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Compactness theorems of fuzzy semantics

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Chinese Science Bulletin 2000年第2期45卷 120-125页

作者： wang Guojun Institute of mathematics, shaanxi normal university, xi’an/710062, china Institute of mathematics Shaanxi Nornial University Xi’an China

The relationship among diverse fuzzy semantics vs. the corresponding logic consequence operators has been analyzed systematically The results that compactness and logical compactness of fuzzy semantics are equivalent to compactness and continuity of the logic consequence operator induced by the semantics respectively have been proved under certain conditions. A general compactness theorem of fuzzy semantics have been established which says that every fuzzy semantics defined on a free algebra with members corresponding to continuous functions is compact.

关键词： fuzzy semantics compactness loglc compactness inductive set point compactness Lukasiewicz semantics.

Quasi-formal deductive system for fuzzy propositional calculus

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Chinese Science Bulletin 1997年第14期42卷 1154-1157页

作者： wang GuojunInstitute of mathematics, shaanxi normal university, xi’an 710062, china Institute of mathematics Shaanxi Normal University Xi’ an China

ELKAN’s report 'The paradoxical success of fuzzy logic' at the 11th IA Annual Conferenceof America in July, 1993 caused quite a great stir in the research field of IA as well as fuzzysystems, fifteen scholars published papers thereupon to refute it and finally Elkan answeredthem with his paper 'The paradoxical controversy over fuzzy logic'. An exclusive analysiscan be found in Prof. Wu wangming’s paper. It seems from this event that fuzzy proposi-

关键词： ΣR-tautology α-mod us ponens α-hypothetical syllogism quasi-formal deductive system.

Implicational lattices and generalization of Stone's representation theorem

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Chinese Science Bulletin 1998年第12期43卷 997-1000页

作者： wang Guojun Institute of mathematics, shaanxi normal university, xi’an 710062, china Institute of mathematics Shaanxi Normal University Xi’an China

Let F(S) be the free algebra of type (,∨,→) generated by the non_empty set S, it is proved that the logical equivalent relation defined by means of R 0_semantics is a congruence relation on F(S) and the corresponding quotient algebra is said to be the R 0_semantic Lindenbaum algebra. Taking R 0_semantic Lindenbaum algebra as a prototype, the concepts of implicational lattices and regular implicational lattices which are generalizations of the concept of Boolean algebras are introduced. Besides, the concept of fuzzy implicational spaces is introduced and the representation theorem of regular implicational lattices is obtained by means of fuzzy implicational spaces. In case of Boolean algebras, the corresponding fuzzy implicational spaces are zero_dimensional compact Hausdorff spaces and herefrom it is proved that the famous Stone’s representation theorem of Boolean algebras is a corollary of the representation theorem of regular implicational lattices.

关键词： Stone’s representation theorem R 0_semantic Lindenbaum algebra implicational lattice fuzzy implicational space representation theorem of regular implicational lattices.

The connectedness of H(λ) unit interval and H(λ) real line

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Chinese Science Bulletin 1997年第23期42卷 2016-2017页

作者： LI ShenggangDepartment of mathematics, shaanxi normal university, xi’an 710062, china Department of mathematics Shaanxi Normal University Xi’an China

JUST like in classical mathematics, fuzzy unit interval I(L) and fuzzy real line R(L), thebasic and important research objects of fuzzy mathematics, also interest a good few *** 1992, wang and Xu, by refining the topology of I(L), defined a new fuzzy unit in-terval——H(λ) unit interval I(L), and established the Stone-Cech compactification theoryfor weakly induced L-fuzzy topological spaces. In 1993, Zhang and Liu introduced the con-cept of weak inducification of L-fuzzy topological space, and in this way, I(L) is exactly

关键词： real line The connectedness of H unit interval and H line

Maximal contractions in Boolean algebras

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Science china(Information Sciences) 2012年第9期55卷 2044-2055页

作者： SHI Huixian & wang GuoJun Institute of mathematics,shaanxi normal university,xi’an 710062,china 1. Institute of mathematics Shaanxi Normal University Xi’an 710062 China

In the present paper,the concepts of deductive element and maximal contraction are introduced in Boolean algebras,and corresponding theories of consistency and maximal contractions are *** algorithm principle is proposed to compute all maximal contractions for a consistent set with respect to its refutation in Boolean *** is pointed out that the quotient algebra of the first-order language with respect to its provable equivalence relation constitutes a Boolean algebra,and hence the computation of R-contractions for closed formulas in first-order languages can be converted into the one in Boolean algebras proposed in this ***,the concept of basic element is introduced in Boolean algebras,which contributes to the definitions of clause and Horn clause transplanted from logic to a special type of Boolean algebras generated by basic *** is also pointed out that the computation of R-contractions for clauses in the classical propositional logic can be converted into the one in Boolean algebras generated by basic elements proposed in this paper.

