Logical Extensions of Aristotle’s Square

Abstract

We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], 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 □α, E is $$\square\neg\alpha$$ , I is $$\neg\square\neg\alpha$$ and O is $$\square\neg\alpha$$ , where α 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 $$\neg$$ is involutive or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of $$\{\alpha,\square\alpha\}$$ 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 [14], 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 figures.