excluded. Generally all models (N, S) with S ∈ Γ2 [P(ω)] will behave correctly for sentences in which the truth predicate is iterated once. Thus as the application of Γ to P(ω) is iterated further and further only more and more refined potential extensions of the truth predicate will be left. So, less formally speaking, the longer a set stays in the sets Γn [P(ω)] (for n as large as possible) the better the set is as an extension of the truth predicate. At the first limit level only those sets S are left that have not been eliminated at any finite level. So the ω-th level could be defined as \ Γω [P(ω)] := Γn [P(ω)]. n∈ω

The sets that survive all finite levels and make it into Γω [P(ω)] could be said to handle all finite iterations of the truth predicate adequately. But there is no such set: \ theorem 14.11. Γn [P(ω)] = Ø n∈ω

This result, though in slightly different terms, is due to Belnap and Gupta (1993). proof. The proof is merely a corollary of McGee’s (1985) ω-inconsistency Theorem 13.9. The result will be established by showing that the assumption \ Γn [P(ω)] 6= Ø n∈ω

leads to a contradiction. So assume there is a set \ S0 ∈ Γn [P(ω)]. n∈ω

Then there is an infinite chain S0 , S1 , S2 ,. . . with Γ(Sn+1 ) = Sn because S is in Γn [P(ω)] for arbitrarily large n. Thus for any k and n the set Γn (Sk ) is Sk−n . As in the proof of McGee’s theorem, I employ the binary primitive recursive function f that yields, when applied to a number n and a sentence ϕ, the sentence ϕ preceded by n truth predicates: f(n, ϕ) := TT. . . . T. ┌ϕ┐ | {z } n occurrences

revision semantics

167

This primitive recursive function is represented in Peano arithmetic by the symbol f . . Using the diagonal lemma I obtain a sentence γ such that the following equivalence is provable in Peano arithmetic: γ ↔ ∃x ¬Tf (x, ┌γ┐) . The equivalence therefore holds in any model of Peano arithmetic with arbitrary extensions of the truth predicate and, in particular, for standard models with one of the sets Sk from the series of sets as its extension: (N, Sk ) ⊨ γ ↔ ∃x ¬Tf (x, ┌γ┐) for all k ∈ ω . From this the next line follows immediately for arbitrary k ∈ ω: (N, Sk ) ⊨ ¬γ iff (N, Sk ) ⊨ ∀x Tf (x, ┌γ┐) . iff ∀n > k (N, Sn ) ⊨ γ

def. γ Lemma 14.9 (ii)

iff ∀n > k (N, Sn ) ⊨ ∃x ¬Tf (x, ┌γ┐) . iff ∀n > k ∃i (N, Sn ) ⊨ ¬Tf (i, ┌γ┐) . iff ∀n > k ∃i (N, Sn ) ⊨ ¬T T. . . . T. ┌γ┐ | {z }

def. γ

iff ∀n > k ∃i (N, Sn+i+1 ) ⊨ ¬γ

Lemma 14.9 (ii)

N standard model def. f

i occurrences

iff ∃n > k (N, Sn ) ⊨ ¬γ From the contradiction in the second and last lines I conclude the negation of the assumption: (N, Sk ) ⊨ γ for all k ∈ ω But this implies the second line and thus a contradiction as well. Hence there cannot be such a chain S0 , S1 , S2 ,. . . with Γ(Sn+1 ) = Sn such T n that Γ [P(ω)] is empty. a n∈ω

So in this sieving procedure every set S is eliminated as a candidate for the extension of the truth predicate after only finitely many levels. Revision theorists have proposed to start with some set S and then to apply the revision operator through all finite steps; at limit stages some clever tricks have been applied to define a new extension. The revision process at successor levels is very well understood and appraisals of the revision theory often focus on the definition of successor levels, while the definition of limit levels is somewhat artificial. From the mathematical point of view, what happens at the limit

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classical symmetric truth

levels is most interesting: all the mathematical complexity is generated at the limit levels, which are less well motivated (see Welch 2003). Here I will not go into a more serious discussion of the transfinite levels of revision semantics, first, because I have serious worries about the adequacy of the rules that are employed to define the extension at limit levels, and second, because I do not know of any appropriate axiomatization of transfinite levels of the revision process. Cantini (1996, p. 394) observes that Turner’s (1990) axiomatization of what is called stable truth, that is, of a certain important limit level, fails to be sound, as McGee’s ω-inconsistency applies to this system. Providing an axiomatization of limit levels of revision semantics thus remains a challenge (recent work by Bruni 2009 may open the way for an axiomatization). Investigations into the complexity of revision semantics reveal that it is much more complicated than Kripke’s theory of truth, which only relies on positive inductive definitions and thus on simple rules for limit levels: at limit stages unions of the previous stages are taken (see Section 15.1). So a measure of the success of an axiomatization of revision semantics would be the extent to which the complexity of the higher levels of the revision process can be captured by an axiomatization that reflects the recursion-theoretic complexity in its proof-theoretic strength. Concentrating on the finite levels of the revision process is worthwhile: one can thereby avoid all the difficult issues concerning limit levels and just capture the chief attractive feature of revision semantics, which is the revision process via the operator Γ. On the semantic approach, however, no definite model is generated by the revision process through all finite levels. Obviously it would be less than natural to focus on any particular set Γn [P(ω)] for some fixed n, and as we have just seen, there is no model at the first limit stage. Corresponding to the steps in the revision process there are theories fsn having exactly the standard models with extensions from Γn [P(ω)] as their standard models. On the proof-theoretic approach one can consistently take the union of all those theories. They will not have a standard model, as the previous theorem shows, but the theory is consistent nevertheless. To make this more precise, I define subsystems of fs with limited numbers of applications of nec and conec. definition 14.12 (fsn ). The system fs0 is pat, that is, Peano arithmetic formulated in the language LT with all induction axioms in LT .

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169

The system fs1 is fsn (that is, fs without nec and conec) with the axiom
∀x SentT (x) ∧ Bewpat (x) → Tx . For n > 1 the systems fsn are just like fs with the exception that in fsn the rule nec may be applied to at most n − 1 distinct sentences and that conec may be applied to at most n−1 distinct sentences. The systems and fs↾n that have only the induction axioms in L are defined analogously. Since the global reflection axiom for pat is a theorem of fs by Lemma 14.5, the union of all systems fsn with n ∈ ω is just fs. The Axiom fs1 is not needed in fs1 as it is a consequence of the global reflection principle for pat. The systems fsn and fs1 are actually very similar. But fs1 does not prove the truth of all identity axioms. For the proof of the global reflection, that is, in the proof of Lemma 14.5, nec is used to show the truth of identity axioms. So fs1 is very slightly stronger than fsn. The definition of the systems fsn is largely independent of the logical calculus in which the systems are formulated. In what follows it is convenient to standardize proofs and to assume that the systems fsn are formulated in a Hilbert-style calculus, that is, in a linear axiomatic calculus. Linear systems have the advantage that a formulae never need occur more than once in a proof, while, for instance, in proofs in tree forms of Natural Deduction subproofs sometimes have to copied to other places in the proof. This means that if fs is formulated in such a proof system with a tree structure, nec might have to be applied more than once to the same sentence; the same holds for conec. So keep some linear axiomatic proof system fixed. In this setting the following observation holds: lemma 14.13. If fsn ` ϕ holds, then there is a proof of ϕ in fsn with at most n−1 applications of nec and at most n−1 applications of conec. The proof is trivial. If nec is applied to ϕ to infer T┌ϕ┐, then the proof of T┌ϕ┐ does not have to be repeated again and can be henceforth be used in the remaining parts of the proof. A similar remark applies to conec. There is another way to reduce the number of applications of nec and conec. Assume, for instance, there are two applications of nec in different subproofs that are independent of each another. Then one can merge the two subproofs so that only one application of nec is needed. Hence in a given proof one need only count consecutive, not parallel, applications of the rules. This is the content of the following lemma.

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classical symmetric truth

lemma 14.14. Assume fsn ` ϕ. Then there is a proof P of ϕ in fsn with at most n−1 applications of nec and at most n−1 applications of conec such that for any two subproofs P1 and P2 of P each ending with an application of nec or each ending with an application of conec either P1 is a subproof of P2 or P2 is a subproof of P1 . proof. Assume that there is a subproof ending with an application of nec to ψ1 and another subproof ending with an application of nec to ψ2 , so that neither is a subproof of the other. Combine the proofs of ψ1 and ψ2 to a proof of ψ1 ∧ ψ2 and apply nec to deduce T┌ψ1 ∧ ψ2┐. Using Axiom fs3 conclude T┌ψ1┐ and T┌ψ2┐. In the new proof the two relevant applications of nec have been replaced with one application of nec. The argument for conec is similar. a The connection between revision semantics and the systems fsn can now be stated as follows: theorem 14.15. The following equivalence obtains for all sets S ⊆ ω: S ∈ Γn [P(ω)] iff (N, S) ⊨ fsn The equivalence also holds for the systems fs↾n in place of fsn . proof. The system fs0 does not impose any restrictions on the extension of the truth predicate; thus any set S of numbers can be used in a model (N, S) of fs0 . This yields the equivalence for n = 0. For n = 1, I reason in the following way. If S ∈ Γ[P(ω)], then in (N, S) the truth predicate is interpreted as the set of sentences true in some model (N, S0 ) with S0 some set of numbers and it is not hard to verify (N, S) ⊨ fs1 . In particular, the global reflection principle for pat ∀x SentT (x) ∧ Bewpat (x) → Tx




holds in (N, S) because all theorems of pat hold in (N, S0 ), as pat is fs0 and any model (N, S0 ) verifies the fs0 -theorems by the case n = 0. To prove the other direction for n = 1, I assume (N, S) ⊨ fs1 and define the following set: S1 := {k : (N, S) ⊨ T┌Tk┐} = {k : Tk ∈ S} The set S1 contains k if and only if there is a closed term t such that Tt is in S and the value of the term t (in the standard model N) is k. For if Tk is in S

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171

and the value of t is k, then Tt is in S as (N, S) verifies the global reflection principle for pat and thus S is closed under identity logic. (Incidentally, this was the chief reason to add the reflection principle to fs1 rather than to define fs1 as fsn.) The identity S = Γ(S1 ) is then proved inductively. For atomic sentences Tt I reason as follows, assuming that k is the value of the term t: Tt ∈ S iff k ∈ S1

def. S1

iff (N, S1 ) ⊨ Tt iff (Tt) ∈ Γ(S1 ) For complex sentences Axioms fs1–fs6 are used. For instance, the induction step for negation can be proved in the following way: ¬ϕ ∈ S iff ϕ ∈ /S

Axiom fs2

iff ϕ ∈ / Γ(S1 )

ind. hyp.

iff ¬ϕ ∈ Γ(S1 )

def. Γ

The other cases are treated in a similar way. Hence S = Γ(S1 ) is established and therefore S is an element of Γ[P(ω)]. For larger n the claim is established inductively, by simultaneously proving both directions of the claim. So I assume the equivalence in the statement of the theorem holds for n. First I show the left-to-right direction of the claim. So let S be an element of Γn+1 ; I need to show (N, S) ⊨ fsn+1 . By Lemma 14.10 (N, S) is already a model of fsn , so it mainly remains to show that (N, S) verifies an additional application of nec and conec. Assuming that the additional application of conec is an application to a sentence T┌ψ┐, I proceed as follows: fsn ` T┌ψ┐ implies ∀S1 ∈ Γn [P(ω)] (N, S1 ) ⊨ T┌ψ┐ implies ∀S1 ∈ Γ

n−1

[P(ω)] (N, S1 ) ⊨ ψ

induction hyp. Lemma 14.8

implies ∀S1 ∈ Γ [P(ω)] (N, S1 ) ⊨ ψ

Lemma 14.10

implies (N, S) ⊨ ψ

S ∈ Γn [P(ω)]

n

Hence S is an element not only of Γn+1 [P(ω)] but also of Γn−1 [P(ω)]. As S ∈ Γn [P(ω)] there must be an S1 with Γ(S1 ) = S. To show that (N, S) verifies an additional application of nec, I assume fsn ` ϕ and show (N, S) ⊨ T┌ϕ┐ in the following way: fsn ` ϕ implies (N, S1 ) ⊨ ϕ implies (N, S) ⊨ T┌ϕ┐

induction hyp., S1 ∈ Γn [P(ω)] Γ(S1 ) = S

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classical symmetric truth

This shows that (N, S) with S ∈ Γn+1 [P(ω)] is a model of fsn with one additional application of nec or conec. A proof in fsn+1 , however, can consist in an fsn -proof plus an additional application of nec and an additional application of conec in arbitrary order. Even then the claim follows as only for nec need one go to sets in Γn+1 [P(ω)]. For conec one does not have to go a level higher. This concludes the proof of the induction step for the left-to-right direction. To show the right-to-left direction of the theorem for n > 1, I assume (N, S) ⊨ fsn+1 and define the set S1 as in the case n = 1 in the following way: S1 := {k : Tk ∈ S} In fact, S1 can be shown to contain only sentences, so S1 is the set {k : T┌ϕ┐ ∈ S} = {ϕ : (N, S) ⊨ T┌T┌ϕ┐┐}. As in the case n = 1, one can show Γ(S1 ) = S. To complete the proof of S ∈ Γn+1 [P(ω)], it suffices to show S1 ∈ Γn [P(ω)]. But from fsn ` ϕ I infer fsn+1 ` T┌ϕ┐ and hence (N, S) ⊨ T┌ϕ┐, and from this (N, S1 ) ⊨ ϕ. Hence (N, S1 ) is a model of fsn and by induction hypothesis S1 is in Γn [P(ω)]. a The proof of Theorem 14.15 for the systems fsn with the rule conec removed is easier than the proof for the full systems. This follows from the observation that I did not make use of conec in the proof of the induction step in the right-to-left direction. I will now list some consequences of this adequacy theorem. corollary 14.16. The system fs is consistent. proof. As any proof in fs can only contain finitely many applications of the rules nec and conec, any proof in fs is a proof in a subsystem fsn . Hence to prove the consistency of fs it will suffice to establish the consistency of each subsystem fsn . To this end fix an arbitrary set S ⊆ ω. By the theorem (N, Γn (S)) is a model of fsn ; this implies the consistency of fsn . a Below a proof-theoretic argument for the consistency of fs will also be provided. Since fs is a symmetric system, closed under nec and conec, this corollary implies together with Lemma 13.4 that the liar sentence is independent from fs: neither the liar sentence nor its negation are provable in fs.

