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Logic in Question pp 435–458Cite as

Birkhäuser

Analogical Proportions and Binary Trees

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Part of the Studies in Universal Logic book series (SUL)

Abstract

Analogical reasoning has been thought for a longtime as something aside, away from logical reasoning. However, it is not exactly so. This chapter in its first part mainly surveys works of the last decade, which have proposed a logical modeling of analogical proportions (i.e., statements of the form “a is to b as c is to d”) among other logically expressible proportions and have shown their use in analogical inference. It also emphasizes the pervasiveness of analogical proportions as soon as we compare situations. The second part of the chapter shows that dichotomous trees built from pairs of mutually exclusive properties have also a reading in terms of Boolean analogical proportions, thus providing another clue of the links existing between analogy and logically expressed taxonomies. This also gives birth to noticeable opposition structures and can be related to formal concept analysis.

Keywords

  • Analogical reasoning
  • Analogical proportion
  • Taxonomic tree
  • Structure of opposition
  • Formal concept analysis
  • Comparison

Mathematics Subject Classification (2000)

  • Primary 03B99; Secondary 03A99
  • 03B05
  • 97F80

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Notes

  1. 1.

    Other patterns can be obtained by combination. This in particular the case of the Joint method of agreement and difference:

    A B C occur together with x y z,

    A D E occur together with x v w also B C occur with y z

    —————————————————————————————

    Therefore A is the cause, or the effect, or a part of the cause of x.

  2. 2.

    This cube is distinct from the cube of opposition obtained as an extension of the traditional square of opposition; see [7, 12, 42].

  3. 3.

    The third conjunct is unfortunately missing in [1].

  4. 4.

    This subsection follows from an idea suggested by Bernhard Ganter to the second author.

References

  1. Barbot, N., Miclet, L., Prade, H.: The analogical cube of opposition. In: Béziau, J.Y., Buchsbaum, A., Vandoulakis, I. (eds.) Handbook of abstracts of the 6th World Congress on the Square of Opposition, Crete, Nov. 1–5, pp. 9–10 (2018)

    Google Scholar 

  2. Barbot, N., Miclet, L., Prade, H.: Relational proportions between objects and attributes. In: Kuznetsov, S.O., Napoli, A., Rudolph, S. (eds.) Proc. 6th Int. Workshop “What can FCA do for Artificial Intelligence?”, co-located with (IJCAI/ECAI 2018), Stockholm, Sweden, July 13. CEUR Workshop Proc., vol. 2149, pp. 33–44 (2018)

    Google Scholar 

  3. Béziau, J.Y.: New light on the square of oppositions and its nameless corner. Logical Investigations 10, 218–233 (2003)

    MATH  Google Scholar 

  4. Bounhas, M., Prade, H., Richard, G.: Analogical classification: A new way to deal with examples. In: ECAI 2014 - 21st European Conference on Artificial Intelligence, 18–22 August 2014, Prague, Czech Republic. Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 135–140. IOS Press (2014)

    Google Scholar 

  5. Bounhas, M., Prade, H., Richard, G.: Analogical classification: Handling numerical data. In: Straccia, U., Calì, A. (eds.) Proc. of Scalable Uncertainty Management - 8th Int. Conf., SUM. LNCS, vol. 8720, pp. 66–79. Springer (2014)

    Google Scholar 

  6. Bounhas, M., Prade, H., Richard, G.: Analogy-based classifiers for nominal or numerical data. Int. J. Approx. Reasoning 91, 36–55 (2017)

    CrossRef  MATH  Google Scholar 

  7. Ciucci, D., Dubois, D., Prade, H.: Structures of opposition induced by relations - The Boolean and the gradual cases. Ann. Math. Artif. Intell. 76(3–4), 351–373 (2016)

    CrossRef  MATH  Google Scholar 

  8. Couceiro, M., Hug, N., Prade, H., Richard, G.: Analogy-preserving functions: A way to extend Boolean samples. In: Proc. 26th Int. Joint Conf. on Artificial Intelligence (IJCAI’17), Melbourne (2017)

    Google Scholar 

  9. Dastani, M., Indurkhya, B., Scha, R.: Analogical projection in pattern perception. J. of Experimental and Theoretical Artificial Intelligence 15(4), 489–511 (2003)

    CrossRef  MATH  Google Scholar 

  10. Davies, T.R., Russell, S.J.: A logical approach to reasoning by analogy. In: McDermott, J.P. (ed.) Proc. of the 10th International Joint Conference on Artificial Intelligence (IJCAI’87), Milan, pp. 264–270. Morgan Kaufmann (1987)

