Skip to main content
SpringerLink
Log in

From the Structures of Opposition Between Similarity and Dissimilarity Indicators to Logical Proportions

A General Representation Setting for Capturing Homogeneity and Heterogeneity

  • Chapter
  • First Online:
Representation and Reality in Humans, Other Living Organisms and Intelligent Machines

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 28))

Abstract

Comparative thinking plays a key role in our appraisal of reality. Comparing two objects or situations A and B, described in terms of Boolean features, may involve four basic similarity or dissimilarity indicators referring to what A and B have in common (positively or negatively), or to what is particular to A or particular to B. These four indicators are naturally organized into a cube of opposition, which includes two classical squares of opposition, as well as other noticeable squares. From the knowledge of one situation A, it is possible to recover the description of another one, B, provided that we have enough information about the comparison between A and B. Then comparison indicators between A and B can be equated with comparison indicators between two other situations C and D. A conjunction of two such comparisons between pairs (A, B) and (C, D) gives birth to what is called a logical proportion. Among the 120 existing logical proportions, 8 are of particular interest since they are independent of the encoding (positive or negative) used for representing the situations. Four of them have remarkable properties of homogeneity, and include the analogical proportion “A is to B as C is to D”, while the four others express heterogeneity by stating that “there is an intruder among A, B, C and D, which is not X” (where X stands for A, B, C or D). Homogeneous and heterogeneous logical proportions are of interest in classification, anomaly detection tasks and IQ test solving.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Notes

  1. 1.

    Note that \(I_{(a,b)}\) (or \(I'_{(a,b)}\)) refers to one element in the set \(\{a \wedge b, \overline{a} \wedge b, a \wedge \overline{b}, \overline{a} \wedge \overline{b} \}\), and should not be considered as a functional symbol: \(I_{(a,b)}\) and \(I_{(c,d)}\) may be indicators of two different kinds. Still, we use this notation for the sake of simplicity.

  2. 2.

    A conditional object b|a can take three truth values: true if \(a \wedge b\) is true, false if \(a \wedge \lnot b\) is true, not applicable if a is false; it may be intuitively thought of as representing the rule “if a then b” [5].

References

  1. Bayoudh, S., Miclet, L., Delhay, A.: Learning by analogy: a classification rule for binary and nominal data. In: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI’07), pp. 678–683 (2007)

    Google Scholar 

  2. Bounhas, M., Prade, H., Richard, G.: Analogical classification: a new way to deal with examples. In: Schaub, T., Friedrich, G., O’Sullivan, B. (eds.) Proceedings of the 21st European Conference on Artificial Intelligence (ECAI’14), Prague, 18–22 Aug. Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 135–140. IOS Press (2014)

    Google Scholar 

  3. Correa, W., Prade, H., Richard, G.: When intelligence is just a matter of copying. In: Proceedings of the 20th European Conference on Artificial Intelligence, Montpellier, 27–31 Aug, pp. 276–281. IOS Press (2012)

    Google Scholar 

  4. Correa Beltran, W., Jaudoin, H., Pivert, O.: Estimating null values in relational databases using analogical proportions. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) Proceedings of the 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’14), Part III, Montpellier, 15–19 July. Communication in Computer and Information Science, vol. 444, pp. 110–119. Springer (2014)

    Google Scholar 

  5. Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. IEEE Trans. Syst. Man Cybern. 24, 1724–1740 (1994)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Dubois, D., Prade, H.: Possibility theory and formal concept analysis: characterizing independent sub-contexts. Fuzzy Sets Syst. 196, 4–16 (2012)

    Article  MathSciNet  Google Scholar 

  8. Faris, J.A.: The Gergonne relations. J. Symbolic Logic 207–231 (1955)

    Google Scholar 

  9. 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 

  10. Gergonne, J.D.: Essai de dialectique rationnelle. Ann. Math. Pures Appl. 189–228 (1817)

    Google Scholar 

  11. Klein, S.: Culture, mysticism & social structure and the calculation of behavior. In: Proceedings of the 5th European Conference in Artificial Intelligence (ECAI’82), Orsay, France, pp. 141–146 (1982)

    Google Scholar 

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

    MATH  Google Scholar 

  13. Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Summer 2015 edn. (2015)

    Google Scholar 

  14. Piaget, J.: Logic and Psychology. Manchester University Press (1953)

    Google Scholar 

  15. Piaget, J.: The Development of Thought: Equilibration of Cognitive Structures (trans. Rosin, A.). Viking, Oxford, UK (1977)

    Google Scholar 

  16. Prade, H., Richard, G.: Logical proportions—typology and roadmap. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) Computational Intelligence for Knowledge-Based Systems Design: Proceedings of the 13th International Conference on Information Processing and Management of Uncertainty (IPMU’10), Dortmund, 28 June–2 July. LNCS, vol. 6178, pp. 757–767. Springer (2010)

    Chapter  Google Scholar 

  17. Prade, H., Richard, G.: Reasoning with logical proportions. In: Lin, F.Z., Sattler, U., Truszczynski, M. (eds.) Proceedings of the 12th International Conference on Principles of Knowledge Representation and Reasoning, KR 2010, Toronto, 9–13 May, pp. 545–555. AAAI Press (2010)

    Google Scholar 

  18. 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.) Proceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR’12), Rome, 10–14 June, pp. 402–412. AAAI Press (2012)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Prade, H., Richard, G.: Homogenous and heterogeneous logical proportions. If CoLog J. Logics Appl. 1(1), 1–51 (2014)

    Google Scholar 

  21. Prade, H., Richard, G.: Proportional reasoning in a Boolean setting. In: Booth, R., Casini, G., Klarman, S., Richard, G., Varzinczak, I.J. (eds.) International Workshop on Defeasible and Ampliative Reasoning (DARe@ECAI’14), Prague, 19 Aug. CEUR Workshop Proceedings, vol. 1212 (2014)

    Google Scholar 

  22. Stroppa, N., Yvon, F.: An analogical learner for morphological analysis. In: Online Proceedings of the 9th Conference on Computational Natural Language Learning (CoNLL-2005), pp. 120–127 (2005)

    Google Scholar 

  23. Stroppa, N., Yvon, F.: Analogical learning and formal proportions: Definitions and methodological issues. Technical report, ENST, June 2005

    Google Scholar 

  24. Tversky, A.: Features of similarity. Psychol. Rev. 84, 327–352 (1977)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRIT, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 09, France

    Henri Prade & Gilles Richard

Authors
  1. Henri Prade
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles Richard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henri Prade .

Editor information

Editors and Affiliations

  1. Gothenburg, Sweden

    Gordana Dodig-Crnkovic

  2. Faculty of Philosophy, Pontifical Lateran University, Vatican City, Italy

    Raffaela Giovagnoli

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Prade, H., Richard, G. (2017). From the Structures of Opposition Between Similarity and Dissimilarity Indicators to Logical Proportions. In: Dodig-Crnkovic, G., Giovagnoli, R. (eds) Representation and Reality in Humans, Other Living Organisms and Intelligent Machines. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-43784-2_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-43784-2_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43782-8

  • Online ISBN: 978-3-319-43784-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation