Boolean considerations on John Buridan's octagons of opposition

History and Philosophy of Logic

2019

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In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric versions of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme.

Section: Introductionmentioning

confidence: 88%

Section: Oresme's Livre Du Ciel Et Du Mondementioning

confidence: 99%

Between Square and Hexagon in Oresme's Livre du Ciel et du Monde

History and Philosophy of Logic

2019

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“…This diagram clearly shows that (the extension/sphere of) Horse is a proper subset of (the extension/sphere of) Animal. 9 We have already seen in Subsection 2.1 that this proper subset-relation essentially amounts to a subalternation 9 One might object that Schopenhauer's words ("completely enclose") commit him to Horse being a subset of Animal, but not necessarily a proper subset. This objection is misguided, for the following two reasons.…”

Section: From An Euler Diagram To a Classical Square Of Oppositionmentioning

confidence: 99%

“…One of the main theoretical insights of logical geometry is that a given family of Aristotelian diagrams can have multiple Boolean subtypes, i.e. it is perfectly possible for two Aristotelian diagrams to exhibit exactly the same configuration of Aristotelian relations among their respective sets of elements, and yet have completely different Boolean properties [9,15]. The first concrete example of this phenomenon was pointed out by Pellissier [26], and concerns the JSB hexagons.…”

Section: The α-Structures and Their Propertiesmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

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From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry

Studies in Universal Logic

2020

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In this paper I explore the connection between Schopenhauer's Euler diagrams and the Aristotelian diagrams that are studied in contemporary logical geometry. One can define the Aristotelian relations in a very general fashion (relative to arbitrary Boolean algebras), which allows us to define not only Aristotelian diagrams for statements, but also for sets. I show that, once this generalization has been made, each of Schopenhauer's concrete Euler diagrams can be transformed into a well-defined Aristotelian diagram. More importantly, I also argue that Schopenhauer had several more general, systematic insights about Euler diagrams, which anticipate general insights and theorems about Aristotelian diagrams in logical geometry. Typical examples include the correspondence between n-partitions and α-structures (a particular class of Aristotelian diagrams), and the fact that many families of Aristotelian diagrams have distinct Boolean subtypes. Because of his various concrete Euler diagrams and, especially, his more systematic observations about them, Schopenhauer can rightly be considered a distant forerunner of contemporary logical geometry.

On the Cognitive Potential of Derivative Meaning in Aristotelian Diagrams

Smessaert

Shimojima

Diagrammatic Representation and Inference

2021

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In this paper we investigate the cognitive potential of Derivative Meaning -defined in terms of Abstraction Tracking (Shimojima 2015) -in order to characterise various families of Aristotelian diagrams. In a first part we consider the notion of subdiagrams -i.e. uniform triangles of contrariety relations and implication relations -inside Aristotelian hexagons and octagons. In a second part we look at different strategies for embedding complete Aristotelian diagrams -i.e. classical and degenerate squares -into Aristotelian hexagons and octagons.