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Tri-simplicial Contradiction: The “Pascalian 3D Simplex” for the Oppositional Tri-segment

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The Exoteric Square of Opposition

Part of the book series: Studies in Universal Logic ((SUL))

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Abstract

In this paper, we deal with the theory of the “oppositional poly-simplexes”, producing the first complete analysis of the simplest of them: the oppositional tri-segment, the three-valued counterpart of the oppositional-geometrical red contradiction segment. The concept of poly-simplex has been proposed by us in 2009 (Moretti A, The geometry of logical opposition. PhD thesis, University of Neuchâtel, Switzerland, 2009), for generalizing the theory of the “oppositional bi-simplexes”, which is the heart of our and Angot-Pellissier’s “oppositional geometry” (Angot-Pellissier R, 2-opposition and the topological hexagon. In: [30], 2012; Moretti A, Geometry for modalities? Yes: through n-opposition theory. In: [27], 2004; Pellissier R, Logica Universalis 2:235–263, 2008). The latter is meant to be the general theory of structures like the logical hexagon (which is a bi-triangle). The poly-simplexes are the most straightforward way to turn any “geometry of oppositions” consequently many-valued, which is otherwise still a desideratum of the field. We start by recalling the general theoretical context: how the field was opened, around 2002, by a reflection on the foundations of paraconsistent negation (Béziau J-Y, Logical Investigations 10:218–233, 2003) and how from that has progressively emerged “oppositional geometry”, the theory of the “oppositional structures”, enabling to model the “oppositional complexity” of “oppositional phenomena”. After recalling how emerged the idea of poly-simplex, we explain why time seems to have now come to explore them for real, since we have two powerful new tools: (1) Angot-Pellissier’s sheaf-theoretical technique (Angot-Pellissier R, Many-valued logical hexagons in a 3-oppositional trisimplex. In: this volume, 2022; Angot-Pellissier R, Many-valued logical hexagons in a 3-oppositional quadrisimplex. Draft, January 2014) for producing the many-valued oppositional vertices of the poly-simplexes (and evaluating their edges) (2) and a generalization of “Pascal’s triangle”, turned here into a more general “Pascalian ND simplex”, whose suited “horizontal (N-1)D sections” provide a much needed and very powerful numerical-geometrical “roadmap”, for constructing and exploring any arbitrary oppositional poly-simplex (Sect. 1). After unfolding successfully the structure of the tri-segment (Sects. 2 and 3), we make an unexpected detour (Sect. 4) by Smessaert and Demey’s “logical geometry” (Smessaert H. and Demey L., Journal of Logic, Language and Information 23:527–565, 2014), composed of two “twin geometries”: one for “opposition” and another for “implication”. We thus develop the “implication geometry” of the tri-segment and so discover that what these authors take for a “bricolage” (called by them “Aristotelian geometry”) is in fact, when considered as general “Aristotelian combination” in oppositional spaces higher than the bi-simplicial one (the one in which they remain tacitly but constantly), the mathematically optimal way, bottom-up, for exploring methodically the poly-simplicial space. We end (Sect. 5) by considering applications of this tri-segment resulted from such a tri-simplicial diffraction of the bi-simplicial contradiction segment (which adds to it paracomplete, i.e., intuitionist, and paraconsistent, i.e., co-intuitionist, features) in many-valued logics, paraconsistent logics, quantum logic, dialectics, and psychoanalysis. In particular, we show that the tri-segment, by its paracomplete substructure, models, better than did anything before it, “Lacan’s square”.

To Jean-Yves, Régis, and Hans

Three structuralist deep minds

Who changed forever my

Creative perception

Of mathematics

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Acknowledgments

I wish to thank here warmly (1) my dear mother, for I owe her incredibly much, since without her love and patience, this study would not have been possible; (2) my friends Roland Bolz and Jean-François Mascari, who patiently read and commented a pre-final version of this study; (3) as well as Hans Smessaert, to which this study is co-dedicated, who tried to help me clarifying some technical points about his and Lorenz Demey’s idea of “logical geometry”. All mistakes remain mine.

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  1. Université degli Studi eCampus, Novedrate, Italy

    Alessio Moretti

  2. SSML di Varese, Varese, Italy

    Alessio Moretti

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  1. Alessio Moretti
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  1. Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Jean-Yves Beziau

  2. The Hellenic Open University, Patras, Greece

    Ioannis Vandoulakis

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Moretti, A. (2022). Tri-simplicial Contradiction: The “Pascalian 3D Simplex” for the Oppositional Tri-segment. In: Beziau, JY., Vandoulakis, I. (eds) The Exoteric Square of Opposition. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90823-2_16

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