Formally Deriving the Third Newton’s Law from a Pair of Nontrivial Assumptions in a Formal Axiomatic Theory “Sigma-V”

The article is devoted to hitherto never undertaken applying an almost unknown logically formalized axiomatic epistemology-and-axiology system called “Sigma-V” to the Third Newton’s Law of mechanics. The author has continued investigating the extraordinary (paradigm-breaking) hypothesis of formal-axiological interpreting Newton’s mathematical principles of natural philosophy and, thus, has arrived to discrete mathematical modeling a system of formal axiology of nature by extracting and systematical studying its proper algebraic aspect. Along with the proper algebraic machinery, the axiomatic (hypothetic-deductive) method is exploited in this investigation systematically. The research results are the followings. 1) The Third Newton’s Law of mechanics has been modeled by a formal-axiological equation of two-valued algebraic system of metaphysics as formal axiology. (Precise defining the algebraic system is provided.) The formal-axiological equation has been established (and examined) in this algebraic system by accurate computing compositions of relevant evaluation-functions. Precise tabular definitions of the evaluation-functions are given. 2) The wonderful formula representing the Third Newton’s Law (in the relevant physical interpretation of the formal theory Sigma-V) has been derived logically in Sigma-V from the presumption of a-priori-ness of knowledge. A precise axiomatic definition of the nontrivial notion “a-priori-ness of knowledge” is given. The formal derivation is implemented in strict accordance with the rigor standard of D. Hilbert’s formalism;hence, checking the formal derivation submitted in this article is not a difficult task. With respect to proper theoretical physics, the formal inference is a nontrivial scientific novelty which has not been discussed and published elsewhere yet.

一种基于规则的语言的公理语义

为了准确描述离散事件控制系统对象之间的逻辑关系和编写控制程序,提出了一种基于规则的语言——逻辑规则描述语言(LRDL)。用EBNF给出了LRDL的语法定义,基于Hoare逻辑的公理系统,形式化地给出并证明了LRDL的公理语义,为用LRDL编写的程序的正确性证明提供了理论依据。

有穷时间投影时序逻辑的完备公理系统

为采用定理证明的方法对并发及交互式系统进行验证,研究了有穷论域下有穷时间一阶投影时序逻辑(projection temporal logic,简称PTL)的一个完备公理系统.在介绍PTL的语法、语义并给出公理系统后,提出了PTL公式的正则形(normal form,简称NF)和正则图(normal form graph,简称NFG).基于NF给出了NFG的构造算法,并利用NFG可描述公式模型的性质证明PTL公式的可满足性判定定理和公理系统的完备性.最后,结合实例展示了PTL及其公理系统在系统验证中的应用.结果表明,基于PTL的定理证明方法可方便用于并发系统的建模与验证.

投影时序逻辑的公理系统与形式验证

为了采用定理证明的方法对并发及交互式系统进行验证,提出了一阶投影时序逻辑的公理系统.利用投影时序逻辑既可描述待验证系统性质和规范,又可描述其实现模型的特点,在同一投影时序逻辑框架可以方便地对待验证系统进行建模和性质描述,并使用公理系统完成系统性质的证明.最后通过一个实例来展示投影时序逻辑及公理系统在系统验证中的应用.

从易经逻辑到数理辩证逻辑

早在6000多年前,中国先人就创造了以爻卦为命题形式,以八卦和六十四卦为体系的辩证逻辑形式演绎系统。此后,又给出了爻辞和卦辞的语义解释系统。后来,又给出了《易经》的语义和语用解释系统。它自古至今,长盛不衰,成为中国的传统逻辑、文化发展的活水源头。以互补结构整体思想的形式结构和推理有效式规律为研究对象的数理辩证逻辑,是以易经逻辑为其辩证逻辑基础的,它又是易经逻辑的一致性扩大系统。它为实现易经逻辑的现代化发展提供了现代逻辑形式化方法。易经逻辑形式公理系统DPA++就是其现代化形式公理系统。

多值逻辑的公理化

给出了一种新的多值逻辑系统，公理化所得到的ＧＭＬ３形式系统与二值逻辑有本质的区别，二值逻辑是它的一个退化．

公理化及其在《高等代数》课程中的体现与作用

公理化方法是现代数学非常重要的思想方法.阐述了公理化方法的发展脉络以及其在《高等代数》课程中的体现与作用.通过对公理化方法深层次的认识,学生可以更好的理解高度抽象的和逻辑严密的《高等代数》知识体系,更好的提高思维能力.

从易经逻辑到数理辩证逻辑

早在公元6000多年前，中国先人就创造了以爻卦为命题形式，以八卦和六十四卦为体系的辩证逻辑形式演绎系统。此后，又给出了爻辞和卦辞的语义解释系统。后来，又给出了《易经》的语义和语用解释系统。它自古至今，长盛不衰，成为中国的传统逻辑、文化发展的活水源头。以互补结构整体思想的形式结构和推理有效式规律为研究对象的数理辩证逻辑，是以易经逻辑为其辩证逻辑基础的，它又是易经逻辑的一致性扩大系统。它为实现易经逻辑的现代化发展提供了现代逻辑形式化方法。本文给出的易经逻辑形式公理系统DPA＋＋就是其现代化形式公理系统。