# Tarski's Semantic theory of truth

ALFRED TARSKI

Alfred Tarski (January 14, 1901 – October 26, 1983) was a Polish logician, mathematician and philosopher. Educated at the University of Warsaw and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics and philosophy, he immigrated to the USA in 1939 where he became a naturalized citizen in 1945, and taught and carried out research in mathematics at the University of California, Berkeley from 1942 until his death.

A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.

His biographers Anita and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models."

SEMANTIC THEORY OF TRUTH

The "Convention T" (also T-schema) standard in his "inductive definition of truth" was an important contribution to symbolic logic, semantics, and the philosophy of language.
In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych", setting out a mathematical definition of truth for formal (logical) languages. The 1935 German translation was titled "Der Wahrheitsbegriff in den formalisierten Sprachen", "The concept of truth in formalized languages", sometimes shortened to "Wahrheitsbegriff". An English translation appeared in 1956 in the first edition of the Logic, Semantics, Metamathematics.

Some recent philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centers on how to read Tarski's condition of material adequacy for a truth definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined:

"p" is true if and only if p.

(where p is the proposition expressed by "p")

The debate amounts to whether to read sentences of this form, such as

"Snow is white" is true if and only if snow is white

as expressing merely a deflationary theory of truth or as embodying truth as a more substantial property (see Kirkham 1992). It is important to realize that Tarski's theory of truth is for formalized languages, so examples in natural language are not illustrations of the use of Tarski's theory of truth.

(Wikipedia)

In 1933, Alfred Tarski published (in Polish) his analysis of the notion of a true sentence. This long paper undertook two tasks: first to say what should count as a satisfactory definition of ‘true sentence’ for a given formal language, and second to show that there do exist satisfactory definitions of ‘true sentence’ for a range of formal languages.

We say that a language is *fully interpreted* if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted.This was the main difference between the 1933 definition and the later model-theoretic definition of 1956.

Tarski described several conditions that a satisfactory definition of truth should meet.

OBJECT LANGUAGE AND METALANGUAGE

If the language under discussion (the object language) is L, then the definition should be given in another language known as the metalanguage, call it M. The metalanguage should contain a copy of the object language (so that anything one can say in L can be said in M too), and M should also be able to talk about the sentences of L and their syntax.

FORMAL CORRECTNESS

The definition of True should be ‘formally correct’. This means that it should be a sentence of the form

For all x, True(x) if and only if φ(x),

where *True* never occurs in φ;

or the definition should be provably equivalent to a sentence of this form. The equivalence must be provable using axioms of the metalanguage that don't contain *True*.

MATERIAL ADEQUACY

The definition should be ‘materially adequate’. This means that the objects satisfying φ should be exactly the objects that we would intuitively count as being true sentences of L, and that this fact should be provable from the axioms of the metalanguage.

Tarski used many sentences of M to express truth, namely all the sentences of the form

φ(*s*) if and only if ψ

whenever *s* is the name of a sentence S of L and ψ is the copy of S in the metalanguage. So the technical problem is to find a single formula φ that allows us to deduce all these sentences from the axioms of M; this formula φ will serve to give the explicit definition of *True*.

Tarski's own name for this criterion of material adequacy was *Convention T*. More generally his name for his approach to defining truth, using this criterion, was *the semantic conception of truth*.

(The Stanford Encyclopedia of Philosophy)

## Semantic theory of truth

A **semantic theory of truth** is a theory of truth in the philosophy of language which holds that truth is a property of sentences.

ORIGIN

The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish logician Alfred Tarski in the 1930s. Tarski, in "On the Concept of Truth in Formal Languages", attempted to formulate a new theory of truth in order to resolve the liar paradox. In the course of this he made several metamathematical discoveries, most notably Tarski's undefinability theorem using the same formal technique as Kurt Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate satisfying convention-T for the sentences of a given language cannot be defined *within* that language.

TARSKI'S THEORY

To formulate linguistic theories without semantic paradoxes like the liar paradox, it is generally necessary to distinguish the language that one is talking about (the object language) from the language that one is using to do the talking (the metalanguage). In the following, quoted text is use of the object language, while unquoted text is use of the metalanguage; a quoted sentence (such as "P") is always the metalanguage's name for a sentence, such that this name is simply the sentence P rendered in the object language. In this way, the metalanguage can be used to talk about the object language; Tarski demanded that the object language be contained in the metalanguage.

Tarski's *material adequacy condition*, also known as *Convention T*, holds that any viable theory of truth must entail, for every sentence "P", a sentence of the following form (known as "form (T)"):

(1) "P" is true if, and only if, P.

For example,

(2) 'snow is white' is true if and only if snow is white.

These sentences (1 and 2, etc.) have come to be called the "T-sentences". The reason they look trivial is that the object language and the metalanguage are both English; here is an example where the object language is German and the metalanguage is English:

(3) 'Schnee ist weiß' is true if and only if snow is white.

It is important to note that as Tarski originally formulated it, this theory applies only to formal languages. He gave a number of reasons for not extending his theory to natural languages, including the problem that there is no systematic way of deciding whether a given sentence of a natural language is well-formed, and that a natural language is closed (that is, it can describe the semantic characteristics of its own elements). But Tarski's approach was extended by Davidson into an approach to theories of meaning for natural languages, which involves treating "truth" as a primitive, rather than a defined concept. (See truth-conditional semantics.)

Tarski developed the theory to give an inductive definition of truth as follows.

For a language L containing ¬ ("not"), ∧ ("and"), ∨ ("or"), ∀ ("for all"), and ∃ ("there exists"), Tarski's inductive definition of truth looks like this:

(1) "A" is true if, and only if, A.

(2) "¬A" is true if, and only if, "A" is not true.

(3) "A∧B" is true if, and only if, A and B.

(4) "A∨B" is true if, and only if, A or B or (A and B).

(5) "∀x(Fx)" is true if, and only if, every object x satisfies the sentential function F.

(6) "∃x(Fx)" is true if, and only if, there is an object x which satisfies the sentential function F.

These explain how the truth conditions of *complex* sentences (built up from connectives and quantifiers) can be reduced to the truth conditions of their constituents. The simplest constituents are atomic sentences. A contemporary semantic definition of truth would define truth for the atomic sentences as follows:

An atomic sentence F(x1,...,xn) is true (relative to an assignment of values to the variables x1, ..., xn)) if the corresponding values of variables bear the relation expressed by the predicate F.

Tarski himself defined truth for atomic sentences in a variant way that does not use any technical terms from semantics, such as the "expressed by" above. This is because he wanted to define these semantic terms in terms of truth, so it would be circular were he to use one of them in the definition of truth itself. Tarski's semantic conception of truth plays an important role in modern logic and also in much contemporary philosophy of language. It is a rather controversial matter whether Tarski's semantic theory should be counted either as a correspondence theory or as a deflationary theory.