[FOM] Understanding Universal Quantification
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sun Feb 23 01:38:35 EST 2003
Since Russell discussed the theory of types in an appendix to his
"Principles of Mathematics," we can say that discussion, in the philosophy
of mathematics and the philosophy of logic, of this topic has been going
for a century!
It seems to me that there are at least two divisions (so four possible
positions?) that we have to worry about.
One is between constructive and non-constructive "understandings."
Since the same universal quantifier rules can be used in both classical and
intuitionistic systems of natural deduction, my gut feeling is that the
proof-theoretic approach, though CLEARLY on the track of SOMETHING, doesn't
give us the whole story: put briefly, the rules underconstrain the meaning.
(Dyed-in-the-wool constructivists will say that the classical
"Understanding" is a fraud. Those more drawn to philosophical realism will
perhaps-- some of them! -- look at the familiar interpretations of
intuitionistic logic in [assorted] MODAL logic[s] and say that the
constructive "understanding" is a substitute notion for use when *real*
universal quantification is inappropriate: I'm thinking, here, of the late
David Lewis, who commented that a modal interpretation of intuitionism made
it sound like a sort of "fictionalism," with the implicit modal operator
being understood along the lines of "according to the mathematical fiction,
it is the case that....")
Another-- I think the two are separable-- is over the question of
whether the "understanding" of universal quantification presupposes some
sort of delimitation of the domain quantified over. One side of this-- I
think it was the dominant view from "Principia Mathematica" (which was
based on a type theory) until sometime after the Second World War (when
Quine's writings on the philosophical side and the development of model
theory and set theory on the mathematical focused attention on First Order
Logic)-- draws its greatest support from the paradoxes: theories about ALL
classes or ALL concepts or ALL propositions have either seemed to be
artificially restricted (why SHOULDN'T proper classes have unit sets?) or
turned out inconsistent (Frege, Russell, Church, and Quine all, at one time
or another, proposed inconsistent theories). On the other side, Quine and
philosophers he influenced have tended to defend the "univocity of being":
there is a completely general notion of ENTITY, and we can quantify over
all of them.
The best recent defense of this universalist view I know is
Richard Cartwright, "Speaking of Everything," in "Noùs," v. 28
(1994), pp. 1-20.
Defenders of univocity can be quite funny about the type-theorist's
claims: "What's to KEEP me from quantifying over (absolutely) everything?
Is there an angel who zaps my brain when I try?" (Samples of the humor--
and first-rate philosophical discussion-- can be found in chapters 2 and 14
of George Boolos's "Logic, Logic, and Logic": I thought the line about the
angel zapping our brains to keep us from conceiving non-type-limited
generality was from Boolos, but I can't find it now.)
My own feeling is that both sides have a point. There's not much you
can say about absolutely every entity: to get mathematically interesting
statements, you have to restrict your universal quantification to (say)
classes. And someone attracted by the type-theoretic intuition could admit
that quantification over all "entities" is understandable, but say that the
restricting clauses, "...is a class" or "...is a meaningful condition," are
ambiguous, and have no MOST INCLUSIVE sense.
Sorry to go on for so long.
---
Allen Hazen
Philosophy Department
University of Melbourne
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