Why the Logical Hexagon?

Logica Universalis 6 (1-2):69-107 (2012)
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The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change.



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Alessio Moretti
Université de Neuchâtel

Citations of this work

The power of the hexagon.Jean-Yves Béziau - 2012 - Logica Universalis 6 (1-2):1-43.
Logical Geometries and Information in the Square of Oppositions.Hans Smessaert & Lorenz Demey - 2014 - Journal of Logic, Language and Information 23 (4):527-565.
Constraints on the lexicalization of logical operators.Roni Katzir & Raj Singh - 2013 - Linguistics and Philosophy 36 (1):1-29.
Color-Coded Epistemic Modes in a Jungian Hexagon of Opposition.Julio Michael Stern - 2022 - In Jean-Yves Beziau & Ioannis Vandoulakis (eds.), The Exoteric Square of Opposition. Birkhauser. pp. 303-332.

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References found in this work

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A Natural History of Negation.Laurence R. Horn - 1989 - University of Chicago Press.
A New Introduction to Modal Logic.M. J. Cresswell & G. E. Hughes - 1996 - New York: Routledge. Edited by M. J. Cresswell.
Conceptual Spaces: The Geometry of Thought.Peter Gärdenfors - 2000 - Tijdschrift Voor Filosofie 64 (1):180-181.
A Natural History of Negation.Laurence R. Horn - 1989 - Philosophy and Rhetoric 24 (2):164-168.

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