内涵的《算术基本规律》

Intensional Grundgesetze

罗素在《算术基本规律》中所发现的悖论是由二阶逻辑的概括公理和公理V造成的,其根源在于,弗雷格持有两个相互冲突的哲学信条:"数的给出包含着概念的断定"和"数是独立自主的对象"。为了在"对象"和"概念"之间建立一一对应,可以把谓词的涵义看作"概念",把专名的涵义视为"对象",谓词的涵义的外延变成专名的涵义,由此可在谓词的涵义和专名的涵义之间建立一一对应。据此思路,设计了一个模态的形式系统IG(Intensional Grundgesetze),由内涵公理V和内涵概括公理构成。最后证明:IG是一致的,并且IG可以解释Robinson算术。

The conflict between Axiom V and Comprehension Axiom in second-order logic leads to a paradox that Russell found in Frege's Grundgesetze.This paper attempts to argue that Russell's paradox results from Frege's two contradictory philosophical doctrines:(1) die Zahlengabe eine Aussage von einem Begriffe enthalte;(2) die Zahlen sind selbstandige Gegenstande.Moreover,it will propose a new interpretation of Frege's distinction between reference and sense,and provide an intensional way to save Frege from the contradiction.At last,this paper will establish a modal system IG,which consists of Intensional Axiom V and Intensional Comprehension Axiom,and show the consistency of IG.