Logical Extensions of Aristotle's Square

International audienceWe start from the geometrical-logical extension of Aristotle's square in [Bla66], [Pel06] and [Mor04], and study them from both syntactic and semantic points of view. Recall that Aristotle's square under its modal form has the following four vertices: A is \Box\alpha, E is \Box\neg\alpha, I is \neg\Box\neg\alpha and O is \neg\Box\alpha, where \alpha is a logical formula and is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether : is involutive or not) modal logic. [Bez03] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [Pel06] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of {\alpha, \Box\alpha} by the logical operations {\neg, \wedge,\vee}, under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle's square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [Mor04], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figure