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173

corollary 14.17. The system fs is ω-inconsistent. proof. By Theorem 14.15 the ω-models of fs are the models (N, S) with S ∈ T n Γ [P(ω)]. Since there are no such models by Theorem 14.11, the system fs

n∈ω

does not have any ω-models. Generally, theories without ω-models are ωinconsistent. I do not prove this general result here, which can be proved via the omitting-types theorem (see Chang and Keisler 1990, p. 82, Proposition 2.2.13 for a proof). a Although fs is ω-inconsistent, it does not prove any incorrect arithmetical claims. corollary 14.18. The system fs is arithmetically sound; that is, if a sentence ϕ without the truth predicate is provable in fs, then ϕ holds in the standard model. Again, I will provide a proof-theoretic argument for this claim later; the model-theoretic argument is a direct consequence of the adequacy theorem. proof. Assume ϕ is a sentence of L that is provable in fs. Then there must be an n such that fsn ` ϕ. It then follows from Theorem 14.15 that (N, Γn (S)) ⊨ ϕ holds for any set S of natural numbers, and thus that the arithmetical sentence is true in the standard model N. a The discussion up to this point has been based on the standard model N of arithmetic. The revision operator Γ is defined with the help of the standard model. One might try to generalize revision semantics to nonstandard models. Here I will only hint at some directions for future research and on the benefits of developing revision semantics for nonstandard models. Nonstandard models of the language LT take the form (𝔐, S), where 𝔐 is a nonstandard model of Peano arithmetic and S is a set of elements of the domain |𝔐| of 𝔐. If revision semantics is to be applied to nonstandard models an analogue of the revision operator Γ is needed. So given a model (𝔐, S) one seeks a new model (𝔐, S1 ) where S1 contains the set of sentences true in (𝔐, S); S1 should also include nonstandard sentences in the sense of 𝔐, so that (𝔐, S1 ) satisfies the axioms fs1–fs6 and, in particular, Axiom fs2 for negation. definition 14.19 (type-free satisfaction class). A type-free satisfaction class for (𝔐, S) is a set S1 ⊆ |𝔐| such that (𝔐, S1 ) ⊨ fs1 and the following condition obtains for all sentences ϕ of LT : If (𝔐, S) ⊨ ϕ, then ϕ ∈ S1

174

classical symmetric truth

So S1 contains all standard sentences that hold in (𝔐, S). If one is interested in models for the system fs, one would impose the additional condition that (𝔐, S1 ) should verify all the induction axioms including those in the language LT . Such type-free satisfaction classes could be called inductive type-free satisfaction classes. For the moment being, I will briefly indicate some features of type-free satisfaction classes that are not necessarily inductive. Since a type-free satisfaction class is also a (typed) satisfaction class, Lachlan’s theorem 8.24 imposes a first difficulty in the development of nonstandard revision semantics: the revision operation cannot be applied to any model (𝔐, S). One might hope to escape this problem by starting with a countable and recursively saturated model (𝔐, S). In such a model all recursive types over (𝔐, S) in the language LT and not only in the language L are realized. The existence of a type-free satisfactionclass S1 for (𝔐, S) could then be established by a proof similar to the one by Kotlarski et al. (1981). This construction, however, does not yield a unique type-free satisfaction class. So in contrast to the standard case, there will be many, in fact uncountably many, different type-free satisfaction classes that can be seen as the result of applying a revision operator. Therefore the revision process could branch. The new model (𝔐, S1 ), where S1 is a type-free satisfaction class for (𝔐, S), may again fail to be recursively saturated; so the revision process would come to a halt. To avoid this, one might try to choose a type-free satisfaction class S1 in such a way that (𝔐, S1 ) is also recursively saturated; this is possible using standard techniques for constructing satisfaction classes. Using revision semantics for nonstandard models it should be possible to prove the conservativity of fs↾ over Peano arithmetic. For this result, however, there is a simpler alternative proof available: the cut-elimination argument of Section 8.1 for the conservativity of ct↾ over Peano arithmetic can be adapted to fs↾. By making use of the techniques developed below in the section on the proof theory of fs. Another application of nonstandard revision semantics could shed some light on the role of the rule conec. Using nonstandard revision semantics the rule conec could be shown to be superfluous in the system fs: every theorem of fs can be derived without invoking conec. One could proceed in the following fashion: If ϕ is not derivable in fs↾n without conec, there is a countable and recursively saturated model (𝔐, S) ⊨ fs↾n − + ¬ϕ

proof theory of the friedman–sheard theory

175

where fs↾n − is the system fs↾n without conec. Now applying nonstandard revision one can prove the existence of a type-free satisfaction class S1 for (𝔐, S) and thereby establish the following claim: (𝔐, S1 ) ⊨ fs↾n+1 − + T┌¬ϕ┐ From this one can conclude fs↾n+1 ⊬ ϕ. Hence the rule conec is dispensable in fs↾. By considering nonstandard revision semantics one might also hope to construct models for the entire theory fs↾. In Theorem 14.11 I showed that no possible extension S ⊆ ω survives the selection procedure generated by the revision operator Γ. Consequently fs↾ and fs do not have ω-models. But fs↾ and fs do have nonstandard models. Perhaps they can be conceived as natural limit models of a revision sequence S0 , S1 , S2 ,. . . based on a nonstandard model. This could yield a more constructive approach to models of fs, avoiding the use of the compactness theorem. Moreover, these limit models could be seen as limits of a revision process that is more direct than the limit constructions for standard models, which have to be very complicated. But so far no results have been obtained on these ‘natural’ models of fs. If the focus is shifted to the theory fs with full induction, the situation becomes even more complicated. Results by Kotlarski and Ratajczyk (1990a) and(1990b) on inductive satisfaction classes may be applicable. 14.2 Proof theory of the Friedman–Sheard theory To analyse the strength of fs I will relate it to other systems. In particular, in order to obtain an upper bound for the strength of fs I will reduce it to other well-understood theories. Now I could try to define the truth predicate of fs in another theory, for instance, in one of the theories rt<γ . There is, however, an obvious obstacle to this approach: no ω-consistent theory can define the truth predicate of fs. In other words, there is no Lconservative relative interpretation of fs in an ω-consistent theory because the ω-inconsistency of fs carries over to target theory via an L-conservative relative interpretation. This yields the following result: lemma 14.20. If the truth predicate of fs↾ or fs is definable in a truth theory, then that theory is ω-inconsistent. To analyse fs, I would like to reduce fs to a system of iterated truth and determine how many levels of iterated classical compositional truth are implicit

176

classical symmetric truth

in fs and how many levels of iterated classical typed truth it takes to match fs. As the systems rt<γ of ramified truth from Section 9 are all ω-consistent, the truth predicate of fs is undefinable in any of them, although, as I shall show, many of them are much stronger than fs in other respects. Because of Lemma 14.20 I must resort to some means other than truthdefinability for the reduction of fs to one of the theories rt<γ . A direct definition of a relative interpretation of fs in an ω-consistent theory does not look easy, as the most straightforward relative interpretations between truth theories are truth-definitions, that is, relative interpretations that do not relativize the quantifiers or affect the arithmetical vocabulary. Instead I propose to break down fs into the subsystems fsn as defined in the previous section. These subsystems are ω-consistent and therefore more amenable to a reduction to the systems rt<γ . In fact, taking ideas from the revision semantics model construction in the previous section I will try to obtain a reduction of fs to the system rt<ω of ramified truth for all finite levels. As defined above, the language Ln contains all the truth predicates Tk with k ≤ n, while L



rt3n ∀x ∀y Sent




rt4n ∀x ∀y Sent
proof theory of the friedman–sheard theory The system rt<ω is the theory

S

n∈ω

177

rt
First I will show how to reduce the systems rt
178

classical symmetric truth

I do not provide a full proof of the induction step; its proof can be broken down into several lemmata covering the connectives and quantifiers. Here I give the negation case as an example. fs1 ` ∀s ∀t SentT (s◦ ) ∧ t◦ = ¬. (s◦ ) → (Tt ↔ ¬Ts)





fs2 ` T┌∀s ∀t SentT (s ) ∧ t = ¬. (s ) → (Tt ↔ ¬Ts) ┐
fs2 ` ∀t ∀s SentT (s◦ ) ∧ t◦ = ¬. (s◦ ) → (TT. t ↔ T¬. T. s)
fs2 ` ∀t ∀s SentT (s◦ ) ∧ t◦ = ¬. (s◦ ) → (TT. t ↔ ¬TT. s) ◦

◦

◦

fs2 nec fs2–fs4, Lem. 14.22 fs2

Assuming the induction hypothesis, I derive the following claim: fs2 ` ∀t ∀s SentT (t) ∧ t◦ = ¬. s◦ → ((TT. s ↔ Ts◦ ) → (TT. t ↔ ¬Ts◦ ))




Using fs2 once more, I obtain the desired clause for the induction step concerning negated sentences: fs2 ` ∀t ∀s SentT (t) ∧ t◦ = ¬. s◦ → ((TT. s ↔ Ts◦ ) → (TT. t ↔ Tt◦ ))




The cases of other connectives and the quantifiers are dealt with in a similar fashion. So by induction on the length of sentences one can infer the desired result as Lfs 0 is just the language L of arithmetic: ◦ ◦ fs2 ` ∀t Sentfs . t ↔ Tt ) 0 (t ) → (TT




The multiple use of nec can be justified by appealing to Lemma 14.14. Induction step n 7→ n+1. By induction hypothesis one can assume the following:
◦ ◦ fsn+2 ` ∀t Sentfs . t) ↔ Tt ) n (t ) → (T(T From this the following line is obtained by Lemma 14.22, an application of nec, and by the axioms of fs1 :
◦ fsn+3 ` ∀s ∀t Sentfs . s → (TT. t ↔ TT. s) n (s ) ∧ t = T Performing a formalized induction on the length of sentence as in the base case, I infer the induction step:
◦ ◦ fsn+3 ` ∀t Sentfs . t ↔ Tt ) n+1 (t ) → (TT

a

The Lemma implies the following generalization of Lemma 14.22, which can be established by an induction on the length of the formula ϕ(x1 , . . . , xn ).

proof theory of the friedman–sheard theory

179

corollary 14.24. For all formulae ϕ(x1 , . . . , xn ) in the language Lfs n the following holds:
fsn+2 ` ∀t1 . . . ∀tn T┌ϕ(t. 1 , . . . , t. n )┐ ↔ ϕ(t1 ◦ , . . . , tn ◦ ) Using Lemma 5.2 I inductively define a primitive recursive translation function hn from the hierarchical system rt 0 the functions are defined in the following way:  k, if k is of the form s = t for terms s, t      0 = 1 if k is not a sentence of Ln     Sent
180

classical symmetric truth

corresponding Axiom fs1. Axiom rt7n can be treated very much like the final Axiom rt8n , which is the only one for which I will give an explicit proof. For Axiom rt8n I begin by observing that for i < k and formula ϕ of L



The following is merely a logical weakening of the theorem above:  ∀t ∀i < k Sent k for all closed terms t under the assumption i < k and Sent
To prove the equivalence of Th. i (t◦ ) and TT. h. i (t) the system fsi+2 is required. As i < k < n obtains, the system fsn will suffice. Next I apply Axiom fs3: ◦ ◦ Sent
Using the definition of the translation functions I infer the following sentence under the same assumptions: Sent
a

proof theory of the friedman–sheard theory

181

So the theory rt<ω of the Tarskian hierarchy for all finite levels, that is, the union of all systems rt
182

classical symmetric truth

In what follows I will make use of the following simple properties of these translation functions: lemma 14.28. The following items hold for all n ∈ ω: (i) gn (k) ∈ L



Sketch of proof. The proof is as for Lemma 8.4. In addition to the atomic formulae of L the language L



Finally the following result yields the reduction of fs to the hierarchical system rt<ω : theorem 14.31. For all n ∈ ω and formulae ϕ of LT , the following implication obtains: If fsn ` ϕ, then rt<2n ` gn (ϕ) Actually I will show a stronger claim, which implies the theorem, namely I will show the following implication for all i ≤ n and all formulae ϕ of LT : If fsi ` ϕ, then rt<2n ` gi (ϕ) ∧ gi+1 (ϕ) ∧ . . . ∧ g2n−i (ϕ)

proof theory of the friedman–sheard theory

183

proof. For the proof it is convenient to assume that the proofs are formulated in a linear axiomatic system and that nec is never applied to the same sentence twice, and also that conec is not applied to the same sentence twice. This assumption ensures that proofs in fsi contain at most i − 1 many applications of nec and at most i − 1 many applications of conec for i ≥ 1. For fixed n I prove the claim by induction on i. Case i = 0. I prove for the following implication for all k ∈ ω: If fs0 ` ϕ, then rt<2n ` gk (ϕ) The theory fs0 , which I also call pat, is Peano arithmetic formulated in the language LT with the truth predicate. The theory does not contain any nonlogical axioms with the truth predicate except for the induction axioms in LT . Hence if all subformulae Tt of ϕ are uniformly substituted with some other formula, the result will be still derivable in pat and thus a fortiori rt<2n ` gk (ϕ) must hold for all k. Case i = 1. For this case the following implication must be established: (14.6)

If ϕ is an axiom of fs1 , then rt<2n ` g1 (ϕ) ∧ . . . ∧ g2n−1 (ϕ)

The claim of the theorem for i = 1 then follows from (14.6) because the translation functions hk preserve the logical structure of formulae and therefore closure under logic is immediate. The translation of the first Axiom fs1 can be derived in the following way for any k < 2n: rt<2n ` ∀s ∀t (Tk (s=. t) ↔ s◦ = t◦ )

Axiom rt1n

rt<2n ` ∀s ∀t (Tk g (s=. t) ↔ s = t ) .k
rt<2n ` gk+1 ∀s ∀t (T(s=. t) ↔ s◦ = t◦ ) ◦

◦

Lemma 14.28 (iv) definition gk+1

Next I turn to the Axiom fs2 for negation, whose translation can be proved in the following fashion. Assume again k < 2n. Then use Axiom rt2n and Lemma 14.28 (ii) in the first line; and then apply the definition of gk :
rt<2n `∀x SentT (x) → (Tk ¬. g (x) ↔ ¬Tk g (x)) .k .k
rt<2n `∀x SentT (x) → (Tk g (¬. x) ↔ ¬Tk g (x)) .k .k

rt<2n `gk+1 ∀x SentT (x) → (T¬. x ↔ ¬T(x)) The cases for the other connectives and the quantifiers, that is, Axioms fs3 and fs6, are handled in an analogous manner.

