Abstract
Comparing two items (objects, images) involves a set of relevant attributes whose values are compared. Such a comparison may be expressed in terms of different modalities such as identity, similarity, difference, opposition, analogy. Recently J.-Y. Béziau has proposed an “analogical hexagon” that organizes the relations linking these modalities. The hexagon structure extends the logical square of opposition invented in Aristotle time (in relation with the theory of syllogisms). The interest of these structures has been recently advocated in logic and in artificial intelligence. When non-Boolean attributes are involved, elementary comparisons may be a matter of degree. Moreover, attributes may not have the same importance. One might only consider most attributes rather than all of them, using operators such as ordered weighted min and max. The paper studies in which ways the logical hexagon structure may be preserved in such gradual extensions. As an illustration, we start with the hexagon of equality and inequality due to Blanché and extend it with fuzzy equality and fuzzy inequality.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Blanché, R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)
Blanché, R.: Structures Intellectuelles. Essai sur l’Organisation Systématique des Concepts, Vrin (1966)
Béziau, J.-Y.: The power of the hexagon. Log. Univers. 6(1–2), 1–43 (2012)
Béziau, J.-Y.: An analogical hexagon. Int. J. Approx. Reason. 94, 1–17 (2018)
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)
Coletti, G., Petturiti, D., Vantaggi, B.: Fuzzy weighted attribute combinations based similarity measures. In: Antonucci, A., Cholvy, L., Papini, O. (eds.) ECSQARU 2017. LNCS (LNAI), vol. 10369, pp. 364–374. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61581-3_33
Dubois, D., Fargier, H., Prade, H.: Beyond min aggregation in multicriteria decision: (ordered) weighted min, discri-min, leximin. In: Yager, R.R., Kacprzyk, J. (eds.) The Ordered Weighted Averaging Operators: Theory and Applications, pp. 181–191. Kluwer Academic Publishers, Dordrecht (1997)
Dubois, D., HadjAli, A., Prade, H.: Fuzziness and uncertainty in temporal reasoning. J. Univers. Comput. Sci. 9, 1168–1194 (2003)
Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Log. Univers. 6, 149–169 (2012)
Dubois, D., Prade, H.: Gradual structures of oppositions. In: Magdalena, L., Verdegay, J.L., Esteva, F. (eds.) Enric Trillas: A Passion for Fuzzy Sets. SFSC, vol. 322, pp. 79–91. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16235-5_7
Dubois, D., Prade, H., Rico, A.: Residuated variants of Sugeno integrals: towards new weighting schemes for qualitative aggregation methods. Inf. Sci. 329, 765–781 (2016)
Dubois, D., Prade, H., Rico, A.: Graded cubes of opposition and possibility theory with fuzzy events. Int. J. Approx. Reason. 84, 168–185 (2017)
Murinová, P., Novák, V.: Analysis of generalized square of opposition with intermediate quantifiers. Fuzzy Sets Syst. 242, 89–113 (2014)
Parsons, T.: The traditional square of opposition. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2008)
Prade, H., Richard, G.: From analogical proportion to logical proportions. Log. Univers. 7, 441–505 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Dubois, D., Prade, H., Rico, A. (2018). Fuzzy Extensions of Conceptual Structures of Comparison. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 853. Springer, Cham. https://doi.org/10.1007/978-3-319-91473-2_60
Download citation
DOI: https://doi.org/10.1007/978-3-319-91473-2_60
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91472-5
Online ISBN: 978-3-319-91473-2
eBook Packages: Computer ScienceComputer Science (R0)