Aristotelian diagrams for semantic and syntactic consequence

Abstract

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the wellknown square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system's soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation. has been a surge of interest in Aristotelian diagrams at the metalogical level, which contain statements or notions belonging to the metatheory of a given logical system. For example,

The Modal Logic of Aristotelian Diagrams

“…Secondly, next to the standard research on Aristotelian diagrams for (formulas coming from) the object language L S of some logical system S, there is also a flourishing tradition of studying Aristotelian diagrams for metalogical notions [25,[60][61][62][63][64][65]. For example, given a logic S, we can say that the Aristotelian relations of S-contrariety and S-subcontrariety are themselves contrary to each other (since two L S -formulas cannot simultaneously be S-contraries and S-subcontraries, while they can be neither S-contraries nor S-subcontraries, e.g., when they are S-contradictories).…”

From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry

“…This is not a coincidence: because of their binary nature, the Aristotelian relations cannot capture the full Boolean complexity that may arise in larger sets [7]. 8 Theorem 2.6.…”

“…The oldest and most well-known example of an Aristotelian diagram is the so-called 'square of opposition' for the categorical statements from syllogistics, but there are also many other (larger, more complex) Aristotelian diagrams, often developed in very different logical systems than traditional syllogistics. In contemporary logic, it has become clear that these diagrams can be fruitfully studied as objects of independent mathematical and philosophical interest, which has led to the burgeoning subfield of logical geometry [6,8,11,16,32,33].…”

“…There also exist Aristotelian families that have more than two Boolean subtypes. For example, the family of Buridan octagons has three Boolean subtypes: one that requires bitstrings of length 4, one that requires bitstrings of length 5, and one that requires bitstrings of length 6[9,15] 8. For an easy illustration from classical propositional logic, note that the 3-element set {p ∨ q, ¬p, ¬q} is inconsistent, while each of its 2-element subsets is consistent.…”

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

“…In the research program of logical geometry, we study Aristotelian diagrams as objects of independent mathematical interest, regardless of the specific details of their concrete applications in philosophy, artificial intelligence and elsewhere [51][52][53][54][55][56][57][58][59][60]. One of the main insights from logical geometry is that Aristotelian diagrams are highly sensitive to the details of the logical system with respect to which they are defined [52].…”