    Google Scholar 

  11. Dorolle, M.: Le Raisonnement par Analogie. PUF, Paris (1949)

    Google Scholar 

  12. Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Universalis 6(1–2), 149–169 (2012)

    CrossRef  MATH  Google Scholar 

  13. Falkenhainer, B., Forbus, K.D., Gentner, D.: The structure-mapping engine: algorithm and examples. Artif. Intell. 41(1), 1–63 (1989)

    CrossRef  MATH  Google Scholar 

  14. Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer-Verlag (1999)

    CrossRef  MATH  Google Scholar 

  15. Gentner, D.: Structure-mapping: A theoretical framework for analogy. Cognitive Science 7(2), 155–170 (1983)

    CrossRef  Google Scholar 

  16. Gentner, D.: The mechanisms of analogical learning. In: Vosniadou, S., Ortony, A. (eds.) Similarity and Analogical Reasoning, pp. 197–241. Cambridge University Press, New York (1989)

    Google Scholar 

  17. Gentner, D., Holyoak, K.J., Kokinov, B.N.: The Analogical Mind: Perspectives from Cognitive Science. Cognitive Science, and Philosophy, MIT Press, Cambridge, MA (2001)

    Google Scholar 

  18. Gust, H., Kühnberger, K., Schmid, U.: Metaphors and heuristic-driven theory projection (HDTP). Theoretical Computer Science 354(1), 98–117 (2006)

    CrossRef  MATH  Google Scholar 

  19. Hesse, M.: On defining analogy. Proceedings of the Aristotelian Society 60, 79–100 (1959)

    CrossRef  Google Scholar 

  20. Hofstadter, D., Mitchell, M.: The Copycat project: A model of mental fluidity and analogy-making. In: Hofstadter, D., The Fluid Analogies Research Group (eds.) Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought. pp. 205–267. Basic Books, Inc., New York, NY (1995)

    Google Scholar 

  21. Hug, N., Prade, H., Richard, G., Serrurier, M.: Analogical classifiers: A theoretical perspective. In: Kaminka, G., Fox, M., Bouquet, P., Hüllermeier, E., Dignum, V., Dignum, F., van Harmelen, F. (eds.) Proc. 22nd Europ. Conf. on Artificial Intelligence (ECAI’16) , The Hague, 29 Aug.-2 Sept. pp. 689–697. IOS Press (2016)

    Google Scholar 

  22. Hwang, S.J., Grauman, K., Sha, F.: Analogy-preserving semantic embedding for visual object categorization. In: Proc. 30th Int. Conf. on Machine Learning - Volume 28. pp. III–639–III–647. JMLR.org (2013)

    Google Scholar 

  23. Jing, L., Lu, Y., Gang, H., Sing Bing, K.: Visual attribute transfer through deep image analogy. In: ACM Transactions on Graphics, (Proc. of Siggraph). vol. 36 (2017)

    Google Scholar 

  24. Kalinowski, G.: Sur l’inférence par analogie. Du catalogue à l’analogue. Recueil de Textes du Groupe de Travail sur l’Analogie n ∘2, 1–8 (UA 962 CNRS, Conseil d’Etat, 1980)

    Google Scholar 

  25. Klein, S.: Analogy and mysticism and the structure of culture (and Comments & Reply). Current Anthropology 24 (2), 151–180 (1983)

    CrossRef  Google Scholar 

  26. Langius, I.C.: Inventvm Novvm Quadrati Logici Vniversalis […] (1714), Gissa Hassorum: Henningius Mllerus

    Google Scholar 

  27. Lemanski, J.: Logic diagrams in the Weigel and Weise circles. History and Philosophy of Logic 39 (1), 3–28 (2018)

    CrossRef  MATH  Google Scholar 

  28. Lepage, Y.: Analogy and formal languages. Electr. Not.Theor. Comp. Sci. 53, 180–191 (2002), proc. joint meeting of the 6th Conf. on Formal Grammar and the 7th Conf. on Mathematics of Language, (L. S. Moss, R. T. Oehrle, eds.)

