A Hexagonal Framework of the Field F4 and the Associated Borromean Logic

The hexagonal structure for ‘the geometry of logical opposition’, as coming from Aristoteles–Apuleius square and Sesmat–Blanché hexagon, is presented here in connection with, on the one hand, geometrical ideas on duality on triangles (construction of ‘companion’), and on the other hand, constructions of tripartitions, emphasizing that these are exactly cases of borromean objects. Then a new case of a logical interest introduced here is the double magic tripartition determining the semiring B3 and this is a borromean object again, in the heart of the semiring Mat3(BAlg). With this example we understand better in which sense the borromean object is a deepening of the hexagon, in a logical vein. Then, and this is our main objective here, the Post-Mal’cev full iterative algebra P4 = P(F4) of functions of all arities on F4, is proved to be a borromean object, generated by three copies of P2 in it. This fact is induced by a hexagonal structure of the field F4. This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting F4 as a ‘boolean manifold’. This structure allows to analyze also P4 as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in F4. It is related to the splitting of paradoxes, to modified logic, to specular logic. It is a setting for a theory of paradoxical sentences, seen as computations of movements on the bi-hexagonal link among the 12 classical logics on a set of 4 values. Mathematics Subject Classification (2010). 00A06, 03B45, 03B50, 03E02, 03G05, 06Exx, 06E25, 06E30, 11Txx, 18xx.