Simple tableaux for two expansions of Gödel modal logic Article Swipe
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· 2024
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.2401.15395
· OA: W4391376056
This paper considers two logics. The first one, $\mathbf{K}\mathsf{G}_\mathsf{inv}$, is an expansion of the Gödel modal logic $\mathbf{K}\mathsf{G}$ with the involutive negation $\sim_\mathsf{i}$ defined as $v({\sim_\mathsf{i}}ϕ,w)=1-v(ϕ,w)$. The second one, $\mathbf{K}\mathsf{G}_\mathsf{bl}$, is the expansion of $\mathbf{K}\mathsf{G}_\mathsf{inv}$ with the bi-lattice connectives and modalities. We explore their semantical properties w.r.t. the standard semantics on $[0,1]$-valued Kripke frames and define a unified tableaux calculus that allows for the explicit countermodel construction. For this, we use an alternative semantics with the finite model property. Using the tableaux calculus, we construct a decision algorithm and show that satisfiability and validity in $\mathbf{K}\mathsf{G}_\mathsf{inv}$ and $\mathbf{K}\mathsf{G}_\mathsf{bl}$ are PSpace-complete.
