Non-commutative EQ-logics and their extensions (original) (raw)
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EQ-logics: Non-commutative fuzzy logics based on fuzzy equality
Fuzzy Sets and Systems, 2011
In this paper, we develop a specific formal logic in which the basic connective is fuzzy equality and the implication is derived from the latter. Moreover, the fusion connective (strong conjunction) is noncommutative. We call this logic EQ-logic.
Constructing some logical algebras from EQ-algebras
Filomat, 2021
EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets in [18] introduced various kinds of EQ-algebras such as good, residuated, and lattice ordered EQ-algebras. In any logical algebraic structures, by using various kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, by means of fantastic filters of EQ-algebras we construct MV-algebras. Also, we study prelinear EQ-algebras and introduce a new kind of filter and named it prelinear filter. Then, we show that the quotient structure which is introduced by a prelinear filter is a distributive lattice-ordered EQ-algebras and under suitable conditions, is a De Morgan algebra, Stone algebra and Boolean algebra.
Some algebraic structures for many-valued logics
1998
Brouwer{Zadeh MV-algebras and de Morgan BZMV-algebras are the result of a natural \pasting" between Brouwer{Zadeh algebras and MValgebras. Such structures are characterized by a splitting of the operations that correspond to the basic logical connectives (not, and, or). In this framework, we prove the categorical equivalence between di erent kinds of structures that have been introduced in order to semantically characterize some many-valued logics. In particular, we investigate the cases of Chang's MV-algebras, Cignoli{Monteiro Lukasiewicz algebras, Stonian MV-algebras.
Eq-Logics with Delta Connective
Iranian Journal of Fuzzy Systems, 2015
In this paper we continue development of formal theory of a special class offuzzy logics, called EQ-logics. Unlike fuzzy logics being extensions of theMTL-logic in which the basic connective is implication, the basic connective inEQ-logics is equivalence. Therefore, a new algebra of truth values calledEQ-algebra was developed. This is a lower semilattice with top element endowed with two binaryoperations of fuzzy equality and multiplication. EQ-algebra generalizesresiduated lattices, namely, every residuated lattice is an EQ-algebra but notvice-versa.In this paper, we introduce additional connective logdeltalogdeltalogdelta in EQ-logics(analogous to Baaz delta connective in MTL-algebra based fuzzy logics) anddemonstrate that the resulting logic has again reasonable properties includingcompleteness. Introducing DeltaDeltaDelta in EQ-logic makes it possible to prove alsogeneralized deduction theorem which otherwise does not hold in EQ-logics weakerthan MTL-logic.
Trends in Logic, 2000
The paper considers the fundamental notions of many-valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics.
Neutrality and Many-Valued Logics
2007
This book written by A. Schumann & F. Smarandache is devoted to advances of non-Archimedean multiple-validity idea and its applications to logical reasoning. Leibnitz was the first who proposed Archimedes' axiom to be rejected. He postulated infinitesimals (infinitely small numbers) of the unit interval [0, 1] which are larger than zero, but smaller than each positive real number. Robinson applied this idea into modern mathematics in and developed so-called non-standard analysis. In the framework of non-standard analysis there were obtained many interesting results examined in [37], [38], [74], [117]. There exists also a different version of mathematical analysis in that Archimedes' axiom is rejected, namely, p-adic analysis (e.g., see: [20], [86], [91], ). In this analysis, one investigates the properties of the completion of the field Q of rational numbers with respect to the metric ρ p (x, y) = |x − y| p , where the norm | · | p called p-adic is defined as follows:
Commutative basic algebras and non-associative fuzzy logics
Archive for Mathematical Logic, 2009
Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. Łukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers and includes the class of MV-algebras. We show that the logic L CBA is very close to the Łukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization.
Distinguished algebraic semantics for -norm based fuzzy logics: Methods and algebraic equivalencies
Annals of Pure and Applied Logic, 2009
This paper is a contribution to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and ∆-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we propose five kinds of distinguished semantics for these logics -namely the class of algebras defined over the real unit interval, the rational unit interval, the hyperreals (all ultraproducts of the real unit interval), the strict hyperreals (only ultraproducts giving a proper extension of the real unit interval) and finite chains, respectively-and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and (strict) hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. §1. Introduction. In his famous book [31], Hájek considers the problem of finding a basic fuzzy logic which is a common fragment of the most important fuzzy logics, namely Lukasiewicz, Gödel and product logics. There, he introduced a logic, named BL, and he proposed it for the role of basic fuzzy logic. Hájek's proposal was greatly supported by , where it was shown that BL is the logic of all continuous t-norms 1 and of their residua. But in the authors observed that the minimal condition for a t-norm to have a residuum, and therefore to determine a logic, is left-continuity (continuity is not necessary). There, they proposed a weaker logic, called MTL (monoidal t-norm based logic), and conjectured that MTL is the logic of left-continuous t-norms and of their residuals. This conjecture was shown to be true in . Thus, it makes sense to propose it (instead of BL) as the real 'basic fuzzy logic' (this claim is also supported by an interesting methodological paper [4]). Another feature of MTL, which adds interest to it, is constituted by its relationship with substructural logics. Indeed, MTL is a logic without contraction (see ) and it can be characterized as FL ew (i.e. Full Lambek calculus plus exchange and weakening, see [55]) plus prelinearity. As the most important substructural logics, MTL can be formulated in a hypersequent calculus which enjoys cut-elimination (see [3]) by adding to the calculus for FL ew Avron's communication rule [1], a rule which yields completeness with respect to linearly ordered (commutative, integral, bounded) residuated lattices. Further generalizations are also possible, (e.g. by removing exchange or weakening), but MTL seems general enough.