La théorie de la quaternalité, telle que Piaget et Gottschalk l'ont appliquée aux connectifs binaires du calcul bivalent, appelle quelques précisions et compléments.Les seize connectifs ne comportent que deux quaternes complets: celui des jonctions et celui des implications. Leurs similitudes formelles ne doivent pas dissimuler une différence dans leur mode de construction. Elle apparaît sur leurs diagrammes (inspirés du “carré logique” traditionnel) par la place de la cellule initiale et par celles des signes barrés du trait vertical de la négation:En (...) effet, la dualité qui se combine avec celle de la négation contradictoire (DN, flèche diagonale) est, pour les jonctions, la dualité “morganienne” (DM, flèche verticale) et pour les implications, la dualité des réciproques (DR, flèche horizontale). Car dans les jonctions, liaisons symétriques, il n'y a point de vraies réciproques; inversement, les implications ne comportent pas (pourvu, bien sûr, qu'on ne les traduise pas en termes de jonctions) de dualité morganienne. Le quatrième terme, obtenu par la combinaison des deux relations génératrices, sera donc, pour les jonctions, une contreduale (DM× DN) et, pour les implications, une contreréciproque (DR× DN). (shrink)

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms (...) and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

. 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 , I is and O is , 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 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 by the logical operations , 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. (shrink)

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences (...) in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth. (shrink)

Our aim is to propose a non-referential semantics for the principle of logical charity: neither logical universalism (one logic, one way of thinking), nor logical relativism (several logics, several ways of thinking) afford an adequate conceptual framework to interpret the meaning of any speech act. But neither of them is totally wrong, either. The point is to know to which extent each of these views is partly right, thus leading to a more consensual but paradoxical-sounding "relative principle of charity". After (...) recalling the theoretical background of logical charity, we suggest a four-valued logic of acceptance and rejection (hereafter: AR4); then we explain how such a non-referential semantics does justice both to the champions of logical charity and its opponents. While endorsing coherence as a precondition for rationality, we argue that such a criterion does not entail that classical logic is a necessary conceptual scheme to interpret the others' beliefs. A better application of charity should take account of the questions implicitly asked by a statement, and we bring these questions out in replacing Quine's truth-functions by Quine’s verdict functions while emphasizing upon their varying degrees of strength. (shrink)