命题逻辑的序结构

The Ordered Structure of Propositional Logic

含n个命题变元的合式公式A(P1,P2,…,Pn)组成的集合M关于命题公式的等值关系 构成的商集M/ ={CA| B∈CA M,A B}关于如下代数运算∨和序关系≤是一个特殊的双格半群,即F格半群:CA∨CB=CA∨B;CA∧CB=CA∧B;CA≤CB当且仅当A的每一成真赋值都是B的成真赋值( CA,CB∈M/ ),这里的∧运算是∨的对偶运算,而M上的∨、∧运算分别是逻辑"或"逻辑"与",同时给出了它的分子结构,并指出该F格半群与n元真值函数集构成的F格半群是同构的.

Let the set M consists of all propositional formaulas A(p_,p_2,…p_n),p_1,p_2,…,p_n be propositional variable,and  a relation of propositional equavalent on the M.Then the the quotien set S=M/={C_A｜B∈C_AM,AB} is a espeial double latice-ordered Semigroup,i.e,F-lattice semigroup concerning following the algebraic operation and the ordered relation ≤:C_A,C_B∈S,C_A∨C_B=C_(A∨B);C_A∧C_B=C_(A∧B);C_A≤C_B if and only if for every obtaining truth-value valution of A is obtaining truth-value valution of B,where the operation ∧ is dual operation of the operation ∨,and operation ∨,∧ on the morespectively is logical 'or ',logical 'and'.Let M(S)={C_(m__i)｜m__i is a minimal term in prorositional logical,i=0,1,2,…,2~n-1} be a set of all molecules in S.It is proved that this F-lattice semigroup (S,∨,≤) and n-ary truth-value functions constituting F-lattice semigroup(L,∨,≤) one another are isomorphic.