命题逻辑

The course contains four sections as follows: mathematical logic (including basic concepts of propositional logic and predicate logic, propositional calculuses and inference theories), set theory (including set algebras, relations, functions and cardinal numbers), algebraic structure (including algebraic systems, semigroups and groups, rings and fields, lattices and Boolean algebras), graph theory (including basic concepts of graph, Euler graphs and Hamiltonian graphs, trees, planar graphs and coloring graphs, some special vertex subsets and edge subsets).

本课程包含四部分内容：数理逻辑（包含命题逻辑与一阶逻辑的基本概念、等值演算以及推理理论），集合论（包含集合代数、二元关系、函数和基数），代数结构（包含代数系统、半群与群、环与域、格与布尔代数），图论（包含图的基本概念、欧拉图与哈密顿图、树、平面图及图的着色、图的某些特殊的顶点子集与边子集）。

The soundness and completeness theorem of this method were proved at last. This part consists of the following points: 1. The properties of the indecomposable extremely simple form of LP were discussed, the theorem shown that every logical formula of LP was equals of a generalized conjunction normal form. 2. The method for judging a logical formula is a indecomposable extremely simple form or not was discussed, the necessary and sufficient conditions of a logical formula with not more than three implicative connective being a indecomposable extremely simple form was given. 3. The α-resolution fields of the indecomposable extremely simple form of LP were discussed, some elementary results were obtained. 4. The automated reasoning method based on the resolution principle of classical logic was discussed, the author given a method to describe the set of clauses by using a matrix, and translated the resolution of two clauses to column operations of this matrix.

二、关于格值命题逻辑系统LP中α-自动推理方法的研究 1、对LP中不可分极简式的性质进行了研究，证明了LP中任何一个格值逻辑公式等值于一个广义合取范式； 2、对LP中不可分极简式的判定方法进行了研究，讨论了含有不超过3个蕴涵运算的格值逻辑式为不可分极简式的充分必要条件； 3、对LP中不可分极简式的α-归结域进行了研究，得到了一些基本结果； 4、对经典逻辑中基于归结原理的自动推理方法进行了研究，给出了用符号矩阵表示子句集合的一种方法，把不同的归结过程转换成矩阵的列变换。

Based on paper [1] and [2], this paper gives two generating algorithms to calculate and output the special conjunctive normal forms and special disjunctive forms of given propositional formulas automatically and formally.

摘要在文[1]和文[2]的基础上，给出了命题逻辑中任一命题公式的主析取范式和主合取范式的自动生成算法，并实现了多个命题公式主范式的同时形式化输出。

Based on the outcome of Xu Yang and Qin Keyun about lattice implication algebra and lattice-valued prepositional logic LP with truth-value in a lattice implication algebra, the author studied the properties of lattice implication algebra and the α-automated reasoning method based on α-resolution principle of LP. The specific contents are as follows: The Study of Lattice Implication Algebra On the basis of previous results of lattice implication algebra, this part consists of the following three points: 1. Some properties of lattice implication algebra L were discussed, and some important results were given if L was a complete lattice implication algebra. 2. The properties of left idempotent elements of lattice implication algebras were discussed, and the conclusion that lattice implication algebra L was equals of the directed sum of the range and dual kernel of a left map constructed by a left idempotent element was proved. 3. The properties of the filters of lattice implication algebra were discussed, the theorem was shown that they satisfy the hypothetical syllogism and substitute theorem of the propositional logic. 4. The concept of weak niters of lattice implication algebras and their properties and structures are discussed. It is proved that all weak filters of a lattice implication algebra form a topology and the the implication isomorphism betweem two lattice implication algebras is a topological mapping between their topological spaces. The Study of α-automated reasoning method based on the lattice-valued propositional logic LP In this part, the author given an a-automated reasoning method based on the lattice-valued propositional logic LP.

本文基于徐扬和秦克云的关于格蕴涵代数和以格蕴涵代数为真值域的格值命题逻辑系统LP的研究工作，对格蕴涵代数以及格值命题逻辑系统LP中基于α-归结原理的自动推理方法进行了系统深入的研究，主要有以下两方面的研究成果：一、关于格蕴涵代数的研究 1、对格蕴涵代数的格论性质进行了研究，得到了当L为完备格蕴涵代数时，关于∨，∧，→运算的一些结果； 2、对格蕴涵代数的左幂等元进行了研究，证明了格蕴涵代数L可以分解为任何一个左幂等元所对应的左映射的像集合与其对偶核的直和； 3、对格蕴涵代数的滤子的性质进行了研究，证明了滤子的结构相似于逻辑学中的Hypothetical syllogism规则和替换定理； 4、给出了格蕴涵代数中弱滤子的概念，对弱滤子的性质个结构进行了研究，证明了格蕴涵代数的全体弱滤子构成一个拓扑结构，格蕴涵代数之间的蕴涵同构是相应的拓扑空间之间的拓扑映射。

Professor Zhang Qing-yu gets paraconsistent modal proposition logic C〓G'on the base ofCosta's paraconsistent proposition logic system C〓.