184

classical symmetric truth

In addition to the Axioms fs1–fs6, fs1 also has the global reflection principle for pat as an axiom. To deal with this axiom I proceed in the following way: Theorem 8.39 states that ct, that is, rt<1 , proves the global reflection principle for Peano arithmetic. This can be generalized for k < 2n: rt<2n ` ∀x Sent<2n (x) ∧ Bewpat (x) → Tk x




Peano arithmetic proves that any gk maps any theorem of pat to a sentence of L



This concludes the proof of (14.6). Case i 7→ i + 1. I assume that the claim holds for i and that a proof of ϕ in fsi+1 is given. Consider the first subproof that is not a proof in fsi . This proof ends either with the nth application of nec or with the nth-application of conec. Assume it ends with an application of nec to the sentence ψ, so fsi ` ψ obtains. Using the induction hypothesis I proceed in the following way: rt<2n `gi (ψ) ∧ gi+1 (ψ) ∧ . . . ∧ g2n−i (ψ) rt<2n `Ti g (┌ψ┐) ∧ Ti+1 g (┌ψ┐) ∧ . . . ∧ T2n−i g (┌ψ┐) .i . i+1 . 2n−i rt<2n `gi+1 (T┌ψ┐) ∧ gi+2 (T┌ψ┐) ∧ . . . ∧ g2n+1−i (T┌ψ┐)

Lemma 14.30

Now suppose that an application of conec to T┌χ┐ follows the application of nec above or the first subproof of the proofs of ϕ that is not a proof in fsi ends with an application of conec to T┌χ┐. In either case the next line obtains (and only if nec is applied first could one add an additional conjunct g2n+1−i (T┌χ┐), and when nec is not applied first one could add a conjunct gi (T┌χ┐); but these additional conjuncts are not needed): rt<2n `gi+1 (T┌χ┐) ∧ . . . ∧ g2n−i (T┌χ┐) rt<2n `Ti g (┌χ┐) ∧ . . . ∧ T2n−i−1 g (┌χ┐) .i . 2n−i−1 rt<2n `gi (χ) ∧ . . . ∧ g2n−(i+1) (χ)

def. of g .k Lemma 14.30

the friedman–sheard axiomatization

185

In any event, whether only nec or only conec is applied n times or whether they are both applied n times in whatever order, the following implication holds for all i ≤ n and all formulae ϕ of LT : If fsi+1 ` ϕ, then rt<2n ` gi+1 (ϕ) ∧ . . . ∧ g2n−(i+1) (ϕ) This completes the proof of the induction step and thereby the proof of the theorem. a Since gi (ϕ) is identical to ϕ if ϕ is arithmetical, it follows from the theorem that rt<ω proves all arithmetical theorems of fs. Together with Theorem 14.26 this yields the following result: corollary 14.32. The systems fs and rt<ω prove the same arithmetical theorems, as does fs bar the rule conec. Since rt<ω is arithmetically sound, the corollary implies the arithmetical soundness of fs, which gives another proof of Corollary 14.18. The proof-theoretic analysis of fs can be used to obtain information about certain variants of this system. For instance, one can embed fs↾ with limited numbers of applications of nec and conec in systems of iterated truth also with only arithmetical induction. Through an iteration of the proof of the consistency of ct↾ in Theorem 8.12 these systems can be shown to be conservative over Peano arithmetic; it follows that fs↾ is conservative over pa as well. 14.3 The Friedman–Sheard axiomatization I will now present the original axiomatization used by Friedman and Sheard (1987). The point of providing this alternative axiomatization is not so much to make the connection with Friedman and Sheard’s original paper explicit but rather to illustrate what can be proved in fs and that fs has another elegant and attractive axiomatization. Friedman and Sheard (1987) work with ¬ and → as primitive connectives, while LT has ¬, ∧, and ∨ as connectives. To facilitate a direct comparison, I will treat Friedman and Sheard’s → as a defined symbol. The function symbol → . stands for the function sending formulae ϕ and ψ to the formula ¬ϕ∨ψ. I have adapted Friedman and Sheard’s axioms to the notation used throughout this book and made some minor changes to facilitate the presentation in my notational framework.

186

classical symmetric truth

definition 14.33. The system fso is given by the following axioms and rules: axioms BaseT pat-Refl T-Cons T-Comp U-Inf E-Inf rules of inference T-Intro T-Elim ¬T-Intro ¬T-Elim

the axioms of Peano arithmetic with full induction in LT
∀x ∀y SentT (x) ∧ SentT (y) → (T(x→ y) → (Tx → Ty))
. ∀x SentT (x) ∧ Bewpat (x) → Tx
∀x SentT (x) → (¬(Tx ∧ T¬. x))
∀x SentT (x) → (Tx ∨ T¬. x)
∀v ∀x SentT (∀. vx) → (∀t T(x(t/v)) → T(∀. vx))
∀v ∀x SentT (∀. vx) → (T(∃. vx) → ∃t T(x(t/v))) from ϕ infer Tϕ (This is nec.) from Tϕ infer ϕ (This is conec.) from ¬ϕ infer ¬Tϕ from ¬Tϕ infer ¬ϕ

Instead of pat-Refl Friedman and Sheard (1987) use the global reflection axiom for a theory pre weaker than pat. The axiom schema pat-Refl proves the consistency of Peano arithmetic while Friedman and Sheard’s reflection principle is provable in Peano arithmetic. Since the resulting system will prove the global reflection principle for pat anyway, I take it as an axiom to avoid additional notational difficulties. To prove the equivalence of the two systems fs and fso I will use the following lemma. lemma 14.34. In the system fso the following sentences are provable:
(i) ∀x ∀y SentT (x∧. y) → (T(x∧. y) ↔ T(x) ∧ T(y))
(ii) ∀v ∀x SentT (∀. vx) → (T(∀. vx) → ∀t T(x(t/v)))
(iii) ∀s ∀t T(s=. t) ↔ s◦ = t◦ proof. I prove the two directions of the equivalence claim (i) separately: fso ` ∀x ∀y SentT (x∧. y) → Bewpat ((x∧. y)→ . x)
→ T((x∧. y)→ . x)
→ (T(x∧. y) → Tx)




def. pat pat-Refl BaseT


The claim ∀x ∀y SentT (x∧. y) → (T(x∧. y) → Ty) for the second conjunct can be proved in a similar way, of course.

the friedman–sheard axiomatization For the other direction I reason as follows: 
 fso ` ∀x ∀y SentT (x∧. y) → Bewpat x→ . (y→ . (x∧. y))
 → T(x→ . (y→ . (x∧. y)))
 → Tx → T(y→ (x∧ y) . .
 → Tx → (Ty → T(x∧. y))
 → Tx ∧ Ty → T(x∧. y) For (ii) I reason in fso as follows: 
 fso ` ∀v ∀x SentT (∀. vx) → ∀t Bewpat ∀. vx→ . x(t/v)
 → ∀t T ∀. vx→ . x(t/v)
 → ∀t T∀. vx → T(x(t/v))
 → T(∀. vx) → ∀t T(x(t/v))

187

def. pat pat-Refl BaseT BaseT logic

def. pat pat-Refl BaseT logic

I prove the two directions of item (iii), that is, Axiom fs1, separately. For the proof of the left-to-right direction I use the completeness of pat with respect to atomic arithmetical sentences. This yields the first line: fso `∀s ∀t ¬s◦ = t◦ → Bewpat (¬. s = t)
∀s ∀t ¬s◦ = t◦ → T(¬. s = t)
∀s ∀t ¬s◦ = t◦ → ¬T(s = t)
∀s ∀t T(s=. t) → s◦ = t◦


pat-Refl T-Cons contraposition

To prove the other direction of (iii) I again employ the fact that pat proves all true closed equations:
fso `∀s ∀t s◦ = t◦ → Bewpat (s=. t)
∀s ∀t s◦ = t◦ → T (s=. t)

pat-Refl

This concludes the proof of (iii).

a

I can now establish the equivalence of Friedman and Sheard’s axiomatization with mine. theorem 14.35. The systems fs and fso prove the same theorems.

188

classical symmetric truth

proof. To show that fs is a subsystem of fso it suffices to show that all axioms of fs1 are theorems of fso because the rules nec and conec of fs are also rules of fso. Axiom fs1 is a theorem of fso by Lemma 14.34 (iii). The Axioms T-Cons and T-Comp together imply Axiom fs2 by Lemma 13.6, which yields the following: fso ` ∀x SentT (x) → (T(¬. x) ↔ ¬Tx)




Axiom fs3 for conjunction is covered by Lemma 14.34 (i); Axiom fs4 for disjunction can be established in a similar way. Axiom fs5, that is, ∀v ∀x SentT (∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v)))




is a consequence of Axiom U-Inf and Lemma 14.34 (ii). The corresponding Axiom fs6 for existentially quantified sentences is proved in a similar way. It remains to prove that all axioms and rules of the Friedman–Sheard axiomatization fso can be derived in fs. By Lemma 14.5 the global reflection principle for the system pat, that is, pat-Refl, can be established in fs. To see how the rule ¬T-Intro can be derived in fs assume that the sentence ¬ϕ has been derived. By nec, that is, T -Intro, I obtain T┌¬ϕ┐, and then using Axiom fs2 I conclude ¬T┌ϕ┐. The rule ¬T-Elim is obtained in a similar way. The Axioms T-Cons and T-Comp are consequences of Axiom fs2 by Lemma 13.6. a 14.4 Expressing necessitation via reflection If the rules nec and conec are dropped from fs, a much weaker system is obtained: the resulting system fsn is only as strong as the theory ct or rt0 of (uniterated) compositional truth. This follows from the proof-theoretic analysis of fs and its subsystems fsn . The full system fs is much stronger than ct or rt0 . It is as strong as rt<ω in the sense explained in Section 14.2. For the embedding of rt<ω in fs, that is, for the proof of Theorem 14.27, I made essential use of the rule nec; but I did not use conec. Hence, if conec is removed from fs, the system remains as strong as the full system fs, at least in the sense that no arithmetical theorems are lost by dropping conec. The use of nec in the reduction of rt<ω to fs is clear: it serves the purpose of adding a further layer of truth, so that an application of nec adds a further level of compositional truth in the Tarskian hierarchy of languages.

necessitation via reflection

189

The rule conec, in contrast, does not so obviously allow one to go beyond fsn: conec seems to be a rule for removing a layer of the Tarskian hierarchy. Sheard (2001), however, proved an ingenious theorem, according to which conec can be very powerful: if nec is deleted from fs the resulting system retains the same arithmetical content as fs itself, although there is no axiom or rule left that could be used for iterating truth in a straightforward way. Leigh and Rathjen (2010) took Sheard’s method even further. So in a certain context, conec does have strong arithmetical consequences. But Sheard also notes that in the context of certain weak axioms for truth conec does not yield nec, cons, or comp. So a theory based on conec can add much arithmetical strength and produce a theory as strong as fs without proving other fs-axioms and rules: conec is very effective for proving arithmetical theorems, but weak at generating truth-theoretic consequences. The rule nec, in contrast, seems much better at producing attractive truththeoretic theorems. For instance, in Theorem 14.23 I used nec to prove further uniform disquotation sentences in fs and one can see from this result that adding nec to the system fsn without the rules nec and conec yields an attractive type-free theory of truth. In what follows I will investigate how and whether one can improve on nec as a truth-theoretic rule. In particular, I will look into the prospects of adding rules or axioms to fs that allow one to prove claims about transfinite iterations of truth. The starting point for the discussion is again the system fsn, that is, fs without nec or conec. Omitting conec will facilitate the following discussion as conec would deserve special treatment as well. Hence, when discussing how the rule nec might be replaced with suitable axioms, I will do so within the context of a theory without conec; and therefore I will consider how nec can be replaced within the theory fs-conec, that is, in the theory fs without conec. The result I will obtain can then also be used to show that the axioms nec will be replaced with can also serve this purpose in the presence of conec. Instead of adding nec to fsn I will add global reflection principles. The first step is to augment fsn with the global reflection principle for this system:
∀x SentT (x) ∧ Bewfsn (x) → Tx Next, this addition of the global reflection principle is iterated. To this end I define fsr1 as fsn and fsrn+1 as fsrn plus the global reflection principle for fsrn :
∀x SentT (x) ∧ Bewfsrn (x) → Tx

190

classical symmetric truth

I assume that the provability predicate is defined in a natural way. As only one axiom is added at every step, this does not pose a problem. Finally, fsr is defined as the union of all the systems fsrn : fsr :=

[

fsrn

n∈ω

Now I can prove that the iterated global reflection principles yield nec: lemma 14.36. The system fsr is closed under nec. proof. Obtaining an application of the rule nec from the reflection principle is straightforward: any proof in fsr is a proof in some system fsrn for some n ∈ ω. From fsrn ` ϕ I infer pa ` Bewfsrn (┌ϕ┐) as Bewfsrn (x) numerates (weakly represents) the property of being provable in fsrn . Using the reflection axiom of fsrn+1 I conclude fsrn+1 ` T┌ϕ┐, so fsr is closed under applications of nec. a lemma 14.37. The system fs-conec proves all reflection principles (14.7)

∀x SentT (x) ∧ Bewfsrn (x) → Tx




for all n ∈ ω. Proof by induction on n. I will only sketch the proof. For the case n = 1, I need to prove in fs-conec that (the universal closures of) all theorems of fsr1 = fsn are true. In Lemma 14.5 I have shown how to prove that all theorems of pat are true and how to prove closure under logic in fs without using conec. So it remains to establish the truth of the Axioms fs1–fs6. Since there are only finitely many axioms, nec can be applied to each of them to conclude that they are all true (actually one application of nec will suffice by Lemma 14.14). For the induction step assume that (14.7) is provable in fs-conec and apply nec to (14.7). Because the truth of all other axioms of fsrn has been established already and closure under logic can be proved as above, it follows that fs-conec proves
∀x SentT (x) ∧ Bewfsrn+1 (x) → Tx . The two lemmata imply the equivalence of fsr and fs-conec. theorem 14.38. The systems fsr and fs-conec have the same theorems.

a

necessitation via reflection

191

Whereas it is not easy to iterate a rule like nec into the transfinite, it is easy to do so for reflection principles. For fsr I do not have to go very far. There is no need to employ the sophisticated machinery of transfinite recursive progressions developed by Turing (1939) and Feferman (1962), for the simple reason that in the current setting iterations into the transfinite face triviality: corollary 14.39. The system fsr plus global reflection, that is, fsr with the axiom
(14.8) ∀x SentT (x) ∧ Bewfsr (x) → Tx . is inconsistent. Sketch of proof. This is a corollary to McGee’s (1985) ω-inconsistency theorem, that is, Theorem 13.9. It is easy to see that fsr, that is, the system fs-conec, satisfies the conditions of McGee’s theorem. Hence by McGee’s theorem there is a formula ϕ(x) of LT with (14.9)

fsr ` ϕ(n) for all n ∈ ω

but also (14.10)

fsr ` ¬∀x ϕ(x).

The proof of (14.9) can be formalized in pa in a straightforward way, which yields the following: (14.11)

pa ` ∀t Bewfsr (┌ϕ┐(t/┌x┐))

Using the reflection principle (14.8), I conclude the following: fsr + (14.8) ` ∀t T┌ϕ┐(t/┌x┐) Using Axiom fs5 I conclude fsr ` T┌∀x ϕ(x)┐. To derive the inconsistency without conec, some additional steps are required. An application of nec to (14.10) gives T┌¬∀x ϕ(x)┐. So one has an internal inconsistency. An application of nec to the tautology in quotes yields the following line: T┌∀x ϕ(x) ∧ ¬∀x ϕ(x) → 0 = 1┐ Using the fsn axioms one may conclude T┌∀x ϕ(x)┐ ∧ T┌¬∀x ϕ(x)┐ → T┌0 = 1┐. Hence one may infer fsr+(14.8) ` T┌0 = 1┐ and from this also fsr+(14.8) ` 0 = 1 by Lemma 14.22. a

192

classical symmetric truth

This shows that iterating the global reflection principles beyond ω yields an outright inconsistency. The corollary is merely a variation of a general phenomenon: adding the uniform reflection principle for a system S to an ω-inconsistent system S yields an outright inconsistency, if certain natural conditions are met. In fact, using the uniform reflection principle ˙ → ϕ(x) ∀x Bewfsr (┌ϕ(x)┐)




instead of the global reflection principle (14.8) in the corollary allows one to arrive at the contradiction without using the truth predicate in its proof. In this setting, the truth-theoretic axioms are only required to produce the ω-inconsistency. 14.5 Without satisfaction If another base theory is used instead of Peano arithmetic, then Axiom fs5 ∀v ∀x SentT (∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v)))




stating that truth commutes with the universal quantifier may fail to be expressible because, for instance, quantification over the values of all arbitrary closed terms will not coincide with quantification over all objects if some objects lack closed terms. In such cases the Friedman–Sheard theory fs can be formulated with a binary satisfaction predicate instead of the unary truth predicate used here, provided arbitrary sequences of objects can be coded in the theory. If bases theories other than Peano arithmetic such as set-theoretic systems are taken into account, so that a satisfaction predicate cannot be easily replaced with a unary truth predicate, systems of the ilk of fs might be better called classical symmetric theories of satisfaction rather than of truth. What happens, however, if satisfaction or quantification over closed terms as in fs5 and fs6 is avoided? That is, what can be said about theories of classical symmetric truth or what one could call de dicto truth as opposed to satisfaction? The formulation of the remaining axioms of fs does not require a satisfaction predicate or quantification of this kind. So the theory obtained by removing Axioms fs5 and fs6 from fs may be seen as a theory of classical symmetric truth. definition 14.40. The system f is the system fs but without Axioms fs5 and fs6.

without satisfaction

193

By model-theoretic means Friedman and Sheard (1987, pp. 17f) prove that a system slightly stronger than f is conservative over Peano arithmetic. theorem 14.41. The system f is conservative over Peano arithmetic. I do not reproduce Friedman and Sheard’s model construction, but rather outline a proof-theoretic argument for the theorem. Using the techniques of the proof of Theorem 14.31, one can embed subsystems of f with a limited number of applications of nec and conec in the ramified systems rt
194

classical symmetric truth

Now assume that there is a sentence ϕ and an n ∈ ω such that f-conecn ⊬ ϕ. So the set f-conecn ∪ {¬ϕ} is consistent and has a maximal propositionally consistent extension S. The model (N, S), where N is the standard model of arithmetic, satisfies all axioms of f1 , which coincides with f-conec1 , because a conjunction ψ ∧ χ will be in S if and only if both conjuncts ψ and χ are in S, and ¬ψ is in S if and only ϕ is not in S, and so on. From f-conecn ` ψ it follows that ψ is in S and thus also that T┌ψ┐ is valid in (N, S). Therefore (N, S) is a model of f-conecn+1 ; and since ¬ϕ is an element of S, (N, S) ⊨ T┌¬ϕ┐ follows, and with Axiom fs2 I conclude (N, S) ⊨ ¬T┌ϕ┐. Hence I have established f-conecn+1 ⊬ T┌ϕ┐. As n was arbitrary, this amounts to f-conec ⊬ T┌ϕ┐. Hence f-conec ` T┌ϕ┐ implies f-conec ` ϕ. a

15 Kripke–Feferman

The Friedman–Sheard system fs is based on the compositional theory ct. The axioms of fs are obtained by relaxing the type restriction on the ct-axioms: while ct states that a conjunction of L-sentences is true if and only if both sentences are true, the type-free system fs postulates this for all sentences including those containing the truth predicate. In addition, to the type-free generalizations of the ct-axioms, fs features two rules nec and conec that force the internal and external logic of fs to coincide. These rules are, metaphorically speaking, the remnants of the following axiom, which can be seen as a type-free generalization of the axiom for atomic sentences without a truth predicate, that is, of Axiom ct1, which cannot be consistently added to the theory fsn: (t-sym)

∀t (TT. t ↔ Tt◦ )

In Section 8.7 an alternative axiomatization of typed compositional truth was given: the axioms of the theory pt are modelled on the positive inductive Definition 8.51 of arithmetical truth, which may be less familiar than the Tarskian non-positive inductive definition, on which ct is based. In the end both definitions – the Tarskian Definition 8.1 and the positive Definition 8.51 – yield the same set of L-sentences and the corresponding theories ct and pt coincide. Just in the absence of certain instances of induction in the language LT both may come apart, as is shown by the fact that the systems ct↾ and pt↾ fail to be equivalent. Instead of using the Tarskian Definition 8.1 of truth and its associated axiomatic theory ct as the starting point for developing a theory of type-free truth, the positive definition, that is Definition 8.51 or Lemma 8.52, and the corresponding theory pt can be employed. In this chapter I will first look at the resulting axiomatic system and then deal with its semantics in the following section. As a first step towards a type-free theory of truth based on pt, I will just restate the axioms of pt with the type restriction relaxed: whereas the quantifiers in the axioms of pt range over sentences of L they are now stipulated to also range over sentences with the truth predicate.

195

196 kf1 ∀s ∀t T(s=. y) ↔ s◦ = t◦

kripke–feferman


kf2 ∀s ∀t T(¬. s=. t) ↔ s◦ 6= t◦




kf3 ∀x SentT (x) → (T(¬. ¬. x) ↔ Tx)




kf4 ∀x ∀y SentT (x∧. y) → (T(x∧. y) ↔ Tx ∧ Ty)





kf5 ∀x ∀y SentT (x∧. y) → (T¬. (x∧. y) ↔ T(¬. x) ∨ T(¬. y))
kf6 ∀x ∀y SentT (x∨. y) → (T(x∨. y) ↔ Tx ∨ Ty)
kf7 ∀x ∀y SentT (x∨. y) → (T¬. (x∨. y) ↔ T(¬. x) ∧ T(¬. y))
kf8 ∀v ∀x SentT (∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v)))
kf9 ∀v ∀x SentT (∀. vx) → (T(¬. ∀. vx) ↔ ∃t T(¬. x(t/v)))
kf10 ∀v ∀x SentT (∃. vx) → (T(∃. vx) ↔ ∃t T(x(t/v)))
kf11 ∀v ∀x SentT (∃. vx) → (T(¬. ∃. vx) ↔ ∀t T(¬. x(t/v))) The resulting system corresponds to fsn of Definition 14.1, that is, to fs without the rules nec and conec: the system based on kf1–kf11 contains type-free generalizations of the pt-axioms for the connectives and quantifiers, just as fsn contains type-free generalizations of the ct-axioms for the connectives and quantifiers. Both theories are type-free in the sense of Definition 10.1, but neither system proves the truth of a sentence of the form Tt, that is, no sentence T┌Tt┐ is a theorem of fsn or the system with the above axioms. So neither this system nor fsn prove any iterated truth claims. In the case of Axioms kf1–kf11 this is a consequence of the typed nature of the system pt, on which they are based: since pt is a theory of truth in the language L, it does not say anything about the truth of sentences not in the base language L. If the base language contained more predicate symbols beyond the identity symbol, the following two axioms matching kf1 and kf2, that is, Axioms pt1 and pt2, would have to be added: (15.1)

∀t TP. t ↔ P t◦




and (15.2)

∀t T¬. P. t ↔ ¬P t◦




Now one can try to treat the truth predicate along these lines as just another unary predicate and formulate the potential axioms (15.1) and (15.2) with the truth predicate in place of the predicate P . In the case of the first Axiom (15.1), substituting P with the truth predicate yields Axiom t-sym above, which can be consistently added to the other axioms kf1–kf11 and the axioms of pat.

kripke–feferman

197

Hence Axiom t-sym will be used as an axiom for the theory kf, the type-free generalization of pt. Replacing the predicate P with the truth predicate in (15.2), however, yields a type-free uniform disquotation sentence that can be used to derive the liar paradox: lemma 15.1. The sentence (15.3)

∀t (T¬. T. t ↔ ¬Tt◦ )

is inconsistent with Peano arithmetic. proof. The Diagonal lemma yields a term l with the following property: (15.4)

pa ` l = ┌¬Tl┐

So ¬Tl is the usual liar sentence. An inconsistency can now be arrived at in the following way: pa + (15.3) ` T┌¬Tl┐ ↔ ¬Tl

(15.3)

↔ ¬T ┌¬Tl┐

(15.4)

Hence the contradiction T┌¬Tl┐ ↔ ¬T ┌¬Tl┐ follows.

a

The lemma is conclusive: (15.3) has to be rejected as an axiom not because it is incompatible with the other Axioms kf1–kf11 but rather because it is inconsistent with the base theory. Hence it is not possible to replace Axioms kf1–kf11 with other truth-theoretic axioms in order to retain (15.3): (15.3) has to be dropped in any case. There is another crucial difference between (15.3) and the other axioms discussed so far in this section: in all other axioms the truth predicate only occurs positively on both sides of the biconditional, while it occurs negatively on the right-hand side in (15.3). This undermines the entire point of an axiomatization in the style of pt, which is to capture a positive inductive definition of truth. So not only is (15.3) inconsistent; it fails to be consonant with the other axioms kf1–kf11, as it violates the positiveness condition. As a replacement for the outright pa-inconsistent (15.3), one might flirt with the idea of replacing it with the following axiom from fs: (fs2)

∀x SentT (x) → (T¬. x ↔ ¬Tx)




198

kripke–feferman

But fs2 and Axiom t-sym yield (15.3). This is established by the following derivation:
∀t T¬. T. t ↔ ¬TT. t
↔ ¬Tt◦ For the first line the lemma pa ` SentT (T. t) is used. Consequently fs2 will not be a theorem of any consistent theory in which Axiom t-sym is provable (and which is consistent with the claim that only sentences are true). In such systems, of which kf will be one, the claim that a sentence is not true and the claim that the sentence’s negation is true will no longer be equivalent for all sentences. If a sentence is said to be false just in case its negation is true, then in such systems being not true and being false are different properties. This is the reason why the compositional axioms come in pairs. Axioms kf4 and kf5, for instance, govern the truth and falsity of conjunctions: Axiom kf4 states the truth condition; Axiom kf5 states the falsity condition. In a system like fs with fs2 as a theorem such a duplication of axioms is not necessary, as the truth condition automatically implies the falsity condition because nontruth and falsity coincide in such systems. In such a context one might consider supplementing the language of the theory with a primitive predicate for falsity. In fact, in most presentations of kf a falsity predicate is used. If I were to use a primitive falsity predicate F, I would have to change the coding to include sentences containing the falsity predicate. Then the sentence
(15.5) ∀x Fx ↔ T¬. x would be added as an axiom defining falsity. By using the falsity predicate the awkward doubling of axioms could be avoided. For instance, instead of Axiom kf5 for negated conjunctions the following axiom could be employed:
∀x ∀y SentTF (x∧. y) → (F(x∧. y) ↔ Fx ∨ Fy) In the axiom SentTF (x) expresses that x is a sentence of the language LT expanded by the new primitive predicate symbol F. So Axioms kf2, kf5, kf7, kf9, and kf10 would be replaced by axioms in which F takes the place of T¬. as in the replacement for kf5 above. Clauses concerning the falsity of negated sentences like
∀x ∀y SentT (x∧. y) → (F¬. (x∧. y) ↔ Tx ∧ Ty)

kripke–feferman

199

are superfluous because using the defining Axiom (15.5) for falsity and the Axiom kf3 for double negation these sentences can be derived. So far the addition of the falsity predicate amounts merely to the addition of a new, explicitly defined predicate. The interesting part comes when the axioms for atomic sentences with T and F are considered. For atomic sentence with T, Axiom t-sym, that is, Axiom kf12 below, can be used, but an axiom about the truth of atomic sentences of the form Ft also needs to be added: (15.6)

∀t (TF. t ↔ Ft◦ )

This axiom is in line with 15.1 for arbitrary predicate symbols. If falsity is treated as a primitive notion, one would also add axioms about the falsity of atomic sentences Tt and Ft: (15.7) (15.8)


∀t FT. t ↔ (Ft◦ ∨ ¬ SentT (t◦ ))
∀t FF. t ↔ (Tt◦ ∨ ¬ SentT (t◦ ))

In many axiomatizations of kf the disjunct ¬ SentT (t◦ ) is omitted. For many of the properties of kf it is irrelevant what is stipulated about the falsity of atomic sentence of the form Tt, but it seems plausible to me to postulate that sentences of the form Tt or Ft are false if t does not denote a sentence. In both axioms there are no negative occurrences of the truth or falsity predicates on either side of the biconditional. So they correspond to a semantic approach where both, truth and falsity, are defined simultaneously by a positive inductive definition, very much in the spirit of Kripke’s (1975) semantic truth theory where the extension and antiextension of the truth predicate are simultaneously defined by positive inductive definitions. The usual formulations of the Kripke–Feferman theory as given by Reinhardt (1986) and Feferman (1991) also feature falsity predicates and axioms similar to t-sym, (15.6), (15.7), and (15.8). Its truth axioms are a generalization of the pt-axioms and as truth does not coincide with falsity – not even in the presence of full induction – a falsity predicate is added and axiomatized using positive clauses. Although this approach is elegant, I do not follow it here because I would like to stay within the framework laid out in the first part of the book. Formulating kf is a language different from that of other truth theories in this book would make comparisons more awkward. Therefore I will not add a further primitive predicate for falsity to the language LT . Fortunately there is no stringent reason to expand the language: one can rely on two different concepts, truth and falsity, for motivating the axioms

200

kripke–feferman

of kf, while only using one predicate in its formulation. This is possible because the falsity predicate can be understood as a defined predicate in the style of (15.5). Similarly the function symbol F. in (15.6) and (15.8) can be understood as standing for T. ¬. . For the definition of a translation function Lemma 5.2 on translations can be used. In such a setting without a primitive falsity predicate, Axioms kf2, kf5, kf7, and kf9 may be viewed as falsity axioms, with the falsity predicate replaced with its definition T¬. . Hence the falsity axioms for complex sentences and atomic arithmetical sentences are already on the list kf1–kf11 under this substitution of F with T¬. and corresponding changes to the coding of sentences. The candidate axiom (15.6) becomes the sentence ∀t (TT. ¬. t ↔ T¬. t◦ ), which is already a consequence of Axiom t-sym, that is, Axiom kf12 below. Sentence 15.7, in contrast, is not a consequence of kf1–kf12 once the substutition is performed, that is, (kf13)

