<Regular Articles>Is truth a logical connective?: A truth theory and the harmony
32 , 2015-03-31 , 京都大学文学部科学哲学科学史研究室
Truth theories like the Friedman-Sheard's truth theory (FS) have two rules, T-in rule and T-out rule, about introduction and elimination of the truth predicate. They look like the introduction rule and the elimination rule of a logical connective. From the proof theoretic semantics viewpoint, one might think that the truth predicate is a logical connective which is governed by these two rules. From this proof theoretic semantics viewpoint, the nature of truth is like deflationist's nature of truth. Additionally one of the most important things is that the truth predicate does not disturb the traceability of the argument from the premises to a conclusion. However, a crucial problem has been known: any criteria to be a logical connective, known as a "harmony" of the introduction rule and the elimination rule, are not satisfied because of the ω-inconsistency of FS. Such ω-inconsistency is caused by the fact that the truth predicate enables us to define paradoxical formulae of seemingly infinite-length. These formulae can be regarded as coinductive objects in terms of computer science. The reason of the failure of the harmony is that these criteria are defined not for coinductively defined paradoxical formulae but for inductively defined formulae. In this paper, we examine how we can extend the criteria for harmony for coinductive formulae.