Logic-Sensitivity and Bitstring Semantics in the Square of Opposition

Abstract

This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. Bitstring semantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two topics have already been studied extensively in logical geometry, and are thus well-understood by themselves. However, the precise details of their interplay turn out to be far more complicated. In particular, the paper describes an elegant and natural interaction between bitstrings and logic-sensitivity, which makes perfect sense when bitstrings are viewed as purely combinatorial entities. However, when we view bitstrings as semantically meaningful entities (which is actually the standard perspective, cf. the term 'bitstring semantics'!), this interaction does not seem to have a full and equally natural counterpart. The paper describes some attempts to address this situation, but all of them are ultimately found wanting. For now, it thus remains an open problem to capture this interaction between bitstrings and logic-sensitivity from a semantic (rather than merely a combinatorial) perspective.

Boolean Subtypes of the U4 Hexagon of Opposition

“…However, this suggestion is mistaken: our point is merely that this conversion closure is not impacted by the choice between the two logical systems that are considered in this paper, i.e., FOL and SYL. If one were to consider other logical systems beyond these two, then the conversion closure of F cat , too, would start displaying similar kinds of 'logic-sensitivity' [40,42,66,67]. For example, consider a non-classical logic S * in which ∧ is no longer commutative.…”

“…Today, Aristotelian diagrams are studied in a variety of areas, including philosophy [24][25][26], linguistics [27][28][29], legal theory [30][31][32], and computer science [33][34][35]. The contemporary research program of logical geometry studies Aristotelian diagrams as objects of independent mathematical and philosophical interest [36][37][38][39][40]. A major (and still ongoing) research effort in this area is the development of a comprehensive typology of Aristotelian diagrams, which allows us to systematically classify these diagrams into various families and subfamilies [41][42][43].…”