Non-deterministic Conditionals and Transparent Truth

Abstract

Theories where truth is a naive concept fall under the following dilemma: either the theory is subject to Curry’s Paradox, which engenders triviality, or the theory is not trivial but the resulting conditional is too weak. In this paper we explore a number of theories which arguably do not fall under this dilemma. In these theories the conditional is characterized in terms of (infinitely-valued) non-deterministic matrices. These non-deterministic theories are similar to infinitely-valued Łukasiewicz logic in that they are consistent and their conditionals are quite strong. The difference is the following: while Łukasiewicz logic is ω-inconsistent, the non-deterministic theories might turn out to be ω-consistent.