The formal study of truth begins in the 1930s, with Alfred Tarski's now classic paper "The Concept of Truth in Formalized Languages". By that time, the concept of truth—and related concepts, like definability—had come to play an important role in mathematical logic. But both philosophers and logicians were worried about the credentials of the notion of truth. In particular, it was well-known the the concept of truth could generate paradoxes, the best known of which is the so-called liar paradox, which concerns such sentences as this one:
The first sentence on the course description for Philosophy 1890D that is set off as a quote is not true.
It is an empirical fact that the first sentence on the course description for Philosophy 1890D that is set off as a quote is, in fact, the sentence "The first sentence on the course description for Philosophy 1890D that is set off as a quote is not true." It is then easy enough to generate a contradiction.
Tarski's announced goal was to show that the concept of truth can consistently be used for the sorts of purposes for which it was then being used in mathematics. To accomplish this goal, Tarski does several things: He produces an analysis of the source of the liar paradox; he constructs a formal theory of truth that is intended to avoid the sources thus identified; and he proves that this theory is consistent---or, better, that it is consistent if a related sort of set theory is consistent.
Tarski's work has inspired a good deal of further work, both philosophical and formal. We'll look at a good deal of such work, including Saul Kripke's revolutionary treatment of the liar paradox, which opened up so much new ground. Toward the end of the semester, we will look at some mathematical applications of definitions and theories of truth to the study of formal theories of arithmetic.