Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

On triangular norms and uninorms definable in ŁΠ12

AbstractIn this paper, we investigate the definability of classes of t-norms and uninorms in the logic ŁΠ12. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0,1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation–annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms

Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

Fuzzy Sets and Formal Logics

The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. © 2015 Elsevier B.V. All rights reserved.partial support by the Spanish projects EdeTRI (TIN2012-39348- C02-01) and 2014 SGR 118.Peer reviewe

ON SOME AXIOMATIC EXTENSIONS OF THE MONOIDAL T-NORM BASED LOGIC MTL: AN ANALYSIS IN THE PROPOSITIONAL AND IN THE FIRST-ORDER CASE

The scientific area this thesis belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case (see [8],[9],[6],[7]). The thesis is divided in two parts: in the first one the necessary background about these logics, with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic (NM, see [8]): in chapter 9 the sets of tautologies of some NM-chains (subalgebras of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have been also presented in papers [1, 4, 5, 2, 3]. ---------------------------------References--------------------------------------------- [1] S. Aguzzoli, M. Bianchi, and V. Marra. A temporal semantics for Basic Logic. Studia Logica, 92(2), 147-162, 2009. doi:10.1007/s11225-009-9192-3. [2] M. Bianchi. First-order Nilpotent Minimum Logics: first steps. Submitted for publication,2010. [3] M. Bianchi. On some logical and algebraic properties of Nilpotent Minimum logic and its relation with G\uf6del logic. Submitted for publication, 2010. [4] M. Bianchi and F. Montagna. Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Log., 48(8), 719-736, 2009. doi:10.1007/s00153-009-0145-3. [5] M. Bianchi and F. Montagna. n-contractive BL-logics. Arch. Math. Log., 2010. doi:10.1007/s00153-010-0213-8. [6] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, and C. Noguera. Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Log., 160(1), 53-81, 2009. doi:10.1016/j.apal.2009.01.012. [7] P. Cintula and P. H\ue1jek. Triangular norm predicate fuzzy logics. Fuzzy Sets Syst., 161(3), 311-346, 2010. doi:10.1016/j.fss.2009.09.006. [8] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy sets Syst., 124(3), 271-288, 2001. doi:10.1016/S0165-0114(01)00098-7. [9] P. H\ue1jek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer Academic Publishers, paperback edition, 1998. ISBN:9781402003707

On some axiomatic extensions of the monoidal T-norm based logic MTL : an analysis in the propositional and in the first-order case

The scientific area this book belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case. The book is divided in two parts: in the first one the necessary background about these logics, with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic: in chapter 9 the sets of tautologies of some NM-chains (subalgebras of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have been also presented in papers

Paraconsistency properties in degree-preserving fuzzy logics

Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature. © 2014, Springer-Verlag Berlin Heidelberg.All the authors have been partially supported by the FP7 PIRSES-GA-2009-247584 project MaToMUVI. Besides, Ertola was supported by FAPESP LOGCONS Project, Esteva and Godo were supported by the Spanish project TIN2012-39348-C02-01, Flaminio was supported by the Italian project FIRB 2010 (RBFR10DGUA_02) and Noguera was suported by the grant P202/10/1826 of the Czech Science Foundation.Peer reviewe

On similarity in fuzzy description logics

This paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics . For this purpose we use concrete domains in the fuzzy description logic IALCEF(D)IALCEF(D) associated either with a left-continuous or with a finite t-norm. We propose to expand this fuzzy description logic by adding a Similarity Box (SBox) including axioms expressing properties of fuzzy equalities. We also define a global similarity between objects from similarities between the values of each object attribute (local similarities) and we prove that the global similarity defined using a t-norm inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). We also prove a result relative to global similarities expressing that, in the context of the logic MTL∀, similar objects have similar properties, being these properties expressed by predicate formulas evaluated in these object