Very and more or less in non-commutative fuzzy logic

In this paper we consider fuzzy subsets of a universe as L-fuzzy subsets instead of [ 0, 1 ]-valued, where L is a complete lattice. We enrich the lattice L by adding some suitable operations to make it into a pseudo-BL algebra. Since BL algebras are main frameworks of fuzzy logic, we propose to consider the non-commutative BL-algebras which are more natural for modeling the fuzzy notions. Based on reasoning with in non-commutative fuzzy logic we model the linguistic modifiers such as very and more or less and give an appropriate membership function for each one by taking into account the context of the given fuzzy notion by means of resemblance L-fuzzy relations.     

多值Łukasiewicz逻辑公式的范式表示和计数问题

将符号化计算树逻辑中的Shannon展开式做了推广,在n值Łukasiewicz逻辑系统Łn中,研究了由逻辑公式导出的n值McNaughton函数的展开式,给出了m元n值McNaughton函数的准析取范式和准合取范式.在此基础上,给出了m元n值McNaughton函数的计数问题,并在n值Łukasiewicz逻辑系统Łn中,给出了m元逻辑公式的构造方法及其逻辑等价类的计数问题.     

Theory of truth degrees of formulas in Łukasiewicz<Emphasis Type="Italic">n</Emphasis>-valued propositional logic and a limit theorem

The concept of truth degrees of formulas in Łukasiewiczn-valued propositional logicL _n is proposed. A limit theorem is obtained, which says that the truth functionτ _n induced by truth degrees converges to the integrated truth functionτ whenn converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Łukasiewicz logic and the continuous valued Łukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.     

Topology on the set of R 0 semantics for R 0 algebras

This paper introduces the concepts of R ₀ valuation, R ₀ semantic, countable R ₀ category , R ₀ fuzzy topological category , etc. It is established in a natural way that the fuzzy topology δ and its cut topology on the set Ω _M consisting of all R ₀ valuations of an R ₀ algebra M, and some properties of fuzzy topology δ and its cut topology are investigated carefully. Moreover, the representation theorem for R ₀ algebras by means of fuzzy topology is given, that is to say the category is equivalent to the category . By studying the relation between valuations and filters, the Loomis–Sikorski theorem for R ₀ algebras is obtained. As an application, K-compactness of the R ₀ logic is discussed.     

A note on the representation of McNaughton lines by basic literals

 We introduce a syntactically simple subclass of formulas of the infinite-valued logic of Łukasiewicz, the class of basic literals, whose associated McNaughton functions are truncated lines. We present some properties of these formulas and an application to states of MV-algebras.     

Basic fuzzy logic and <Emphasis Type="Italic">BL</Emphasis>-algebras II

 Three new (easy) results about the computational complexity of basic propositional fuzzy logic BL are presented. An important formula of predicate logic is shown 1-true in all interpretations over saturated BL-chains but is not a BL-1-tautology, i.e. is not 1-true in a safe interpretation over a non-saturated BL-algebra. Partial support of the grant No. A1030004/00 of the Grant Agency of the Academy of Science of the Czech Republic is acknowledged.     

Cut-free proof systems for logics of weak excluded middle

 In this work we perform a proof-theoretical investigation of some logical systems in the neighborhood of substructural, intermediate and many-valued logics. The common feature of the logics we consider is that they satisfy some weak forms of the excluded-middle principle. We first propose a cut-free hypersequent calculus for the intermediate logic LQ, obtained by adding the axiom *A∨**A to intuitionistic logic. We then propose cut-free calculi for systems W _n , obtained by adding the axioms *A∨(A ⊕ ⋯ ⊕ A) (n−1 times) to affine linear logic (without exponential connectives). For n=3, the system W _n coincides with 3-valued Łukasiewicz logic. For n>3, W _n is a proper subsystem of n-valued Łukasiewicz logic. Our calculi can be seen as a first step towards the development of uniform cut-free Gentzen calculi for finite-valued Łukasiewicz logics.     

On very true operators on pocrims

Hájek introduced the logic enriching the logic BL by a unary connective vt which is a formalization of Zadeh’s fuzzy truth value “very true”. algebras, i.e., BL-algebras with unary operations, called vt-operators, which are among others subdiagonal, are an algebraic counterpart of Partially ordered commutative integral residuated monoids (pocrims) are common generalizations of both BL-algebras and Heyting algebras. The aim of our paper is to introduce and study algebraic properties of pocrims endowed by “very-true” and “very-false”-like operators. Research is supported by the Research and Development Council of Czech Government via project MSN 6198959214.     

n-Fold implicative basic logic is G?del logic

We prove that Haveshki’s and Eslami’s n-fold implicative basic logic is G?del logic and n-fold positive implicative basic logic is a fragment of ukasiewicz logic.     

Observations on non-commutative fuzzy logic

The paper presents some results on the logic psBL (pseudo-basic fuzzy logic, the generalization of BL not assuming commutativity of conjunction) and on the analogous logic psMTL – a non-commutative version of the monoidal t-norm logic MTL of Esteva and Godo.Partial support of the project No LN00A056 (ITI) of the Ministry of Education (MMT) of the Czech Republic is acknowledged. Thanks are due to F. Esteva and the anonymous referee for their comments on a draft of this paper.     

