形式系统的可靠性和完全性问题

语法完全性仅与一个形式系统的语法构造相关,经典命题逻辑的有代入规则的系统是语法完全的,无代入规则的系统不是语法完全的。将全称概括规则作为初始规则的系统是否限制对该规则的使用可导致系统相对于通常的语义解释是否具有强可靠性。对必然化规则和模态逻辑系统来说有类似的结果。

On Formal Systems's oundness and Completeness

Syntactical completeness is only concerned with syntactics. The classical proposition logic systems with the rule of substitution are syntactically complete, while the ones without the rule are not. Whether the classical predicate logic systems with the primitive rule of universal generalization have the strong semantic soundness depends on the restriction of the rule. There exists the similar relation between the necessitation and modal logic.