【作者】 关晓红；

【导师】 王国俊；

【作者基本信息】 陕西师范大学 ， 基础数学， 2007， 硕士

【摘要】 1997年，王国俊教授基于蕴涵算子R₀提出了修正的Kleene系统W，W，W_k，又于1998年引入了广义重言式的概念，对修正的Kleene系统的广义重言式类进行了深刻而细致的讨论，建立了广义重言式理论，为模糊逻辑提出了新的研究方向．此后，吴洪博教授在修正的Kleene系统中把广义重言式的概念进行了扩充和推广，并在此系统中引入广义语义MP规则与广义HS规则，取得了一些有意义的研究成果，使得我们对修正的Kleene系统有了更清楚的认识．在多值逻辑系统的三种逻辑运算非（→），并（∨），蕴涵（→）中，蕴涵算子的选取至关重要，不同的蕴涵算子从语义上构成不同的多值逻辑系统，其中左连续t-模的伴随是具有较好性质的蕴涵算子．众所周知，与R₀型蕴涵算子相对应的R₀型t-模是左连续的，兰蓉基于R₀型t-模给出了一类带参数的[0，1]上的t-模*_α。及与之伴随的新的蕴涵算子H_α（0≤α≤1），进而建立了多值逻辑系统H_α．多值逻辑系统H_α（0＜α＜1）推广了修正的Kleene系统，并且这些系统之间是彼此同构的．因此我们主要研究当α=1/2时的多值逻辑系统H_1/2，以H_1/2为赋值域利用广义重言式概念得到公式集F（S）的一个分划，接着讨论了系统H_1/2中的各类广义语义MP规则与广义语义HS规则，并把系统H_1/2中的相关结论推广到H_α（0＜α＜1）中，之后讨论了基于系统H_α的蕴涵算子的三I算法以及还原性问题．1994年美国数学家Foulis和Bennett引进了效应代数的概念，推广了正交模格，被看作量子逻辑的数学模型．这种抽象的效应代数虽然历史不长，然而它却引起了数学工作者和理论物理学工作者的极大兴趣．本文第三章讨论了效应代数与多种模糊逻辑代数系统的关系，使我们对效应代数有了进一步的了解，并且通过对效应代数的研究，从蕴涵的角度出发引入了偏蕴涵代数，证明了偏蕴涵代数与效应代数是两种不同形式的等价的代数结构，并讨论了它与MV代数之间的关系．更多还原

【Abstract】 In 1997, based on R₀ implication operator professor Wang Guojun proposed the revised Kleene system. Again in 1998, professor Wang proposed the concept of generalized tautology and discussed the classes of generalized tautologies, which gave a new research direction in fuzzy logic. Subsequently, professor Wu Hongbo generalized the concept of generalized tautologies and proposed generalized semantic MP rules and generalized semantic HS rules in revised Kleene system, so we had more clear understanding about this system.Among the three kind of logic operations in many-valued logic system: negtion（→）, union（V） and implication（→）. It is very important to choose the implication operator because different implication operator forms different many-valued logic system. The residuated operators of left-continuous t-norms have better properties among many implication operators. It was well known that R₀ t-norm which R₀ implication operatior residuated to was left-continuous. Based on R₀ t-norm, Lan Rong proposed a new class of t-norms *_αand corresponding resituted implication operatiors H_αwith a parameterαin [0,1] and obtained many-valued logic system H_α.Many-valued logic system H_α（0≤α≤1） generalized revised Kleene system. Arbitray one system of H_αis ismorphic with another, so we mainly study the logic system H_1/2. In the logic system H_1/2, a new partion of F（S） can be obtained by utilizing the concept of generalized tautologies, moreover, a variety of generalized semantic MP and generalized semantic HS rules are obtained in H_1/2. After that, the correspoding resluts in H_1/2 are generalized to the systems H_α（0＜α＜1）. At last, based on the implication operators of systems H_α, Triple I method and reversibility properties were discussed.Effect algebras were introduced as a mathematical model of quantum logic by American mathematicians Foulis and Bennett in 1994, which generalized orthomodular lattices. Many researchers, in the fields of mathematics and physics, were attracted to study them. In the third chapter the relation of effect algebra and many fuzzy logic algebra systems are discussed. Through the study of effect algebra, the partial implication algebra was proposed from the point of implication. It is proved that partial implication algebra and effect algebra are equivalent. Then we talk about the relation of partial implication algebra and MV-algebra.更多还原