模糊BIK~+-逻辑与非可换模糊逻辑(英文)

为了建立各种可换和非可换模糊逻辑的公共基础(蕴涵片段),提出了一个新的蕴涵逻辑,称为模糊BIK+-逻辑。证明了这一新的蕴涵逻辑的可靠性和弱完备性定理,同时讨论了模糊BIK+-逻辑与各种模糊逻辑之间的关系,以及与它们配套的代数结构之间的关系。     

模糊逻辑代数的Loomis—Sikorski表现定理

通过研究MV-代数、Π-代数、G-代数、R0-代数等模糊逻辑代数的赋值(从模糊逻辑代数L到单位区间[0,1]的同态)与滤子之间的关系,建立了MV-代数、Π-代数、G-代数、R0-代数等模糊逻辑代数的Loomis-Sikorski表现定理.     

R0-代数的格蕴涵表示定理

通过对模糊命题演算系统∧*及相应的Lindenbaum代数的研究,给出了R0-代数的格蕴涵表示形式,极大地简化了R0-代数的定义形式,使得R0-代数从定义形式上更加符合逻辑代数的特征,突出了R0-代数和其它逻辑代数的区别与联系,为进一步研究R0-代数及其和其它逻辑代数的关系提供了一个强有力的工具。     

可换逻辑的代数语义综述

主要介绍九种可换逻辑的语义系统,它们是布尔代数, MV-代数, BL-代数, MTL-代数,剩余格, Hoops,半Hoops, EQ-代数和相等代数,并给出相应的例子.进而结合作者的工作介绍了这些代数系统在概率、格序群和拓扑中的研究进展,同时给出如下看法:布尔代数是经典逻辑;从代数角度讨论了经典逻辑与模糊逻辑的区别.最后给出值得进一步研究的公开问题.     

Kleene逻辑函数—一类广义模糊逻辑函数

引进了一类广义模糊逻辑函数－Ｋｌｅｅｎｅ逻辑函数，讨论了这类函数的基本性质，并在这个函数类与正则三值逻辑函数之间建立起同构关系。     

一种度量结构在四种逻辑代数上的共性

对MV逻辑代数、Godel逻辑代数、乘积逻辑代数、R0逻辑代数在度量方面的性质进行了进一步研究。首先根据四种逻辑代数的共同性质,在它们的单位区间[0,1]中建立了一种逻辑度量结构:其次对度量结构在四种逻辑代数中的共有性质进行了讨论,并分别在四种逻辑代数中给出了这种度量结构的具体形式;最后证明了四种逻辑代数中的基本运算关于度量结构的连续性。     

格蕴涵代数的蕴涵表示定理

在对格蕴涵代数和模糊蕴涵代数研究的基础上,给出了格蕴涵代数的三个蕴涵表示定理。极大地简化了格蕴涵代数的定义形式,使得格蕴涵代数在形式上更加突出逻辑代数的特征及其与其它逻辑代数之间的联系与区别。为进一步研究格蕴涵代数及其与其它逻辑代数的关系提供了一个有力的工具。     

基于完备剩余格值逻辑的BCI-代数的三种模糊理想

通过完备剩余格值逻辑中一元模糊谓词，将经典BCI-代数中的p-理想、q-理想和a-理想进行重新刻画，引入了BCI-代数的l-值模糊p-理想、l-值模糊q-理想和l-值模糊a-理想的概念。利用完备剩余格值逻辑的语义方法，研究这三种l-值模糊理想的性质及关系，推广了经典模糊情形下相应的现有结论。     

MTL-代数上的广义赋值

本文研究了MTL-代数上的几类广义赋值,讨论了MTL-代数上广义赋值、态以及滤子之间的关系,获得了MTL-代数上广义赋值成为(正)关联广义赋值的等价刻画,并基于广义赋值构造的同余关系研究了MTL-代数的商结构.所得结果推广了基于三角模的模糊逻辑代数上广义赋值的相关理论,进一步丰富了基于三角模的模糊逻辑代数上概率测度的代数结论.     

关于BCI逻辑的一种偏序代数

通常,人们认为Kiyoshi Iséki在20世纪60年代引入的BCI-代数是组合逻辑中BCI逻辑的代数对等物。然而这种广为人知的断言却是有问题的,因为BCI逻辑关于BCI代数是不完备的。在本文中,我们引入一种称为MPE的偏序代数。在MPE中的每个不等式对应BCI逻辑中的一个重言式且反之亦然,从而MPE代数是与BCI逻辑完备的代数类。     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

Continuity, proof systems and the theory of transfinite computations

 We use the theory of domains with totality to construct some logics generalizing ω-logic and β-logic and we prove a completenes theorem for these logics. The key application is E-logic, the logic related to the functional ³ E. We prove a compactness theorem for sets of sentences semicomputable in ³ E. Received: 21 January 1998 / Published online: 2 September 2002     

Cut-elimination and interpolation for Ω-logic

In 1978, Girard introduced-logic to generalize-logic. The basic category of-logic is the categoryON of ordinals. For geometric structure reasons, Girard changed the basic categoryON into the more general categoryWF of well-founded orders (1983). The logic he obtained was called-logic. Here, we extend (unpublished) results of-logic to-logic.     

A Criterion for Admissibility of Inference Rules in Some Class of S4-Logics Without the Branching Property

We consider a finitely approximable modal S4-logic without the branching property. Although Rybakov's criterion is inapplicable, using his method we manage to obtain an algorithmic criterion for admissibility of inference rules in a given logic.     

Generalized Bosbach states: part I

States have been introduced on commutative and non-commutative algebras of fuzzy logics as functions defined on these algebras with values in [0,1]. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0,1], in this paper we introduce Bosbach states defined on residuated lattices with values in residuated lattices. We are led to two types of generalized Bosbach states, with distinct behaviours. Properties of generalized states are useful for the development of an algebraic theory of probabilistic models for non-commutative fuzzy logics.     

Spectrum topology of a residuated lattice

In this paper, given a non-commutative residuated lattice L, a topological space is constructed using certain fuzzy subsets of L. Indeed, we show that the set of all prime fuzzy filters of a non-commutative residuated lattice L forms a topological space. Particularly, we show that this space is compact and a T ₀-space and its certain subspaces are Hausdorff spaces. Finally, we show that the set of all prime filters of L is also a Hausdorff space.     

Foundation of credibilistic logic

In this paper, credibilistic logic is introduced as a new branch of uncertain logic system by explaining the truth value of fuzzy formula as credibility value. First, credibilistic truth value is introduced on the basis of fuzzy proposition and fuzzy formula, and the consistency between credibilistic logic and classical logic is proved on the basis of some important properties about truth values. Furthermore, a credibilistic modus ponens and a credibilistic modus tollens are presented. Finally, a comparison between credibilistic logic and possibilistic logic is given.     

Algebraic properties of complex fuzzy events in classical and in quantum information systems

Baer¹-semigroups are regarded as the main abstract structures for an algebraic analysis of complex fuzzy events in generalized probability theory. This assumption is verified in the case of classical probability theory in the framework of measure and integration theory. The corresponding fuzzy language is extended to the non-commutative probability theory based on operators in Hilbert space.Starting from a quantum information system a quantum probability space is constructed, which is naturally embedded in a classical information system. In this last both exact than fuzzy quantum events are represented as classical fuzzy events. Lastly, the classical fuzzy events which correspond to exact quantum events are characterized by some minimality properties.