New Light on the Square of Oppositions and its Nameless Corner

It has been pointed out that there is no primitive name in natural and formal languages for one corner of the famous square of oppositions. We have all, some and no, but no primitive name for not all. It is true also in the modal version of the square, we have necessary, possible and impossible, but no primitive name for not necessary. I shed here a new light on this mysterious non-lexicalisation of the southeast corner of the square of oppositions, the so-called O-corner, by establishing some connections between negations and modalities. The E-corner, impossible, is a paracomplete negation (intuitionistic negation if the underlying modal logic is S4) and the O-corner, not necessary, is a paraconsistent negation. I argue that the three notions of opposition of the square of oppositions (contradiction, contrariety, subcontrariety) correspond to three notions of negation (classical, paracomplete, paraconsistent). We get a better understanding of these relations of opposition in the perspective of Blanché’s hexagon and furthermore in the perspective of a three dimensional object, a stellar dodecahedron of oppositions that I present at the end of the paper.