Axiomatic truth theories and reflection principles

Abstract

The concept of truth occupies a curious place in logic. On the one hand it is central; showing completeness and correctness for a given deductive system is a matter of showing that what is true lines up with what is provable. On the other hand truth is almost always a meta-theoretical notion, which is not present within the deductive systems considered. Ever since the work of Tarski, we know that straightforwardly internalizing the concept of truth is impossible, due to the Liar paradox. Axiomatic theories of truth are an attempt at circumventing this paradox, by introducing a primitive truth predicate, and suitable axioms governing it.The chief aim of this thesis is to give a self-contained introduction to truth deflationism, the philosophical position that the concept of truth has only a minor linguistic role to fulfill, and the axiomatic truth theories that underlie it. In particular, an account of the conservativity debate is given, as exemplified by the axiomatic truth theories TB and CT. Critics of truth deflationism have pointed out that an axiomatic truth theory can’t be both conservative, roughly meaning that it does not provide genuinely new mathematical knowledge (like TB), and validate truth-theoretical reasoning (like CT).Recently, a reply for the deflationist has been proposed that rests on an implicit com- mitment to certain reflection principles, which (partially) express the soundness of a deductive system. If one is committed to a reflection principle for a conservative truth theory, like TB, then one can obtain stronger systems like CT. The different technical results are presented, and the philosophical motivation for this implicit commitment is evaluated. It is argued that reflection principles are indeed acceptable, and crucially, can be justified without recourse to the strong concept of truth which they are supposed to recover for the truth deflationist. This leaves the truth deflationist in a position to withstand the criticism: the concept of truth can be conservative, in an appropriate sense, and linguistically useful too.