Truth as a logical connective Shunsuke Yatabe Center for Applied Philosophy and Ethics, Graduate School of Letters, Kyoto University I swear, in the kingdom of generalities, you could be imperius rex. (Haruki Murakami “A Wild Sheep Chase”) 1 Introduction Some truth theories allow to represent and prove generalized statements as “all that you said is true” or “all theorems of PA are true” in the sense of deflation- ism. But these theories are ω-inconsistent [HH05] caused by McGee’s paradox [M85], Yablo paradox [Yb93] and so on. In this paper, we examine the relation- ship between generality and ω-inconsistency in terms of proof theoretic seman- tics. It is done by means of regarding the truth as a logical connective. Philo- sophically speaking, we do not try to assert that the truth conception should be represented by a logical connective instead of the predicate though some authors suggests the deflationistic truth is a logical expression [Gl16]: only we want to do here is to provide a new perspective to analyze the behavior of the truth predicate in ω-inconsistent truth theories. The object of this analysis is Freidman-Sheared’s truth theory FS [FS87] which is known to be ω-inconsistent. It consists of all axioms and schemata of PA, and the following special rules: – Formal Commutativity (FC): for any logical connective ◦ and quantifier Q, Tr(x ˙ ◦y) ≡ Tr(x) ◦ Tr(y) Tr(Qz(x(z))) ≡ QzTr(x(z)) – two inference rules NEC , CONEC : φ Tr(⌈φ⌉) NEC Tr(⌈φ⌉) φ CONEC where ⌈φ⌉ is a Godel code of φ. Since NEC looks like the introduction rule and CONEC looks like the elimination rule of Tr, it is natural to ask the following question from a viewpoint of a naive proof theoretic semantics [Hj12]: Is it possible to think the truth predicate Tr of FS as a logical con- nective? It is because — from the naive proof theoretic semantics viewpoint — to be a logical connective is to have its introduction and elimination rule.