We analyze the use of the composition of mappings as a fuzzy conjunction between indexes of inclusion. Instead of the general approach of the φ-index of inclusion, we consider a fresh approach that computes the φ-index of inclusion when restricted to a join-subsemilattice of indexes of inclusion. Under this restriction, we identify a certain join-subsemilattice which has a biresiduated structure when composition is interpreted as conjunction. The main consequence of this biresiduated structure is a representation theorem of biresiduated lattices on the unit interval in terms of the composition and subsets of indexes of inclusion.

This work extends our research on fuzzy relational Galois connections, previously established in the context of complete Heyting algebras, to the broader framework of arbitrary residuated lattices. In this context, we study the properties of fuzzy closure relations and fuzzy closure systems, and the relationship with fuzzy relational Galois connections. The main result of the paper is the generalization of the necessary and sufficient conditions for the existence of a right adjoint for a fuzzy relation linking a fuzzy transitive directed graph to an unstructured set.

We continue our research line on the analysis of the properties of the f-indexes of inclusion and contradiction; in this paper, specifically, we show that both notions can be related by means of the, conveniently reformulated, Aristotelian square of opposition. We firstly show that the extreme cases of the f-indexes of inclusion and contradiction coincide with the vertexes of the Aristotelian square of opposition in the crisp case; then, we allocate the rest of f-indexes in the diagonals of the extreme cases and we prove that the Contradiction, Contrariety, Subcontrariety, Subalternation and Superalternation relations also hold between the f-indexes of inclusion and contradiction.

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.

We continue studying the properties of the f-index of inclusion and show that, given a fixed pair of fuzzy sets, their f-index of inclusion can be linked to a fuzzy conjunction which is part of an adjoint pair. We also show that, when this pair is used as the underlying structure to provide a fuzzy interpretation of the modus ponens inference rule, it provides the maximum possible truth-value in the conclusion among all those values obtained by fuzzy modus ponens using any other possible adjoint pair.

In this article, we study and classify a family of adjoint pairs and adjoint triples for Atanassov$L$-fuzzy framework based on a complete residuated lattice satisfying the double negation law.

We continue the study of (isotone) Galois connections, also called adjunctions, in the framework of fuzzy preordered structures, which generalize fuzzy preposets by considering underlying fuzzy equivalence relations. Specifically, we present necessary and sufficient conditions so that, given a mapping f : A → B from a fuzzy preordered structure A = 〈A, ≈A,ρA〉 into a fuzzy structure 〈B, ≈B〉, it is possible to construct a fuzzy relation ρBthat induces a suitable fuzzy preorder structure on B and such that there exists a mapping g : B → A such that the pair (f, g) constitutes an Galois connection.

The problem of studying the existence of a right adjoint for a mapping defined between sets with different fuzzy structure naturally leads to the search of new notions of adjunction which fit better with the underlying structure of domain and codomain. In this work, we introduce a version of relational fuzzy adjunction between fuzzy preposets which generalizes previous approaches in that its components are fuzzy relations. We also prove that the construction behaves properly with respect to the formation of quotient with respect to the symmetric kernel relation.

A two-fold general approach to the theory of formal concept analysis is introduced by considering intuitionistic fuzzy sets valued on a residuated lattice as underlying structure for the construction.

In this paper, we present a way to represent contradiction between fuzzy sets. This representation is given in terms of the notion of f-weak-contradiction. Unlike other approaches, we do not define contradiction just by using one of the relations of f-weak-contradiction but by considering the whole set of relations. This consideration avoids the need to fix an operator beforehand in order to take into account all the information between two fuzzy sets. As a result, we characterize the contradiction between fuzzy sets and define a family of measures of contradiction satisfying four interesting properties: symmetry; antitonicity; if the intersection is empty, then the measure is one; and if there is an element in the intersection with degree of membership 1, then the measure is zero.

