The paper discusses analogical proportions in relation with the square of oppositions, a classical structure in Ancient logic which is related to the different forms of statements that may be involved in deductive syllogisms. The paper starts with a short reminder on the logical modeling of analogical proportions, viewed here as Boolean expressions expressing similarities and possibly differences between four items, as in the statement “a is to b as c is to d”. The square of oppositions and its hexagon-based extension is then restated in a knowledge representation perspective. It is observed that the four vertices of a square of oppositions form a constrained type of analogical proportion that emphasizes differences. In fact, the different patterns making an analogical proportion true can be covered by a square of oppositions or by a “square of agreement”, leading to disjunctive expressions of the analogical proportion. Besides, an “analogical octagon” is shown to capture the general construction of an analogical proportion from two sets of properties. Since the square of oppositions offers a common setting relevant for syllogisms and analogical proportions, it also provides a basis for the discussion of the possible interplay between deductive arguments and analogical arguments.


Truth Table Boolean Variable Analogical Reasoning Formal Concept Analysis Syllogistic Reasoning 
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  1. 1.
    Béziau, J.-Y.: New light on the square of oppositions and its nameless corner. Logical Investigations 10, 218–233 (2003)Google Scholar
  2. 2.
    Blanché, R.: Structures Intellectuelles. Essai sur l’Organisation Systématique des Concepts. Vrin, Paris (1966)Google Scholar
  3. 3.
    Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Univers. 6, 149–169 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Lepage, Y.: De l’analogie rendant compte de la commutation en linguistique. Habilit. à Diriger des Recher. Univ. J. Fourier, Grenoble (2003)Google Scholar
  5. 5.
    Miclet, L., Bayoudh, S., Delhay, A.: Analogical dissimilarity: definition, algorithms and two experiments in machine learning. JAIR 32, 793–824 (2008)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Miclet, L., Prade, H.: Handling analogical proportions in classical logic and fuzzy logics settings. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS (LNAI), vol. 5590, pp. 638–650. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Fall 2008 Edition) (2008)Google Scholar
  8. 8.
    Prade, H., Richard, G.: Analogy-making for solving IQ tests: A logical view. In: Ram, A., Wiratunga, N. (eds.) ICCBR 2011. LNCS (LNAI), vol. 6880, pp. 241–257. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Prade, H., Richard, G.: Cataloguing/analogizing: A non monotonic view. Int. J. Intell. Syst. 26(12), 1176–1195 (2011)CrossRefGoogle Scholar
  10. 10.
    Prade, H., Richard, G.: Homogeneous logical proportions: Their uniqueness and their role in similarity-based prediction. In: Brewka, G., Eiter, T., McIlraith, S.A. (eds.) Proc. 13th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2012), Roma, June 10-14, pp. 402–412. AAAI Press (2012)Google Scholar
  11. 11.
    Prade, H., Richard, G.: From analogical proportion to logical proportions. Logica Universalis 7(4), 441–505 (2013)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Russell, S.J.: The use of Knowledge in Analogy and Induction. Pitman, UK (1989)zbMATHGoogle Scholar
  13. 13.
    Stroppa, N., Yvon, F.: Analogical learning and formal proportions: Definitions and methodological issues. Technical report (June 2005)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Laurent Miclet
    • 1
  • Henri Prade
    • 2
  1. 1.University of Rennes 1Irisa, LannionFrance
  2. 2.CNRS/IRITUniversity of ToulouseToulouseFrance

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