Crisp Bi-Gödel modal logic and its paraconsistent expansion

0577158 - ÚI 2025 US eng J - Článek v odborném periodiku
Bílková, Marta - Frittella, S. - Kozhemiachenko, D.
Crisp Bi-Gödel modal logic and its paraconsistent expansion.
Logic Journal of the IGPL. Online First 28 September 2023 (2024), č. článku jzad017. ISSN 1367-0751. E-ISSN 1368-9894
Grant CEP: GA ČR(CZ) GA22-01137S
Institucionální podpora: RVO:67985807
Klíčová slova: Paraconsistent logics * Gödel modal logic * correspondence theory * axiomatic systems * complexity
Obor OECD: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Impakt faktor: 1, rok: 2022
Způsob publikování: Omezený přístup
https://dx.doi.org/10.1093/jigpal/jzad017

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 $ eg $, 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 $ eg $. 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 $\mathfrak {G}\mathfrak {K}^{c}$.
Trvalý link: https://hdl.handle.net/11104/0346391