逻辑性和弗雷格算术与简单类型论的逻辑主义  

Logicality and the Logicism of Frege Arithmetic and Simple Type Theory

作　　者：石伟军 Weijun Shi(Department of Philosophy,Renmin University of China;Department of Philosophy,Humboldt-Universität zu Berlin)

机构地区：[1]中国人民大学哲学系 [2]柏林洪堡大学哲学系

出　　处：《逻辑学研究》2020年第3期62-81,共20页Studies in Logic

摘　　要：弗雷格和罗素的逻辑主义由两个部分构成:可证明性论题和可定义性论题。可以很确信地说,可证明性论题并不能得到完全的辩护。但是,为了向两者,特别是可定义性论题,提供辩护或者拒斥之,我们需要逻辑性的标准来决定,除了其它常元的逻辑性之外,表示数的常元和表示属于关系的常元的逻辑性。我将采用的逻辑性标准是塔尔斯基和谢尔提出的同构不变量标准和费弗曼提出的同态不变量标准。塔尔斯基和谢尔在不同的地方已经指出罗素的表示属于关系的常元是同构不变量。在本文中,我将证明如下结论:第一,表示属于关系的常元是同态不变量;第二,弗雷格的表示数的常元既不是同构不变量也不是同态不变量;第三,如果逻辑性是同构不变量或者同态不变量,弗雷格的逻辑主义(弗雷格算术)的可定义性论题不成立;第四,如果逻辑性是同构不变量,罗素的逻辑主义(简单类型论)的可定义论题成立,但若逻辑性是同态不变量,这个论题则不成立。The logicism of Frege and Russell consists of two-fold components: the provability thesis and the definability thesis. It is safe to say that the provability thesis cannot be completely upheld. However, to justify or dismiss them, in particular, the definability thesis, one needs a criterion for logicality to determine whether the constants for the concept "the number of" and the membership relation, among others, are logical. The criteria that I shall adopt are logicality as isomorphism-invariance on the part of Tarski and Sher and logicality as homomorphism invariance on the part of Feferman. Tarski and Sher have pointed out that Russell’s constant for the membership relation is isomorphism-invariance on different occasions. I shall demonstrate the following conclusions in the article: First, this constant is also homomorphism invariance;Second, the constant for the concept "the number of" is neither isomorphism invariance nor homomorphism-invariance;Third, if logicality is isomorphism invariance or homomorphism invariance, then the definability thesis of Frege ’s logicism(here Frege arithmetic) does not get justified;Forth, if logicality is isomorphism invariance, the thesis of Russell’s logicism(here simple type theory) is fully justified, whereas it is not so provided that logicality is homomorphism invariance.

关 键 词：逻辑主义 弗雷格 常元 同态 可定义性 塔尔斯基 同构 谢尔 

分 类 号：B81-05[哲学宗教—逻辑学]