Regular non-normal modal classicalities
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910031128035328 |
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| author | Freire, Alfredo Roque Martins, Manuel António |
| author_facet | Freire, Alfredo Roque Martins, Manuel António |
| contents | We present a novel investigation into the consistency operator ($\circ$), traditionally associated with paraconsistent logics, as a means of capturing non-normal modal classicalities within the Kripke framework. By semantically reinterpreting $\circ$ as an operator that distinguishes top and bottom values from other values in the algebra, we extend its applicability beyond paraconsistency into classical and modal logics. We introduce the logic $\mathcal{B}_4^\circ$, a four-valued Boolean logic augmented with the consistency operator, and provide a sound and complete axiomatization. Building on this foundation, we extend the semantics to the modal domain using the many-logics modal logic (MLML) framework. Specifically, we construct Kripke frames based on an eight-valued Boolean algebra that contains three distinct four-valued subalgebras, each representing a different world type. Our analysis reveals that the resulting modal logic exhibits a normal local consequence relation alongside a non-normal global consequence relation. Consequently, the characterization of frame properties --such as transitivity, reflexivity, and Euclideanness --deviates from modal logic $K$, requiring novel semantic tools. We further identify new modal formulas in an extended language that capture previously unavailable kinds of accessibility, leading to frame characterizations unattainable in traditional modal frameworks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20287 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regular non-normal modal classicalities Freire, Alfredo Roque Martins, Manuel António Logic We present a novel investigation into the consistency operator ($\circ$), traditionally associated with paraconsistent logics, as a means of capturing non-normal modal classicalities within the Kripke framework. By semantically reinterpreting $\circ$ as an operator that distinguishes top and bottom values from other values in the algebra, we extend its applicability beyond paraconsistency into classical and modal logics. We introduce the logic $\mathcal{B}_4^\circ$, a four-valued Boolean logic augmented with the consistency operator, and provide a sound and complete axiomatization. Building on this foundation, we extend the semantics to the modal domain using the many-logics modal logic (MLML) framework. Specifically, we construct Kripke frames based on an eight-valued Boolean algebra that contains three distinct four-valued subalgebras, each representing a different world type. Our analysis reveals that the resulting modal logic exhibits a normal local consequence relation alongside a non-normal global consequence relation. Consequently, the characterization of frame properties --such as transitivity, reflexivity, and Euclideanness --deviates from modal logic $K$, requiring novel semantic tools. We further identify new modal formulas in an extended language that capture previously unavailable kinds of accessibility, leading to frame characterizations unattainable in traditional modal frameworks. |
| title | Regular non-normal modal classicalities |
| topic | Logic |
| url | https://arxiv.org/abs/2602.20287 |