Abstract
Structures of opposition, such as the hexagon and different cubes, derived from the square of opposition of ancient logic, have for more than a decade shown their interest in the analysis of various frameworks for the representation and processing of information (possibly pervaded with uncertainty). The use of a renewed and less constrained vision of the structures of opposition leads in this article to consider a general cube and a hypercube of opposition applicable to binary or gradual settings, which is here exemplified on Sugeno integrals and related integrals
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Obviously, dealing with a ternary expression R(A, B, C), and h applying to argument C would lead to a natural and straightforward extension.
References
Béziau, J.Y.: New light on the square of oppositions and its nameless corner. Logical Invest. 10, 218–233 (2003)
Béziau, J.-Y.: The power of the hexagon. Logica Universalis 6(1–2), 1–43 (2012)
Blanché, R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)
Blanché, R.: Structures Intellectuelles. Essai sur l’Organisation Systématique des Concepts. Librairie philosophique J. Vrin, Paris (1966)
Ciucci, D., Dubois, D., Prade, H.: Structures of opposition induced by relations. The Boolean and the gradual cases. Ann. Maths Artif. Intel. 76, 351–373 (2016)
Dubois, D., Faux, F., Prade, H., Rico, A.: Qualitative capacities and their informational comparison. In: Proceedings 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2021), 19–24 September 2021, Bratislava (2021)
Dubois, D., Marichal, J.-L., Prade, H., Roubens, M., Sabbadin, R.: The use of the discrete Sugeno integral in decision-making: a survey. Int. J. Uncert. Fuzz. Knowl. Syst. 9(5), 539–561 (2001)
Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Univers. 6, 149–169 (2012)
Dubois, D., Prade, H., Rico, A.: The cube of opposition: a structure underlying many knowledge representation formalisms. In: Proceedings 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), Buenos Aires, AAAI Press, pp. 2933–2939 (2015)
Dubois, D., Prade, H., Rico, A.: Representing qualitative capacities as families of possibility measures. Int. J. Approximate Reasoning 58, 3–24 (2015)
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. Approximate Reasoning 84, 168–185 (2017)
Dubois, D., Prade, H., Rico, A.: 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. SFSC, vol. 344, pp. 27–45. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-40314-4_2
Dubois, D., Prade, H., Rico, A.: Structures of opposition and comparisons: Boolean and gradual cases. Logica Univ. 14, 115–149 (2020)
Dubois, D., Prade, H., Rico, A.: Le tesseract de l’intégrale de Sugeno. In: Actes 29èmes Rencontres Francophones sur la Logique Floue et ses Applications (LFA 2020), Sète, 15–16 October, Cépaduès (2020)
Dubois, D., Prade, H., Rico, A., Teheux, B.: Generalized qualitative Sugeno integrals. Inf. Sci. 415, 429–445 (2017)
Gottschalk, W.H.: The theory of quaternality. J. Symb. Logic. 18, 193–196 (1953)
Holčapek, M., Rico, A.: A note on the links between different qualitative integrals. In: Fuzz-IEEE (2020)
Johnson, W.E.: Logic. Cambridge University Press, Part I (1921)
Keynes, J.N.: Studies and Exercises in Formal Logic, 3rd edn. MacMillan (1894)
Moretti, A.: The geometry of standard deontic logic. Logica Univ. 3, 19–57 (2009)
Moyse, G.: Résumés linguistiques de données numériques: interprétabilité et périodicité de séries. Thèse Univ, Paris (2016)
Moyse, G., Lesot, M.-J., Bouchon-Meunier, B.: Oppositions in fuzzy linguistic summaries. Int. Conf. Fuzzy System (Fuzz-IEEE 2015), Istanbul (2015)
Murray, F.B. (ed.): Critical Features of Piaget’s Theory of the Development of Thought. University of Delaware Press, Newark (1972)
Nilsson, J.F.: A cube of opposition for predicate logic. Logica Univ. 14, 103–114 (2020)
Piaget, J.: Traité de logique. Essai de logistique opératoire, Armand Colin (1949)
Parsons, T.: The traditional square of opposition. In: The Stanford Encyclopedia of Philosophy (2008)
Pfeifer, N., Sanfilippo, G.: Probabilistic squares and hexagons of opposition under coherence. Int. J. Approximate Reasoning 88, 282–294 (2017)
Pizzi, C.: Contingency logics and modal squares of opposition. In: Beziau, J.Y., Gan-Krzywoszynska, K. (eds.) Handbook of Abstracts of the 3rd World Congress on the Square of Opposition, Beirut, 26–30 June, pp. 29–30 (2012)
Reichenbach, H.: The syllogism revised. Philos. Sci. 19(1), 1–16 (1952)
Westerståhl, D.: Classical vs. modern squares of opposition, and beyond. In: Payette, G., Béziau, J.-Y. (eds.) The Square of Opposition. A General Framework for Cognition, Peter Lang, pp. 195–229 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Dubois, D., Prade, H., Rico, A. (2021). Towards a Tesseract of Sugeno Integrals. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_38
Download citation
DOI: https://doi.org/10.1007/978-3-030-86772-0_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-86771-3
Online ISBN: 978-3-030-86772-0
eBook Packages: Computer ScienceComputer Science (R0)