Axiomatizing Truth: Why and How?

Broadly speaking there are two kinds of theories of truth, philosophical and logical. The philosophical theories of truth go back to the Greeks and forward to the present day. Among these are the correspondence, coherentist, pragmatist, deflationary and primitivist theories of truth. Logical theories of truth are roughly of two kinds, semantical (or definitional) and axiomatic. Tarski inaugurated semantical theories in the mid 1930s with his definition of truth for a logically circumscribed language within a metalanguage for it, i.e. in a typed setting in order to avoid inconsistency. However, the ordinary use of truth in natural language is untyped, and so beginning in the 1960s, attempts were made to obtain useful consistent untyped semantical theories by giving up some of Tarski’s basic assumptions. One of the most successful was due to Kripke in 1975, who defined a notion of truth for an untyped three-valued language. My axiomatization of Kripke’s model a few years later inaugurated a considerable body of work on a variety of axiomatic theories that continues to be actively pursued. In this paper general considerations are presented as to why one should axiomatize theories of truth and what criteria should be applied to them. These are then illustrated with three examples from my own work as to how one might try to go about meeting these criteria.