2n-布尔值逻辑:2-值命题逻辑的自然推广

将2-值命题逻辑的语义理论推广到了有限布尔代数,得到了2n-布尔值命题逻辑.同时本文给出了2n-布尔值命题逻辑应用的一些例子.     

Natural Selection: Interactive Subset Creation

Abstract

Visualization is a critical technology for understanding complex, data-rich systems. Effective visualizations make important features of the data immediately recognizable and enable the user to discover interesting and useful results by highlighting patterns. A key element of such systems is the ability to interact with displays of data by selecting a subset for further investigation. This operation is needed for use in linked views systems and in drill-down analysis. It is a common manipulation in many other systems and is as ubiquitous as selecting icons in a desktop graphical user interface (GUI). It is therefore surprising to note that little research has been done on how selection can be implemented. This article addresses this omission, presenting a taxonomy for selection mechanisms and discussing the interactions between branches of the taxonomy.     

Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic (FNL). We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.     

Logic and Colour

In this paper evidence will be provided that Wittgenstein??s intuition about the logic of colour relations is to be taken near-literally. Starting from the Aristotelian oppositions between propositions as represented in the logical square of oppositions on the one hand and oppositions between primary and secondary colors as represented in an octahedron on the other, it will be shown algebraically how definitions for the former carry over to the realm of colour categories and describe very precisely the relations obtaining between the known primary and secondary colours. Linguistic evidence for the reality of the resulting isomorphism will be provided. For example, the vertices that resist natural single-item lexicalization in logic (such as the O-corner, for which there is no natural lexicalization *nall (=not all)) are not naturally lexicalized in the realm of colour terms either. From the perspective of the architecture of cognition, the isomorphism suggests that the foundations of logical oppositions and negation may well be much more deeply rooted in the physiological structure of human cognition than is standardly assumed.     

Abstract Logics, Logic Maps, and Logic Homomorphisms

What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2]. This research was supported by the grant CNPq/FAPESB 350092/2006-0.     

Logic and Religion

We first start by describing the happening of the 1st World Congress on Logic and Religion. We then explain the motivation for developing the interaction between logic and religion. In a third part we discuss some papers presented at this event published in the present special issue.     

计量逻辑学中的雪崩逻辑公式

将密码学中满足严格雪崩准则的布尔函数的概念引入到计量逻辑学之中,提出了雪崩逻辑公式的概念,并研究了雪崩逻辑公式的真度及其性质。证明了至少含有三个原子公式的雪崩逻辑公式的真度之集为H1={k/2n-12n-3≤k≤3×2n-3;n=3,4,…},在此基础上,通过引入函数ξ建立了n(n≥3)元雪崩布尔函数个数的表达式,给出了雪崩逻辑公式的构造方法。最后,研究了反射变换下k阶雪崩逻辑公式的性质。     

Logic and probabilistic systems

Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set of the sentences that get limit probability 1 is a theory, in fact can be a complete set. We prove incompleteness results for this setting, by showing for instance that for every there are true sentences that get limit probability less than . No set as above can contain the set of all true sentences, although we exhibit some containing all the true sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent. Received March 20, 1995     

Categorical Abstract Algebraic Logic: Ordered Equational Logic and Algebraizable PoVarieties

A syntactic apparatus is introduced for the study of the algebraic properties of classes of partially ordered algebraic systems (a.k.a. partially ordered functors (pofunctors)). A Birkhoff-style order HSP theorem and a Mal’cev-style order SLP theorem are proved for partially ordered varieties and partially ordered quasivarieties, respectively, of partially ordered algebraic systems based on this syntactic apparatus. Finally, the notion of a finitely algebraizable partially-ordered quasi-variety, in the spirit of Pałasińska and Pigozzi, is introduced and some of the properties of these quasi-povarieties are explored in the categorical framework.      

Natural and Doubly Natural Dualities

Representable dualities between categories of modules satisfying natural closure properties are investigated and shown to be determined by homological properties shared by various classical and recently investigated types of dualities.     

Logic Regression

Logic regression is an adaptive regression methodology that attempts to construct predictors as Boolean combinations of binary covariates. In many regression problems a model is developed that relates the main effects (the predictors or transformations thereof) to the response, while interactions are usually kept simple (two- to three-way interactions at most). Often, especially when all predictors are binary, the interaction between many predictors may be what causes the differences in response. This issue arises, for example, in the analysis of SNP microarray data or in some data mining problems. In the proposed methodology, given a set of binary predictors we create new predictors such as “X₁, X₂, X³, and X⁴ are true,” or “X⁵ or X⁶ but not X⁷ are true.” In more specific terms: we try to fit regression models of the form g(E[Y]) = b₀ + b₁ L₁ + · · · + b_n L_n , where L_j is any Boolean expression of the predictors. The L_j and b_j are estimated simultaneously using a simulated annealing algorithm. This article discusses how to fit logic regression models, how to carry out model selection for these models, and gives some examples.     

支撑模糊集合转换的模糊逻辑是伪逻辑

模糊逻辑系统用模糊集合描述模糊信息、用模糊集合转换处理模糊信息.阐述支撑模糊集合转换的模糊逻辑为什么是伪逻辑的原因.指出定义在论域一个空间上的模糊集合,因为破坏了集合中元素的"不可分割性",所以模糊集合描述的模糊信息不能用数学计算通过模糊集合转换处理.实际应用中的模糊信息定义在论域与状态空间两个空间上,其正确表达方式是满足"归一化"条件的隶属度向量;处理的正确途径是,研究基于状态转移矩阵的隶属度转换;支撑隶属度转换的是近似推理逻辑,目的是使构建的隶属度转换模型是当前条件下人们可能构建的"最优"近似模型.     

Perfect MV-algebras and their Logic

In this paper, after recounting the basic properties of perfect MV-algebras, we explore the role of such algebras in localization issues. Further, we analyze some logics that are based on Łukasiewicz connectives and are complete with respect to linearly ordered perfect MV-algebras.      

Rudimentary Languages and Second-Order Logic

The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second-order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing a direct logical characterization of rudimentary languages and a representation result of second-order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity.