命題演算的公理系統

<正> §1. 問題的提出 對於傳統的二值邏輯系統(以後叫做系統M)所作的公理系統,優點最多的可說是Hilbert-Bernays[1]Ⅰ册66頁上所载的(一名Munster派公理,以後即用此名).這個公理系統共有兩個模式(又名原則)及五组公理,模式即代入原則

SOME AXIOM SYSTEMS FOR PROPOSITIONAL CALCULUS

Among various axiom systems for the traditional two-valued logical system, the one given by Hilbert-Bernays in Grundlagen der Mathematik perhaps is the best. It possesses the following advantages: first, it divides the axioms into five groups according to the connectors involved and shows the essential properties of each of them; second, it makes the distinction between the three important logical systems (the traditional system, the intuitionistic system and the minimalkalkul) quite clear. However, it has a shortage that no group of axioms is sufficient. By the sufficiency of groups of axioms we mean that if a proposition can be deduced in the whole system then it can be deduced by means of the groups of axioms involving the same connectors only. The present paper is to remedy this shortage. To meet various requirements we give several systems. The difference of them is either by the number of axioms in each group or the method (adding or strengthening axioms) for distinguishing the three logical systems mentioned above. They may be shown in the following table: Among the main results we have:If we add the proposition "CCCpqpp" to an axiom system of the intuitionistic system we get an axiom system of the traditional system. If every group of its axioms is sufficient the same for the result system.If in a consistent system we can deduce the following proposition and rules: CpCqp; Cαβ→CCβΥCαΥ, CCΥαCΥβ; β, CαCβΥ→CαΥ then any finite number of the rest organic axioms of the form "Cαβ" may be combined into a single organic axiom, provided α may turn into an asserted proposition after a suitable substitution.