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arXiv · 2303.14198

Non-standard modalities in paraconsistent G\"{o}del logic

Marta Bilkova·Sabine Frittella·Daniil Kozhemiachenko
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Abstract

We introduce a paraconsistent expansion of the G\"{o}del logic with a De Morgan negation $\neg$ and modalities $\blacksquare$ and $\blacklozenge$. We equip it with Kripke semantics on frames with two (possibly fuzzy) relations: $R^+$ and $R^-$ (interpreted as the degree of trust in affirmations and denials by a given source) and valuations $v_1$ and $v_2$ (positive and negative support) ranging over $[0,1]$ and connected via $\neg$. We motivate the semantics of $\blacksquare\phi$ (resp., $\blacklozenge\phi$) as infima (suprema) of both positive and negative supports of $\phi$ in $R^+$- and $R^-$-accessible states, respectively. We then prove several instructive semantical properties of the logic. Finally, we devise a tableaux system for branching fragment and establish the complexity of satisfiability and validity.

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Marta Bilkova, Sabine Frittella, Daniil Kozhemiachenko. 2023-03-24. Non-standard modalities in paraconsistent G\"{o}del logic. https://doi.org/10.1007/978-3-031-43619-2_29

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