The oppositional hexagon
bi-simplex of dim m
(or logical hexagon)
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
Remark that the logical hexagon has been used by
(1982) for building a new model of narratology.
Gallais, P.: (1982)
Pellissier, R.: " "Setting" n-opposition" (2008)
Pellissier, R.: "2-opposition
and the topological hexagon" (forthcoming)
Last update: 08th/04/2012