Logic of proofs with substitution

The substitution operation in logic of proofs is axiomatized. For the system constructed, symbolic semantics is introduced and a completeness theorem is proved.     

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。     

The Logic with Truth and Falsehood Operators from a Point of View of Universal Logic

The logic with independent truth and falsehood operators TFL is proposed. In TFL(→) standard truth-conditions for the implication are adopted. Nevertheless the laws of classical logic are not valid. In this language more then 10⁷ different binary connectives can be defined. So this logic can be treated as universal logic relatively to the class of sentential logics.     

Common Belief with the Logic of Individual Belief

The logic of common belief does not always reflect that of individual beliefs. In particular, even when the individual belief operators satisfy the KD45 logic, the common belief operator may fail to satisfy axiom 5. That is, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. We identify the intersubjective restrictions on individual beliefs that are incorporated in axiom 5 for common belief.     

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.     

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.     

模糊综合评判加权平均模型的误导性阐释

模糊综合评判在不能界定隶属度转换是线性转换条件下,硬性用加权平均线性模型实现隶属度转换.通过揭示指标隶属度中通常包含确定目标隶属度不起作用的非线性冗余值,建立区分权滤波的冗余理论,用冗余理论界定隶属度转换不是线性转换,并构建近似推理逻辑支撑的隶属度转换模型.阐述所建模型为什么是当前条件下人们能构建的最优近似模型.当指标隶属度只取1或0两个值时模型将退化成模糊综合评判的"加权平均"模型.