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 Warsaw School suspected truth to be a concept that should be avoided in good philosophy (see Woleński and Simons 1989 for a historical account).

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.

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.

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. 194 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 Ë 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.

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 language-level – 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.

The reader may wonder why I have left the discussion of type-free disquotation systems of truth until the end of the part on type-free theories, given that in the part on typed theories the disquotational theories come before all other typed theories.

In both parts I have roughly followed the strategy of discussing simpler and weaker theories first before proceeding to the stronger systems. But while typed disquotational theories such as TB and UTB are weaker than compositional theories such as CT and PT, type-free disquotational systems of truth are not necessarily weaker than type-free compositional theories. This is not to say that type-free disquotational theories are all very strong. Rather the type-free disquotational truth theories do not form a very homogenous class of theories: the strength of the consistent systems of type-free disquotation truth ranges from theories conservative over Peano arithmetic to theories of arbitrary strength, as will be shown below.

A reason for the disparity between type-free disquotational systems is that it is not always clear what the good systems are. Type-free disquotational system are not easily obtained by generalizing typed disquotational theories: generalizing the theory TB to a type-free theory is not as easy as generalizing CT to FS or PT to KF.

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. So it seems that I need to defend myself for mainly considering theories of truth formulated in classical logic rather than in a paraconsistent or some other nonclassical logic.

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.

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.

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 a sentence and its negation cannot both 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.

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 8.35, 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.