New family of triangular norms via contrapositive symmetrization of residuated implications

We present a family of algebraic structures, called rectangular bilattices, which serve as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs). These structures are useful on one hand to clarify the exact nature of the relationship between the above two common extensions of fuzzy sets, and on the other hand provide an intuitively attractive framework for the representation of uncertain and potentially conflicting information. We also provide these structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Application potential and intuitive appeal of the proposed framework are illustrated in the context of preference modeling.

show abstract

“…This follows from a result of Jenei,52 who proves this identity in general for Girard monoids, of which MV-algebras are a specific case. In Ref.…”

Section: Definition 19mentioning

confidence: 71%

A Bilattice-Based Framework for Handling Graded Truth and Imprecision

Deschrijver

Arieli

Cornelis

et al. 2007

Int. J. Unc. Fuzz. Knowl. Based Syst.

View full text Add to dashboard Cite

show abstract

“…This property was already studied by Trillas-Valverde in 1981 ( [19]) and Fodor in 1995 ( [10]) for the cases when the negation N is strong, and it was subsequently studied by Jenei in 2000 ( [12]) and Balasubramaniam in 2006 ([5]). The equation (1) comes from the classical law of contraposition and it plays an important role in applications like approximate reasoning, deductive systems and formal methods of proof.…”

Section: I(n (Y) N (X)) = I(x Y) For All X Y ∈ [0 1] (1)mentioning

confidence: 99%

“…It is worth to point up that the study of the contrapositivisation of the residuated implications made in [10] gave rise to the t-norm Nilpotent minimum, the first known left-continuous (but not continuous) t-norm, which produced the so important and prolific study of the left-continuous t-norms (see, for instance, [7], [12], [13] and the references therein).…”

Section: I(n (Y) N (X)) = I(x Y) For All X Y ∈ [0 1] (1)mentioning

confidence: 99%

N-contrapositivisation of fuzzy implication functions

Aguiló¹,

Suñer²,

Torrens³

2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

View full text Add to dashboard Cite

The law of contraposition with respect to a negation (usually strong) is one of the most studied properties in the theory of fuzzy implication functions. We already know some methods for modifying an implication with the aim that the new implication satisfies this property, these methods are called contrapositivisation. In this paper we present two new methods of contrapositivisation with respect to any strong negation and we study their properties. Along this study we will see that these new methods not only preserve the usual properties preserved by the already known methods, but they also have some additional property.

show abstract

“…The general answer is "no" (although the contrary was "proved" in [41]). Therefore, the following problem was studied in [26].…”

Section: Left-continuous Triangular Normsmentioning

confidence: 99%

Binary Operations on Fuzzy Sets: Recent Advances

Fodor

2003

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. The main aim of this paper is to summarize recent advances on operations on fuzzy sets. First a detailed description of our present knowledge on left-continuous t-norms is presented. Then some new classes of associative operations (uninorms, nullnorms, t-operators) are reviewed. Finally we demonstrate the role of the evaluation scales (especially the case of totally ordered finite sets) in the choice of operations.

show abstract

New family of triangular norms via contrapositive symmetrization of residuated implications

Cited by 79 publications

References 6 publications

A Bilattice-Based Framework for Handling Graded Truth and Imprecision

A Bilattice-Based Framework for Handling Graded Truth and Imprecision

N-contrapositivisation of fuzzy implication functions

Binary Operations on Fuzzy Sets: Recent Advances

Contact Info

Product

Resources

About