發(fā)布時(shí)間：2016-11-21 07:57

  本文關(guān)鍵詞：三種邏輯代數(shù)的等價(jià)刻畫和模糊模態(tài)邏輯，，由筆耕文化傳播整理發(fā)布。

三種邏輯代數(shù)的等價(jià)刻畫和模糊模態(tài)邏輯

三種邏輯代數(shù)的等價(jià)刻畫和模糊模態(tài)邏輯

模糊邏輯是對(duì)經(jīng)典命題邏輯的改善和推廣,它更能適應(yīng)現(xiàn)實(shí)生活的需求.常見的模糊邏輯系統(tǒng)有邏輯系統(tǒng)Lukasiewicz,乘積邏輯系統(tǒng)∏,G（o|¨）del邏輯系統(tǒng)G,以及王國(guó)俊教授提出的L*系統(tǒng).而與上述系統(tǒng)相匹配的代數(shù)結(jié)構(gòu)分別是MV代數(shù),∏代數(shù),G代數(shù),R0代數(shù).一般而言,MV代數(shù),R0代數(shù)和Boole代數(shù)均是建立在格序框架之下的,這不便于我們?cè)诟訉挿旱捏w系下研究它們與其它邏輯代數(shù)之間的關(guān)系.一個(gè)自然的不足就是可否放棄格序前提分別給出上述三種代數(shù)的等價(jià)刻畫,以便于進(jìn)一步研究他們與其他邏輯代數(shù)之間的聯(lián)系呢?本文對(duì)此進(jìn)行了研究并作出了回答.另外,王國(guó)俊教授通過在系統(tǒng)L,Luk以及L*中引入了公式真度的概念,將數(shù)理邏輯與數(shù)值計(jì)算有機(jī)結(jié)合起來,并提出了計(jì)量邏輯學(xué).使得經(jīng)典意 義下既非重言式又非矛盾式的公式有了評(píng)價(jià)其真?zhèn)纬潭鹊臉?biāo)準(zhǔn).2007年,傅麗在經(jīng)典命題邏輯系統(tǒng)L中,通過把賦值域由{0,1}擴(kuò)充到Boole代數(shù)引入了B-賦值的概念,并且以有限Boole代數(shù)為前提建立了公式的B-真度理論.另一方面,在B-賦值語義下系統(tǒng)L是否完備?同一公式的真度值與B-真度值之間有什么關(guān)系?這些不足尚未及討論,本文將作出解答.模態(tài)邏輯屬于非經(jīng)典邏輯的范疇,而模態(tài)語言則是從內(nèi)在的局部觀點(diǎn)來表達(dá)關(guān)系結(jié)構(gòu)的.從語構(gòu)的觀點(diǎn)來看,模態(tài)邏輯只不過是在經(jīng)典命題邏輯中的連接詞→與→之外又添加了一些模態(tài)詞的邏輯系統(tǒng)而已.它在知識(shí)表示和知識(shí)推理等領(lǐng)域均有廣泛的應(yīng)用.模態(tài)邏輯的語義一般是建立在Kripke模型基礎(chǔ)之上的.Kripke模型是一個(gè)三碩博在線論文網(wǎng)組M=（W,R,V）,其中W,R,V分別表示集合,二碩博在線論文網(wǎng)關(guān)系和映射.一般來講,模型中的R,V都是分明集合,那么能否將R和V分別模糊化來建立語義理論?能否給模態(tài)邏輯賦予代數(shù)語義?和語構(gòu)和諧嗎?本文對(duì)此展開了研究并得到了一些結(jié)果.本文的主要結(jié)論如下:（1）在非格序框架下,給出了Boole代數(shù),MV代數(shù)以及R0代數(shù)的等價(jià)刻畫.證明了Boole代數(shù)等價(jià)刻畫中各條公理是相互獨(dú)立的.并證得Boole代數(shù)與正則的HFI代數(shù)是等價(jià)的.（2）證明了真度不變性定理,即對(duì)同一個(gè)公式A而言,A的真度值與B-真度值相同.（3）在B-賦值語義下,系統(tǒng)L是完備的.（4）引入了MR0代數(shù)的概念.討論了它的一些重要性質(zhì),給出了MR0代數(shù)的同構(gòu)定理.（5）構(gòu)建了模態(tài)系統(tǒng)K1,證明了在MR0代數(shù)語義下該系統(tǒng)是完備的.（6）通過將Kripke模型中的賦值V模糊化,建立了模態(tài)邏輯系統(tǒng)K2,并證明了系統(tǒng)K2是可靠的;通過將Kripke模型中的二碩博在線論文網(wǎng)關(guān)系R模糊化,建立了模態(tài)邏輯系統(tǒng)K3,并證明了系統(tǒng)K3是完備的.

【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.

【關(guān)鍵詞】 邏輯代數(shù)； 等價(jià)刻畫； 真度不變性； MR0代數(shù)； 模態(tài)邏輯；
【Key words】 logic algebra； equivalent characterization； invariabal of truth degree； MR0 algebra； modal logic；

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  本文關(guān)鍵詞：三種邏輯代數(shù)的等價(jià)刻畫和模糊模態(tài)邏輯，由筆耕文化傳播整理發(fā)布。

本文編號(hào)：184275