Model-Theoretic versus Axiomatic Approach

Two basic methods of characterising the notion of truth for formal languages are prevalent in the contemporary literature: model-theoretic and axiomatic. This chapter contains a description and evaluation of these methods.

When applying the model-theoretic method, we work in a metatheory and consider a concrete, well-defined, formal language L. In the first step, a general notion of a model of L is defined. The conditions for being a model of L are usually fairly liberal and consist basically in the model's having similar structure (or ‘signature’, as it is sometimes called) as L itself. In the next stage, we provide a definition of ‘truth in M’ – a binary relation between a model M and the sentences of L. Finally we single out a concrete model as the standard or the intended one and declare that truth simpliciter (of sentences of L) should be understood as truth in this model. In effect, with a model-theoretic approach, truth becomes a defined notion.

When using the axiomatic method, our approach is quite different. Given a language L, we extend it (if necessary) to the language LT by adding a new one-place predicate ‘T’, which will express our notion of truth – that is at least the intention. Then we specify the set of basic axioms or rules in the language LT. The idea is that some of these axioms/rules, containing ‘T’, will play the role of ‘meaning postulates’ – basic principles characterising the content of the notion of truth. It is exactly these principles, and not some external interpretation, that give the meaning to the truth predicate.

In what follows, examples of both types of characterisation will be presented and discussed. Starting with the model-theoretic approach, two classical constructions will be sketched: one due to Tarski and one proposed by Kripke. I assume the familiarity of the reader with both the Tarskian notion of truth in a model and with Kripkean fixed-point semantics. Accordingly, in each of these two cases I will omit the technical details, providing a rough sketch only and concentrating on how both of these approaches can help us to understand the notion of truth simpliciter. In contrast, full definitions of some axiomatic truth theories will be given here.

How useful to the deflationist are conservative theories of truth? In this chapter, I will discuss one particular strategy of employing the conservativity condition in order to foster the deflationist's philosophical goals. The main idea is to combine two attractive properties of truth theories: syntactic conservativeness and maximality.

As emphasised in the last paragraph of the previous chapter, conservative truth theories are very attractive to the deflationist in certain important respects. Now, maximality can be viewed as another desirable property. To begin with, imagine the disquotationalist who wants to extend his base theory of syntax with some chosen instantiations of the T-schema. Obviously, the set of instantiations has to be limited in some way in order to avoid the liar-type paradoxes. Unfortunately, this is easier said than done. Which instantiations should be adopted? Which of them should be rejected? What sort of choices can be made without inviting the charge of arbitrariness? Paul Horwich gives the following answer:

In view of this, one option which suggests itself consists in trying to include as many instances of the T-schema as possible in an attempt to build a maximal consistent extension of our base theory.

This possibility has been investigated by McGee (1992). At the start, McGee observes that it is indeed possible to extend a chosen base theory (say, Peano arithmetic) in such a way. However, in the end he notices two problems.

Firstly, there are continuum many maximal consistent extensions of Peano arithmetic. In effect, some additional principles will be required to select a truth theory of our choice. In short, maximality is not enough. We still have to “make comparative judgments about which instances of (T) we regard as essential and which we are willing to relinquish”.