Semantic Theory of Truth (Tarski)

When a language is closed, i.e., able to make statments about its own semantic elements, we can write sentences that reference themselves. This leads to paradoxes such as:

Tarski's theory of truth specifies an object language \(L\) and a meta language \(ML\). Then, the truth of a sentence in \(L\) is discussed in \(ML\). The meta-language also contains the object language.

Tarski wanted his theory of truth to be materially adequate that is, it should entail for every sentence 'P' in the object language,

Here, 'P' is the name for the (object language) sentence in the meta language and P is a copy (rendered in the meta-language) of the (object language) sentence. Sentences of the above form are called T-scheme sentences. I think of a T-scheme sentence of saying "The sentence P is true if and only if … <insert something that intuitively describes what is needed to make P true>"

Tarski believed that the concept of "satisfaction" from math could be used to build a theory of truth. An open formula (see below) can either be satsified or not satisfied by an assigment of values to free variables. Then, given a domain \(D\), an open formula can be satisfied by some, none, or all of the elements in \(D\). For example, let \(D=\{\text{London}, \text{Thames}\}\). Then "\(x\) is a city" is satisfied only by "London".

So, the concept of satisfaction is related to the concept of truth that we are trying to define. But what about closed formulas? Really, a sentence like "London is city" is a closed formula, because it doesn't have free variables, only constants. Here, the move is to consider closed-formulas as a special case of open-formulas, with no free variables. Then, a sentence \(S\) is true iff. for all elements in \(D\), \(S\) is satisfied.

This allows us to say which atomic sentences are true. Tarski then gives a way of inductively getting more true sentences. For example, the sentence '\(F \wedge G\)' is true iff. 'F' is true and 'G' is true.

In math, formulas can be closed or open. Closed formulas have no free variables, e.g. \(\exists x. P x\) (related: Combinators). Open formulas do have free variables, e.g. \(\sum_{i=1}^{n} i = 10\) where \(n\) is a free variable. The truth of an open formula depends on the assignment of value to the free variables.

Tarski's formulation uses two types of symbols:

• constants: logical constants (e.g. =, \(\wedge\),…) and symbols of fixed meaning (i.e. "London")
• variables, which are used for quantification (e.g. \(\forall x P x\)

Model Theory makes a distinction between three types of symbols in a language:

• variables (same as above)
• the logical symbols, e.g. (=, \(\wedge\),…)
• non-logical symbols, which are interpreted in a given structure \(A\). These include functions, relations, predicates, (e.g. +, *, "London")

How should Tarski's theory of truth be applied to model-theoretic languages? The truth value of a sentence depends on the model being used. So the truth of a sentence in the object language is:

1. described in the meta-language, and …
2. depends on how the non-logical constants are interpreted according to a model.

Note that the non-logical constants are not to be understood as variables. In a sentence, a non-logical constant such as "London" has a definite reference, given by the model.

• Internet Encyclopedia of Philosophy on Tarski's theory of truth
• Stanford Encyclopedia of Philosophy entry
• Wikipedia entry