逻辑系统L_3中公式的随机真度及近似推理

利用赋值集的随机化方法,在三值Lukasiewicz命题逻辑系统中引入公式的随机真度,证明了随机真度的MP规则、HS规则及交推理规则;同时引入公式间的随机相似度和伪距离,建立了随机逻辑度量空间,推导出随机相似度的若干性质,证明了随机逻辑度量空间中逻辑运算的连续性;并在随机逻辑度量空间中提出了三种不同类型的近似推理模式,证明了三种近似推理模式的等价性.     

n值Lukasiewicz命题逻辑系统中公式的随机真度及近似推理

利用赋值集的随机化方法,在n值Lukasiewicz命题逻辑系统中引入公式的随机真度,证明了随机真度的MP规则、HS规则及交推理规则;同时引入公式间的随机相似度和随机伪距离,建立了随机逻辑度量空间,推导出随机相似度的若干性质,证明了随机逻辑度量空间中逻辑运算的连续性;并在随机逻辑度量空间中提出了三种不同类型的近似推理模式,证明了三种近似推理模式的等价性.     

关于三值R0命题逻辑系统的若干注记

在三值R0命题逻辑系统中证明了随机真度的MP、HS和交推理规则;提出了随机开放度,指出随机开放度与随机发散度是从两个不同的角度刻画了理论的相容程度,并得出对同一个理论而言,二者取值相等的结论.     

逻辑系统G3中随机真度的推理规则及近似推理

在三值Godel命题逻辑系统中,推出了公式随机真度的推理规则,证明了随机逻辑度量空间中逻辑运算的连续性;研究了随机逻辑度量空间理论的发散度,提出了三种不同类型的近似推理模式,并证明了三种推理模式的等价性.这将进一步完善三值Gōdel逻辑系统中随机真度和随机逻辑度量空间的理论.     

经典命题逻辑中公式的Γ-随机真度与近似推理

利用概率空间的无穷乘积,在经典二值命题逻辑中引入了公式的Γ-随机真度概念以及公式间的Γ-相似度概念.进而导出了全体公式集上的一种伪距离,建立了逻辑度量空间.最后提出了基于Γ-随机真度的三种不同的近似推理模式,并且证明了这三种近似推理模式之间是相互等价的.     

有限Boole语义中基于前提信息的随机真度

本文研究了有限Boole语义中基于前提信息的随机化问题.利用赋值集的随机化方法,提出了公式的基于前提信息Γ的Γ-随机真度,得到了Γ-随机真度的MP规则,HS规则和交推理规则.通过引入公式间的Γ-随机相似度和Γ-伪距离,建立了Γ-随机逻辑度量空间,在有限Boole语义上推广了基于前提信息的近似推理理论.     

落影表现理论中的Fuzzy集运算

关于Fuzzy集运算与Fuzzy推理关系的基本定义,迄今已经有了几十种不同的表达式。它们的产生多少带有点经验色彩,缺乏理论上的依据。面对着这众多的定义,人们不知从何选择,不知依据什么去作选择。本文运用落影表现理论对Fuzzy集运算及Fuzzy推理关系给出了统一的定义,根据Fuzzy集之间的不同关联情况,推导出了包括Zadeh表达式在内的各种具体公式,并对它们的适用场合进行了初步的分析。     

经典推理模式的随机化研究及其应用（Ⅱ）

证明了D-逻辑度量空间中三种近似推理模式是等价的,指出了全体原子公式之集在D-逻辑度量空间中未必是全发散的.     

基于Lukasiewicz蕴涵算子的反向三I算法

三Ⅰ算法是针对模糊推理的FMP与FMT模型的一种新的推理方法。本文针对Lukasiewicz蕴涵算子讨论FMP模型及FMT模型的反向三Ⅰ推理及反向α-三Ⅰ推理,借助该蕴涵算子的性质,给出相应的推理算法。     

核心素养导向下初中数学代数推理能力培养探析

推理是数学核心素养之一,是数学基本的思维方式.初中阶段的推理按类型分为几何推理和代数推理.很多初中数学教师注重几何推理,而忽视代数推理,从而导致在教学过程中缺少对学生代数推理能力的培养.为后面更长远的学习带来阻碍,也无法提升学生的数学核心素养.本文从代数推理的现状、推理之间的关系及代数推理能力的培养策略三部分进行探析.     

An integral with respect to probabilistic-valued decomposable measures

Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach to integration of non-negative real-valued functions with respect to probabilistic-valued decomposable (sub)measures. Basic properties of the corresponding probabilistic integral are investigated in detail. It is shown that certain properties, among them linearity and additivity, depend on the properties of the underlying triangle function providing (sub)additivity condition of the considered (sub)measure. It is demonstrated that the introduced integral brings a new tool in approximate reasoning and uncertainty processing with possible applications in several areas.     

