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

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.

Logic-Sensitivity and Bitstring Semantics in the Square of Opposition

“…On the other hand, bitstring semantics is a combinatorial technique that allows us to systematically compute compact representations of the formulas that appear in a given diagram, thus providing a concrete grip on its logical behavior. The wideranging significance of the present paper should be clear, since logic-sensitivity and bitstring semantics are not only among the most important research topics in logical geometry today [8,13,15,38], but are also regularly discussed across the broad field of philosophical logic [2,16,23,27,28,31,32]. These two topics are well-understood by themselves, but the precise details of their interaction turn out to be far more complicated.…”

“…Logic-sensitivity is a well-known phenomenon in logical geometry [8,13] and beyond [2,28,31,32]. In order to properly describe this phenomenon, we first need to introduce the notions of Aristotelian relations, diagrams and isomorphisms.…”

“…More precisely, given a fragment F ⊆ L and different logical systems S 1 and S 2 (which share the same language L), we have a case of logic-sensitivity iff the identity function id F is not an Aristotelian isomorphism between (F, S 1 ) and (F, S 2 ). We can distinguish between two types of cases of logic-sensitivity [13]: in the 'blatant' cases, there simply does not exist an Aristotelian isomorphism between (F, S 1 ) and (F, S 2 ) whatsoever, while in the more 'subtle' cases, such an Aristotelian isomorphism does exist, but id F itself is not an Aristotelian isomorphism between (F, S 1 ) and (F, S 2 ).…”

“…The square of opposition has a rich and well-documented history in logic, and nowadays it is also used in various other disciplines that are concerned with reasoning, such as linguistics, psychology and computer science [13,30,37]. Among the oldest and most well-known examples are the squares of opposition for the categorical statements (from assertoric syllogistics) and for the modal expressions, which can both be traced back to the logical works of Aristotle.…”

The Modal Logic of Aristotelian Diagrams

“…Furthermore, historical scholarship has shown that Aristotelian diagrams for modal logic can be reconstructed from the works of many earlier authors, such as Theophrastus [6,7], Chrysippus [8,9], and Avicenna [10,11]. Today, Aristotelian diagrams not only appear in well-known textbooks on modal logic [12,13], but they are also used in applications of modal logic to a variety of philosophical and logical topics, such as paraconsistency [14,15], logic-sensitivity [16][17][18], and theories of truth [19,20]. It should be emphasized that all of these applications manifest a wide diversity of Aristotelian diagrams: next to the "ordinary" squares of oppositions, we also find several types of hexagons, octagons, and even more complex diagrams [21].…”