Tarski proceeded to solve the problem for formal languages. He defended a correspondence theory of truth in The Concept of Truth in Formalized Languages (1933) and The Semantic Conception of Truth and the Foundations of Semantics (1944). According to Tarski, any proposed definition of truth must entail as a consequence all equivalences of the following form:
(a) A sentence S is true in some language L, if and only if p; where p represents a translation of S in a second-order or meta language.
For this condition which Tarski calls 'Convention T', an example would be:
(b) "Schnee ist weiss" is true in German, if and only if snow is white.
It is equally true in English: (c) "Snow is white"; if and only if snow is white.
For Tarski, what is important for any proposed definition of truth is the distinction between an 'object language' and a 'meta-language.'
The complete sentences in (a), (b) and (c) are all sentences enclosed in a meta-language, in other words, used to mention and assert something of another sentence. In the case of (c) clearly the meta-language and object language are both English. Natural languages such as English and German are their own meta-languages which allows them to both use and mention their own sentences. Tarski calls these types of languages ‘semantically closed’. Formal languages such as those found in logic and mathematics may be ‘semantically open’ because no sentence which mentions another sentence in the same language counts as a well formed formula.
For Tarski, the difference between a 'semantically open' and
'semantically closed' language is imperative. First, because only
semantically open languages can have a definition of truth; and second,
because when in natural languages the object language and the
meta-language are identical, paradoxes such as the liar paradox can be
generate which are undecidable. For example:
(d) This sentence is false.
The above sentence is undecidable because in referring to itself if it is true, it is false and if it is false, then it is true. Thus, Tarski states that truth can only be completely defined for ‘open’ languages, that is, languages where truth is ascribed from outside of the language (in a meta-language).
Therefore, according to Tarski, since truth is a property of sentences, as opposed to the world or of states of affairs, then any definition of truth must ascribe that property to a sentence as long as that sentence tells how things stand in the world.
Tarski's view of the truth is therefore in line with the 'classical' concept of truth as it corresponds to language and the world. However, Tarski's account has brought about much discussion and work in an attempt to solve the problem of defining truth in natural or 'closed' languages.
Ultimately, Alfred Tarski has been recognized as 'the man who defined truth'. His work on the concepts of truth and logical consequence set the stage for modern logic, influencing developments in mathematics, philosophy, linguistics, and computer science. Tarski promoted his view of logic as the foundation of all rational thought.