Modal Propositional Truth Logic Tr and its Completeness

Abstract

In this paper Sobochin_ski’s four-valued modal logic $\textbf{V2}$ (extension of $ \textbf{S5} $) is considered. The emergence of that logic, some its interesting properties and different equivalent formulations are presented. Its algebraic models are of particular interest: as the extension of De Morgan algebra by boolean negation $\neg$ and as the extension of Boolean algebra by the endomorphism $\textit{g}$, which is interpreted then as the propositional truth operation $T$. The logic corresponding to the last case is denoted by $Tr$. The attention is paid to the application of $Tr$ in Fitting’s theory of truth. The axiomatization of $Tr$ in language ($\rightarrow, \neg, T$) is considered. The completeness of logic $Tr$ is proved with use of Sahlqvist’s powerful theorem, which gives the sufficient condition of Kripke completeness for normal modal logics. Algebraic completeness of logic $Tr$ is also proved.