对偶三角模-余模逻辑的语义理论与应用

It's difficult to judge truth or fake of proposition (predicate) for the existing uncertainty of information. So, we need to give a scientific method for measuring of degree of belief that proposition (predicate) is true ，in short truth degree. Then classical logic needs to be extended to nonclassical logic. The existing nonclassical logics includes multi-valued logic, probabilistic logic (called dual Product-Additionlogic)and uncertain logic (called dual Infimum-Supremum logic). These logics display characteristic superiority in dealing with uncertainty. However, they need further improvement. Particularly, the study of uncertain logic is in its initial stage. The members of the project have studied these logics, and have presented multi-dimensions and three layer logics about fuzzy propositions by abstracting the process to group decision making. The researchful contents of the project are as follows: 1. Study dual Product-Addition logic and Infimum -Supremum logic, and establish generalized theories of logic and reasoning using all dual t-norms and t-conorms. 2. Establish the theory of many-dimensions and three layers logic based on dual t-norms and t-conorms using the thought of many-dimensions and three logics of fuzzy propositions. 3. Study practical application about the above new logics and reasoning. The project belongs to intersected category of mathematic theory with uncertainty and nonclassical logic.

信息的不确定性常常导致命题（谓词）的真假无法确定。因此，需要给出度量命题（谓词）真的可能程度(简称真度)的科学方法，即需要将命题（谓词）的二值逻辑推广到命题（谓词）的非经典逻辑。人们提出了多值逻辑、概率逻辑（称对偶乘-加逻辑）及不确定逻辑（称对偶下-上确界逻辑）。这三种逻辑在处理不确定性方面显示出独特的优势。然而，其自身理论并不完善。特别是不确定逻辑仅处于研究的初始阶段。本项目成员已经研究了这三种逻辑，并通过抽象人们实际中群决策的过程，利用连续三角模逻辑，提出了模糊命题的多维三层逻辑。本项目的研究内容是： 一、继续研究对偶乘-加逻辑和下-上确界逻辑，并就所有三角模和对偶的三角余模建立一般性的逻辑和推理理论. 二、利用模糊命题的多维三层逻辑的思想，建立多维三层对偶三角模-余模逻辑理论。 三、研究新建逻辑的实际应用。 该项目属于不确定性的数学理论和非经典逻辑的交叉范畴。

本项目围绕非经典数理逻辑的理论与应用展开研究，主要工作如下。首先，把概率逻辑、模糊逻辑和不确定逻辑等纳入到统一框架之下，初步建立了更一般的对偶三角模-三角模余模逻辑；其次，探讨了新逻辑在不确定规划中的应用，特别是建立了带有不确定性的保险和机器加工排序等问题的优化模型；再次，研究了不确定微分方程的稳定性理论；最后，在多值逻辑框架下，探讨了格值拓扑的收敛理论和格值粗糙集的公理化问题。在本项目中，我们一方面发展和丰富了非经典数理逻辑的理论框架；还将相关理论应用于金融和工程等领域的研究。因此，我们的工作不仅有重要的理论意义，而且还有很大的应用价值。