We assume the reader is familiar with first-order logic. Nevertheless, we feel it worthwhile to collect the main definitions and results in the present chapter so that we may refer to them throughout the remainder of the book. Doing so will serve the dual purpose of making the book self-contained and preparing the reader for what lies ahead.

Syntax

Let {x0, x1, x2, …} be a countably infinite set of variables. We will use x, y, z, … as meta-variables ranging over the variables in this set. The symbols U, V, W, … range over finite sets of variables.

Definition 3.1 A vocabulary is a set of relation symbols and function symbols. Each symbol is associated with a natural number, called its arity, that indicates the number of arguments the symbol accepts. Nullary function symbols are called constant symbols, and will often be treated separately. Unary relation symbols are sometimes called predicates.

The function symbols in a vocabulary can be combined with variables to form more complicated expressions called terms.

Definition 3.2 Let L be a vocabulary. The set of L-terms is generated by the finite application of the following rules:

• Every variable is an L-term.

• Every constant symbol in L is an L-term.

• If f is an n-ary function symbol in L and t1, …, tn are L-terms, then f(t1, …, tn) is an L-term.

Many basic properties of IF logic were observed by Hodges while he was developing trump semantics [32, 33]. Subsequently, Caicedo and Krynicki attempted to prove a prenex normal form theorem for IF logic [10], but they failed to properly account for the subtleties of signaling. Later Caicedo, Dechesne, and Janssen succeeded in proving many logical equivalences for IF logic, including a prenex normal form theorem [9]. Additional equivalences and entailments first appeared in [39] and were later published in [40]. Basic model-theoretic properties of IF logic have been investigated in [24, 49–51].

Basic properties

A team of assignments encodes a player's knowledge about the current assignment. When we write, ℕ, X ⊧+ ϕ we are asserting that Eloise has a winning strategy for the semantic game for ϕ as long as she knows the initial assignment belongs to X. Now imagine that, at the beginning of the game, an oracle informs Eloise that the initial assignment belongs to a subteam Y ⊂ X. Then Eloise gains an advantage because she has fewer possibilities to consider. Thus smaller teams represent more information about the current assignment. At the extremes, a singleton team represents having perfect information about the current assignment, while the team of all possible assignments with a given domain represents having no information about the current assignment.

The following propositions record two important properties of assignment teams: (1) every subteam of a winning team of assignments is winning, and every subteam of a losing team of assignments is losing; (2) the empty team of assignments is both winning and losing for every IF formula, and it is the only team of assignments that can be both winning and losing for the same formula.

In our final chapter we briefly discuss two topics we feel deserve to be mentioned in spite of space constraints that prevent us from giving them fuller treatment. In the first half of the chapter, we address the debate concerning the possibility of giving a compositional semantics for IF logic. In the second half, we investigate the effect of introducing imperfect information to modal logic.

Compositionality

The original game-theoretic semantics for IF logic assigned meanings only to IF sentences [30]. Thus IF logic was immune to a common complaint lodged against Tarski's semantics for first-order logic, namely that truth is defined in terms of satisfaction, rather than truth alone. However, it also meant that one was not able to analyze IF sentences by looking at the meanings of their subformulas. Furthermore, Hintikka famously claimed that there could be no compositional semantics for IF logic:

Hintikka's assertion inspired Hodges to develop his trump semantics, which gives meanings to all IF formulas [32, 33]. In Chapter 4, we defined two other semantics for IF formulas: game-theoretic semantics and Skolem semantics. In order to emphasize the similarities between IF logic and first-order logic, we introduced both semantics in terms of single assignments.

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.

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.

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.

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.

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.

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.