On the empty O-corner of the Aristotelian Square: A view from Singapore English

Ziegeler, Debra

doi:10.1016/j.pragma.2017.03.008

Cited by 3 publications

(2 citation statements)

References 24 publications

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“…Several of these diagrams have a long history, which can be traced back anywhere from a few decades [8][9][10][11][12][13][14][15] to several centuries [16][17][18][19][20][21][22][23]. 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].…”

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

Axioms

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“…They are also widely used to study families of logics such as modal logic (Rini and Cresswell 2012), relevant logic (Beall et al 2006), deontic logic (Barcan Marcus 1966) and probabilistic logic (Pfeifer and Sanfilippo 2017). Furthermore, because of the ubiquity of the logical relations that they visualize, Aristotelian diagrams are nowadays also frequently used beyond philosophy and logic, in disciplines such as computer science (Ciucci et al 2016), law (Vranes 2006) and linguistics (Ziegeler 2017).…”

Section: Introductionmentioning

confidence: 99%

Aristotelian diagrams for semantic and syntactic consequence

Demey

2018

Synthese

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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,

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Computing the maximal Boolean complexity of families of Aristotelian diagrams

Demey

2018

Journal of Logic and Computation

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On the empty O-corner of the Aristotelian Square: A view from Singapore English

Cited by 3 publications

References 24 publications

Boolean Subtypes of the U4 Hexagon of Opposition

Boolean Subtypes of the U4 Hexagon of Opposition

Aristotelian diagrams for semantic and syntactic consequence

Computing the maximal Boolean complexity of families of Aristotelian diagrams

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