State operators on GMV algebras

Flaminio and Montagna recently introduced state MVMV algebras as MVMV algebras with an internal state in the form of a unary operation. Di Nola and Dvurečenskij further presented a stronger variation of state MVMV algebras called state-morphism MVMV algebras. In the paper we present state GMVGMV algebras and state-morphism GMVGMV algebras which are non-commutative generalizations of the mentioned algebras.     

The completeness and applications of the formal system ℒ<Superscript>*</Superscript>

Since the formal deductive system ℒ^* was built up in 1997, in has played important roles in the theoretical and applied research of fuzzy logic and fuzzy reasoning. But, up to now, the completeness problem of the system ℒ^* is still an open problem. In this paper, the properties and structure ofR ₀ algebras are further studied, and it is shown that every tautology on theR ₀ interval [0,1] is also a tautology on anyR ₀ algebra. Furthermore, based on the particular structure of ℒ^*-Lindenbaum algebra, the completeness and strong completeness of the system ℒ^* are proved. Some applications of the system ℒ^* in fuzzy reasoning are also discussed, and the obtained results and examples show that the system ℒ^* is suprior to some other important fuzzy logic systems.     

On copulas, quasicopulas and fuzzy logic

We investigate (quasi)copulas as possible truth functions of fuzzy conjunction which is not necessarily associative and present some axiom systems for such fuzzy logics. In particular, we study an expansion of Łukasiewicz (infinite valued propositional) logic by a new connective interpreted as an arbitrary quasicopula (and also by a new connective interpreted as the residuum of the copula). Main results concern standard completeness.     

A Generalization of Local Fuzzy Structures

The class of bounded residuated lattice ordered monoids Rl-monoids) contains as proper subclasses the class of pseudo BL-algebras (and consequently those of pseudo MV-algebras, BL-algebras and MV-algebras) and of Heyting algebras. In the paper we introduce and investigate local bounded Rl-monoids which generalize local algebras from the above mentioned classes of fuzzy structures. Moreover, we study and characterize perfect bounded Rl-monoids.     

Fuzzy prime ideals of pseudo-<Emphasis Type="Italic">MV</Emphasis> algebras

The aim of this paper is to introduce the notion of fuzzy prime ideals of pseudo-MV algebras and investigate some of its properties.      

On BCK algebras: Part II: New algebras. The ordinal sum (product) of two bounded BCK algebras

Since all the algebras connected to logic have, more or less explicitly, an associated order relation, it follows, by duality principle, that they have two presentations, dual to each other. We classify these dual presentations in “left” and “right” ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as “left” algebras or as “right” algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the “left” presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the properties satisfied. In this work (Parts I–V) we make an exhaustive study of these algebras—with two bounds and with one bound—and we present classes of finite examples, in bounded case. In Part II, we continue to present new properties, and consequently new algebras; among them, bounded α γ algebra is a common generalization of MTL algebra and divisible bounded residuated lattice (bounded commutative Rl-monoid). We introduce and study the ordinal sum (product) of two bounded BCK algebras. Dedicated to Grigore C. Moisil (1906–1973).     

Pseudo-t-norms and pseudo-BL algebras

BL algebras were introduced by Hájek as algebraic structures for his Basic Logic, starting from continuous t-norms on [0,1]. MV algebras, product algebras and Gödel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-groups. We have generalized the BL algebras and pseudo-MV algebras, introducing the pseudo-BL algebras. In this paper we introduce weak-BL algebras and weak-pseudo-BL algebras. We also introduce non-commutative t-norms (we call them pseudo-t-norms) and use them in constructing pseudo-BL algebras and weak-pseudo-BL algebras.     

Finiteness in Infinite-Valued Łukasiewicz Logic

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic _m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in _m. We also reduce the notion of logical consequence in to the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic.     

Interval-valued fuzzy n-ary subhypergroups of n-ary hypergroups

This paper provides a continuation of ideas presented by Davvaz and Corsini (J Intell Fuzzy Syst 18(4):377–382, 2007). Our aim in this paper is to introduce the concept of quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set. This concept is a generalized concept of quasicoincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup of a n-ary hypergroup. This newly defined interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup is a generalization of the usual fuzzy n-ary subhypergroup. Finally, we consider the concept of implication-based interval-valued fuzzy n-ary subhypergroup in an n-ary hypergroup; in particular, the implication operators in ￡ukasiewicz system of continuous-valued logic are discussed.     

Basic fuzzy logic and BL-algebras

 The many-valued propositional logic BL (basic fuzzy logic) is investigated. It is known to be complete for tautologies over BL-algebras (particular residuated lattices). Each continuous t-norm on [0,1] determines a BL-algebra; such algebras are called t-algebras. Two additional axioms B1, B2 are found such that BL+(B1,B2) is complete for tautologies over t-algebras. It remains open whether B1, B2 are provable in BL.