∀t (T¬. T. t ↔ (T¬. t◦ ∨ ¬ SentT (t◦ )))

is not implied by kf1–kf12. Hence I will add kf13 as an axiom. It is motivated by the above considerations about falsity. Finally, (15.8) becomes a derived theorem under the substitution, as can be shown in the following way: ∀t (T(¬. T. ¬. t) ↔ (T¬. ¬. t◦ ∨ ¬ SentT ((¬. t)◦ )))

kf13

∀t (T(¬. T. ¬. t) ↔ (Tt ∨ ¬ SentT (t )))

kf3

◦

◦

The last sentence is (15.8) with F replaced by T¬. and F. replaced by T. ¬. . The last line uses the equivalence of SentT ((¬. t)◦ ) and SentT (t◦ ), which is provable in pa using assumptions from Section 5 about ¬. , namely the assumption that applying the negation function to a number that does not code a sentence yields 0 and the assumption that 0 does not code a sentence. The entire excursus on the falsity predicate as a primitive predicate has helped to motivate Axiom kf13, and I have sketched a proof of the equivalence of my formulation of kf with the usual formulation with different primitive predicates for truth and falsity. Finally I summarize the discussion by giving the full list of kf-axioms used here. definition 15.2 (kf). The system kf is given by all the axioms of pat and the following axioms:

kripke–feferman kf1 ∀s ∀t T(s=. t) ↔ s◦ = t◦

201




kf2 ∀s ∀t T(¬. s=. t) ↔ s◦ 6= t◦




kf3 ∀x SentT (x) → (T(¬. ¬. x) ↔ Tx)




kf4 ∀x ∀y SentT (x∧. y) → (T(x∧. y) ↔ Tx ∧ Ty)




kf5 ∀x ∀y SentT (x∧. y) → (T¬. (x∧. y) ↔ T¬. x ∨ T¬. y)
kf6 ∀x ∀y SentT (x∨. y) → (T(x∨. y) ↔ Tx ∨ Ty)





kf7 ∀x ∀y SentT (x∨. y) → (T¬. (x∨. y) ↔ T¬. x ∧ T¬. y)
kf8 ∀v ∀x SentT (∀. vx) → (T(∀. vx) ↔ ∀t T(x(t/v))) kf9 ∀v ∀x SentT (∀. vx) → (T(¬. ∀. vx) ↔ ∃t T(¬. x(t/v)))
kf10 ∀v ∀x SentT (∃. vx) → (T(∃. vx) ↔ ∃t T(x(t/v)))




kf11 ∀v ∀x SentT (∃. vx) → (T(¬. ∃. vx) ↔ ∀t T(¬. x(t/v)))




kf12 ∀t (T(T. t) ↔ Tt◦ ) kf13 ∀t T¬. T. t ↔ (T¬. t◦ ∨ ¬ SentT (t◦ ))




In the literature, the Kripke–Feferman theory has been advocated by various authors; Reinhardt (1986), Feferman (1991), McGee (1991), Soames (1999), Halbach and Horsten (2006), and Burgess (2009) offer a thorough discussion; also, Maudlin (2004) defends a theory very similar to the Kripke–Feferman theory. I will now begin the analysis of the Kripke–Feferman theory with a generalization of a result about pt, which is equivalent to ct. The theory utb of uniform Tarski-biconditionals is a subtheory of the Compositional theory ct of truth as ct proves all typed uniform T-sentences. This result can be strengthened for kf. Before stating the result, I define the class of t-positive formulae. definition 15.3. A formula ϕ of LT is t-positive if and only if T does not occur in ϕ in the scope of an odd number of negation symbols. For this definition it is important that only ¬, ∧, and ∨, and not connectives like →, are connectives of LT . lemma 15.4 (Cantini 1989). All disquotation sentences ∀t1 . . . ∀tn T┌ϕ(t. 1 , . . . , t. n )┐ ↔ ϕ(t1 ◦ , . . . , tn ◦ )




are provable in kf for all t-positive formulae ϕ(x1 , . . . , xn ). The proof is very much like the proof of Theorem 8.4, that is, of the claim that

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utb↾ is a subtheory of ct↾. The disquotation sentences cannot be proved for ϕ of the form ¬Tt (see Lemma 15.1); this forces the restriction to t-positive formulae. 15.1 Fixed-point semantics The Kripke–Feferman system is seen as an axiomatization of Kripke’s (1975) semantic theory of truth. Starting from well behaved models, Kripke showed how to arrive at an interpretation of the truth predicate that has some attractive features. The interpretation of the truth predicate is provided by two sets, the extension and the antiextension of the truth predicate. In Kripke’s semantics T┌ϕ┐ holds if and only if the sentence ϕ (or its code) is in the extension; and ¬T┌ϕ┐ holds if and only if ϕ is in the antiextension of the truth predicate. If the antiextension of the truth predicate coincides with the complement of its extension with respect to the set of all sentences, that is, if a sentence is in the antiextension of the truth predicate if and only if it is not in its extension, then Kripke’s construction would be a classical model. But there may be – and because of the paradoxical sentences there will be – sentences neither in the extension nor the antiextension of the truth predicate; the alternative resolution of the paradoxes, an overlapping of the extension and the antiextension is ruled out by Kripke. Consequently there are sentences, among them the liar sentence, such that neither these sentences nor their negations hold in Kripke’s construction; whence one might call Kripke’s construction a partial model as some sentences do not receive a classical truth value. If such sentences are combined with other sentences, there must be some rule in place that determines the truth value (or the absence of a classical truth value). Kripke did not settle on one such method; rather he considered different evaluation schemata, that is, extensions of classical logic to a logic in which a sentence may fail to hold together with its negation. Kripke’s method works for a wide variety of evaluation schemata; among them are the so-called Strong Kleene evaluation schema, the Weak Kleene evaluation schema, and supervaluations. I will return to the later, but will first consider the Strong Kleene evaluation schema, in a slightly generalized form due to Woodruff (1984a) and Visser (1984), in which the extension and the antiextension of the truth predicate are permitted to overlap. If the Strong Kleene schema for handling partial languages is adopted, one can present Kripke’s construction in a guise in which it appears as a straightforward extension of the positive inductive definition of truth, that is, of Definition 8.51.

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definition 15.5. A set S ⊆ ω is a Kripke truth-set if and only if it satisfies the following condition: n ∈ S if and only if (i) there are closed terms s and t such that n is s = t and the value of s is identical to the value of t; or (ii) there are closed terms s and t such that n is ¬s = t and the value of s is different from the value of t; or (iii) there is an LT -sentence ϕ such that n is ¬¬ϕ and ϕ ∈ S; or (iv) there are LT -sentences ϕ and ψ such that n is ϕ ∧ ψ and ϕ ∈ S and ψ ∈ S; or (v) there are LT -sentences ϕ and ψ such that n is ¬(ϕ ∧ χ) and (¬ϕ ∈ S or ¬ψ ∈ S); or (vi) there are LT -sentences ϕ and ψ such that n is ϕ ∨ ψ and (ϕ ∈ S or ψ ∈ S); or (vii) there are LT -sentences ϕ and ψ such that n is ¬(ϕ ∨ ψ) and ¬ϕ ∈ S and ¬ψ ∈ S; or (viii) there is an LT -sentence ∀vχ such that n is ∀v χ and for all closed terms t, χ(t/v) ∈ S; or (ix) there is an LT -sentence ∀vχ such that n is ¬∀vχ and for some closed terms t, ¬χ(t/v) ∈ S; or (x) there is an LT -sentence ∃vχ such that n is ∃vχ and for some closed term t, χ(t/v) ∈ S; or (xi) there is an LT -sentence ∃vχ such that n is ¬∃vχ and for all closed terms t, ¬χ(t/v) ∈ S; or (xii) there is a closed term t such that n is Tt and the value of t is an LT sentence in S; or (xiii) there is a closed term t such that n is ¬Tt and the value is not an LT sentence or the value of t is an LT -sentence ξ with ¬ξ ∈ S. This definition agrees with the positive inductive definition of L-truth, that is, Definition 8.51, with the exception that now all clauses (iii)–(xi) are generalized to LT -sentences and that the last two clauses are added. As in the original definition of L-truth, the predicate expression ∈ S appears only positively in the right-hand side of the equivalence, so this definition is positive inductive as well. The existence of a set S follows from the general theory of inductive definitions sketched in Section 8.7. The following lemma relates to the above definition in the same way the positive inductive Definition 8.51 of L-truth relates to Lemma 8.52. It is basically the same definition split into several clauses.

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lemma 15.6. A set S ⊆ ω is a Kripke truth-set if and only if it contains only LT -sentences and it satisfies the following conditions, assuming that s and t are closed terms and that ϕ, ψ, and ∀vχ are sentences of LT : (i) s = t ∈ S iff s and t coincide in their values. (ii) s 6= t ∈ S iff s and t differ in their values. (iii) ¬¬ϕ ∈ S iff ϕ ∈ S. (iv) ϕ ∧ ψ ∈ S iff ϕ ∈ S and ψ ∈ S. (v) ¬(ϕ ∧ ψ) ∈ S iff ¬ϕ ∈ S or ¬ψ ∈ S. (vi) ϕ ∨ ψ ∈ S iff ϕ ∈ S or ψ ∈ S. (vii) ¬(ϕ ∨ ψ) ∈ S iff ¬ϕ ∈ S and ¬ψ ∈ S. (viii) ∀x χ ∈ S iff for all n, χ(n) ∈ S. (ix) ¬∀x χ ∈ S iff there is a n with ¬χ(n) ∈ S. (x) ∃x χ ∈ S iff there is a n with χ(n) ∈ S. (xi) ¬∃x χ ∈ S iff for all n, ¬χ(n) ∈ S. (xii) Tt ∈ S iff the value of t is an LT -sentence in S. (xiii) ¬Tt ∈ S iff the value of t is not an LT -sentence or it is an LT -sentence whose negation is in S. The axioms of kf have been obtained by turning the clauses in this lemma into axioms. Definition 15.5 gives rise to a monotone operator Φ, as explained in Section 8.7. According to Definition 15.5, a Kripke truth set is a set S satisfying an equivalence of the following form for all numbers n: n ∈ S iff ζ(n, S) Here ζ(n, S) is the right-hand side of the equivalence in Definition 15.5, in which S only occurs positively. The operator Φ associated with this positive inductive definition is then introduced in the following way: Φ(S) := {n : ζ(n, S)} Thus, a set S is a Kripke truth set if and only if it is a fixed point of this operator, that is, if and only if Φ(S) = S. It follows from the general theory of positive inductive definitions that there is a minimal fixed point IΦ satisfying the equivalence in Definition 15.5. That is, there is a smallest set IΦ with Φ(IΦ ) = IΦ . The minimal fixed point can be obtained, as explained on p. 118, by closing the empty set under the operation Φ. However, in contrast to the definition for truth in L, there is no unique Kripke truth-set. The minimal fixed point IΦ is consistent. This can be proved by induction on the generation procedure for IΦλ .

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lemma 15.7. The minimal fixed point IΦ is consistent, that is, there is no sentence of LT such that the sentence and its negation are in IΦ . As has been mentioned above, the operator Φ has further fixed points besides the minimal one. For instance, one can start from the singleton of a truth teller sentence, that is, from {τ}, and then close this set under Φ, where τ is a sentence satisfying pa ` τ ↔ T┌τ┐ obtained by diagonalization. This yields a (consistent) fixed point different from the minimal one, which does contain τ. However, if one begins with a set containing the liar sentence one will arrive at an inconsistent fixed point. It can be shown that there are uncountably many fixed points of Φ. The characteristic properties of all fixed points of Φ is that they are closed under Strong Kleene logic, as will be described below, and that a sentence is in the fixed point if and only if the claim that the sentence is true is in the fixed point as well: lemma 15.8. If Φ(S) = S holds, then T┌ϕ┐ ∈ S if and only if ϕ ∈ S obtains. As pointed out above, Kripke’s original approach in (1975) differs from Definition 15.5. In the above definitions the Strong Kleene evaluation schema is implicit; if the Kripke truth-sets are defined in Kripke’s original way, the Strong Kleene schema is made explicit. In what follows I will establish the equivalence of an approach explicitly based on this evaluation schema with the one presented here. For this purpose I will introduce the Strong Kleene schema, in a generalized version which also allows for truth-value gluts as well as truth-value gaps. Technically this will be achieved by allowing the extension and the antiextension of the truth predicate to overlap. Strong Kleene models, or sk-models for short, take the form (𝔐, S1 , S2 ), where 𝔐 is a model of L, and S1 and S2 are subsets of the domain of 𝔐. For the time being, I will only be concerned with models in which the arithmetical vocabulary is interpreted in the standard way. So I will define the Strong Kleene evaluation schema for 𝔐 = N. Whether a sentence holds in an sk-models under the Strong Kleene evaluation schema is defined by induction on what I call the positive complexity of a formula. definition 15.9 (positive complexity). Atomic formulae and negated atomic formulae have positive complexity 0. The positive complexity of ¬¬ϕ, ∀x ϕ, ¬∀x ϕ, ∃x ϕ, and ¬∃x ϕ is the positive complexity of ϕ plus 1. The positive complexity of ϕ ∧ ψ, ¬(ϕ ∧ ψ), ϕ ∨ ψ, and ¬(ϕ ∨ ψ) is the maximum of the positive complexities of ϕ and ψ plus 1.

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The Strong Kleene evaluation schema is now defined in the following way: definition 15.10. For sets S1 , S2 ⊆ ω, (N, S1 , S2 ) ⊨ sk is defined in the following way for all closed terms s, t, LT -sentences ϕ, ψ and LT -formulae χ(x) with exactly x free. The induction is on the positive complexity of sentences: (i) (N, S1 , S2 ) ⊨sk s = t iff s and t coincide in their values. (ii) (N, S1 , S2 ) ⊨sk ¬s = t iff s and t differ in their values. (iii) (N, S1 , S2 ) ⊨ sk Tt iff t is a closed term whose value is an LT -sentence in S1 . (iv) (N, S1 , S2 ) ⊨ sk ¬Tt iff t is a closed term whose value is an LT -sentence in S2 or whose value is not an LT -sentence. (v) (N, S1 , S2 ) ⊨sk ¬¬ϕ iff (N, S1 , S2 ) ⊨sk ϕ (vi) (N, S1 , S2 ) ⊨sk ϕ ∧ ψ iff (N, S1 , S2 ) ⊨sk ϕ and (N, S1 , S2 ) ⊨sk ψ (vii) (N, S1 , S2 ) ⊨sk ¬(ϕ ∧ ψ) iff (N, S1 , S2 ) ⊨sk ¬ϕ or (N, S1 , S2 ) ⊨sk ¬ψ (viii) (N, S1 , S2 ) ⊨sk ϕ ∨ ψ iff (N, S1 , S2 ) ⊨sk ϕ or (N, S1 , S2 ) ⊨sk ψ (ix) (N, S1 , S2 ) ⊨sk ¬(ϕ ∨ ψ) iff (N, S1 , S2 ) ⊨sk ¬ϕ and (N, S1 , S2 ) ⊨sk ¬ψ (x) (N, S1 , S2 ) ⊨sk ∀xϕ(x) iff for all n ∈ ω (N, S1 , S2 ) ⊨sk ϕ(n) (xi) (N, S1 , S2 ) ⊨sk ¬∀xϕ(x) iff for at least one n ∈ ω (N, S1 , S2 ) ⊨sk ¬ϕ(n) (xii) (N, S1 , S2 ) ⊨sk ∃xϕ(x) iff for at least one n ∈ ω (N, S1 , S2 ) ⊨sk ϕ(n) (xiii) (N, S1 , S2 ) ⊨sk ¬∃xϕ(x) iff for all n ∈ ω (N, S1 , S2 ) ⊨sk ¬ϕ(n) The claim (N, S1 , S2 ) ⊨ sk n does not hold for any n that is not the code of an LT -sentence. The definition can easily be rewritten as a positive inductive definition of the set of all LT -sentences ϕ with (N, S1 , S2 ) ⊨ sk ϕ. The sets S1 and S2 serve as parameters in such a definition. As I do not exclude truth-value gluts, that is, sentences that hold together with their negation in an sk-model, the logic described here is not Strong Kleene logic proper, which does not admit truth-value gluts, but rather a generalization of it. The following two equivalences are immediate consequences of the definition: (N, S1 , S2 ) ⊨sk T n iff n ∈ S1 (N, S1 , S2 ) ⊨sk ¬T n iff n ∈ S2 If n 6∈ S1 ∪ S2 , that is, if n is neither in the extension nor the antiextension, then neither T n nor ¬T n holds in the model (N, S1 , S2 ). If n is in both, the extension and the antiextension, that is, if n ∈ S1 ∩ S2 , then one has both (N, S1 , S2 ) ⊨sk T n and (N, S1 , S2 ) ⊨sk ¬T n.

