弗雷格《概念文字》理解的两点注记

Two Notes on Frege's Begriffsschrift

文章旨在简要地讨论弗雷格《概念文字》,指出其中的两个重要但被一些国内学者误解或忽略的贡献:首先我们指出,根据Boolos等人的论证,弗雷格《概念文字》中的逻辑本质上是带完整二阶存在概括规则的二阶逻辑,这点在国内一些学者的著作与文章中存在误解;其次,我们讨论弗雷格如何用遗传性概念来定义祖先关系,进而定义自然数或有穷数,并使得数学归纳法仅根据自然数的定义就得以成立,这也为弗雷格把算术还原为逻辑奠定了基础。

This review points out that the logic of Frege＇s Begriffsschrift is essentially full second-order logic according Boolos＇s argument, and then discusses Frege＇s definition of ancestral relation with hereditary property. Based on this definition, we discuss his definition of finite number and infinite number. At last, we discuss the way Frege cope with the principle of mathematical induction, point out mathematical induction principle can be inferred from these definitions with simple logic.