Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

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Lorenz Demey
Demey, L. (2021) Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics. Axioms, 10, 128. https://doi.org/10.3390/axioms10030128

Abstract : 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.
Keywords : Aristotelian diagram; non-normal modal logic; square of opposition; logical geometry; neighborhood semantics; bitstring semantics

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