The Interaction Between Logic and Geometry in Aristotelian Diagrams

Smessaert

2017

Symmetry

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Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B 4 , viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Smessaert

2017

Symmetry

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“…the contemporary, systematic study of Aristotelian diagrams. For example, logical geometry typically makes the simplifying logical assumption that Aristotelian diagrams are closed under negation Smessaert 2016c, 2018b;Smessaert andDemey 2014, 2017b), together with the simplifying geometrical assumption that the diagrams are highly regular, symmetric polygons/polyhedra (Demey and Smessaert 2016b, 2018a. It was already known that there exist Aristotelian diagrams which do not satisfy these simplifying assumptions, but these were treated as exceptions, and dealt with in an ad hoc fashion.…”

Section: Resultsmentioning

confidence: 99%

“…This symmetry can mathematically be described in various ways, using tools from Euclidean geometry, group theory and graph theory(Demey and Smessaert 2014, 2016b, 2018aSmessaert and Demey 2016).…”

mentioning

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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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.

“… The visual‐diagrammatic properties of Aristotelian diagrams are studied more systematically in Demey and Smessaert , and .…”

mentioning

confidence: 99%

Using Syllogistics to Teach Metalogic

2017

Metaphilosophy

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This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the distinction between contradictories and contraries, and the square of opposition. This strategy helps to improve students understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics.