The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon

Logica Universalis 6 (1-2):171-199 (2012)
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Abstract
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 vertical relations in the classical square are unidirectional, whereas in the modern square they are bidirectional. The present paper argues that these differences become even bigger when two more operators are added, namely the U ( ${{\equiv} {\rm A}\,{\vee} \,{\rm E} }$ , all or no) and Y ( ${\equiv{\rm I} \,{\wedge} \,{\rm O}}$ , some but not all) of Blanché (Structures Intellectuelles, 1969). In the resulting Aristotelian hexagon the two extra nodes are perfectly integrated, yielding two interlocking triangles of CR and SCR. In the duality hexagon by contrast, they do not enter into any relation with the original square, but constitute a independent pair of their own, since they are their own SNEGs. Hence, they not only stand in a relation of external NEG, but also in one of duality. This reflexive nature of the SNEG will be shown to result in defective monotonicity configurations for the pair, namely the absence of right-monotonicity (on the predicate argument). In the second half of the paper, we present an overview of those hexagonal structures which are both Aristotelian and Duality configurations, and those which are only Aristotelian
Keywords Logical square  logical hexagon  aristotelian relations of opposition  duality relations  external versus internal negation  monotonicity properties  modal logic
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DOI 10.1007/s11787-011-0031-8
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References found in this work BETA

Generalized Quantifiers and Natural Language.John Barwise & Robin Cooper - 1981 - Linguistics and Philosophy 4 (2):159--219.
Mathematical Methods in Linguistics.Barbara H. Partee, Alice ter Meulen & Robert E. Wall - 1992 - Journal of Symbolic Logic 57 (1):271-272.
Generalized Quantifiers and Natural Language.Jon Barwise - 1980 - Linguistics and Philosophy 4:159.
On the 3d Visualisation of Logical Relations.Hans Smessaert - 2009 - Logica Universalis 3 (2):303-332.

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Citations of this work BETA

Logical Geometries and Information in the Square of Oppositions.Hans Smessaert & Lorenz Demey - 2014 - Journal of Logic, Language and Information 23 (4):527-565.
Was Lewis Carroll an Amazing Oppositional Geometer?Alessio Moretti - 2014 - History and Philosophy of Logic 35 (4):383-409.

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