斯多葛命题逻辑系统不完全性的新证明

A new proof for the incompleteness of stoic propositional logic system

斯多葛学派的命题逻辑理论是古希腊人对逻辑学的第二次伟大贡献。从现代逻辑的角度来看,应该把它理解为一个有且仅有5条推演规则的自然演绎系统而不是公理系统。陈志美和胡泽洪在承认它是自然演绎系统的基础上,运用算术解释方法证明了它的不完全性。然而,他们所证明的是添加了两条“元逻辑规则”之后的系统,而且证明过程还存着一些不严谨之处。为此,本文采用现代逻辑中更为常用的语义比较方法直观且严谨地证明了斯多葛命题逻辑系统的不完全性,并给出了寻找该系统所缺失的规则从而使其具有完全性的方法,进一步深化了我们对斯多葛命题逻辑的认识。

The Stoic theory of propositional logic is the second great contribution of the Ancient Greeks to logic.From the perspective of modern logic,it would be more natural to understand it as a natural deductive system rather than an axiomatic system.Zhimei Chen and Zehong Hu(2001) used arithmetic interpretation method to prove its incompleteness.However,what they proved was the system after adding two “meta-logic rules”,and there are still some imperfections in the proof process.This paper adopts the Semantic Comparison Method,a widely used approach in modern logic,to intuitively and rigorously demonstrate the incompleteness of the Stoic propositional logic system.It also offers a methodology for identifying the missing rules that could render the system complete,thereby enhancing our comprehension of Stoic propositional logic.