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The semantic properties of the connectives are summarized in the truth tables below. In the tables the entry t for a sentence ϕ means that ϕ holds in the model, the entry f means that the negation of ϕ holds. That neither ϕ nor its negation hold is indicated by a dash, which marks a truth-value gap. If both ϕ and its negation ¬ϕ hold in a model, then this is marked by t, f, which is called a truth-value glut.

ϕ t f − t, f

¬ ϕ f t − t, f

ϕ t t t t f f f f − − − − t, f t, f t, f t, f

ψ t f − t, f t f − t, f t f − t, f t f − t, f

ϕ∧ψ t f − t, f f f f f − f − f t, f f f t, f

ϕ t t t t f f f f − − − − t, f t, f t, f t, f

ψ t f − t, f t f − t, f t f − t, f t f − t, f

ϕ∨ψ t t t t t f − t, f t − − t t t, f t t, f

It is not hard to derive these truth tables from the inductive definition of (N, S1 , S2 ) ⊨sk , that is Definition 15.10 above. If the extension S1 and the antiextension S2 do not overlap, there are no truth value gluts, that is, lines with an entry t, f cannot arise. definition 15.11. An sk-model (N, S1 , S2 ) is consistent if and only if there is no LT -sentence in S1 ∩ S2 . This definition is similar to the notion of consistency used in Lemma 15.7. lemma 15.12. If an sk-model (N, S1 , S2 ) is consistent, then there is no sentence ϕ with (N, S1 , S2 ) ⊨sk ϕ and (N, S1 , S2 ) ⊨sk ¬ϕ. proof. The proof is by induction on the length of ϕ. If ϕ is atomic, the claim follows directly from the consistency of the model. For complex formulae, I consider the case of conjunction as an example. I make the following two assumptions:

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(i) (N, S1 , S2 ) ⊨sk ϕ ∧ ψ (ii) (N, S1 , S2 ) ⊨sk ¬(ϕ ∧ ψ) Using Definition 15.10 I conclude (N, S1 , S2 ) ⊨ from (i). But (ii) implies that

sk

ϕ and (N, S1 , S2 ) ⊨

sk

ψ

either (N, S1 , S2 ) ⊨sk ¬ϕ or (N, S1 , S2 ) ⊨sk ¬ψ holds. This, however contradicts the induction hypothesis, which says that neither ϕ and ¬ϕ nor ψ and ¬ψ can hold simultaneously in a consistent skmodel. a Classical models are pairs (𝔐, S) while sk-models are triples (𝔐, S1 , S2 ). But classical models can also be seen as special cases of sk-models where S2 is the complement of S1 . So classical models can be seen as sk-models satisfying the following two conditions: (i) S1 ∪ S2 = ω (completeness) (ii) S1 ∩ S2 = Ø (consistency) So far no conditions have been imposed on the extension S1 and the antiextension S2 . The next task is to define suitable extensions and antiextensions. To this end an operator Λ on pairs of sets of natural numbers is defined in the following way: Λ(S1 , S2 ) := ({ϕ ∈ LT : (N, S1 , S2 ) ⊨sk ϕ}, NSent ∪ {ϕ ∈ LT : (N, S1 , S2 ) ⊨sk ¬ϕ}) As above, NSent is the set of numbers that are not (codes of) sentences of LT . The theory of positive inductive definitions can be applied to the operator Λ. One difference to the account developed in Section 8.7 is that Λ operates on pairs of sets rather than on a single set. Generalizing the account of positive inductive definitions does not pose a problem. The definition of Λ relies on the definition of (N, S1 , S2 ) ⊨ sk . As remarked above, the set {n : (N, S1 , S2 ) ⊨ sk n} is definable by a positive inductive definition with S1 and S2 as parameters. I do not give a detailed proof for the existence of the fixed points because fixed points of Λ can be recovered from fixed points of Φ by Lemma 15.14 below. A fixed point of Λ is a pair (S1 , S2 ) of sets of numbers satisfying Λ(S1 , S2 ) = (S1 , S2 ). If (S1 , S2 ) is a fixed point of Λ, the two following identities follow immediately from the definition of Λ: (15.9)

S1 = {ϕ ∈ LT : (N, S1 , S2 ) ⊨sk ϕ}

(15.10)

S2 = NSent ∪ {ϕ ∈ LT : (N, S1 , S2 ) ⊨sk ¬ϕ}

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Fixed-points models have the following attractive feature: lemma 15.13. If (S1 , S2 ) is a fixed point of Λ the following equivalence holds for all sentences of LT : (N, S1 , S2 ) ⊨sk T┌ϕ┐ iff (N, S1 , S2 ) ⊨sk ϕ proof. (N, S1 , S2 ) ⊨sk T┌ϕ┐ iff ϕ ∈ S1

Def. 15.10 of ⊨sk

iff (N, S1 , S2 ) ⊨sk ϕ

(15.9)

a

lemma 15.14. If (S1 , S2 ) is a fixed point of Λ, then S1 is a fixed point of Φ. Conversely, if a set S1 is a fixed point of Φ and S2 := NSent ∪ {ϕ ∈ LT : ¬ϕ ∈ S1 }, then (S1 , S2 ) is a fixed point of Λ. Sketch of proof. Assume that (S1 , S2 ) is a fixed point of Λ. It suffices to prove that S1 is a Kripke truth-set in the sense of Lemma 15.6. Then S1 is a set of LT sentences by Definition 15.10. It remains to prove the following equivalence for all sentences ϕ of LT : ϕ ∈ S1 iff ϕ ∈ Φ(S1 )

(15.11)

To prove the clause (xii) of Lemma 15.6 for S1 , one can reason as follows, using the assumption that (S1 , S2 ) is a fixed point of Λ in the first line: Tt ∈ S1 iff (N, S1 , S2 ) ⊨sk Tt

(15.9)

iff the value of t is an LT -sentence in S

Def. 15.10 (iii)

Clause (ii) can be proved in a similar way, and clauses (i) and (ii) are trivial. So claim (15.11) is proved for sentences of positive complexity 0. For sentences with positive complexity greater than 0, claim (15.11) is established by induction on the positive complexity. For instance, for (iii) one can proceed in the following way: ¬¬ϕ ∈ S1 iff (N, S1 , S2 ) ⊨sk T┌¬¬ϕ┐

Def. 15.10 of ⊨sk

iff (N, S1 , S2 ) ⊨sk ¬¬ϕ

Lemma 15.13

iff (N, S1 , S2 ) ⊨sk ϕ

Def. 15.10 of ⊨sk

iff (N, S1 , S2 ) ⊨sk T┌ϕ┐

Lemma 15.13

iff ϕ ∈ S1

Def. 15.10 of ⊨sk

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To prove the second part of the lemma assume that S1 is a fixed point of Φ, that is, a Kripke truth set. The antiextension S2 is then defined as indicated above. Assume furthermore the following: Λ(S1 , S2 ) = (S01 , S02 ) To show that (S1 , S2 ) is a fixed point of Λ, I will prove S01 = S1 and S02 = S2 . As for the previous direction, I use an induction on the positive complexity of ϕ to demonstrate the following two equivalences: (15.12)

ϕ ∈ S01 iff ϕ ∈ S1

(15.13)

ϕ ∈ S02 iff ϕ ∈ S2

As an example I consider the case that ϕ is of the form ¬Tt and that an LT sentence ψ is the value of the term t: ¬Tt ∈ S02 iff (N, S1 , S2 ) ⊨sk ¬¬Tt

def. S02

iff (N, S1 , S2 ) ⊨sk Tt

Def. 15.10 (v)

iff ψ ∈ S1

Def. 15.10 (iii)

iff Tt ∈ S1

Φ(S1 ) = S1

iff ¬¬Tt ∈ S1

Lemma 15.6 (iii)

iff ¬Tt ∈ S2

def. S2

The remaining cases are proved in a similar way.

a

As has already been mentioned, the operators Φ and therefore also Λ have more than one fixed point. Arguably, Kripke’s main contribution was not so much the construction of the smallest fixed point of Λ but rather his classification of the different consistent fixed points and the discussion of their use for discriminating between ungrounded sentences, paradoxical sentences, and so on. For instance, a sentence is defined as paradoxical if and only if it is not contained in any consistent fixed point.1 Therefore the liar sentence is classified as paradoxical because it is not contained in any consistent fixed point of Φ. The truth teller sentence, in contrast, is contained in some consistent fixed points but not in others; moreover the negation of the truth teller is contained in other consistent fixed points. The classifications of fixed points and sentences have been discussed by Kripke (1975) and other authors, among 1 Kripke (1975) considered only consistent fixed points. Therefore a sentence is paradoxical in Kripke’s sense if and only if it is not in any fixed point.

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them McGee (1991) and Belnap and Gupta (1993). Instead of going further into these semantic considerations, I return to the axiomatic approach and the Kripke–Feferman axiomatization of Kripke’s semantic theory. The axioms of the Kripke–Feferman theory are just the clauses in the definition of a Kripke truth-set turned into axioms. Feferman (1991) developed the standard semantics for kf (see also McGee 1991, p. 93, Theorem 4.3) and showed how his theory is related to Kripke’s construction. The fixed points of Φ (with the possible addition of non-sentences) are exactly the possible extensions of the truth predicate over the standard model. theorem 15.15 (adequacy of kf). For all sets S ⊆ ω the following obtains: (N, S) ⊨ kf + ∀x (Tx → Sent(x)) iff Φ(S) = S Thus, for a set S of LT -sentences, (N, S) is a model of kf if and only if S is a Kripke truth-set. Sketch of proof. If S is a fixed point of Φ, it is easy to see using Lemma 15.6 that all axioms of kf and the sentence ∀x (Tx → Sent(x)) are verified by (N, S). Hence the right-to-left direction of the claim is established. Now assume that (N, S) is a model of kf plus the axiom ∀x (Tx → Sent(x)) stating that only sentences are true. Using the appropriate axioms of kf it is straightforward to show that the extension of the truth predicate is a Kripke truth-set in accordance with Lemma 15.6. a The Kripke–Feferman theory is an axiomatization of the fixed points of the operator Φ, that is, of a Kripkean fixed-point semantics with the Strong Kleene evaluation schema with truth-value gluts. Of course, no axiomatization of these fixed points can be complete in the sense of fully describing the standard models, that is, the Strong Kleene fixed points of Kripke’s construction over the standard model of Peano arithmetic. Incompleteness in this sense cannot be avoided as even the set of arithmetical sentences valid in the standard model of Peano arithmetic cannot be recursively axiomatized, but these sentences are exactly the arithmetical sentences that hold in any of the standard models. But if the standard model is fixed as the underlying model for the language L of the base theory, then kf fully characterizes the fixed points of Kripke’s theory with the Strong Kleene schema. This is the content of Theorem 15.15. Thus kf is a theory about all fixed points of Φ, not just the minimal one.

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A fixed point S of Φ, as defined in the previous section, is inconsistent if and only if there is a sentence ϕ such that both ϕ and ¬ϕ are in S. In particular, if the liar sentence is in a fixed point S, then its negation is also in S. Consistent fixed points contain neither the liar sentence nor its negation. As has been mentioned already, Kripke (1975) only admitted truth value-gaps and not gluts. Many other authors have followed him in this. In the recent philosophical literature on the liar paradox, the question whether truth value gluts and/or gaps should be admitted has been hotly debated. Truth-value gluts correspond to a so-called dialethic conception of truth; excluding gluts and admitting only gaps leads to a conception of what is usually called partial truth. In axiomatic theories such as kf, truth value gluts are excluded by the Consistency axiom: (cons)

∀x SentT (x) → ¬(Tx ∧ T¬. x)




It states that no sentence is both true and false. lemma 15.16. kf + cons ` ∀x (Tx → Sent(x)) proof. I reason in kf + cons in the following way: ∀t ¬ SentT (t◦ ) → T¬. T. t




from kf13

→ ¬TT. t




cons

→ ¬Tt ) ◦

kf12

The claim then follows by contraposition.

a

While cons excludes truth-value gluts, gaps are excluded by the Completeness axiom (comp)


∀x SentT (x) → (Tx ∨ T¬. x) .

