logicism and Frege

[Fernando Ferreira]

In modern language, the claim that Frege’s theory is a *logical* system is based on two assumptions: that second-order impredicative logic is logic, and that taking unrestricted extensions is logically acceptable. In the Foundations of Arithmetic, Frege wrote

“even the mathematician cannot create things at will, any more than the geographer can; he too can only discover what is there and give it a name.”

and he has an interesting discussion on extensions of concepts in the second volume of Grundgesetze

“It has thus become plausible that creating proper is not available to the mathematician (…) Against this, it could be pointed out that in the first volume we ourselves created new objects, namely [extensions]. What did we in fact do there? Or to begin with: what did we not do? We did not list properties and then say: we create a thing that has these properties.”

Frege goes on saying that we have the right to convert a general equality (co-extensionality of concepts) into an identity between extensions (of the said concepts). Frege claims that this must be regarded as a law of logic. It is not, as Russell showed with a bang.

The blame for the inconsistency is usually attributed to the unrestricted formation of extensions. Properly speaking, there is no single culprit. In the early nineties, Michael Dummett suggested that if one restricts Frege’s logic from impredicative to predicative quantifications, then the system (with unrestricted formation of extensions) is consistent. Riki Heck proved that this is the case in 1996. Even though some nontrivial mathematics can be formalized in this predicative system, the system is well below primitive recursive arithmetic.

I recently wrote a paper “Zigzag and Fregean arithmetic” (in The Philosophers and Mathematics, Springer, 2018) in which I consider a “Fregean” impredicative system in which one can only take extensions of predicative concepts. This system has a very good theory of finite sets. One can prove in this theory the axiom of finite reducibility: if a concept is true only of finitely many elements than it is co-extensional with a predicative concept (and, therefore, it has an extension). Based on this, it is shown that one has an operative Fregean arithmetic, sufficient to develop second-order arithmetic. This system can be considered a (consistent) restriction of Frege’s system: the second-order logic is impredicative (as in Frege) but only extensions of predicative concepts are allowed (unlike Frege).

Fernando Ferreira


Fernando Ferreira
Departamento de Matemática
Faculdade de Ciências
Universidade de Lisboa
Campo Grande, Edifício C6, Gabinete 6.2.13
P-1749-016 Lisboa
Portugal

http://webpages.fc.ul.pt/~fjferreira/