Abstract
In the literature, there are several forms of extensions of the classical bi-implication for the fuzzy logic, as for example, the axiomatization proposed by Fodor and Roubens [1]. Another way to obtain a generalization is to provide a definition based on the classical equivalence \(\phi \iff \psi \equiv (\phi \Rightarrow \psi )\wedge (\psi \Rightarrow \phi )\), in which the classical operators of conjunction and implication are replaced, respectively, by a t-norm (T) and a fuzzy implication (I). In this paper, we investigate a particular class of fuzzy bi-implications \(B(x,y)=T(I(x,y),I(y,x))\), in which I is a fuzzy (T, N)-implication introduced by Bedregal [2]. We study several properties satisfied by (T, N)-bi-implications, such as the sufficient conditions that they must satisfy in order to be a f-bi-implication.
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Notes
- 1.
If \(T:[0,1]^2\rightarrow [0,1]\) is a t-norm and \(N:[0,1]\rightarrow [0,1]\) is a fuzzy negation, then we say that the pair (T, N) satisfies the law of non-contradiction if \(T(x,N(x))=0\), for all \(x\in [0,1]\) (this law is equivalently stated in [11, p. 55]).
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This work is partially supported by Universidade Federal Rural do Semi-Árido - UFERSA (Project PIH10002-2018).
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Farias, A.D.S., Callejas, C., Marcos, J., Bedregal, B., Santiago, R. (2019). Fuzzy Bi-implications Generated by t-norms and Fuzzy Negations. In: Kearfott, R., Batyrshin, I., Reformat, M., Ceberio, M., Kreinovich, V. (eds) Fuzzy Techniques: Theory and Applications. IFSA/NAFIPS 2019 2019. Advances in Intelligent Systems and Computing, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-030-21920-8_53
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