Tarski's truth definition is very important for philosophical semantics: Since Frege many philosophers interested in the nature of meaning saw a close connection between meaning and truth. After all, the following 'most certain principle' (as Max Cresswell called it) is highly suggestive:
- For all declarative sentences s, s' and situations w: If s and s' have the same meaning, then s is true in w iff s' is true in w.
So whatever meanings are, it is at least partially clear what meanings do: They determine truth conditions. To get a formal theory of meaning started and formalize this property of meanings philosophers and logicians such as David Lewis, Richard Montague, Max Cresswell and others set out to rigorously define the assignment of truth conditions to possibly complex natural language sentences. This assignment could be conceived as a function from (the syntactic analysis of) sentences to truth conditions. But how to define this function?
This is where Tarski's importance comes in: Tarski showed us how to define such a function by recursion over the complexity of a very simple language (FOL): Define a class of models to interpret the non-logical vocabulary and on this basis recursively define the notion of truth in a model. Lewis et al. solved the problem mentioned above by semantically treating (fragments of) natural languages in the same model theoretic way in which Tarski treated FOL: Define a class of appropriate models that interpret the lexical items of the language and on this basis define the notion of truth of sentences in a model.
Tarski's definition was of course restricted to a quite impoverished language that could not represent a host of natural language phenomena such as intensionality, vagueness etc. But it provided the paradigm that could easily be combined with a more appropriate notion of model, essentially that stemming from higher-order modal logic.