Logica Universalis

, Volume 6, Issue 1, pp 119–147

# A Hexagonal Framework of the Field $${\mathbb{F}_4}$$ and the Associated Borromean Logic

Article

DOI: 10.1007/s11787-011-0033-6

Guitart, R. Log. Univers. (2012) 6: 119. doi:10.1007/s11787-011-0033-6

## Abstract

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 semi-ring $${\mathcal{B}_3}$$ and this is a borromean object again, in the heart of the semi-ring $${{\rm Mat}_{3}(\mathbb{B}_{\rm Alg})}$$ . 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 $${\mathbb{P}_4 = \mathbb{P}(\mathbb{F}_4)}$$ of functions of all arities on $${\mathbb{F}_4}$$ , is proved to be a borromean object, generated by three copies of $${\mathbb{P}_2}$$ in it. This fact is induced by a hexagonal structure of the field $${\mathbb{F}_4}$$ . This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting $${\mathbb{F}_4}$$ as a ‘boolean manifold’. This structure allows to analyze also $${\mathbb{P}_4}$$ as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in $${\mathbb{F}_4}$$ . 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

### Keywords

Square of oppositions hexagon of opposition borromean object boolean algebra specular logic modality many-valued logics finite fields Aristotle Apuleius Sesmat Blanché