Tableaux for epistemic Gödel logic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908716634210304 |
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| author | Bílková, Marta Ferguson, Thomas Kozhemiachenko, Daniil |
| author_facet | Bílková, Marta Ferguson, Thomas Kozhemiachenko, Daniil |
| contents | 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 Gödel fuzzy logic with an involutive negation and multiple $\mathbf{S5}$-like modalities. As already Gödel 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tableaux for epistemic Gödel logic Bílková, Marta Ferguson, Thomas Kozhemiachenko, Daniil Logic 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 Gödel fuzzy logic with an involutive negation and multiple $\mathbf{S5}$-like modalities. As already Gödel 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. |
| title | Tableaux for epistemic Gödel logic |
| topic | Logic |
| url | https://arxiv.org/abs/2510.04642 |