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In 2003, the French-Swiss logician and philosopher Jean-Yves Béziau (1965/), working on Sesmat–Blanché‘s logical hexagon (in order to defend paraconsistent logics against an heavy attack by H. Slater in 1995), remarked that in it the “bare modalities” (or “null modalities”), i.e. the logical atoms (or their negations) prefixed by no modality at all, were not represented. So he proposed to take them into account in the geometry of the logical oppositions as being two new points to be added to the logical hexagon.
However, as this solution seemed to him to be a little bit clumsy (and heuristically not too fertile), he proposed rather to express the until then forgotten null modalities by way of two new suited logical hexagons, which he discovered. As an anecdote, they were useful to him in order to investigate – against Slater – the paraconsistent and the paracomplete (i.e. intuitionist) properties of the negation operator.
Now, because in these three logical hexagons some vertices are present two times, this gave him the revolutionary idea to look for some whole geometrical arrangement of them in the three-dimensional space. Counting the different vertices (without repetitions there are 12 of them) he proposed to see them as forming together a “stellar dodecahedron of opposition” (or “Escher’s solid”, the three-dimensional star admitting twelve vertices).
Remark that in his own papers Béziau never made himself the 3D drawing. Later on, in 2004, it was shown by Moretti that another solution was to be preferred (an Archimedian solid, the “logical cuboctahedron”). Further deepenings were brought by Smessaert and by Pellissier. For this idea of looking for a three-dimensional solid of oppositions made of logical hexagons, Béziau is (with Blanché and his model made of two joined logical hexagons) one of the two forerunners of the idea of logical “β-structure“. Moreover, considering that the logical square expresses 2-opposition and that the logical hexagon expresses 3-opposition (Béziau’s own terms are “dichotomy” and “tritomy”) he also looked for some possible quaternary, 4-oppositional (Béziau’s term was “quadritomy”) successor of the logical hexagon: his proposal was a polygon made of two nested squares. As it happens, the right solution to Béziau’s problem was found later by Moretti as being a “logical bi-tetrahedron”, or “logical cube”. For this revolutionary idea of looking for a successor of the logical hexagon, Béziau is (along, this time, with Sesmat, who proposed a “tetrahedric structure”, and with Joerden, who discovered – for deontic logic – a pentadic structure which happens to express perfectly 5-opposition) one of the three forerunners of the notion of logical “α-structure“.
- Béziau, J.-Y., “New Light on the Square of Oppositions and its Nameless Corner“, …, 2003
- Béziau, J.-Y., “The Power of the Hexagon”, Logica Universalis, 6 (1-2), 2012
- Moretti, A., The Geometry of Logical Opposition, PhD Thesis, Université de Neuchâtel, Switzerland, 2009
- Moretti, A., « The Critics of Paraconsistency and of Many-Valuedness and the Geometry of Opposition », Logic and Logical Philosophy, 19, 1-2, 2010
- one of his websites