In this paper we investigate the logic MTL∼ obtained by extending Esteva and Godo’s logic MTL with an involutive negation ∼ not dependent on the t-norm and with the Baaz’s operator ∆. Moreover, we also introduce and study the related predicate calculus MTL∀∼. Algebraic and standard completeness results are shown to hold for both MTL∼ and MTL∀∼.

In this paper, we introduce the notion of fuzzy associative I-ideals in an IS-algebra with respect to a t-conorm and investigate some of its properties. We discuss the properties of homomorphic image and inverse image of S-fuzzy associative I-ideals in an IS-algebras. Relationship between T -fuzzy associative I-ideals and S-fuzzy fuzzy associative Iideals of an IS-algebra is introduced. Connect...

Similarity: measure of how close to each other two instances are. The “closer” the instances are to each other, the larger is the similarity value. Dissimilarity: measure of how different two instances are. Dissimilarity is large when instances are very different and is small when they are close. Proximity: refers to either similarity or dissimilarity Distance metric: a measure of dissimilarity...

The realization of this monograph has needed some permissions to use copyrighted material. Hence I would like to thank Springer (chapters 6, 7, 8 are derived from papers [ABM09a, BM09, BM10]) and Professors S. Burris and H.P. Sankappanavar (some of the figures in the appendix are taken from the electronic version of their book [BS81]) to have granted their permissions. I profit also to thank Pr...

It is shown how stochastic Itô-Taylor schemes for stochastic ordinary differential equations can be embedded into standard concepts of consistency, stability and convergence. An appropriate choice of function spaces and norms, in particular a stochastic generalization of Spijker’s norm (1968), leads to two-sided estimates for the strong error of convergence under the usual assumptions.

In the literature on social choice with fuzzy preferences, a central question is how to represent the transitivity of a fuzzy binary relation. Arguably the most general way of doing this is to assume a form of transitivity called max-star transitivity. The star operator in this formulation is commonly taken to be a triangular norm. The familiar max-min transitivity condition is a member of this...

We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these re...

Let (Ω,Σ) be a measurable space and let B(Σ) denote the Banach space of bounded, Σ-measurable functions on Ω equipped with the sup-norm. Ω is the set of states of nature and B(Σ) is the set of all (measurable) risks (see Artzner et al. [2]). A measure of risk is a mapping ρ : B(Σ) −→ R. Coherent risk measures were introduced in [5] (under the name of "upper expectations") and further studied in...

The notion of filter is established in fuzzy class theory. Graded properties of filters, prime filters and related concepts are investigated. In particular we study properties of filters relativized with respect to the basic three t-norms: Gödel, product, and Lukasiewicz. It is shown that our approach gives rise to a broad class of models including probability and necessity measures on fuzzy se...

In this paper, by applying the countable extensions of a t-norm, we have proved a coincidence point theorem for the fuzzy Nadler type of contraction mappings. Also, a coincidence point theorem in a fuzzy metric spaces for an implicit relation is given.