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.
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References
Barwise J., Cooper R.: Generalized quantifiers and natural language. Linguist. Philos. 4, 159–219 (1981)
Blanché, R.: Structures Intellectuelles. Essai sur l’organisation systématique des concepts. Librairie Philosophique J. Vrin, Paris (1969)
Deschamps K., Smessaert H.: The logical-semantic structure of legislative sentences. Comparative legilinguistics. Int. J. Legal Commun. 1, 73–87 (2009)
Gamut L.T.F.: Logic, Language and Meaning. Intensional Logic and Logical Grammar, vol. 2. University of Chicago press, Chicago (1991)
Gottschalk W.H.: The theory of quaternality. J. Symb. Log. 18, 193–196 (1953)
Horn, L.R.: Hamburgers and truth: why Gricean explanation is Gricean. In: Hall, K., et al. (eds.) Proceedings of the Sixteenth Annual Meeting of the Berkeley Linguistics Society, pp. 454–471. Berkeley Linguistics Society, Berkeley (1990)
Jacoby P.: A triangle of opposites for types of propositions in Aristotelian logic. New Scholast. 24, 32–56 (1950)
Keenan, E.L.: The semantics of determiners. In: Lappin, S. (ed.) The Handbook of Contemporary Semantic Theory, pp. 41–63. Blackwell, Oxford (1996)
Keenan, E.L.: How much logic is built into natural language? In: Dekker, P., et al. (eds.) Proceedings of 15th Amsterdam Colloquium, pp. 39–45. ILLC, Amsterdam (2005)
Kratzer, A.: Modality. In: von Stechow, A., Wunderlich, D. (eds.) Semantics: An International Handbook of Contemporary Research, pp. 639–650. Walter de Gruyter, Berlin (1991)
Löbner S.: Wahr neben Falsch. Duale Operatoren als die Quantoren natürlicher Sprache. Max Niemeyer Verlag, Tübingen (1990)
McNamara, P.: Deontic logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. (http://plato.stanford.edu/entries/logic-deontic/) (2006/2010)
Moretti, A.: The geometry of logical opposition. PhD Thesis, University of Neuchâtel, Switzerland (2009)
Moretti A.: The geometry of standard deontic logic. Log. Univ. 3/1, 19–57 (2009)
Moretti, A.: The geometry of oppositions and the opposition of logic to it. In: Bianchi, I., Savardi, U. (eds.) The Perception and Cognition of Contraries, pp. 29–60. McGraw-Hill, Milan (2009)
Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. (http://plato.stanford.edu/entries/square/) (1997/2006)
Partee B.H., ter Meulen A.G.B., Wall R.E.: Mathematical Methods in Linguistics. Kluwer, Dordrecht (1990)
Pellissier R.: Setting n-opposition. Log. Univ. 2/2, 235–263 (2008)
Peters S., Westerståhl D.: Quantifiers in Language and Logic. Clarendon Press, Oxford (2006)
Piaget, J.: Traité de logique. Essai de logistique opératoire. Dunod, Paris (1949/1972)
Sesmat A.: Logique II. Les Raisonnements. La syllogistique. Hermann, Paris (1951)
Seuren P.: Language from Within Volume II: The Logic of Language. Oxford University Press, Oxford (2010)
Smessaert H.: Monotonicity properties of comparative determiners. Linguist. Philos. 19/3, 295–336 (1996)
Smessaert, H.: A three-valued approach to (non-)conservativity and (co-)intersectivity with (not) all and (not) only. Talk at the Logic Now and Then Conference, H.U.Brussel, 5–7 Nov 2008 (2008)
Smessaert H.: On the 3D-visualisation of logical relations. Log. Univ. 3/2, 303–332 (2009)
Smessaert, H.: On the fourth Aristotelian relation of opposition: subalternation versus non-contradiction. Talk at the Second NOT-(N-Opposition Theory)-Workshop, K.U.Leuven, 23 Jan 2010 (2010)
Smessaert, H.: Duality and reversibility relations beyond the square. Talk at the Third NOT-(N-Opposition Theory)-Workshop, 22–23 Jun 2010. Université de Nice (2010)
Smessaert, H.: On the hybrid nature of the Aristotelian square and the Sesmat-Blanché hexagon. Talk at the 14th Congress of Logic, Methodology and Philosophy of Science, 19–26 Jul 2011. CLMPS14, Université de Nancy (2011)
van Benthem J.: Essays in Logical Semantics. Foris, Dordrecht (1986)
van Eijck J.: Generalized quantifiers and traditional logic. In: Benthem, J., ter Meulen, A. (eds) Generalized Quantifiers in Natural Language,, pp. 1–19. Foris, Dordrecht (1985)
Westerståhl, D.: Classical vs. modern squares of opposition, and beyond. In: Béziau, J-Y., Payette, G. (eds.) New Perspectives on the Square of Opposition. Proceedings of the First International Conference on The Square of Oppositions, Montreux, Switzerland, 1–3 Jun 2007. Peter lang, Bern (2011, to appear)
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Preliminary versions of this paper were presented at the First NOT-workshop (N-Opposition Theory) on the Geometry of Oppositions (Université de Neuchâtel, March 14, 2009) and at the Second World Conference on The Square of Oppositions (University of Corsica, Corte, June 17–20, 2010).
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Smessaert, H. The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon. Log. Univers. 6, 171–199 (2012). https://doi.org/10.1007/s11787-011-0031-8
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DOI: https://doi.org/10.1007/s11787-011-0031-8