三种逻辑代数的等价刻画和模糊模态逻辑
本文关键词：三种逻辑代数的等价刻画和模糊模态逻辑，，由笔耕文化传播整理发布。
三种逻辑代数的等价刻画和模糊模态逻辑
三种逻辑代数的等价刻画和模糊模态逻辑
模糊逻辑是对经典命题逻辑的改善和推广,它更能适应现实生活的需求.常见的模糊逻辑系统有逻辑系统Lukasiewicz,乘积逻辑系统∏,G（o|¨）del逻辑系统G,以及王国俊教授提出的L*系统.而与上述系统相匹配的代数结构分别是MV代数,∏代数,G代数,R0代数.一般而言,MV代数,R0代数和Boole代数均是建立在格序框架之下的,这不便于我们在更加宽泛的体系下研究它们与其它逻辑代数之间的关系.一个自然的不足就是可否放弃格序前提分别给出上述三种代数的等价刻画,以便于进一步研究他们与其他逻辑代数之间的联系呢?本文对此进行了研究并作出了回答.另外,王国俊教授通过在系统L,Luk以及L*中引入了公式真度的概念,将数理逻辑与数值计算有机结合起来,并提出了计量逻辑学.使得经典意 义下既非重言式又非矛盾式的公式有了评价其真伪程度的标准.2007年,傅丽在经典命题逻辑系统L中,通过把赋值域由{0,1}扩充到Boole代数引入了B-赋值的概念,并且以有限Boole代数为前提建立了公式的B-真度理论.另一方面,在B-赋值语义下系统L是否完备?同一公式的真度值与B-真度值之间有什么关系?这些不足尚未及讨论,本文将作出解答.模态逻辑属于非经典逻辑的范畴,而模态语言则是从内在的局部观点来表达关系结构的.从语构的观点来看,模态逻辑只不过是在经典命题逻辑中的连接词→与→之外又添加了一些模态词的逻辑系统而已.它在知识表示和知识推理等领域均有广泛的应用.模态逻辑的语义一般是建立在Kripke模型基础之上的.Kripke模型是一个三硕博在线论文网组M=（W,R,V）,其中W,R,V分别表示集合,二硕博在线论文网关系和映射.一般来讲,模型中的R,V都是分明集合,那么能否将R和V分别模糊化来建立语义理论?能否给模态逻辑赋予代数语义?和语构和谐吗?本文对此展开了研究并得到了一些结果.本文的主要结论如下:（1）在非格序框架下,给出了Boole代数,MV代数以及R0代数的等价刻画.证明了Boole代数等价刻画中各条公理是相互独立的.并证得Boole代数与正则的HFI代数是等价的.（2）证明了真度不变性定理,即对同一个公式A而言,A的真度值与B-真度值相同.（3）在B-赋值语义下,系统L是完备的.（4）引入了MR0代数的概念.讨论了它的一些重要性质,给出了MR0代数的同构定理.（5）构建了模态系统K1,证明了在MR0代数语义下该系统是完备的.（6）通过将Kripke模型中的赋值V模糊化,建立了模态逻辑系统K2,并证明了系统K2是可靠的;通过将Kripke模型中的二硕博在线论文网关系R模糊化,建立了模态逻辑系统K3,并证明了系统K3是完备的.
【Abstract】 Fuzzy logic is improvement and spread of classical two-valued logic.It is more adaptabal to human life.There are several common fuzzy logic systems,such as logic system Lukasiewicz,product logic systemП,G（o|¨）del logic system G and L* system introduced by professor Wang Guojun.The corresponding algebra of the above mentioned systems are MV algebra,G algebra,R0 algebra respectively.Generally speaking,MV algebra,R0 algebra and Boolean algebra are all on the basis of lattice and order frame.This is not convenient for us to study the connection between them and other logic algebras in a broader system.One natural question may be: can we give up the premise condition about the lattice and order,then get the equivalent characterization of the three above-mentioned algebras.So that we can have a further study on the connection between them and other logic algebra.The paper focuses on the careful study of this question.Besides,via the introduction of the concept about the truth degree of formular, professer Wang Guojun combines the mathematical logic and numerical calculation together and advances quantitative logic.So that,there is a criteria to judge the formula,which is of neither tautologie nor contradictory in the classical logic.In 2007,by FU li,spreading the domain of evaluation from {0,1} into Boolean algebra brought about the concept of B-evaluation and formed B-truth degree theroy of formula on the premise of finite Boolean algebra.However,is system L complete under the B-evaluation semantics? What’s the connection between the truth degree and B-truth degree of the same formula? The answer to these questions will be found in this paper.Modal logic belongs to the category of nonclassical logic,and modal language shows the relation and structure through interal and partial views.Concerning syntactical system,modal logic is simple a logic system which adds some modal words besides the connection words→and→in classical two-valued logic.The semantic of modal logic is often based on Kripke model.Which is triple M= （W,R,V）.W,R,V stands for set,binary relationship and mapping respectively. Generally speaking,R,V are classical sets.So,can R and V be fuzzified respectively to form a semantic theory? Can we give the algebra scmantic to modal logic? Is semantic system and syntactical system harmony? This paper mainly focuses on this question. The main conclusions arrived at in this paper are as follow:（1） Under the non-lattice frame,equivalent characterization of Boolean algebra, MV algebra,and R0 algebra are given.It is proved in the paper that the axioms in the characterization of Boolean algebra are independent of each other.In addition, It proves that Boolean algebra is equivalent to regular HFI algebra.（2） The invariable theorem of truth degree is proved.To be more specific,for the same formular A,the value of truth degree and B-truth degree are equal.（3） Under the B-evaluation semantic,system L is complete.（4） The concept of M R0 algebra is introduced here,some major properties of which are discussed.Also,isomorphism theorems of M R0 algebra is given.（5） Modal logic system K1 is formed,which proves to be a complete system under M R0 semantics.（6） Modal logic system K2 which proved soundness is formed through the fuzzi-fying of the evaluation V in Kripke model.Also,Modal logic system Ks is formed and proved to be complete through the fuzzifying of binary relationship R in Kripke model.
【关键词】 逻辑代数； 等价刻画； 真度不变性； MR0代数； 模态逻辑；【Key words】 logic algebra； equivalent characterization； invariabal of truth degree； MR0 algebra； modal logic；
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本文关键词：三种逻辑代数的等价刻画和模糊模态逻辑，由笔耕文化传播整理发布。
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