Classical Propositional Logic And Modal Propositional Logic

Modal logic, as a non-classical logic, is different from classical logic. Mainling thestudy of modern logic has two aims: one is to solve the logical problems inmathematical deduction and theoretic system building, that is, to give mathematicallogical base; the other is to take general inferences as objects, thus involve someinferences in special fields and solve the logical problems in it. The former,mathematical logic in norrow meaning, is usually called mathematical logic and can beregarded as logic on mathematics. The latter, under the ideas and methods of modernwidely applied mathematics, can be looked on as applied mathematics logic, that is,mathematical logic in broad meaning, or symbolic logic. Classical logic is put forwardso as to build the base of mathematics. In fact, the modern logics firstly established areon mathematics, while modern modal logics are based on the latter, and devided intotwo sides: syntax and semantics. This article will compare modern modal propositionallogics and classical propositional logics according to syntax and semantics. The whole article falls into six parts. The first part is the introduction to the article. It gives a simple view to the historyof modal logic. The second part, the first chapter of the article, retrospects the theoretical origins ofclassical logic and modal logics, the necessity, possibility and forming processes . The second chapter develops into the third part.In this part, the author comparesthe differences between modal function and truth function: one can't make sure the truthof modal function only according to the truth connectives. The fourth part falls on the third chapter. The author first presents the formal axiomsystems, then compares the differences and similarities between the formal axiomsystem of classical propositional logic and its counterparts in modal propositional logic.This comparison is carried out from two aspects. The first is the comparison aboutformal language. The results of comparison between L_PM and L_P: the introduction of â–¡leads to the extensions of primitive symbols and formation rules, and in turn, leads tothe more expansions of LPM's well-formed formulas, atomic formulas, the variables andconstants of metalanguage than LP's counterparts. The second aspect is the comparisonof deductive bases. The results of comparison: the number of axioms and transformationrules increases; and the effective range of the same deductive rule becomes bigger;thenumber of theorems greatly increases; deductive abilities become stronger. In a word,K,D,T,B,S4 and S5 are P's real extensions. Meanwhile, the author discusses in detailone of the important features of modal propositional logic systems-reduction whichincludes the reduction of iterated modalities and the reduction of embedded modalities. The fifth part is the fourth chapter of the article. This part introduces theformations of natural deductive systems of both classical propositional logic and modalpropositional logic, and compares TN, S4N, S5N and PN as regards formal languages,deductive rules, theorems, deductive ability and so on. The author draws a conclusionfrom the comparison: formal language expands; deductive rules increase and theeffective range of the same deductive rules expands; the number of theorems becomesbigger; the deductive ability becomes stronger. The last part is the fifth chapter. The author here introduces the semanticses ofclassical proposional logic and modal propositional logic and compares between them.The paper first compares the differences and similarities of truth valuation, amongwhich it compares the elements of their semanticses and the effective ranges of truthvaluations. After that, the paper compares the differences and similarities of the validformulas obtained from the definition of validity, that is, the comparison ofexplanational abilities. According to the comparison, the paper reveals that: theelements of semantics increases, and two more elements â–¡and R appear; meanwhile,the effective range of the same element V expands; the number...