    Google Scholar 

  29. Lorrain, F.: Réseaux Sociaux et Classifications Sociales. Essai sur l’Algèbre et la Géométrie des Structures Sociales. Hermann, Paris (1975)

    Google Scholar 

  30. Miclet, L., Bayoudh, S., Delhay, A.: Analogical dissimilarity: Definition, algorithms and two experiments in machine learning. J. Artif. Intell. Res. (JAIR) 32, 793–824 (2008)

    Google Scholar 

  31. Miclet, L., Delhay, A.: Relation d’analogie et distance sur un alphabet défini par des traits. Tech. Rep. 1632, IRISA (July 2004)

    Google Scholar 

  32. Miclet, L., Nicolas, J.: From formal concepts to analogical complexes. In: Ben Yahia, S., Konecny, J. (eds.) Proc. 12th Int. Conf. on Concept Lattices and Their Applications, Clermont-Ferrand, Oct.13–16. CEUR Workshop Proc., vol. 1466, pp. 159–170 (2015)

    Google Scholar 

  33. Miclet, L., Prade, H.: Handling analogical proportions in classical logic and fuzzy logics settings. In: Proc. 10th Eur. Conf. on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’09),Verona. pp. 638–650. Springer, LNCS 5590 (2009)

    Google Scholar 

  34. Mitchell, M.: Analogy-Making as Perception: A Computer Model. MIT Press, Cambridge MA (1993)

    Google Scholar 

  35. Mitchell, M.: Analogy-making as a complex adaptive system. In: Segel, L., Cohen, I. (eds.) Design Principles for the Immune System and Other Distributed Autonomous Systems. Oxford University Press (2001)

    Google Scholar 

  36. Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2017)

    Google Scholar 

  37. Prade, H., Richard, G.: Cataloguing/analogizing: A non monotonic view. Int. J. Intell. Syst. 26(12), 1176–1195 (2011)

    CrossRef  Google Scholar 

  38. Prade, H., Richard, G.: From analogical proportion to logical proportions. Logica Universalis 7(4), 441–505 (2013)

    CrossRef  MATH  Google Scholar 

  39. Prade, H., Richard, G.: Homogenous and heterogeneous logical proportions. IfCoLog J. of Logics and their Applications 1(1), 1–51 (2014), reprinted in Handbook of Philosophical Logic - Volume 18 (D. M. Gabbay and F. Guenthner, series editors), Springer Verlag, 51–103, 2018

    Google Scholar 

  40. Prade, H., Richard, G.: Analogical proportions: From equality to inequality. Int. J. Approx. Reasoning 101, 234–254 (2018)

    CrossRef  MATH  Google Scholar 

  41. Prade, H., Richard, G.: Multiple-valued logic interpretations of analogical, reverse analogical, and paralogical proportions. In: Proc. 40th IEEE Int. Symp. on Multiple-Valued Logic (ISMVL’10). pp. 258–263 (Barcelona, 2010)

    Google Scholar 

  42. Reichenbach, H.: The syllogism revised. Philosophy of Science 19 (1), 1–16 (1952)

    CrossRef  Google Scholar 

  43. Russell, S.J.: The Use of Knowledge in Analogy and Induction. Pitman, UK (1989)

    MATH  Google Scholar 

  44. Stuart Mill, J.: A System of Logic, Ratiocinative and Inductive, being a connected view of the principles of evidence and the methods of scientific investigation. John W. Parker, London (1843), 2 vols.: XVI + 580, XII + 624 pp.; republished by University of Toronto Press, Routledge & Kegan Paul, 1974.

    Google Scholar 

  45. Barbot, N., Miclet, l.: La proportion analogique dans les groupes. Applications aux permutations et aux matrices. [Research Report] PI 1914, 2008, pp. 27. 〈inria-00366929〉

    Google Scholar 

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Author information

Authors and Affiliations

  1. Univ. Rennes CNRS, IRISA, Lannion, France

    Nelly Barbot

  2. ENSSAT, Lannion, France

    Laurent Miclet

  3. Institut de Recherche en Informatique de Toulouse (IRIT) CNRS & Université Paul Sabatier, Toulouse, France

    Henri Prade

  4. Institut de Recherche en Informatique de Toulouse (IRIT), Université Paul Sabatier, Toulouse, France

    Gilles Richard

Authors
  1. Nelly Barbot
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  2. Laurent Miclet
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  4. Gilles Richard
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Corresponding author

Correspondence to Henri Prade .

Editor information

Editors and Affiliations

  1. Philosophy, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Jean-Yves Béziau

  2. Mathematics, Sorbonne University, Paris, France

    Jean-Pierre Desclés

  3. Ragnar Nurkse Department of Innovation and Governance, Tallinn University of Technology, Tallinn, Estonia

    Amirouche Moktefi

  4. Laboratory of Sciences and Techniques of Information, Communication and Knowledge, University of Western Brittany, Brest, France

    Anca Christine Pascu

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Barbot, N., Miclet, L., Prade, H., Richard, G. (2022). Analogical Proportions and Binary Trees. In: Béziau, JY., Desclés, JP., Moktefi, A., Pascu, A.C. (eds) Logic in Question. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-94452-0_22

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