Crisp Bi-Gödel modal logic and its paraconsistent expansion

In this paper, we provide a Hilbert-style axiomatization for the crisp bi-Gödel modal logic $\textbf {K}\textsf {biG}$. We prove its completeness w.r.t. crisp Kripke models where formulas at each state are evaluated over the standard bi-Gödel algebra on $[0,1]$. We also consider a paraconsistent expansion of $\textbf {K}\textsf {biG}$ with a De Morgan negation $\neg $, which we dub $\textbf {K}\textsf {G}^{2}$. We devise a Hilbert-style calculus for this logic and, as a consequence of a conservative translation from $\textbf {K}\textsf {biG}$ to $\textbf {K}\textsf {G}^{2}$, prove its completeness w.r.t. crisp Kripke models with two valuations over $[0,1]$ connected via $\neg $. For these two logics, we establish that their decidability and validity are $\textsf {PSPACE}$-complete. We also study the semantical properties of $\textbf {K}\textsf {biG}$ and $\textbf {K}\textsf {G}^{2}$. In particular, we show that Glivenko’s theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in $\textbf {K}$ (the classical modal logic) and the crisp Gödel modal logic $\mathfrak {G}\mathfrak {K}^{c}$. We show that, among others, all Sahlqvist formulas and all formulas $\phi \rightarrow \chi $ where $\phi $ and $\chi $ are monotone define the same classes of frames in $\textbf {K}$ and $\mathfrak {G}\mathfrak {K}^{c}$.