# Semantic Theory of Truth (Tarski)

Tarski gives a theory of truth as a property of sentences.

## 1. Liar paradox

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:

(S): This sentence (S) is false

## 2. Theory of truth

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,

'P' is true, if and only if, P

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>"

## 3. Satisfaction

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.

## 4. Composition

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.

## 5. Closed formulas vs open formulas

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.

## 6. Model

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:

- described in the meta-language, and …
- 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.