FOM: 2nd-order logic

[Raatikainen Panu A K]

I agree with Marcin that it must have been Branching quantifiers 
that Boolos meant (and I can suggest the references he gave).

> George reported to me that Quine regarded this result as 
> answering his principal objections to second-order logic.

This seems to me to be an overstatement, for Quine considered 
such an extension of the standard first-order logic explicitly in his 
writings (even twice),  and found it problematic. His objection is 
that although one does not explicitly quantify over sets and 
functions, (just as with SOL) one fails to have a complete proof 
procedure for validity and for inconsistency. (fails extremely 
strongly, I would add.)

Quine: Existence and quantification, in Ontological Relativity, p. 
108-113.
Quine: Philosophy of Logic, p. 89-91

By the way,  the proof of  equivalence with a SOL sentence 
requires the Axiom of Choice, so it may be problematic to say that 
they are "logically equivalent".

All the best

Panu Raatikainen
Dept. of Philosophy
University of Helsinki