Abstract
The paper shows that a cube of opposition, a structure that generalizes the square of opposition invented in ancient logic, can be generated from the composition of a binary relation with a subset, by the effect of set complementation on the subset, on the relation, or on the result of the composition. Since the composition of relations is encountered in many areas, the structure of opposition exhibited by the cube of opposition has a universal flavor. In particular, it applies to information processing-oriented settings such as rough set theory, possibility theory, or formal concept analysis. We then discuss how this structure extends to a fuzzy relation and a fuzzy subset, and the graded cube of opposition thus obtained provides an organized view of the different existing compositions of fuzzy relations. The paper concludes by pointing out areas of research where the cube of opposition, or its graded version are of interest.
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References
Amgoud, L., Prade, H.: A formal concept view of abstract argumentation. In: van der Gaag, L.C. (ed.) Proceedings 12th Eur. Conf. Symb and Quant. Appr. to Reas. with Uncert. (ECSQARU’13), Utrecht, July 8-10, LNCS 7958, pp. 1–12. Springer (2013)
Bandler, W., Kohout, L.: Fuzzy power sets and fuzzy implication operators. Fuzzy Set. Syst. 4, 13–30 (1980)
Béziau, J.Y.: New light on the square of oppositions and its nameless corner. Logical Investigations 10, 218–233 (2003)
Béziau, J.Y.: The power of the hexagon. Logica Universalis 6(1-2), 1–43 (2012)
Béziau, J.Y., Gan-Krzywoszyńska, K. (eds.): Handbook of Abstracts of the 2nd World Congress on the Square of Opposition, Corte, Corsica (2010)
Béziau, J.Y., Gan-Krzywoszyńska, K. (eds.): Handbook of Abstracts of the 3rd World Congress on the Square of Opposition, Beirut, Lebanon (2010)
Béziau, J.Y., Gan-Krzywoszyńska, K. (eds.): Handbook of Abstracts of the 4th World Congress on the Square of Opposition, Roma, Vatican (2014)
Blanché, R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)
Blanché, R.: Structures Intellectuelles. Essai sur l’Organisation Systématique des Concepts. Vrin, Paris (1966)
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, UK (1980)
Ciucci, D., Dubois, D., Prade, H.: Oppositions in rough set theory. In: Li, T., Nguyen, H.S., Wang, G., Grzymala-Busse, J.W., Janicki, R., Hassanien, A.E., Yu, H. (eds.) Proceedings 7th International Conference on Rough Sets and Knowledge Technology RSKT’12, Chengdu, Aug. 17 20, LNCS, vol 7414, pp 504–513 (2012)
Ciucci, D., Dubois, D., Prade, H.: Rough sets, formal concept analysis, and their structures of opposition extended abstract. In: 4th Rough Set Theory Workshop, Dalhousie University, Halifax, p 1 (2013)
Ciucci, D., Dubois, D., Prade, H.: The structure of oppositions in rough set theory and formal concept analysis - Toward a new bridge between the two settings. In: Beierle, C., Meghini, C. (eds.) Proceedings 8th Int. Symp. on Foundations of Information and Knowledge Systems FoIKS’14, Bordeaux, LNCS, vol. 8367, pp. 154–173. Springer, Berlin (2014)
De Baets, B., Kerre, E.: Fuzzy relational compositions. Fuzzy Set. Syst 60, 109–120 (1993)
De Morgan, A.: On the structure of the syllogism. Trans. Camb. Phil. Soc. VIII, 379–408 (1846). Reprinted in on the syllogism and other logical writings. In: Heath, P. (ed.) . Routledge and Kegan Paul, London (1966)
Dekker, P.J.: Not only barbara. J. Log. Lang. Inf 24(2), 95–129 (2015)
Demey, L.: Algebraic aspects of duality diagrams. In: Cox, P.T., Plimmer, B., Rodgers, P.J. (eds.) Proceedings 7th International Conference on Diagrammatic Representation and Inference Diagrams’12, Canterbury, LNCS, vol. 7352, pp 300–302. Springer (2012)