Mathematical reasoning in teachers’ presentations

This paper presents a study of the opportunities presented to students that allow them to learn different types of mathematical reasoning during teachers’ ordinary task solving presentations. The characteristics of algorithmic and creative reasoning that are seen in the presentations are analyzed. We find that most task solutions are based on available algorithms, often without arguments that justify the reasoning, which may lead to rote learning. The students are given some opportunities to see aspects of creative reasoning, such as reflection and arguments that are anchored in the mathematical properties of the task components, but in relatively modest ways.     

Symmetric implicational method of fuzzy reasoning

Fuzzy reasoning should take into account the factors of both the logic system and the reasoning model, thus a new fuzzy reasoning method called the symmetric implicational method is proposed, which contains the full implication inference method as its particular case. The previous full implication inference principles are improved, and unified forms of the new method are respectively established for FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) to let different fuzzy implications be used under the same way. Furthermore, reversibility properties of the new method are analyzed from some conditions that many fuzzy implications satisfy, and it is found that its reversibility properties seem fine. Lastly, the more general α-symmetric implicational method is put forward, and its unified forms are achieved.     

基于S蕴涵算子的区间值模糊推理的三I算法

针对S蕴涵算子讨论了FM P模型及FM T模型的区间值模糊集三I推理,借助S蕴涵算子的性质,给出相应的推理算法,并讨论了算法的还原性。     

Triple I algorithms based on Schweizer–Sklar operators in fuzzy reasoning

In this paper, we focus on the flexible inference method with parameters, that is the parametric triple I method by the combination of Schweizer–Sklar operators and triple I principles for fuzzy reasoning. Because the Schweizer–Sklar parameter m reflects the interaction between propositions in reasoning processes, the new parameterized triple I algorithms are closer to human reasoning in daily life. Also some properties of the new algorithms such as the reductivity, continuity and approximation are discussed. It is shown that some existing results are special cases of the new algorithms given here and in view of the variability of the parameter m the new algorithms have excellent flexibility in reasoning processes.     

Theory of T-norms and fuzzy inference methods

In this paper, the theory of T-norm and T-conorm is reviewed and the T-norm, T-conorm and negation function are defined as a set of T-operators. Some typical T-operators and their mathematical properties are presented. Finally, the T-operators are extended to the conventional fuzzy reasoning methods which are based on the and operators. This extended fuzzy reasoning provides both a general and a flexible method for the design of fuzzy logic controllers and, more generally, for the modelling of any decision-making process.     

Unclad emperors: A case of mistaken identity

I am ready to rest my case. In summary, I will simply provide the two paragraphs of the conclusion of my “Trivial Mathematics” paper that Koblitz did not quote. In the social science literature, much reasoning takes place about magnitudes without any help from mathematical formalisms, but solely in terms of ordinary language. As the illustrations [in “Trivial Mathematics”] suggest, a good deal of this reasoning makes implicit use of the properties of ordinal variables and monotonic transformations. In both the social science and natural science literatures, much reasoning is also done from diagrams, without concern for the exact forms of the functions depicted or the cardinal values of variables. This reasoning appears also to make use of only the ordinal properties of the magnitudes depicted in the diagrams. One can argue that a great deal of everyday “commonsense” reasoning, when it is concerned with magnitudes, is, implicitly, reasoning about ordinal relations.     

A framework for reasoning under uncertainty based on non-deterministic distance semantics

In this paper, we introduce a general and modular framework for formalizing reasoning with incomplete and inconsistent information. Our framework is composed of non-deterministic semantic structures and distance-based considerations. This combination leads to a variety of entailment relations that can be used for reasoning about non-deterministic phenomena and are inconsistency-tolerant. We investigate the basic properties of these entailments, as well as some of their computational aspects, and demonstrate their usefulness in the context of model-based diagnostic systems.     

Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets

Two basic inference models of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). The Triple I method is a very important method to solve the problems of FMP and FMT. The aim of this paper is to extend the Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets. In the paper, we first investigate the algebra operators' properties on the lattice structure of intuitionistic fuzzy information and provide the unified form of residual implications which indicates the relationship between intuitionistic fuzzy implications and fuzzy implications. Then we present the intuitionistic fuzzy reasoning version of the Triple I principles based on the models of intuitionistic fuzzy modus ponens (IFMP) and intuitionistic fuzzy modus tollens (IFMT) and give the Triple I method of intuitionistic fuzzy reasoning for residual implications. Moreover, we discuss the reductivity of the Triple I methods for IFMP and IFMT. Finally, we propose α-Triple I method of intuitionistic fuzzy reasoning.