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.
Highlights
Aristotelian diagrams, such as the so-called square of opposition, visualize a number of formulas from some logical system, as well as certain logical relations holding between them
It is without any doubt that the second oldest and most widely used Aristotelian diagram is the modal square of opposition, for statements such as ‘it is necessary that p’ and ‘it is possible that p’
These four sections share roughly the same structure: first, we introduce the relevant type of logic-sensitivity and provide a concrete example from the realm of normal modal logic; we provide more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which do turn out to display the relevant type of logic-sensitivity once we turn to non-normal systems of modal logic
Summary
We discuss one final type of logic-sensitivity, viz., with respect to Boolean subfamilies. The type of logic-sensitivity we will discuss is not just another subtype of logic-sensitivity with respect to Aristotelian families, and does represent a fundamentally new phenomenon. The idea is that Aristotelian diagrams for one and the same fragment with respect to two different logical systems might be completely identical, except for their Boolean properties. Using more classification-oriented terminology, the diagrams for (F , S1 ) and (F , S2 ) belong to one and the same Aristotelian family, but they belong to different Boolean subfamilies of this family
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