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  1. Logical Geometries and Information in the Square of Oppositions.Hans Smessaert & Lorenz Demey - 2014 - Journal of Logic, Language and Information 23 (4):527-565.
    The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...)
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  • Logical Extensions of Aristotle’s Square.Dominique Luzeaux, Jean Sallantin & Christopher Dartnell - 2008 - Logica Universalis 2 (1):167-187.
    . 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 (...)
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  • The power of the hexagon.Jean-Yves Béziau - 2012 - Logica Universalis 6 (1-2):1-43.
    The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not correspond to the name used for it. In the case of the O-corner, the problem is not a wrong-name problem but a no-name (...)
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  • Constraints on the lexicalization of logical operators.Roni Katzir & Raj Singh - 2013 - Linguistics and Philosophy 36 (1):1-29.
    We revisit a typological puzzle due to Horn (Doctoral Dissertation, UCLA, 1972) regarding the lexicalization of logical operators: in instantiations of the traditional square of opposition across categories and languages, the O corner, corresponding to ‘nand’ (= not and), ‘nevery’ (= not every), etc., is never lexicalized. We discuss Horn’s proposal, which involves the interaction of two economy conditions, one that relies on scalar implicatures and one that relies on markedness. We observe that in order to express markedness and to (...)
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  • A Triangle of Opposites for Types of Propositions in Aristotelian Logic.Paul Jacoby - 1950 - New Scholasticism 24 (1):32-56.
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  • Avicenna on Possibility and Necessity.Saloua Chatti - 2014 - History and Philosophy of Logic 35 (4):332-353.
    In this paper, I raise the following problem: How does Avicenna define modalities? What oppositional relations are there between modal propositions, whether quantified or not? After giving Avicenna's definitions of possibility, necessity and impossibility, I analyze the modal oppositions as they are stated by him. This leads to the following results: The relations between the singular modal propositions may be represented by means of a hexagon. Those between the quantified propositions may be represented by means of two hexagons that one (...)
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  • Contrariety.Storrs McCall - 1967 - Notre Dame Journal of Formal Logic 8 (1-2):121-132.
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  • “Setting” n-Opposition.Régis Pellissier - 2008 - Logica Universalis 2 (2):235-263.
    Our aim is to show that translating the modal graphs of Moretti’s “n-opposition theory” (2004) into set theory by a suited device, through identifying logical modal formulas with appropriate subsets of a characteristic set, one can, in a constructive and exhaustive way, by means of a simple recurring combinatory, exhibit all so-called “logical bi-simplexes of dimension n” (or n-oppositional figures, that is the logical squares, logical hexagons, logical cubes, etc.) contained in the logic produced by any given modal graph (an (...)
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  • The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon.Hans Smessaert - 2012 - Logica Universalis 6 (1-2):171-199.
    Peters and Westerståhl (Quantifiers in Language and Logic, 2006), and Westerståhl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the “classical” Aristotelian squares of opposition and the “modern” Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality relation of internal negation (SNEG). Furthermore, the (...)
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  • Logical relations.Lloyd Humberstone - 2013 - Philosophical Perspectives 27 (1):175-230.
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  • Logic and colour.Dany Jaspers - 2012 - Logica Universalis 6 (1-2):227-248.
    In this paper evidence will be provided that Wittgenstein’s intuition about the logic of colour relations is to be taken near-literally. Starting from the Aristotelian oppositions between propositions as represented in the logical square of oppositions on the one hand and oppositions between primary and secondary colors as represented in an octahedron on the other, it will be shown algebraically how definitions for the former carry over to the realm of colour categories and describe very precisely the relations obtaining between (...)
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  • Remarques sur la Théorie de L'Hexagone logique de Blanché.Pierre Sauriol - 1968 - Dialogue 7 (3):374-390.
    En cet article nous montrons en premier lieu que la théorie de l'hexagone logique de Blanché n'est pas, comme il le pense, le résultat d'une réflexion philosophique, mais qu'elle relève véritablement de la logique scientifique, puisqu'elle s'insère tout naturellement dans la structure d'ensemble des liaisons uninaires de la logique trivalente des propositions. Cette démonstration nous conduit, en second lieu, à renverser le jugement défavorable que E. J. Lemmon avait porté sur la toute première ébauche de cette théorie, et ainsi à (...)
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  • On the 3d visualisation of logical relations.Hans Smessaert - 2009 - Logica Universalis 3 (2):303-332.
    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 (...)
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  • (1 other version)The theory of quaternality.W. H. Gottschalk - 1953 - Journal of Symbolic Logic 18 (3):193-196.
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  • The octagon of opposition.Edward A. Hacker - 1975 - Notre Dame Journal of Formal Logic 16 (3):352-353.
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  • Quantity, modality, and other Kindred systems of categories.Robert Blanche - 1952 - Mind 61 (243):369 - 375.
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  • Introduction to Logic.William of Sherwood & Norman Kretzmann - 1967 - Philosophy of Science 34 (3):295-296.
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  • Sur l'opposition des concepts.Robert Blanche - 1953 - Theoria 19 (3):89-130.
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  • On certain peculiarities of singular propositions.Tadeusz Czeżowski - 1955 - Mind 64 (255):392-395.
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