Geometry of Modalities ? Yes : Through n-Opposition Theory
Alessio Moretti · reroDoc Digital Library · 2004
We present here some new results about the fundamental relations between modal logic and geometry. The key of our approach is a general renewed theory of “logical n-opposition”, a strong geometrical generalisation of Aristotle’s classical “opposition theory”. The question of the possible relations between (modal) logic and geometry, in some sense brightly foreseen by the russian logician N.A. Vasil’ev around 1910, happens to be still quite a mysterious one and can be shown to be of great relevance for contemporary research in many fields (for instance, but not only, in cognitive science). The most famous (seminal) entanglement between logic and geometry is given by Aristotle’s “logical square of oppositions”, a very poor structure in terms of further mathematical developments. Concrete applications of this structure, although not uninteresting, have been rather mean as well (cf. J. Piaget, J.A. Greimas and J. Lacan). Some recent results, however, have clearly shown that this sad and poor field (opposition theory as a limited and unclear encounter of modal logic and elementary geometry) is in fact much bigger than it was thought. This is strongly and mainly suggested, as we will recall, by the surprising (and almost unnoticed) discovery in 1953 inside “opposition theory” of a “logical hexagon” (Blanche), and then by that, in 2003, of a 3-dimensional “logical tetradecahedron” (Beziau and Moretti), which both are logical “implicational” structures (expressing opposition relations) furnished with a notably high degree of geometrical symmetries. These discoveries are clearly open to further development and generalization, as we will briefly sketch and demonstrate (we will present here many new such structures). The problem thus arising nowadays is a lack of comprehension of the kind of (new) modal “language” we are speaking while developing such structures. Haven’t we entered a radically new logical field? The bulk of this paper will consist in establishing that, in fact, this mysterious emerging field is really a totally new one, and that it happens to be huge (it has to do with science, not with history), and (at least partly) already ∗This paper is the written version of a conference I made in Neuchâtel (Switzerland) the 8th October 2003 at an international workshop on Universal Logic (http://www.unine.ch/unilog/workshop.html). It appeared in: J.-Y. Beziau, A. Costa Leite and A. Facchini (eds.), Aspects of Universal Logic, Centre de Recherches Semiologiques, Neuchâtel, December 2004, pp. 102-145.