Tarski’s theory of truth

• 2.1 Sentences as truth-bearers

• 2.2 Convention T

• 2.3 Recursive definition of truth

• 2.4 Reference and satisfaction

https://plato.stanford.edu/entries/truth/#TarTheTru

2. Tarski’s theory of truth

Modern forms of the classical theories survive. Many of these modern theories, notably correspondence theories, draw on ideas developed by Tarski.

In this regard, it is important to bear in mind that his seminal work on truth (1935) is very much of a piece with other works in mathematical logic, such as his (1931), and as much as anything this work lays the ground-work for the modern subject of model theory – a branch of mathematical logic, not the metaphysics of truth. In this respect, Tarski’s work provides a set of highly useful tools that may be employed in a wide range of philosophical projects. (See Patterson (2012) for more on Tarski’s work in its historical context.)

Tarski’s work has a number of components, which we will consider in turn.

2.1 Sentences as truth-bearers

In the classical debate on truth at the beginning of the 20th century we considered in section 1, the issue of truth-bearers was of great significance. For instance, Moore and Russell’s turn to the correspondence theory was driven by their views on whether there are propositions to be the bearers of truth. Many theories we reviewed took?beliefs?to be the bearers of truth.

In contrast, Tarski and much of the subsequent work on truth takes?sentences?to be the primary bearers of truth. This is not an entirely novel development: Russell (1956) also takes truth to apply to sentence (which he calls ‘propositions’ in that text). But whereas much of the classical debate takes the issue of the primary bearers of truth to be a substantial and important metaphysical one, Tarski is quite casual about it. His primary reason for taking sentences as truth-bearers is convenience, and he explicitly distances himself from any commitment about the philosophically contentious issues surrounding other candidate truth-bearers (e.g., Tarski, 1944). (Russell (1956) makes a similar suggestion that sentences are the appropriate truth-bearers “for the purposes of logic” (p. 184), though he still takes the classical metaphysical issues to be important.)

We will return to the issue of the primary bearers of truth in section 6.1. For the moment, it will be useful to simply follow Tarski’s lead. But it should be stressed that for this discussion, sentences are?fully interpreted?sentences, having meanings. We will also assume that the sentences in question do not change their content across occasions of use, i.e., that they display no context-dependence. We are taking sentences to be what Quine (1960) calls ‘eternal sentences’.

In some places (e.g., Tarski, 1944), Tarski refers to his view as the ‘semantic conception of truth’. It is not entirely clear just what Tarski had in mind by this, but it is clear enough that Tarski’s theory defines truth for sentences in terms of concepts like reference and satisfaction, which are intimately related to the basic semantic functions of names and predicates (according to many approaches to semantics). For more discussion, see Woleński (2001).?

2.2 Convention T

Let us suppose we have a fixed language?L?whose sentences are fully interpreted. The basic question Tarski poses is what an adequate?theory of truth for L?would be. Tarski’s answer is embodied in what he calls?Convention T:

An adequate theory of truth for L?must imply, for each sentence?? of L

? ? ? is true if and only if ?.

(We have simplified Tarski’s presentation somewhat.) This is an adequacy condition for theories, not a theory itself. Given the assumption that L?is fully interpreted, we may assume that each sentence ? in fact has a truth value. In light of this, Convention T guarantees that the truth predicate given by the theory will be?extensionally correct, i.e., have as its extension all and only the true sentences of L.

Convention T draws our attention to the biconditionals of the form

??????? is true if and only if ????,

which are usually called the?Tarski biconditionals?for a language L.

2.3 Recursive definition of truth

Tarski does not merely propose a condition of adequacy for theories of truth, he also shows how to meet it. One of his insights is that if the language L?displays the right structure, then truth for L?can be defined recursively. For instance, let us suppose that L?is a simple formal language, containing two atomic sentences ‘snow is white’ and ‘grass is green’, and the sentential connectives ?and ?.

In spite of its simplicity, L?contains infinitely many distinct sentences. But truth can be defined for all of them by recursion.

1. Base clauses:

1. ‘Snow is white’ is true if and only if snow is white.

2. ‘Grass is green’ is true if and only if grass is green.

2. Recursion clauses. For any sentences ?and ψ of?L:

1. ?is true if and only if ????is true or?is true.

2. is true if and only if it is not the case that?is true.

This theory satisfies Convention T.

2.4 Reference and satisfaction

This may look trivial, but in defining an extensionally correct truth predicate for an infinite language with four clauses, we have made a modest application of a very powerful technique.

Tarski’s techniques go further, however. They do not stop with atomic sentences. Tarski notes that truth for each atomic sentence can be defined in terms of two closely related notions:?reference?and?satisfaction. Let us consider a language L′, just like L?except that instead of simply having two atomic sentences, L′?breaks atomic sentences into terms and predicates.?L′?contains terms ‘snow’ and ‘grass’ (let us engage in the idealization that these are simply singular terms), and predicates ‘is white’ and ‘is green’. So L′?is like?L, but also contains the sentences ‘Snow is green’ and ‘Grass is white’.)

We can define truth for atomic sentences of L′?in the following way.

1. Base clauses:

1. ‘Snow’ refers to snow.

2. ‘Grass’ refers to grass.

3. a?satisfies ‘is white’ if and only if a?is white.

4. a satisfies ‘is green’ if and only if a?is green.

2. For any atomic sentence?:???is???is true if and only if the referent of ??satisfies ??.

One of Tarski’s key insights is that the apparatus of satisfaction allows for a recursive definition of truth for sentences with?quantifiers, though we will not examine that here. We could repeat the recursion clauses for?L?to produce a full theory of truth for L′.

Let us say that a Tarskian theory of truth is a recursive theory, built up in ways similar to the theory of truth for ?L′. Tarski goes on to demonstrate some key applications of such a theory of truth. A Tarskian theory of truth for a language?L?can be used to show that theories in?L?are consistent. This was especially important to Tarski, who was concerned the Liar paradox would make theories in languages containing a truth predicate inconsistent.

For more, see Ray (2018) and the entries on?axiomatic theories of truth, the?Liar paradox, and?Tarski’s truth definitions.