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Organizing Families of Aggregation Operators into a Cube of Opposition

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Granular, Soft and Fuzzy Approaches for Intelligent Systems

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 344))

Abstract

The cube of opposition is a structure that extends the traditional square of opposition originally introduced by Ancient Greek logicians in relation with the study of syllogisms. This structure, which relates formal expressions, has been recently generalized to non Boolean, graded statements. In this paper, it is shown that the cube of opposition applies to well-known families of idempotent, monotonically increasing aggregation operations, used in multiple criteria decision making, which qualitatively or quantitatively provide evaluations between the minimum and the maximum of the aggregated quantities. This covers weighted minimum and maximum, and more generally Sugeno integrals on the qualitative side, and Choquet integrals, with the important particular case of Ordered Weighted Averages, on the quantitative side. The main appeal of the cube of opposition is its capability to display the various possible aggregation attitudes in a given setting and to show their complementarity.

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Notes

  1. 1.

    This example is specially dedicated to Ron Yager in remembrance of a dinner in Lyon in a traditional restaurant, which took place at the occasion of the CNRS Round Table on Fuzzy Sets organized by Robert Féron [31] in Lyon on June 23–25, 1980 [32]. This Round Table was an important meeting for the development of fuzzy set research, because most of the active researchers of the field were there. Interestingly enough, Robert Féron had the remarkable intuition to invite Gustave Choquet in the steering committee, at a time where no fuzzy set researcher was mentioning Choquet integrals! This meeting also included, as usual, some nice moments of relaxation and good humor. In particular, at the above-mentioned dinner, to which quite a number of people took part (including two of the authors of this paper), Ron enjoyed very much a pigs feet dish. He was visibly very happy with his choice, so Lotfi Zadeh told him, “Ron, you should have been a pig in another life”, to which Ron replied “no, Lotfi, it is in this life”, while continuing to suck pigs’ bones with the greatest pleasure.

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Authors and Affiliations

  1. IRIT, CNRS & University of Toulouse, 31062, Toulouse Cedex 9, France

    Didier Dubois & Henri Prade

  2. ERIC, Université Claude Bernard Lyon 1, 69100, Villeurbanne, France

    Agnès Rico

Authors
  1. Didier Dubois
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  2. Henri Prade
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  3. Agnès Rico
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Correspondence to Didier Dubois .

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Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

  2. Research and Innovation Center, Ford Motor Company, Dearborn, Michigan, USA

    Dimitar Filev

  3. Deakin University, Burwood, Victoria, Australia

    Gleb Beliakov

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Dubois, D., Prade, H., Rico, A. (2017). Organizing Families of Aggregation Operators into a Cube of Opposition. In: Kacprzyk, J., Filev, D., Beliakov, G. (eds) Granular, Soft and Fuzzy Approaches for Intelligent Systems. Studies in Fuzziness and Soft Computing, vol 344. Springer, Cham. https://doi.org/10.1007/978-3-319-40314-4_2

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  • DOI: https://doi.org/10.1007/978-3-319-40314-4_2

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