Logicality and the Logicism of Frege Arithmetic and Simple Type Theory

Studies in Logic, Vol. 13, No. 3 (2020): 62–81            PII: 1674-3202(2020)-03-0062-20

Weijun Shi

Abstract. 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.