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.

Ciucci, D., Dubois, D., Prade, H. (2016). Structures of opposition induced by relations: The Boolean and the gradual cases. ANNALS OF MATHEMATICS AND OF ARTIFICIAL INTELLIGENCE, 76(3-4), 351-373 [10.1007/s10472-015-9480-8].

Structures of opposition induced by relations: The Boolean and the gradual cases

CIUCCI, DAVIDE ELIO
Primo
;
2016

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.
Articolo in rivista - Articolo scientifico
Composition of relations; Fuzzy relation; Multiple-valued logic; Square of opposition;
Square of opposition;Composition of relations; Fuzzy relation; Multiple-valued logic
English
2016
76
3-4
351
373
open
Ciucci, D., Dubois, D., Prade, H. (2016). Structures of opposition induced by relations: The Boolean and the gradual cases. ANNALS OF MATHEMATICS AND OF ARTIFICIAL INTELLIGENCE, 76(3-4), 351-373 [10.1007/s10472-015-9480-8].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/89859
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