Non-standard modalities in paraconsistent Gödel logic

Abstract

We introduce a paraconsistent expansion of the Gödel logic with a De Morgan negation $\neg$ and modalities $■$ and $⧫$. 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 $■\varphi$ (resp., $⧫\varphi$) as infima (suprema) of both positive and negative supports of $\varphi$ 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.

arXiv admin note: text overlap with arXiv:2303.14164

References in corpus (2)

• Qualitative reasoning in a two-layered framework
• Crisp bi-Gödel modal logic and its paraconsistent expansion

Cited by in corpus (1)

• Fuzzy bi-Gödel modal logic and its paraconsistent relatives