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.

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.

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.

Some proponents of deflationism with respect to truth contrast deflationism with various definitional theories of truth – such as definitions of truth in terms of correspondence or coherence – and claim that there is no hope of attaining an explicit definition of truth (see, for instance, Horwich 1990). The axiomatic approach to truth seems to be a hallmark of deflationism, although some deflationists flinch from the word axiomatization and prefer to call their axiomatization of truth an implicit definition. Presumably not all philosophers who reject (non-trivial) explicit definitions of truth qualify as deflationist; Donald Davidson's axiomatic account of truth, for instance, is usually not classified as deflationist. But since an axiomatization of truth seems to be a component of many deflationary conceptions of truth, the discussion of deflationism and the work on axiomatic theories of truth are closely related. So far the more formal contributions to the discussion about deflationism are based on typed axiomatic systems of truth. This applies, for instance, to the extensive debate about deflationism and conservativity. The concentration on typed systems of truth in this context seems to be borne out of the desire to avoid the intricacies of type-free systems and settle for a putatively widely accepted solution of the liar paradox. As I will argue, the focus on typed theories is misleading because in the context of type-free systems general claims about disquotational and therefore deflationist accounts of truth are no longer tenable.

The Kripke–Feferman theory is based on Strong Kleene logic as an evaluation schema. Kripke (1975) formulated his semantic construction is such a way that other evaluation schemata can be used as well. In the definition of Λ on p. 208 the relation ⊧SK of being valid in a model under Strong Kleene logic can be replaced with other notions of validity in a model. The relation replacing ⊧SK must satisfy a certain condition; otherwise the operation corresponding to Λ may lack fixed points. Here I will not go into general results on admissible evaluation schemata; rather I shall focus on axiomatic theories akin to kf but based on two alternative evaluation schemata.

First, I consider the standard Weak Kleene logic with only truth-value gaps and no gluts. In Strong Kleene logic a disjunction of two sentences is true if at least one disjunct is true, even when the other disjunct lacks a truth value. In Weak Kleene logic a sentence is evaluated as neither true nor false if one of its components lacks a truth value. The truth tables for Weak Kleene logic are thus easily described: whenever there is a truth-value gap among the entries of a line the value of the complex sentence will also be a gap. All other lines coincide with the lines of the truth tables of classical logic. One can easily develop Kripke's semantic construction for Weak Kleene logic.

Nonclassical logics have played an important role in formal theories of truth. In fact, the development of many nonclassical logics has been motivated by the hope that they can facilitate a resolution of the semantic paradoxes. Strong Kleene logic and supervaluations and their use in the theory of truth have been mentioned already. Recently dialethic theories have somewhat superseded the partial approaches to truth: on the usual dialethic account, the liar sentence is both true and false. If the liar is accepted together with its negation, classical logic must be abandoned to avoid triviality and various paraconsistent logics have been proposed to block the derivation of arbitrary sentences from a contradiction. More recently, Field's book Saving Truth From Paradox (2008) has sparked an increased interest in nonclassical axiomatic truth theories.

Most of the axiomatic theories I have discussed in the previous parts of this book, however, are formulated in classical logic. The only exception is the system pkf, an axiomatization of Kripke's theory in Strong Kleene logic.

Given the extensive use of nonclassical logics in the literature on formal theories of truth, the reader might wonder why I do not dedicate more space to the analysis and discussion of nonclassical truth theories. Actually, a referee of an early version of this book proposed that it should be entitled Classical Axiomatic Theories of Truth, because nonclassical theories are largely ignored by it.

Throughout this book Peano arithmetic has been used as the base theory. Like many other philosophers, I see the theory of truth for the language of arithmetic as the starting point for developing a theory of truth for other, usually more comprehensive languages as base languages and perhaps eventually for natural languages.

When applying the axiomatic theories of truth discussed in this book to base theories other than Peano arithmetic, one is confronted with at least two kinds of problems: first one needs to settle on a sort of truth theory – choosing between one based on the disquotation sentences, or the compositional axioms, a typed or untyped theory, and so on – such that the chosen sort of theory is both suitable for the base theory in question and consonant with its underlying philosophical motivation; and second, once a kind of truth theory has been chosen, the formulation of this kind of truth theory with the new base theory may not be straightforward: there may be different ways of applying the chosen axiomatic conception of truth to a base theory; moreover, as I will show, some ways have unwanted consequences and may even lead to inconsistencies.

Both kinds of problems are beyond the scope of this book. But in this chapter I will show how some of the formal results about axiomatic theories of truth obtained in this book or closely related results can shed at least some light on the problems.

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.

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.

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.

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.

Proof-theoretic reductions of various kinds are often seen as ontological reductions. For instance, the observation that Peano arithmetic is relatively interpretable in (and also reducible in other senses to) Zermelo–Fraenkel set theory is taken by many philosophers to be a reduction of numbers to sets.

Here I will only touch upon some of the issues raised by the results about axiomatic truth theories in this book and will not enter into a general discussion about ontological reduction (but see Bonevac 1982; Feferman 1998; Hofweber 2000; Niebergall 2000 for further discussion). I will proceed under the assumption that ontological commitments to numbers, sets, and other abstract objects are made by accepting theories about those objects. So, for instance, one makes an ontological commitment to numbers by accepting a theory such as Peano arithmetic. This assumption is far from unproblematic, but here I do not attempt to justify it as the general theory of ontological commitment goes far beyond the scope of this book.

If proof-theoretic reductions can be understood as ontological reductions, then in particular proof-theoretic reductions of mathematical theories to truth theories can be seen as such. An example is Theorem 8.42, which shows that the theory aca of sets of natural numbers which are arithmetically definable (with second-order parameters) is reducible to the compositional theory ct of truth. Both systems aca and ct contain Peano arithmetic and thus both bring a commitment to natural numbers, but the ontological commitment to arithmetically definable sets of numbers is eliminated by interpreting aca in ct.