Tableaux for epistemic G\"{o}del logic

Abstract

We propose a multi-agent epistemic logic capturing reasoning with degrees of plausibility that agents can assign to a given statement, with $1$ interpreted as "entirely plausible for the agent" and $0$ as "completely implausible" (i.e., the agent knows that the statement is false). We formalise such reasoning in an expansion of G\"{o}del fuzzy logic with an involutive negation and multiple $\mathbf{S5}$-like modalities. As already G\"{o}del single-modal logics are known to lack the finite model property w.r.t. their standard $[0,1]$-valued Kripke semantics, we provide an alternative semantics that allows for the finite model property. For this semantics, we construct a strongly terminating tableaux calculus that allows us to produce finite counter-models of non-valid formulas. We then use the tableaux to show that the validity problem in our logic is $\mathsf{PSpace}$-complete when there are two or more agents, and $\mathsf{coNP}$-complete for the single-agent case.