The Completeness axioms states that every sentence is either true or false. By the definition above, a fixed point S of ϕ is consistent if and only if no sentence ϕ and its negation ¬ϕ are in S. A fixed point is complete if for every sentence of LT either ϕ or ¬ϕ is in S. The Consistency axiom forces consistent fixed points of Φ as interpretations of the truth predicate; the Completeness axiom forces complete fixed points. Thus Theorem 15.15 can be refined in the following way:

completeness and consistency

213

lemma 15.17. Assume S is a set of natural numbers. Then the following two claims obtain: (i) S is a consistent fixed point of Φ if and only if (N, S) ⊨ kf + cons (ii) S is a complete fixed point of Φ if and only if (N, S) ⊨ kf + comp + ∀x (Tx → SentT (x)) In (i) the sentence ∀x (Tx → SentT (x)) can be dropped by Lemma 15.16. This result can be generalized to apply to nonstandard models instead of (N, S) if Φ is suitably defined for these models and kf is replaced with the system kf↾, which only has induction for L-formulae. In some axiomatizations of the Kripke–Feferman theory the Consistency axiom cons is included. Like Feferman (1991), I have omitted cons from the definition of kf because it is of a different nature to the other axioms of kf; moreover cons does not contribute to the proof-theoretic strength of kf. However, besides Lemma 15.16 the Consistency axiom does have some convenient consequences: lemma 15.18. The Consistency axiom and the axiom stating that truth distributes over the conditional are equivalent over kf, that is, the following obtains:
kf ` cons ↔ ∀x ∀y SentT (x→ . y) → (T(x→ . y) → (Tx → Ty)) proof. As the conditional is not a primitive symbol of LT , x→ . y is an abbreviation for ¬. x ∨ y. To show the left-to-right direction I proceed as follows:
kf + cons ` ∀x ∀y SentT (x→ kf7 . y) → (T(¬. x∨. y) → (T¬. x ∨ Ty))
→ (T(¬. x∨. y) → (¬Tx ∨ Ty)) cons
→ (T(¬. x∨. y) → (Tx → Ty)) def. → To derive cons, only an instance (rather than the general claim) of the righthand side sentence is required. To begin with, I show the following:
(15.14) kf ` ∀x T(x→ . ┌0 = 1┐) ↔ T¬. x Using the definitions above this can be shown in the following way:
kf ` ∀x T(¬. x∨. ┌0 = 1┐) ↔ (T¬. x ∨ T┌0 = 1┐) kf6
↔ T¬. x

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The last line follows from kf ` ¬T┌0 = 1┐, which holds by kf1. To derive the right-to-left direction of the claim, I reason in kf together with the righthand side of the equivalence as follows, beginning with an instance of the right-hand side of the equivalence:
∀x ∀y SentT (x→ . ┌0 = 1┐) → (T(x→ . ┌0 = 1┐) → (Tx → T┌0 = 1┐))
→ (T¬. x → (Tx → T┌0 = 1┐))
→ (T¬. x → ¬Tx)

(15.14)

For the last line I use kf ` ¬T┌0 = 1┐ again. Finally the claim follows from an application of Lemma 13.6. a It is not hard to see that cons does not yield the reverse direction. That is, the following sentence fails to be a theorem of kf + cons:
∀x ∀y SentT (x→ . y) → ((Tx → Ty) → T(x→ . y)) It follows from Friedman and Sheard’s inconsistency result Theorem 13.8 that kf + cons cannot be consistently closed under nec and conec because kf + cons satisfies the premisses of this result by Lemma 15.18 (see also Lemma 15.20 below). Actually kf + cons cannot even be consistently closed under nec alone as kf + cons proves the schema T┌ϕ┐ → ϕ for all sentences ϕ of LT , and this schema yields an inconsistency together with nec by Montague’s theorem 13.1. In fact, over kf the uniform version of the schema is equivalent with the Consistency axiom cons. So there is yet another neat equivalent axiomatization of cons. lemma 15.19. The following three systems prove the same theorems: (i) kf + cons (ii) kf plus all sentences of the form
∀t1 . . . ∀tn T┌ϕ(t. 1 , . . . , t. n )┐ → ϕ(t1 ◦ , . . . , tn ◦ ) (iii) kf + ∀t (T¬. T. t → ¬Tt◦ ) The step from (i) to (ii) is well known. See, for instance, Cantini (1996, p. 54, Theorem 8.8 (i)). For proof of (ii) from (i) I will use the notion of positive complexity from Definition 15.9. Proof of Lemma 15.19. I follow the proof in Halbach and Horsten (2006, p. 686). First I will prove that kf + cons implies the schema in (ii) by induction on the positive complexity of ϕ(x); more precisely I will show that kf + cons implies
∀t1 . . . ∀tn T┌ϕ(t1 , . . . , tn )┐ → ϕ(t1 ◦ , . . . , tn ◦ ) .

completeness and consistency

215

If ϕ(t1 , . . . , tn ) is of positive complexity 0, that is, atomic or negated atomic, the claim follows from kf1 ,kf2, kf12, and kf13. In the latter case, cons is also required: that is, if the formula is of the form ¬Tt, the claim is proved in the following way:
kf + cons ` ∀t T¬. T. t → T¬. t◦ ∨ ¬ SentT (t◦ ) kf13
◦ → ¬Tt cons The disjunct ¬ SentT (t◦ ) can be dropped because kf + cons proves ∀x (Tx → Sent(x)) by Lemma 15.16. In the conjunction case I reason as follows: kf + cons ` ∀t T┌ϕ(t. 1 , . . . , t. n ) ∧ ψ(t. 1 , . . . , t. n )┐
→ T┌ϕ(t. 1 , . . . , t. n )┐ ∧ T ┌ψ(t. 1 , . . . , t. n )┐
→ ϕ(t1 ◦ , . . . , tn ◦ ) ∧ ψ(t1 ◦ , . . . , tn ◦ )

kf4

The last line follows from the induction hypothesis. Of course, ϕ(t1 , . . . , tn ) and ψ(t1 , . . . , tn ) need not contain all the terms t1 , . . . , tn . If, for instance, both ϕ(t1 , . . . , tn ) and ψ(t1 , . . . , tn ) contain a different single variable, their conjunction will contain two variables. The case for negated conjunctions is similar: kf + cons ` ∀t T┌¬(ϕ(t. 1 , . . . , t. n ) ∧ ψ(t. 1 , . . . , t. n ))┐
→ (T┌¬ϕ(t. 1 , . . . , t. n )┐ ∨ T┌¬ψ(t. 1 , . . . , t. n )┐ kf5
→ ¬ϕ(t1 ◦ , . . . , tn ◦ ) ∨ ¬ψ(t1 ◦ , . . . , tn ◦ ) ind. hyp.
◦ ◦ ◦ ◦ → ¬(ϕ(t1 , . . . , tn ) ∧ ψ(t1 , . . . , tn )) The other cases are similar. Item (iii) of Lemma 15.19 is simply an instance of (ii). Thus the implication from (ii) to (iii) is trivial. To prove (i) from (iii) I reason in the following way, using kf13 in the first line and the additional axiom ∀t (T¬. T. t → ¬Tt◦ ) in the second: kf + ∀t (T¬. T. t → ¬Tt◦ ) `∀x (T¬. x → T¬. T. x) ˙ ∀x (T¬. x → ¬Tx)

a

It follows from this lemma that the liar sentence with pat ` λ ↔ ¬T┌λ┐ is decided in kf + cons: From kf + cons ` T┌λ┐ → λ, which follows by the above lemma, one obtains kf + cons ` ¬T┌λ┐ and therefore kf + cons ` λ. So over kf the Consistency axiom cons implies the liar sentence.

216

kripke–feferman

As pointed out above, the lemma implies that kf + cons cannot be consistently closed under nec. The system kf without cons can be closed under conec, as conec is merely a weakening of the axiom schema T┌ϕ┐ → ϕ and kf + cons is consistent. Similarly, kf can be consistently closed under nec. The question now arises whether kf (without cons) can be consistently closed under both rules nec and conec. This question is tantamount to the question of whether kf can be made symmetric n the sense of Definition 13.3, that is, whether the internal and the external logic can be forced to coincide, by adding nec and conec. The system kf in itself fails to be symmetric: an easy semantic argument shows that T┌λ ∨ ¬λ┐ is not a theorem of kf, if λ is the liar sentence, while λ ∨ ¬λ is a classical tautology and therefore a theorem of kf. The following observation from Halbach and Horsten (2006, p. 688) shows that kf cannot be made symmetric. Therefore kf is essentially asymmetric: lemma 15.20. The result of closing kf under the two rules nec and conec is inconsistent. proof. Let λ be the liar sentence again, that is, assume λ satisfies the following: pat ` λ ↔ ¬T┌λ┐ The right-to-left direction can be rephrased as a disjunction: pat ` λ ∨ T┌λ┐ By applying nec and distributing the truth predicate over ∨ using Axiom kf6 I obtain the following: kf + nec ` T┌λ┐ ∨ T┌T┌λ┐┐ Now using kf12 and its consequence T┌λ┐ ↔ T┌T┌λ┐┐ this can be simplified to (15.15)

kf + nec ` T┌λ┐.

From pa ` λ ↔ ¬T┌λ┐ and (15.15) I infer kf + nec ` ¬λ. An application of nec yields kf + nec ` T┌¬λ┐. Combining this with (15.15) and then using Axiom kf4 implies the internal inconsistency of kf + nec: (15.16)

kf + nec ` T┌¬λ ∧ λ┐

Finally an application of conec can be used to derive a contradiction.

a

proof theory of the kripke–feferman system

217

The last line (15.16) shows that closing kf under nec forces truth-value gluts. This leaves three consistent variants of kf: the system kf with neither the Consistency nor the Completeness axiom, kf plus cons, and finally kf plus comp. It can be shown that the three systems do not differ in their arithmetical theorems (Cantini 1989; Feferman 1991). I will focus on the pure variant without cons and comp as these two axioms are of a different nature to the other truth-theoretic axioms of kf. Also the fact that cons and comp decide the liar sentence makes them less attractive. Much of what follows, however, can also mutatis mutandis be applied to kf + cons and kf + comp. 15.3 Proof theory of the Kripke–Feferman system Compared to most of the theories discussed so far in this book, the Kripke– Feferman theory is deductively very strong. In particular, it is much stronger than its competitor fs, which is based on a completely classical notion of truth. There are several senses in which fs is reducible to kf, but in other senses fs is not reducible to kf. Here is a list of some results: 1. All arithmetical sentences provable in fs are also provable in kf. 2. kf relatively interprets fs. 3. The truth predicate of fs is not definable in kf, although the truth predicate of any of the restricted systems fsn is definable in kf. That the truth predicate of fs is not definable in kf follows from a direct application of Lemma 14.20, which states that any theory that defines the truth predicate of fs is ω-inconsistent since kf is ω-consistent as there are ω-models of kf by Theorem 15.15. Feferman determined the proof-theoretic strength of kf in (1991) and in earlier versions of this paper. Cantini (1989) gave a more direct proof-theoretic analysis of kf and some of its subsystems. Most of these results rely on more profound proof-theoretic theorems and techniques than are used in this book, and I am not going to provide a full analysis of kf here. What can be achieved quite easily is an embedding of the systems of ramified truth into kf. More precisely the truth predicates of rt<0 can be defined in kf. It follows from this that kf proves all arithmetical sentences provable in rt<0 and therefore also all arithmetical sentences provable in fs by Corollary 14.32. So while the Friedman–Sheard theory yields ω iterations of the

218

kripke–feferman

compositional theory ct, the Kripke–Feferman theory yields 0 -many iterations. The ordinal 0 has been introduced on p. 126 above. It is the limit of ω the ordinals 1, ω, ωω , ωω ,. . . More formally, define ω0 := 1 and ωn+1 := ωωn . Then 0 is the limit of these ωn for n ∈ ω. The proof sketch showing that the truth predicates of rt<0 can be defined in kf will be broken into several lemmata. theorem 15.21. The system pat, that is, Peano arithmetic in the language LT proves transfinite induction for any ordinal up to but not including 0 . That is, for any LT -formula ϕ(x) and any α < 0 , the following is provable in pat and therefore in kf: (TIα )


∀α ∀β ≺ α ϕ(β) → ϕ(α) → ∀ξ ≺ α ϕ(ξ)

Here ∀α and so on range over codes of ordinals. A coding of ordinals has been in fixed in Section 9. Again, the expression ≺ represents the relation on natural numbers that obtains between n and k if and only if n is the code for an ordinal α, k the code for β, and α < β. In what follows I will use symbols like + to represent the obvious operations on the codes of ordinals. It is essential to the proof that pat contains all induction axioms including those with the truth predicate. The proof of the theorem is Gentzen’s (1943) classic proof: the presence of the additional truth predicate does not force any modification, as no axioms specific to T are contained in pat. This is in contrast to truth theories like kf with specific truth-theoretic axioms. As kf contains pat, transfinite induction up to 0 can be proved for all formulae including those containing the truth predicate. If only formulae without the truth predicate are considered, then kf proves transfinite induction for much larger ordinals. Here I reproduce the usual proof of Theorem 15.21 for reference in later sections breaking it into several lemmata. As I will try to perform the proof in a nonclassical system in a later section, some attention to detail will be required. Assume for the proof that ϕ(x) is an LT -formula. For any formula ψ(x) the expression Prog(ψ) abbreviates that the formula ψ(x) is progressive, that is, it abbreviates the following formula: (Prog(ψ))

∀α ∀β ≺ α ψ(β) → ψ(α)




To begin with, the following claim will be established:

proof theory of the kripke–feferman system

219

lemma 15.22. The following holds for all formulae ϕ(x) of LT : pat ` Prog(ϕ) → Prog(∀ξ (∀η ≺ ξ ϕ(η) → ∀η ≺ ξ + ωx ϕ(η))) | {z } ϕ+ (x) In what follows, ϕ+ (x) abbreviates the underbraced formula. proof. I will reason informally in pat. To prove the lemma, assume its antecedent Prog(ϕ):
(15.17) ∀α ∀β ≺ α ϕ(β) → ϕ(α) From this I will show Prog(ϕ+ ), that is, ∀α (∀β ≺ α ϕ+ (β) → ϕ+ (α)). So assume furthermore the following formula for an arbitrary ordinal α: (15.18)

∀β ≺ α ϕ+ (β)

From this I have to show ϕ+ (α). So I fix an ordinal ξ and also assume (15.19)

∀η ≺ ξ ϕ(η).

Given an ordinal η ≺ ξ + ωα , I distinguish two cases: First case: α = 0. Since ωα = ω0 = 1, I can infer η ≼ ξ from the assumption η ≺ ξ + ωα . If η ≺ ξ, then ϕ(η) follows from (15.19). If η = ξ, then ϕ(η) follows from (15.17) and (15.19). Second case: α  0. I invoke a fact derived from Cantor’s Normal Form theorem about ordinals, which is provable in Peano arithmetic (see, for instance, Pohlers 1989, p. 71). Peano arithmetic proves the following claim (for the chosen natural ordinal notation system):
∀η ∀α ∀ξ η ≺ ξ + ωα ∧ α 6= 0 → ∃ν ≺ ω ∃α0 ≺ α η ≺ ξ + ωα0 ·ν Applying this claim to the assumptions above, I infer there is α0 ≺ α and ν ≺ ω satisfying (15.20)

η ≺ ξ + ωα0 ·ν.

By induction on ν I will establish the implication (15.21)

η ≺ ξ + ωα0 ·ν → ϕ(η).

For ν = 0 the claim η ≺ ξ → ϕ(η) follows from (15.19). For the induction step assume as induction hypothesis (15.21). Since α0 ≺ α, (15.18) yields ϕ+ (α0 ), that is,
∀ξ ∀η ≺ ξ ϕ(η) → ∀η ≺ ξ + ωα0 ϕ(η) .

