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Journal Articles Logica Universalis Year : 2008

Logical Extensions of Aristotle's Square

Abstract

We 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 figures
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Dates and versions

lirmm-00274314 , version 1 (17-04-2008)

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Dominique Luzeaux, Jean Sallantin, Christopher Dartnell. Logical Extensions of Aristotle's Square. Logica Universalis, 2008, 2, pp.167-187. ⟨10.1007/s11787-007-0022-y⟩. ⟨lirmm-00274314⟩
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