Abstract
Interval Fuzzy Logic and Interval-valued Fuzzy Sets have been widely investigated. Some Fuzzy Logics were algebraically modelled by Peter Hájek as BL-algebras. What is the algebraic counterpart for the interval setting? It is known from literature that there is a incompatibility between some algebraic structures and its interval counterpart. This paper shows that such incompatibility is also present in the level of BL-algebras. Here we show both: (1) the impossiblity of match imprecision and the correctness of the underlying BLimplication and (2) some facts about the intervalization of BL-algebras.
Highlights
The motivation of using intervals instead of exact values can be perceived from the fact that the amount of imprecision can be codified through intervals in terms of its width
BL-algebras – which were introduced by Hajek [9] – are an algebraic counterpart to Basic Logic (BL) which generalizes the three most commonly used logics in the theory of fuzzy sets; namely: Łukasiewicz logic, product logic and Godel logic [7, 8]
This together with the fact that intervalvalued fuzzy set theory has been revealed as an increasingly promising extension of usual fuzzy sets [4, 5, 6, 14] – namely: the usual membership degrees are replaced by closed intervals in [0, 1] – lead us to consider the investigation on the intervalization of BL algebras
Summary
The motivation of using intervals instead of exact values can be perceived from the fact that the amount of imprecision can be codified through intervals in terms of its width. Given a BL-algebra: L, ∧, ∨, ∗, →, 0L, 1L in which L is a complete lattice, we define the following binary operations on I(L):. It is possible to obtain interval BL-algebras from BL-algebras, the theorem shows that none of them will provide correct implications. This is informally stated in [16]. Given a BL-algebra L, ∧, ∨, ∗, →, 0L, 1L there is no interval binary operator correct with respect to binary operator → such that I(L), , , , , 0, 1 is an BL-algebra. In both cases the interval binary operator is not correct with respect to binary operator →
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