Logical Extensions of Aristotle’s Square

Abstract

We start from the geometrical-logical extension of Aristotle's square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle's square under its modal form has the following four vertices: A is α, E is ¬α, I is ¬ ¬α and O is ¬ α, where α is a logical formula and is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether ¬ is involutive or not) modal logic.[3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks.[15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of {α, α} by the logical operations {¬, ∧, ∨}, under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle's square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

“…This diagram was first used by Sauriol [85] in his research on the logical geometry of the propositional connectives and was later also adopted by Luzeaux et al [25] and Pellissier [64] in their research on modal logic. (Although these latter authors [25,64] erroneously call their diagram a 'tetraicosahedron'; cf. infra.)…”

“…(As far as we know, this polyhedron is not systematically studied in the geometrical literature on polyhedra; the term 'tetraicosahedron' only seems to occur in the logic-oriented research of Moretti, Pellissier and Luzeaux et al Furthermore, this term has not been used entirely consistently in the literature; for example, Pellissier [64] and Luzeaux et al [25] draw a tetrakis hexahedron, but call it a 'tetraicosahedron', while Moretti [68], conversely, draws a tetraicosahedron, but calls it a 'tetrahexahedron'.) Just like the tetrakis hexahedron, the tetraicosahedron has 14 vertices, 36 edges and 24 faces; even though this polyhedron is not convex, it thus still satisfies the Euler formula V − E + F = 2.…”

“…However, because of the ubiquity of the logical relations that they visualize, Aristotelian diagrams are now also used in other scientific and engineering disciplines, such as cognitive science [5,6], neuroscience [7], natural language processing [8], law [9][10][11], linguistics [12][13][14][15] and artificial intelligence. Within the latter field, Aristotelian diagrams have been used to study various logic-based approaches to knowledge representation, including fuzzy logic [16][17][18][19][20], modal-epistemic logic [21][22][23][24][25] and probabilistic logic [26][27][28]. Furthermore, Aristotelian diagrams are also used extensively to study (the connections between) other types of knowledge representation formalisms, such as formal argumentation theory [29][30][31][32], fuzzy set theory [33][34][35][36], formal concept analysis and possibility theory [37][38][39], rough set theory [37,40,41], multiple-criteria decision-making [42][43][44] and the theory of logical and analogical proportions [45]…”

“…This case study is especially relevant in the context of artificial intelligence, since each of these four visualizations has recently been used in AI-related research. In particular, the rhombic dodecahedron has been used in research on modal logic and dynamic epistemic logic [22,61], the tetrakis hexahedron and tetraicosahedron have been used to investigate modal logic [25,60,64,65], and the nested tetrahedron has been used to study (connections between) the knowledge representation formalisms of rough set theory, formal concept analysis and possibility theory [38,40]. (However, it should also be noted that our focus on B 4 precludes us from studying Aristotelian diagrams for knowledge representation formalisms that go beyond a Boolean setup, such as fuzzy logic and set theory [16][17][18][19][20][33][34][35][36].…”

Logical Geometries and Information in the Square of Oppositions

“…28 The correspondence established above is certainly not the only connection between the binary, truth-functional connectives and logical geometry. For example, several authors have noted that these connectives can be used to decorate a rhombic dodecahedron (Zellweger 1997;Kauffman 2001) and related diagrams (Sauriol 1968;Luzeaux et al 2008;Moretti 2009a;Dubois and Prade 2012). Such diagrams visualize the Aristotelian relations that hold between propositions of the form p • q and p • q, where • and • are binary, truth-functional connectives.…”