28 research outputs found
A logical approach to fuzzy truth hedges
The starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truth-stressing and-depressing hedges as expansions of Hájek's BL logic by new unary connectives. They showed that their logics are chain-complete, but standard completeness was only proved for the expansions over Gödel logic. We propose weaker axiomatizations over an arbitrary core fuzzy logic which have two main advantages: (i) they preserve the standard completeness properties of the original logic and (ii) any subdiagonal (resp. superdiagonal) non-decreasing function on [0, 1] preserving 0 and 1 is a sound interpretation of the truth-stresser (resp. depresser) connectives. Hence, these logics accommodate most of the truth hedge functions used in the literature about of fuzzy logic in a broader sense. © 2013 Elsevier Inc. All rights reserved.The authors acknowledge partial support of the MICINN projects TASSAT (TIN2010-20967-C04-01) and ARINF (TIN2009-14704-C03-03), and the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Carles Noguera also acknowledges support of the research contract “Juan de la Cierva” JCI-2009-05453.Peer Reviewe
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
Logics of formal inconsistency arising from systems of fuzzy logic
This article proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in
this article we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency
in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition
of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics,
expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.The authors have been partially supported by the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Coniglio was also supported by FAPESP (Thematic Project LogCons 2010/51038-0), and by a research grant from
CNPq (PQ 305237/2011-0). Esteva and Godo also acknowledge partial support by the MINECO project TIN2012-39348-C02-01Peer 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
Some Types of Generalized Fuzzy n
Fuzzy filters and their generalized types have been extensively studied
in the literature. In this paper, a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences is established, a quotient residuated lattice with respect to generalized fuzzy filter is induced, and several types of generalized fuzzy n-fold filters such as generalized fuzzy n-fold positive implicative (fantastic and Boolean) filters are introduced; examples and results are provided to demonstrate the relations among these filters