张清宇教授在科斯塔的弗协调命题逻辑系统C 的基础上得到弗协调模态命题逻辑系统CnG＇。

Furthermore, we discuss the truth degree of the propositional formula in two-valued propositional logic system.

并对命题公式的蕴涵问题进行等价描述；对二值命题逻辑系统的程度化研究中命题公式的真度问题进行讨论。

At the basic of knowledge implication propositional logic, a system of knowledge implicational model prepositional logic system has been built. It can describe inference mechanism of inconsistent knowledge system which includes in modal information.

在知识蕴涵命题逻辑的基础上，构建了一个知识蕴涵模态命题逻辑系统，它可以描述包含模态信息的不协调知识系统的推理机制。

Based on the production of other researchers such as professor Xu Yang and professor Qin Keyun, this paper discusses the structure and properties of lattice implication algebra, tautologies in some lattice-valued systems, automated reasoning methods, lattice-valued propositional logic system.

本文的工作是在徐扬教授、秦克云教授等研究成果的基础上，对格蕴涵代数的性质、结构、格值命题逻辑系统中的重言式、自动推理方法、格值命题逻辑系统等进行了一些研究。

The theory of ∑-α-tautologies of fuzzy propositional logic was combined with the theory of truth degree in metrology of logic introduced by professor G.J.Wang, and the theory of ∑Γ－fuzzy truth degrees of formula relative to the finite theory in propositional logic system n was introduced.

将模糊命题逻辑系统中的∑-理论与计量逻辑学中的真度理论相结合，在n-值Lukasievicz模糊命题逻辑系统Ln中引入了公式相对于有限理论的∑Γ－模糊真度理论，讨论了其中的主要性质。

With the background of application of fuzzy control techniques to a rather typical complex control system, gas-burning water heater, theories, applications and techniques of fuzzy information processing have been studies. The obtained results include:(1) For lattice-valued logic systems based on lattice implication algebras, two dynamical lattice-valued propositional logic Lvpl and lattice-valued first-order logic Lvfl have been constructed, and some theories and methods of fuzzy reasoning have been proposed based on fuzzy set theory and multi-valued logic.(2) Some methods for designing and optimizing fuzzy control systems and some corresponding control algorithms have been proposed, and some new theories and methods for fuzzy control rule base and fuzzy control system performance analysis have been introduced.(3) 6 fuzzy control systems have been developed, and some difficult problems on craft of manufacture, batch consistency, quality stability, installation and main

本项目以模糊控制技术用于燃气热水器这一较典型的复杂控制系统为应用背景，在模糊信息处理的理论、应用和技术三个层次开展了相关的研究工作：(1)进一步研究了基于格蕴涵代数的格值逻辑系统，建立了两种动态的格值命题逻辑Lvpl和格值一阶逻辑Lvfl，并基于模糊集理论和多值逻辑提出了多种模糊推理的理论与方法；(2)提出了模糊控制系统的多种设计及优化方法与控制算法，建立了模糊控制规则库和模糊控制系统性能分析的一些新理论与新方法；(3)研制完成了6个模糊控制系统，还特别解决了模糊控制燃气热水器和智能控制燃气热水器在产业化过程中的生产工艺、批量一致、质量稳定、安装维修和性能价格等一系列难题，做好了批量生产前的技术准备，现已小批量生产。

Cynanchum Lingtai apricot production in the average weight 65 grams, the brightly-colored fruit, juicy rich, sweet-sour taste, sweet from the nucleolus, when the late Qing Dynasty famous Shaanxi, Gansu provinces, the Qing imperial court Tongzhi tribute for years.

灵台生产的牛心杏平均单果重65克，果实色泽鲜艳，汁多味浓，甜酸适口，离核仁甜，清末时就驰名陕、甘两省，清同治年间曾为朝廷贡品。

Chenopodium album,Solanum nigrum, and Amaranthus retroflexus were very susceptible to the herbicides. Polygonum persicaria and Abutilon theophrasti were relatively less susceptible to the herbicides, and Lycopersicon esculentum was not susceptible to it. The relationship between reduction rates of weed biomass and PPM values of weed leaves 2,4, and 6 days after treatment was established.

供试的6种杂草对该混剂的敏感性存在显著差异：红心藜Chenopodium album、龙葵Solanum nigrum和反枝苋Amaranthus retroflexus对该混剂最敏感，ED90值分别为47.65、71.67和29.17g/hm2；春蓼Polygonum persicaria和苘麻Abutilon theophrasti敏感，ED90值分别为96.91、114.20g/hm2；而番茄不敏感。