“Setting” n-Opposition

Logica Universalis 2 (2):235-263 (2008)
Abstract
Our aim is to show that translating the modal graphs of Moretti’s “n-opposition theory” (2004) into set theory by a suited device, through identifying logical modal formulas with appropriate subsets of a characteristic set, one can, in a constructive and exhaustive way, by means of a simple recurring combinatory, exhibit all so-called “logical bi-simplexes of dimension n” (or n-oppositional figures, that is the logical squares, logical hexagons, logical cubes, etc.) contained in the logic produced by any given modal graph (an exhaustiveness which was not possible before). In this paper we shall handle explicitly the classical case of the so-called 3(3)-modal graph (which is, among others, the one of S5), getting to a very elegant tetraicosahedronal geometrisation of this logic
Keywords Opposition theory  classical modal logic  Blanché’s logical hexagon  Aristotle’s square  logical bisimplexes  logical cube   β-structure  tetraicosahedron   n-opposition   n-partition of the true  strong n-opposition  weak n-opposition  Moretti’fs modal graph  set translation of modal graphs
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