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# The oppositional hexagon (or logical hexagon)

( square - hexagon - cube - bi-simplex of dim m ) - up
The logical hexagon seems to have been discovered independently by Sesmat (1951) and Blanché (1953), by adding to the classical "AEOI" square (according to the medieval terminology) the "U" and the "Y" positions ("U" is the logical disjunction of "A" and "E", whereas "Y" is the logical conjunction of "I" and "O"). It is the first known proper avatar of the logical square (the "AUEOYI" hexagon contains 3 logical squares, among which the classical "AEOI" one). In this respect, remark that it admits all the kinds of decorations (quantificational, modal, order-theoretical, etc.) which were admitted by the logical square. But despite of its formal, mathematical strength (look, for instance, at the 3 symmetry axes of the logical hexagon, whereas the logical square has only one), its discovery has not been much noticed (neither by pure logicians, nor by analytical philosophers), at least until Béziau (2003) used it in a controversy he held (with others) against a famous argument by Slater (1995), about the question of knowing if logical paraconsistency is a possible formal behaviour. By taking into account the 2 "null modalities" (the "zero modality" and its negation) Béziau discovered the existence of 2 new alethic decorations of the logical hexagon. Later Moretti and Smessaert discovered further alethic decorations and Pellissier discovered the existence of so-called "broken" (or "weak") logical hexagons. In 2004 Moretti has shown that the logical square and the logical hexagon are followed by a logical cube and that the three of them (seen respectively as a 2-, a 3- and a 4-opposition) are particular instances of a more general structure, that of the logical bi-simplexes of dimension m (seen as a n-opposition, with n=m+1). More precisely, in the light of Moretti's later theory of the logical poly-simplexes of dimension m (2009), the logical hexagon is a "logical bi-triangle", whose possible immediate avatars are, therefore, the logical bi-tetrahedron (this is the aforementioned logical cube) and the "logical tri-triangle". Remark that the logical hexagon has been used by Gallais (1982) for building a new model of narratology.
Béziau (2003)
Blanché (1953)
Blanché (1957)
Blanché (1966)
Gallais, P.: (1982)
Gottschalk (1953)
Kalinowski (1972)
Moretti (2004)
Moretti (Melbourne)
Pellissier, R.: " "Setting" n-opposition" (2008)
Pellissier, R.: "2-opposition and the topological hexagon" (forthcoming)
Sesmat (1951)
Smessaert (2009)
Smessaert (draft)

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