220

kripke–feferman

Substituting ξ + ωα0 ·ν for ξ gives the following:
∀ξ ∀η ≺ ξ + ωα0 ·ν ϕ(η) → ∀η ≺ ξ + ωα0 ·ν + ωα0 ϕ(η)
→ ∀η ≺ ξ + ωα0 ·(ν + 1) ϕ(η) Hence I have proved (15.21) with ν + 1 in place of ν from (15.21), establishing the induction step. The induction on ν is actually an induction on ordinal notations. But a suitable instance of the induction schema of pat can be used because there is a primitive recursive function (and thus a pat-representable function) that maps any natural number n to the ordinal notation for n. a lemma 15.23. Assume α is a coded ordinal. Then, if the claim pat ` Prog(ϕ) → ∀ξ ≺ α ϕ(ξ) | {z } TIα obtains for all formulae ϕ(x) of LT , then the following also obtains for all ϕ(x) of LT : pat ` Prog(ϕ) → ∀ξ ≺ ωα ϕ(ξ) | {z } TIωα proof. Assume that TIα is provable in pat for all formulae ϕ(x). Then in particular TIα is provable for ϕ+ (x): pat ` Prog(ϕ+ ) → ∀ξ ≺ α ϕ+ (ξ)

(15.22)

I now reason in pat as follows: pat ` Prog(ϕ) → Prog(ϕ+ )

Lemma 15.22

+

→ ∀x ≺ α ϕ (ξ)

(15.22)

+

→ ϕ (α) → ∀ξ ∀η ≺ ξ ϕ(η) → ∀η ≺ ξ + ωα ϕ(η)

first line


defin. of ϕ+

→ ∀η ≺ 0 ϕ(η) → ∀η ≺ 0 + ωα ϕ(η) → ∀η ≺ ωα ϕ(η) Hence assuming (15.22) I have established Prog(ϕ) → ∀η ≺ ωα ϕ(η) in pat, that is, TIωα . a Transfinite induction up to 0 can now be easily derived:

proof theory of the kripke–feferman system

221

Proof of Theorem 15.21. Assume α < 0 . Then there is n with α < ωn , where ωn is as defined on p. 218 above. The claim TI0 is trivially derivable in pat. Applying Lemma 15.23 n+1 times yields pat ` TIωn . Since α < ωn (and this is provable in pat), the following formula is provable in pat: Prog(ϕ) → ∀ξ ≺ α ϕ(ξ) So TIα is provable in pat for any α < 0 .

a

The next task will be to show how the hierarchical truth predicates can be defined in the type-free language LT . First I introduce sublanguages of LT by recursion on the ordinals up 0 . These sublanguages are obtained by restricting the truth predicate to the previous languages: in the sentences of the language LTγ the truth predicate only occurs in contexts Tt ∧ SentTβ (t) with β < γ, where t is some term and SentTβ (x) expresses the property of being a sentence of LTβ ; LT0 is the language L without the truth predicate. I will write ϑβ (x) for the formula Tt∧SentTβ (t) in what follows. lemma 15.24. For any ordinal β < 0 the system kf proves the following sentences:
(i) ∀s ∀t ϑβ (s=. t) ↔ s◦ = t◦
(ii) ∀x SentTβ (x) → (ϑβ (¬. x) ↔ ¬ϑβ x)
(iii) ∀x ∀y SentTβ (x∧. y) → (ϑβ (x∧. y) ↔ ϑβ (x) ∧ ϑβ (y))
(iv) ∀x ∀y SentTβ (x∨. y) → (ϑβ (x∨. y) ↔ ϑβ (x) ∨ ϑβ (y))
(v) ∀v ∀x SentTβ (∀. vx) → (ϑβ (∀. vx) ↔ ∀t ϑβ (x(t/v)))
(vi) ∀v ∀x SentTβ (∃. vx) → (ϑβ (∃. vx) ↔ ∃t ϑβ (x(t/v)))
(vii) ∀t SentTα (t◦ ) → (ϑβ (ϑ. α t) ↔ ϑα t◦ for β > α
(viii) ∀t ∀α ≺ β SentTα (t◦ ) → (ϑβ (ϑ. α t) ↔ ϑβ t◦ proof. Theorems (i) and (iii)–(vi) follow directly from the respective axioms of kf as they are simply restrictions of these axioms. Items (v) and (vi) are consequences of kf8 and kf10. There is no kf axiom corresponding directly to the negation clause (ii); it can be proved by a main induction on β and a side induction on the complexity of the sentence x.

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I reason informally in kf considering the case of an atomic L-sentence first: kf ` ∀s ∀t ϑβ (¬. s=. t) ↔ s◦ 6= t◦ ↔ ¬ϑβ (s=. t)

kf2


kf1

The other cases are also straightforward, for instance, the case of a sentence beginning with two negation symbols is as follows: kf ` ∀x SentTβ (x) → (ϑβ (¬. ¬. x) ↔ ϑβ x)




kf3

↔ ¬¬ϑβ (x)




↔ ¬ϑβ (¬. x)




ind.hyp.

According to the definition of the languages LTβ the truth predicate T always appears together with a restriction to LTα sentences with α < β.  kf ` ∀x ∀α ≺ β SentTβ (ϑ. α (t)) →
 T ϑβ (¬. ϑ. α (t)) ↔ ϑβ (¬. (T. t∧. Sent def. ϑα (x) . α (t))) 
T ↔ ϑβ (¬. T. t) ∨ ϑβ (Sent kf5 . α (t)) 
T ↔ ϑβ (¬. t◦ ) ∨ ϑβ (¬. Sent kf13 . α (t)) 
T ↔ ¬ϑβ (t◦ ) ∨ ¬ϑβ (Sent ind. hyp. . α (t)) 
T ↔ ¬ϑβ (T. t) ∨ ¬ϑβ (Sent kf12 . α (t)) 
T ↔ ¬ϑβ (T. t∧. Sent kf4 . α (t)) 
↔ ¬ϑβ (ϑ. α (t)) def. ϑα (x) The fourth line from bottom requires TIβ , which is provable in pat by Theorem 15.21 as the formula T ϑβ (¬. t◦ ) ∨ ϑβ (¬. Sent . α (t))

in this line is equivalent to SentTα (t◦ ) → ϑβ (¬. t◦ ) because SentTα (x) is a formula of L, and thus the induction hypothesis is applied to sentences LTα with α < β. To prove (vii) one can prove the typed uniform typed biconditionals. For sentence (viii) Axiom kf12 can be used. a

proof theory of the kripke–feferman system

223

The items (i)–(viii) in Lemma 15.24 correspond to the axioms for ramified truth in Definition 9.2. However, the truth predicates ϑβ (x) do not amount to definitions of the truth predicates Tβ because they apply to syntactically different sentences: the truth predicates apply to sentences of LT while the truth predicates Tβ apply to sentences in the language Lβ . In order for the truth predicates ϑβ (x) to define the ramified truth predicates one can stipulate that a sentence of Lβ is true if and only if the result of replacing each truth predicate Tα with ϑα (x) is true in the sense of ϑβ . It is necessary to replace not only the truth predicates occurring in the sentence but also in the sentences to which the ramified truth predicates are applied. This can be achieved by defining a recursive function h in the following way, using Lemma 5.2:   n, if n is of the form s = t for terms s and t     or if n is not a sentence of LTβ       ¬h(ϕ), if n = (¬ϕ)    h(ϕ) ∧ h(ψ), if n = (ϕ ∧ ψ) h(n) =  h(ϕ) ∨ h(ψ), if n = (ϕ ∨ ψ)      ∀xh(ψ), if n = (∀xϕ)     ∃xh(ψ), if n = (∃xϕ)     ϑ h(t), if n = (Tα t) for a term t α. For the notation see Lemma 5.2. The function h sends each formula of Lβ to a formula of LTβ . This can be proved in Peano arithmetic. Using Lemma 15.24 one can now establish the following implication for all β < 0 and formulae ϕ of LTβ : If rt<β ` ϕ, then kf ` h(ϕ) This shows that ϑα (x) is a definition of the truth predicate Tα in kf. Therefore the following result is established: theorem 15.25. The system kf defines all truth predicates of rt<0 . The obvious question is whether kf can define more ramified truth predicates at the level 0 and beyond. The answer to this question is negative: kf does not define the truth predicate T0 of rt0 . To prove this negative result, a reduction of kf to the ramified system rt<0 can be used. It is fairly obvious that no system rtβ with β < 0 defines the truth predicate of kf, otherwise all truth predicates Tα with α < 0 would be definable in rtβ by the transitivity of the relation of truth-definability.

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So there is no global definition of the truth predicate of kf by a truth predicate Tβ of a ramified system if β < 0 . But one might hope that the truth predicate used in a given kf-proof be definable by a typed truth predicate Tα with a sufficiently high index α < 0 . In general, it is not easy to read off the required ordinal from a given proof. For cut-free proofs, however, this becomes feasible. Cantini (1989) sketched how to embed kf first into an infinitary system, from which cuts can be eliminated. The truth predicate in such a cut-free proof can then be defined by a suitable typed truth predicate Tα with an index α < 0 although positive and negative occurrences are treated in different ways. A detailed elaboration of the proof including an account of the steps involving familiar techniques of ordinal analysis would be lengthy and I will not attempt to give it here. An upper bound for the proof-theoretic strength of kf was given first by Feferman (1991) (and much earlier unpublished work), who gave a very elegant indirect argument by reducing kf to the system Σ11 -ac or Σ11 -choice whose proof-theoretic strength is known. At any rate, the system kf proves the same arithmetical sentences as rt<0 , which in turn is equivalent to the system ra<0 of ramified analysis up to 0 , which is, roughly speaking, the system aca iterated 0 times, that is, the system with 0 times iterated elementary comprehension (and suitable axioms or rules for limit levels).2 It is far from obvious how systems analogous to kf but built over alternative base theories behave. The proof of Theorem 15.25 depends on a result that does not make use of any specific properties of the truth predicate: once transfinite induction up to 0 had been established it was not too hard to prove that kf can define 0 -many levels of Tarski’s hierarchy, that is, 0 many iterations of compositional truth. Hence one might conjecture that the strength of a system obtained by adding kf-style truth axioms depends to a large extent on the mathematical properties of the base theory. Feferman (1991) outlined how his approach to truth might be applied to base theories other than Peano arithmetic. If the Friedman–Sheard system is formulated with a base theory different from Peano arithmetic, one will be able to adapt the proof-theoretic analysis of fs in Section 14.2 to other base theories: under usual circumstances the effect of adding truth axioms and the rules of necessitation and conecessitation 2 For more information on ramified analysis see, for instance, Buchholz et al. (1981), Feferman (1964), and Schütte (1977).

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will be similar to the effect of adding ω-many levels of compositional truth. In the case of fs the number of iterations of the truth predicate is determined by the rule of necessitation; and since nec is a rule that can be applied only finitely often in any given proof, one will not be able to define transfinite levels of the Tarskian hierarchy of languages in an fs-like system. In the case of kf, the number of definable levels of the Tarskian hierarchy is determined by mathematical properties of the base theory. If a base theory different from Peano arithmetic is used less or more levels of the Tarskian hierarchy may be definable, depending mainly on how much transfinite induction is provable in the base theory and its schemata are applied to the extended language with the truth predicate. The proof-theoretic properties of the ramified system, that is, of iterated compositional truth, also depend on the base theory, but at any rate it seems that the properties of kf-like truth are much more entangled in the properties of the base theory than is the case for classical symmetric truth, that is, for fslike systems. Whether this dependence is a welcome or problematic feature of kf depends on one’s perspective: if the truth theory is used, as it is in Feferman (1991), to make explicit the commitments implicit in the acceptance of the base theory, then this stark dependence of the proof-theoretic features of the truth theory on properties of the base theory it is based on will be seen as a strength of Kripke–Feferman-like systems. For then one is mainly interested in certain features of the base theory and less in properties of the notion of truth. If the main focus is on the notion of truth, however, then one will prefer axioms and rules whose effects are less sensitive to the underlying base theory, as is the case with the Friedman–Sheard theory. 15.4 Extensions In logic, axiom schemata are often understood as sets of formulae of a similar form. The induction schema of Peano arithmetic is often identified with the set of all L-formulae ϕ(0) ∧ ∀x (ϕ(x) → ϕ(Sx)) → ∀x ϕ(x). But there is also another way of thinking about schemata: one may conceive of the induction schema as the single expression (15.23)

P 0 ∧ ∀x (P x → P Sx) → ∀x P x

with a unary schematic predicate letter P . Peano arithmetic is then axiomatized by the usual axioms with the set of all induction axioms replaced with

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the single axiom (15.23) and a rule of substitution licensing the substitution of P with any formula of the language L. Of course, in formulating a schema one always makes use of a schematic letter. But usually this letter belongs to the metalanguages and one adds ‘for ϕ a formula in the language so-and-so’. In (15.23), P is a symbol of a formal language. The approach involving schematic letters and rules of substitution for them has the advantage that the application of a schema to a given language is easy to define: applying a schema to a language simply means licensing the substitution of the schematic letter (or letters) with formulae from the language in question. This might be seen as a formally explicit version of the usual formulation. When one is generalizing the notion of an application of a schema to a language, the formulation with a schematic letter can be convenient. It is still not mandatory to go through the axiomatization of the induction schema with a schematic letter and a corresponding rule of substitution. The use of a schematic letter and an accompanying rule of substitution, however, can make a difference if one turns to theories of truth that permit one to reason about the truth of schemata, that is, about the truth of sentences containing the schematic letter P . In these theories P is not only used but also mentioned. Formulae containing the schematic symbol P will be coded along with the other expressions of the language LT . These schematic theories can be developed for any system of truth I have considered, but in many cases such as the Friedman–Sheard theory fs a schematic formulation makes little difference as Fujimoto (2010a) has shown, extending some remarks by Feferman (1991). In the case of Kripke–Feferman and some related systems, however, it makes a marked difference. I am going to sketch some features of Feferman’s schematic version of kf, although this system goes beyond the scope of the book as it is a theory for a language properly extending LT . But the result is significant as it shows that the strength of truth theories can be highly sensitive to the treatment of axiom schemata, not only because one might admit the truth predicate in the induction schema or not, but also because one might reason in the theory about the truth of schemata. The so-called schematic reflective closure kf∗ of Peano arithmetic is formulated in the extension of the language LT that adds a new unary predicate symbol P , allowing for what could be called schematic reasoning. The schematic reflective closure of Peano arithmetic then has all axioms of kf, including all instances of induction in the extended language, the obvious truth

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axiom for atomic sentences with P , (15.24)

∀t (TP. t ↔ P t◦ ),

and a substitution rule for P and any formula ψ, (P -Subst)

ϕ(P ) , ϕ(ψ) ˆ

where ϕ(P ) is a formula not containing the truth predicate (but of course possibly containing P ), and ϕ(ψ) ˆ is the result of substituting atomic subformulae of ϕ(P ) of the form P s with ψ(s), where s is some term. As s may contain free variables it may be necessary to rename variables in ψ to avoid unintended variable bindings. The induction axioms of kf∗ can be replaced with the single schematic axiom (15.23), because all instances of this schema can be derived from the schematic form using P -Subst. The reason for restricting P -Subst to formulae ϕ(P ) without the truth predicate is the following problem: if one could apply the rule P -Subst to the additional truth axiom (15.24), one could derive ∀t (TP. t ↔ χ(t◦ )) for any formulae χ thereby making the system inconsistent as all formulae would become equivalent. Because of this restriction to P -Subst, kf∗ should not be seen as a system about the truth of kf-schemata but as a system about the truth of schemata in the original language without the truth predicate (but with the schematic letter P ). Feferman (1991) showed that the schematic reflective closure kf∗ of pa is significantly stronger than what Feferman calls the ordinary reflective closure, that is, than kf. While kf defines the compositional ramified truth predicates up to 0 , the schematic truth theory kf∗ yields all the truth predicates up to much higher levels. More precisely, kf∗ defines the truth predicates of all levels up to the Feferman–Schütte ordinal Γ0 (see p. 126), which is much larger than 0 . The system rt<Γ0 of ramified truth up to Γ0 is proof-theoretically equivalent to many other known systems. In particular, it is equivalent to some systems that have been proposed as explica