Combining and Relating Aristotelian Diagrams

Vignero, Leander

doi:10.1007/978-3-030-86062-2_20

Cited by 5 publications

(2 citation statements)

References 5 publications

(9 reference statements)

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Boolean Subtypes of the U4 Hexagon of Opposition

Demey,

Erbas

2024

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This paper investigates the so-called ‘unconnectedness-4 (U4) hexagons of opposition’, which have various applications across the broad field of philosophical logic. We first study the oldest known U4 hexagon, the conversion closure of the square of opposition for categorical statements. In particular, we show that this U4 hexagon has a Boolean complexity of 5, and discuss its connection with the so-called ‘Gergonne relations’. Next, we study a simple U4 hexagon of Boolean complexity 4, in the context of propositional logic. We then return to the categorical square and show that another (quite subtle) closure operation yields another U4 hexagon of Boolean complexity 4. Finally, we prove that the Aristotelian family of U4 hexagons has no other Boolean subtypes, i.e., every U4 hexagon has a Boolean complexity of either 4 or 5. These results contribute to the overarching goal of developing a comprehensive typology of Aristotelian diagrams, which will allow us to systematically classify these diagrams into various Aristotelian families and Boolean subfamilies.

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Section: Introductionmentioning

confidence: 99%

Boolean Subtypes of the U4 Hexagon of Opposition

Demey,

Erbas

2024

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“…1 Over the past decade, it has become increasingly clear that squares of opposition (and other, more complex diagrams) can be fruitfully studied as objects of independent interest. This has given rise to the flourishing research program of logical geometry, which today has its own research topics (e.g., informational optimality [37]), its own mathematical tools and techniques (e.g., Aristotelian isomorphisms [9,41]), and also its own internal dynamics, including -as will become clear in this paper -some challenging open problems.…”

Section: Introductionmentioning

confidence: 99%

Logic-Sensitivity and Bitstring Semantics in the Square of Opposition

Demey,

Frijters

2023

J Philos Logic

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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.

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A Database of Aristotelian Diagrams: Empirical Foundations for Logical Geometry

Demey

Smessaert

2022

Lecture Notes in Computer Science

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Combining and Relating Aristotelian Diagrams

Cited by 5 publications

References 5 publications

Boolean Subtypes of the U4 Hexagon of Opposition

Boolean Subtypes of the U4 Hexagon of Opposition

Logic-Sensitivity and Bitstring Semantics in the Square of Opposition

A Database of Aristotelian Diagrams: Empirical Foundations for Logical Geometry

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