Logics of truth.
Raymond Turner · Notre Dame Journal of Formal Logic · 1990
This paper surveys three recent semantic theories of truth and compares them from the perspective of their underlying logics.In particular, the underlying logic of the Gupta-Herzberger theory is investigated, and an analysis of modal logics of truth arising from this semantic theory is given. / Theories of truthIn recent years there has been a revival in the development of semantic theories of truth.They are all attempts to develop theories of truth for languages which contain their own truth predicates and moreover they are all semantic theories in that they are grounded in some semantic interpretation of the truth predicate.In this paper we shall compare these theories from the perspective of their underlying logics of truth.We shall concentrate on those theories cast within the framework of classical logic, since this is where the notion of truth is most at home.Three of the most influential theories are those of Scott [10]-Aczel [1], Kripke [9]-Gilmore [4]-Feferman [3], and Gupta [5]-Herzberger [6].In the case of the first two kinds of semantic theories the logic of the system is explicit.The main objective of this paper is to explore the underlying logic of the last kind of theory and to explore its connections with the other two. LI Frege structuresAczel [1] introduces the notion of a Frege structure in an effort to capture the consistent subtheory of Frege's Grundgesetze der Λrithmetik.Aczel formulates his theory within a model of the untyped Lambda Calculus and develops a theory of classes or types based upon a theory of truth and propositions.Frege structures are models of the Lambda Calculus enriched with two subsets: a set of propositions and a set of true propositions.These sets satisfy very natural closure conditions with respect to the logical connectives: on