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June 2012, Volume 6, Issue 1–2, pp 171–199 | Cite as

The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon



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.


Primary 03B45 03B10 03B65 03G05 Secondary 05C99 47H05

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).


Logical square logical hexagon aristotelian relations of opposition duality relations external versus internal negation monotonicity properties modal logic





© Springer Basel AG 2011



Smessaert, H. Log. Univers. (2012) 6: 171. https://doi.org/10.1007/s11787-011-0031-8



