Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics
Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on n...
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MDPI AG
2021-06-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/3/128 |
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| author | Lorenz Demey |
| author_facet | Lorenz Demey |
| author_sort | Lorenz Demey |
| collection | DOAJ |
| description | Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic. |
| first_indexed | 2024-03-10T10:10:53Z |
| format | Article |
| id | doaj.art-a1ad2f7446ff494d9a969645f63f60ed |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T10:10:53Z |
| publishDate | 2021-06-01 |
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| record_format | Article |
| series | Axioms |
| spelling | doaj.art-a1ad2f7446ff494d9a969645f63f60ed2023-11-22T01:14:26ZengMDPI AGAxioms2075-16802021-06-0110312810.3390/axioms10030128Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal LogicsLorenz Demey0Center for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, BelgiumAristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic.https://www.mdpi.com/2075-1680/10/3/128Aristotelian diagramnon-normal modal logicsquare of oppositionlogical geometryneighborhood semanticsbitstring semantics |
| spellingShingle | Lorenz Demey Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics Axioms Aristotelian diagram non-normal modal logic square of opposition logical geometry neighborhood semantics bitstring semantics |
| title | Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics |
| title_full | Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics |
| title_fullStr | Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics |
| title_full_unstemmed | Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics |
| title_short | Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics |
| title_sort | logic sensitivity of aristotelian diagrams in non normal modal logics |
| topic | Aristotelian diagram non-normal modal logic square of opposition logical geometry neighborhood semantics bitstring semantics |
| url | https://www.mdpi.com/2075-1680/10/3/128 |
| work_keys_str_mv | AT lorenzdemey logicsensitivityofaristoteliandiagramsinnonnormalmodallogics |