The phrase "a truth definition" is correct and usable in written English.
It can be used when discussing the concept of defining what truth means in a philosophical or logical context.
Example: "In philosophy, a truth definition is essential for understanding the nature of reality and belief."
Alternatives: "a definition of truth" or "a truth criterion".
In his 1933 paper Tarski went on to show that many fully interpreted formal languages do have a truth definition that satisfies his conditions.
Tarski's own conclusion was that a truth definition for a language L has to be given in a metalanguage which is essentially stronger than L.
We would like to be able to give a truth definition for set theory; but by Tarski's result this truth definition can't be given in set theory itself.
In 1956 he and his colleague Robert Vaught published a revision of one of the 1933 truth definitions, to serve as a truth definition for model-theoretic languages.
At around the same time as he was writing the 1933 paper, Tarski gave a truth definition by quantifier elimination for the first-order language of the field of real numbers.
Later he gave a fuller account, emphasising that his method provided not just a truth definition but an algorithm for determining which sentences about the real numbers are true and which are false.
Possibilistic versions of situation semantics typically start out with a recursive truth definition that relates utterances of sentences to the sets of possible situations in which the utterances are true, the propositions expressed.
Essentially the same devices allow Jaakko Hintikka to give an internal truth definition for his independence friendly logic; this logic shares the second and third properties of Levy's classes of formulas.
With a little adjustment this truth definition an be made to fit IF logic too.
It turns out that there is one for the slightly different logic introduced by Hodges 1997, and it leads to a Tarski-style truth definition for the language of that logic.
By the late 1940s it had become clear that a direct model-theoretic truth definition was needed.
