F(S_n)中公式集Γ的全体结论之集D(Γ)的结构

The structure of D(Γ) which is the set of all conclusions of a given formula set Γ in F(S_n)

探讨原子公式集为有限集Sn时,二值命题逻辑F(Sn)中公式集Γ的所有结论之集D(Γ)的结构.利用F(Sn)中公式在可证等价意义下的一般表示,通过建立一个特殊映射,将F(Sn)中公式之间的合取∧、析取∨及否定的运算转化为在可证等价意义下某集合的子集之间的求交∩,求并∪,求补′的运算.进而得到(1)对于A∈F(Sn),ΓF(Sn),给出了A∈D(Γ)是否成立的充要条件;(2)对于ΓF(Sn),给出了Γ是否相容的充要条件;(3)对于ΓF(Sn),在可证等价意义下,给出了D(Γ)中所含F(Sn)中公式的个数;(4)对于Γ1,Γ2 F(Sn),给出了D(Γ1)与D(Γ2)之间的关系;(5)对于A∈F(Sn),ΓF(Sn),在可证等价意义下,给出了D(Γ)中与A距离最近的公式.

Under the condition of limit set Sn consisting of atomic formulas,the structure of D（Г） which is the set of all conclusions of a given formula set F in two-valued propositional logic is discussed. By using the result, the general representation of formulas in F （ Sn ） in the meaning of provable equivalence, and making a special mapping, the operations of ∧,∨, among formulas change into operations of ∧,∨,＇on a special set in the meaning of provable equivalence, and then several results below are acquired. （ 1 ） for A ∈ F（ Sn）, F F（ Sn ）, the necessary and sufficient condition of whetherA∈D（F） is given; （2） for Г F（Sn） ,the necessary and sufficient condition of whether F is consistent is given; （3） for Г F（Sn） ,the number of formulas of F（Sn） in D （Г） in the meaning of provable equivalence is given ; （4） for Г1, Г2 F（ Sn ）, the relation between D （Г1 ） and D（Г2） is given;（5） for A∈F（Sn）,F F（Sn） , in the meaning of provable equivalence, the formula which is the nearest one to A in D（Г） is given.