关键词： deductive element consistency maximal contraction minimal subtraction basic element clause

Lattice-valued modal propositional logic and its completeness

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Science china(Information Sciences) 2010年第11期53卷 2230-2239页

作者： SHI Huixian & wang GuoJun1 Institute of mathematics, shaanxi normal university, xi’an 710062, china 1. Institute of mathematics Shaanxi Normal University Xi’an 710062 China

Based on the concept of the complete lattice satisfying the first and second infinite distributive laws, the present paper introduces the semantics of the lattice-valued modal propositional logic. It is pointed out that this semantics generalizes the semantics of both classical modal propositional logic and [0,1]-valued modal propositional logic. The definition of the QMR0-algebra is proposed, and both the Boole-typed latticevalued modal propositional logic system B and the QMR0-typed lattice-valued modal propositional logic system QML* are constructed by use of Boole-algebras and QMR0-algebras, respectively. The main results of the paper are the completeness theorems of both the system B and QML*.

关键词： latticed-valued modal propositional logic modal model QMR0-algebra validity completeness

Borel probabilistic and quantitative logic

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Science china(Information Sciences) 2011年第9期54卷 1843-1854页

作者： ZHOU HongJun & wang GuoJun College of mathematics and Information Science,shaanxi normal university,xi’an 710062,china 1. College of mathematics and Information Science Shaanxi Normal University Xi’an 710062 China

The present paper introduces the notion of the probabilistic truth degree of a formula by means of Borel probability measures on the set of all valuations,endowed with the usual product topology,in classical two-valued propositional *** approach not only overcomes the limitations of quantitative logic,which require the probability measures on the set of all valuations to be the countably infinite product of uniform probability measures,but also remedies the drawback that probability logic behaves only *** is proved that the notions of truth degree,random truth degree in quantitative logic and the probability of formulas in probability logic can all be brought as special cases into the unified framework of the probabilistic truth *** quantitative logic and probability logic are *** also proves a one-to-one correspondence between deductively closed theories and topologically closed subsets of the space of all valuations,and a one-to-one correspondence between probabilistic truth degree functions and Borel probability measures on the space of all *** second part of the present paper proposes an axiomatic definition of the probabilistic truth degree,and it is finally proved that each probabilistic truth degree function is represented by a unique Borel probability measure on the space of all valuations in the way given in the first *** a theory which we call probabilistic and quantitative logic in the framework of classical propositional logic is established.

关键词： probabilistic truth degree finite separation property probability logic quantitative logic probabilistic and quantitative logic

The nonlinear standing wave inside the space of liquid

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Science china(Physics,Mechanics & Astronomy) 2010年第3期53卷 496-503页

作者： wang ChengHui & LIN ShuYu Institute of Applied Acoustics, shaanxi normal university, xi’an 710062, china 1. Institute of Applied Acoustics Shaanxi Normal University Xi’an 710062 China

Based on the basic equations of hydrodynamics, the nonlinear acoustic wave equation is obtained. By taking into account the boundary condition and properties of nonlinear standing wave, the equation is solved through perturbation method, and the stable expressions of fundamental wave and second harmonic are presented. The sound pressures in an ultrasonic cleaner are measured by hydrophones, and the relationship between the received voltages of hydrophones and the output voltages of the ultrasonic generator is researched. The study shows the existence of the nonlinear effect of liquid and analyzes the frequency spectrum of the received signals by hydrophones, by which the fundamental wave, second and high order harmonics are found coexisting in the bounded space filled with liquids. The theory and experimental results testify the existence of the nonlinear standing wave in liquid. Owing to the restricted applicability of perturbation method, the theoretical results of the fundamental wave and second harmonic are good only for the weak nonlinear phenomenon.

关键词： nonlinear acoustic wave equation nonlinear standing wave perturbation method fundamental wave second harmonic

Relations among the category of topological molecular lattices and certain related categories

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Chinese Science Bulletin 1997年第8期42卷 636-640页

作者： wang Guojun and LI Yongming1. Institute of mathematics, shaanxi normal university, xi’an 710062, china 2. Department of Basic Science, Northwestern Institute of Architecture Engineering, xi’an 710061, china Institute of mathematics Shaanxi Normal University Xi’an China Department of Basic Science Northwestern Institute of Architecture Engineering Xi’an China

TAKING the theory of fuzzy topology and point set topology as background,the theory of topo-logical molecular lattices was first advanced in 1979,and after several generalizationsrefs.[4,5] are its general *** topology and point set topology are both special casesof topological molecular lattices as pointed out in ref.[4],whereas it seems not clear whetherone can develop the theory of topological molecular lattices by means of methods in

关键词： category pick-up-one principle.