We propose a parameterized framework based on a Heyting algebra and Lukasiewicz negation for modeling uncertainty for belief. We adopt a probability theory as mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the form of belief types: as a single probability, as an interval (lower and upper boundary for a p...

We study generalized deduction rules of cut and weakening in the context of equational fragment of Pavelkastyle fuzzy logic using complete residuated lattices as the structures of truth degrees. The deduction rules in question are parameterized by a truth stresser, an additional unary operation on the structure of truth degrees. It is shown that the deductive system of fuzzy Horn logic can be r...

The reticulation of an algebra was first defined for commutative rings by Simmons [19] and it was extended by Belluce to non-commutative rings [3]. As for the algebras of fuzzy logics, Belluce also constructed the reticulation of an MV-algebra [2], G. Georgescu defined the reticulation of a quantale [8] and L. Leuştean made this construction for BL-algebras [13, 14]. In each of the papers cited...

This contribution focuses on inverse fuzzy transforms (shortly inverse F-transforms) over residuated lattices introduced by I. Perfilieva and their approximation properties. We will try to reduce some requirements used in the original work to prove Approximation theorem. Moreover, we show in which sense F-transforms are the best approximations. Keywords— Fuzzy transform, Approximation, Extensio...

Commutative, integral and bounded GBL-algebras form a subvariety of residuated lattices which provides the algebraic semantics of an interesting common fragment of intuitionistic logic and of several fuzzy logics. It is known that both the equational theory and the quasiequational theory of commutative GBL-algebras are decidable (in contrast to the noncommutative case), but their complexity has...

The paper deals with fuzzy attribute logic (FAL) and shows its completeness over all complete residuated lattices. FAL is a calculus for reasoning with if-then rules describing particular attribute dependencies in objectattribute data. Completeness is proved in two versions: classical-style completeness and graded-style completeness.

A t-norm is a binary operation on [0, 1] that is associative, commutative, with identity 1 and non-decreasing in both argument. The notion of t-norm is very important in the context of fuzzy logic, for it is used in modeling a general form of the propositional conjunction. The basic fuzzy logic (BL logic) was introduced by Hájek in [9,10] and it was inspired by a continuous t-norm and its resid...

By a symmetric residuated lattice we understand an algebra A = (A,∨,∧, ∗,→,∼, 1, 0) such that (A,∨,∧, ∗,→, 1, 0) is a commutative integral bounded residuated lattice and the equations ∼∼ x = x and ∼ (x ∨ y) =∼ x∧ ∼ y are satisfied. The aim of the paper is to investigate properties of the unary operation ε defined by the prescription εx :=∼ x → 0. We give necessary and sufficient conditions for ...

We investigate a relationship between extensionality of fuzzy relations and their Lipschitz continuity on generalized metric spaces. The duality of these notions is shown, and moreover, two particular applications of the extensionality property in the field of approximation are given.

A conjunction T ties an implication operator A if the identity A(a,A(b, z)) = A(T (a, b), z) holds [A.A. Abdel-Hamid, N.N. Morsi, Associatively tied implications, Fuzzy Sets and Systems 136 (2003) 291–311]. We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contain...