[FOM] Understanding universal quantification

[Peter Apostoli]

GR:

It seems thus that  universal
quantification would  not commit one to the assertion of the ontological
existence of any totality (omega, power set).

PA:
Conceded for the sake of argument.

GR:

It is sometimes claimed that universal quantification  is meaningful
even without reference to the domain of quantification of the variable,
the intended meaning of the quantification being conveyed by one's
understanding of the introduction and elimination rules of a (natural
deduction, sequents) logical system.

PA:

Well, no quantification without identity (Frege,Quine) and no identity
without indiscernibility (Leibniz, Cantor, Frege), so quantification at
least presupposes a prior quantization of the domain of discourse (as the
old saying goes, why do you think they call it quantification . . .).
However this nuance escapes formal notice (but not Geach's) so we (i.e.
proof theorists) live in a world indiscernible from a world in which the
claim you report is true.

Quantification is fundamentally a semantic notion. All Kaplan aside, accept
no syntactic surrogates my friend, for all theory is gray, but the tree of
life is green.

I hope that clears things up :-)