Dubois, D., Dupin de Saint-Cyr, F., Prade, H.: A possibility-theoretic view of formal concept analysis. Fundamenta Informaticae 75(1-4), 195–213 (2007)
Dubois, D., Hajek, P., Prade, H.: Knowledge-driven versus data-driven logics. J. Log. Lang. Inf 9, 65–89 (2000)
Dubois, D., Prade, H.: Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York (1988)
Dubois, D., Prade, H.: Aggregation of possibility measures. In: Kacprzyk, J., Fedrizzi, M. (eds.) Sets, Multiperson Decision Making Using Fuzzy and Possibility theory, pp 55–63. Kluwer, Dordrecht (1990)
Dubois, D., Prade, H.: Possibility theory: Qualitative and quantitative aspects. In: Gabbay, D.M., Smets, P. (eds.) Quantified Representation of Uncertainty and Imprecision, Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1, pp. 169–226. Kluwer, Norwell (1998)
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: Esteva, F., Magdalena, L., Verdegay, J.L. (eds.) Enric Trillas: Passion for Fuzzy Sets, Studies in Fuzziness and Soft Computing, vol. 322, pp 79–91. Springer (2015)
Fodor, J.C.: Contrapositive symmetry of fuzzy implications. Fuzzy Sets Syst 69, 141–156 (1995)
Franklin, C.L.: Some proposed reforms in common logic. Mind 15(57), 75–88 (1890)
Hacker, E.: The octagon of opposition. Notre Dame Journal of Formal Logic 16, 352–353 (1975)
Keynes, J.N.: Studies and exercises in formal logic. MacMillan, London. Part II, chap. IV, p 144 (1884)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer, Berlin (2000)
Kohout, L.J., Bandler, W.: Relational-product architectures for information processing. Inform. Sci 37, 25–37 (1985)
Miclet, L., Prade, H.: Analogical proportions and square of oppositions. In: Laurent, A. et al. (eds.) Proceedings 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Montpellier, CCIS, vol. 443, pp 324–334. Springer, Berlin (2014)
Murinová, P., Novák, V.: The analysis of the generalized square of opposition. In: Montero, J., Pasi, G., Ciucci, D. (eds.) Proceedings 8th Conference of the European Society for Fuzzy Logic and Technology EUSFLAT’13, Milano, Atlantis Press (2013)
Murinová, P., Novák, V.: Analysis of generalized square of opposition with intermediate quantifiers. Fuzzy Set. Syst. 242, 89–113 (2014)
Parsons, T.: The traditional square of opposition (2008)
Pellissier, R.: “setting” n-opposition. Logica Universalis 2(2), 235–263 (2008)
Piaget, J.: Traité de Logique : Essai de Logistique Opératoire. Dunod (1949)
Pizzi, C.: Contingency Logics and Modal Squares of Opposition. In: Beziau, J.Y., Gan-Krzywoszyńska, K. (eds.) Handbook of Abstracts of the 3rd World Congress on the Square of Opposition, Beirut, pp. 29–30 (2012)
Reichenbach, H.: The syllogism revised. Philos. Sci 19(1), 1–16 (1952)
Ruspini, E.H.: A new approach to clustering. Inf. Control 15(1), 22–32 (1969)
Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland (1983)
Smessaert, H., Demey, L.: Logical geometries and information in the square of oppositions. J. Log. Lang. Inf 23(4), 527–565 (2014)
Stepnicka, M., Holcapek, M.: Fuzzy relational compositions based on generalized quantifiers. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) Proceedings 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’14, Part II, Montpellier, July 15-19, Communications in Computer and Information Science, vol. 443, pp 224–233. Springer (2014)
Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338–353 (1965)
Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets & Systems 1, 3–28 (1978)
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Ciucci, D., Dubois, D. & Prade, H. Structures of opposition induced by relations. Ann Math Artif Intell 76, 351–373 (2016). https://doi.org/10.1007/s10472-015-9480-8
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DOI: https://doi.org/10.1007/s10472-015-9480-8