[FOM] Understanding universal quantification

Peter Apostoli apostoli at cs.toronto.edu
Sun Feb 23 06:03:33 EST 2003

Dear Steve,

> Actually, the way quantification got *started* was as a purely
> operation: The prefix generalization of the hitherto dyadic infix
> of sum and product, disjunction and conjunction, to arbitrarily many
> was introduced by William Stanley Jevons in his "Principles of Science"
> which also redefined "logical sum" from the  'exclusive-or' of Boole to
> 'inclusive-or' and its attendant dual-symmetries, which survive to this
> day. The
> term "quantifier", and the use of (capital) sigma and pi for existential
> universal quantifiers was initiated in the works of Charles Saunders
> The *semantic* and ontological analyses of quantification appear to be of
> recent occurrence.

Well, I wasn't speaking on behalf of the history of the concept so much as
on behalf of the truth (my favorite semantic notion, at least when there's
any to be had). But if you like there is another route to the same
conclusion through Cantors notorious definition of a set as a collection of
distinguishable elements that can be thought of together as a totality.
Ignoring the charge of psychologism (a bum rap, anyways), one may discern in
this characterization two components: (a) that the elements be
distinguishable (=discernable)  (b) that collectively they form an object or
totality. Part of Frege's genius was to have recognized the Leibnizian
indiscernibility lurking in (a).

Is the domain of your quantification a set in this sense? It seems plausible
to reject component (b) of Cantor's definition when answering: the domain of
quantification is a collection of objects but need not itself form an object
falling within the range of the quantifiers*. So perhaps, we say it is a
concept, not an object. But not component (a). Without a principle of
discernment - what the vulgar call a critieria of identity - the Fregean
concept lacks the property of quantity characterisitic of count nouns.
"Without indiscernibility, quantification goes by the board" [Bobby
Quine]**. However, I concede that our intuition of this elusive property of
quantity has grammatical roots, so I defer to your syntactic intuitions
about quantification. But Geach's point is that, due to the formal
incompletensss of logical syntax, identity theory will always be incomplete
as a formal theory. The good news is that identity is not a formal theory,
or at least it shouldn't be. But here again I speak on behalf of the truth
and not on behalf of what passes as common sense in our day.

*Indeed on what passes as common sense in our day it will not.

*My favorite American Philosopher is a character from the short story Burnng
Chrome who is briefly mentioned in the novela Neuromancer.

Peter A.

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