The theory of fuzzy property-oriented concept lattices is a formal tool for modeling and processing incomplete knowledge in information systems. This paper relates this research topic to that of mathematical morphology, a theory whose scope is to process and analyze images and signals. Consequently, the theory developed in the concept lattice framework can be used in these particular settings.

In this work we elaborate on the notion of contradiction between fuzzy sets introduced by Trillas et al in a fuzzy logic context. Our approach is parametric in that the operator used to define contradiction is rather a variable than a constant introduced prior to the analysis of contradiction. We give several motivations to consider weaker operators than the usual involutive negations, and obtain some preliminary results which validate this proposal.

An L-fuzzy generalization of the so-called Chu correspondences between formal contexts forms a category called L-ChuCors. In this work, we show that this category naturally embeds ChuCors and prove that it is *ast;-autonomous. We also focus on the di

We introduce the syntax, semantics, and an axiom system for a PDL-based extension of the logic for order of magnitude qualitative reasoning, developed in order to deal with the concept of qualitative velocity, which together with qualitative distance and orientation, are important notions in order to represent spatial reasoning for moving objects, such as robots. The main advantages of using a PDL-based approach are, on the one hand, all the well-known advantages of using logic in AI, and, on the other hand, the possibility of constructing complex relations from simpler ones, the flexibility for using different levels of granularity, its possible extension by adding other spatial components, and the use of a language close to programming languages.

Recent approaches have shown that the measurement of the amount of inconsistent information contained in a logic theory can be useful to infer positive information. This paper deals with the definition of measures of inconsistency in the residuated-logic-programming paradigm under the fuzzy answer set semantics. This fuzzy framework provides a soft mechanism to control the amount of information inferred and, thus, controlling the inconsistencies by modifying slightly the truth values of some rules.

Inconsistency in the framework of general resid-uated logic programs can be, somehow, decomposed into two notions: incoherence and instability. In this work, we focus on the measure of instability of normal residuated programs. Some measures were already provided and initial results obtained in terms of the amount of information that have to be discarded in order to recover stability; in this paper, our interest is focused precisely on the case in which stability can be recovered by adding information to our program.

In this paper we continue analyzing the introduction of negation into the framework of residuated logic programming [18], [19]; specifically, we focus on extended programs, in which strong negation is introduced. The classical approach to extended logic programs consists in considering negated literals as new, independent, ones and, then apply the usual monotonic approach (based on the fix-point semantics and the TPoperator); if the least fix-point so obtained is inconsistent, then the approach fails and no meaning is attached to the program. This paper introduces several approaches to considering consistence (under the term coherence) into a fuzzy setting, and studies some of their properties.

Unlike monotone single-valued functions, multivalued mappings may have zero, one, or (possibly infinitely) many minimal fixed-points. The contribution of this work is twofold. First, we overview and investigate the existence and computation of minimal fixed-points of multivalued mappings, whose domain is a complete lattice and whose range is its power set. Second, we show how these results are applied to a general form of logic programs, where the truth space is a complete lattice. We show that a multivalued operator can be defined whose fixed-points are in one-to-one correspondence with the models of the logic program.

Residuated logic programs have shown to be a generalisation of a number of approaches to logic programming under uncertain or vague information, including fuzzy or annotated or probabilistic or similarity-based logic programming frameworks. Various computational approaches have been developed for propositional residuated logic programs: on the one hand, there exists a bottom-up neural-like implementation of the fixed-point semantics which calculates the successive iterations of the immediate consequences operator; on the other hand, a goal-oriented top-down tabulation procedure has recently been introduced. In this paper, we introduce a sound and complete tabulation-based proof procedure for the first-order extension of residuated logic programs.

Multi-adjoint logic programs generalise monotonic and residuated logic programs in that simultaneous use of several implications in the rules and rather general connectives in the bodies are allowed. As our approach has continuous fixpoint semantics, in this work, a procedural semantics is given for the paradigm of multi-adjoint logic programming and a completeness result is proved. Some applications which could benefit from this theoretical approach, such as threshold computation, fuzzy databases and general fuzzy resolution, are commented on.