weak α2-structure (logical square)

[ logical square (i.e. weak α2-structure)  | weak α3-structure | weak α4-structure | weak α5-structure | … | weak αn-structure ] – up to α-fragments


NOT.04.LogicalSquares The logical square (or square of opposition) seems to be the oldest known opposition structure after Parmenides and Plato‘s non-graphical enquiries on oppositions. If there is no evidence that Aristotle himself, who discovered its underlying logic, discovered this geometrical-logical object (the first known occurence of it appears much later, in the writings of the Latin philosopher and logician Apuleius), there seems to be in Aristotle’s On the interpretation some kind of prefiguration of it, called ὑπογραφή (“hypographe”, that is “sketched drawing”). Remark that this logical square (or “square of opposition”), which summarises very elegantly many of the fundamental laws of Aristotelian logic, was very important during the Middle Ages, where it constituted one of the basis of the education of the students, notably in the universities. Remark that, as shows the figure here, the square admits different readings (OG speaks of “decorations”): quantificational and modal (in fact there exist many more possible decorations). However, after a long success, the square knew a significant decay, notably with respect to the new logic and to the “analytical philosophers” inspired by (and promoting) this new logic: it is in this context that the discovery of a remarkable extension of it, the logical hexagon (1950) was almost not perceived by the scientific community. Nevertheless, the square was much prised, in the 20th century, by the so-called “structuralist” thinkers, for their theories were very much concerned with the fundamental oppositions ruling their respective disciplines (the structuralist studies founded many disciplines, especially – but not only – in the humanities). Nowadays, by developing the general mathematical theory of which the logical square is only a very particular case, OG seems to be restituting to it its deserved interest, for the logical square constitutes the first step of the special branch of mathematics dealing with oppositional structures. Remark that in the framework of OG Pellissier has demonstrated that the square is a weak (not a strong) 2-opposition. Moretti has shown that the square is, with respect of the general theory of the logical poly-simplexes, a logical bi-segment (so that it can be generalised both in terms of a “logical bi-triangle” – this is Jacoby‘s, Sesmat‘s and Blanché‘s logical hexagon – and of a “logical tri-segment”).

Remark as well that the logical square admits a strange relative, that is Piaget’s and Gottschalk’s duality-operational square.


The logical square has recently become the object of a series of “World Congresses on the Square of Opposition” (organised every two years by J.-Y. Béziau)


  • Aristotle, ,
  • Aristotle, ,
  • Angot-Pellissier, R., “2-opposition and the topological hexagon”, in: Béziau J.-Y. and Payette G. (eds.), The Square of Opposition. A General Framework for Cognition, Peter Lang, Bern, 2012
  • Béziau, J.-Y., “New Light on the Square of Oppositions and its Nameless Corner”, …, 2003
  • Chatti S. and Schang F., “The Cube, the Square and the Problem of Existential Import”, History and Philosophy of Logic, 34, 2, 2013
  • Englebretsen
  • Jaspers
  • Jaspers D. And Seuren P., “The Square of Opposition in Catholic hands: a chapter in the history of 20th-century logic”, Logique & Analyse, (forthcoming)
  • Luzeaux D., Sallantin J. and Dartnell C., “Logical Extensions of Aristotle’s Square”, Logica Universalis, 2, 1, 2008
  • Mélès, B., “No Group of Opposition for Constructive Logics: The Intuitionistic and Linear Cases”, in: Béziau J.-Y. and Jacquette D. (eds.), Around and Beyond the Square of Opposition, Birkäuser, Basel, 2012
  • Moretti, A., “Geometry for Modalities? Yes: Through N-Opposition Theory”, in: Béziau J.-Y., Costa-Leite A. and Facchini A. (eds.) Aspects of Universal Logic, N.17 of Travaux de logique, Université de Neuchâtel, December 2004
  • Moretti, A., The Geometry of Logical Opposition, PhD Thesis, Université de Neuchâtel, Switzerland, 2009
  • Parsons, T., “The Traditional Square of Opposition”, Stanford Encyclopedia of Philosophy, 2012 (1997)
  • Pellissier, R., “‘Setting’ n-opposition”, (pre-final draft), Logica Universalis, 2, 2, 2008
  • Seuren, P., The Victorious Square, …
  • Smessaert, H., ,Smessaert H. and Demey L., “Logical Geometries and Information in the Square of Oppositions”, Journal of Logic, Language and Information, 23, 4, 2014
  • Westerstahl, D., “On the Aristotelian Square of Opposition”, …,
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