FOM: Geese and Cheerios
ccocos at julian.uwo.ca
Fri Mar 12 15:54:40 EST 1999
Charles Silver wrote:
> To me, the major point that separates (intuitive) quantification
> from set theory is that you can say "for all x..." without implying that
> the x's must be *in* something. For example, take: "All Canada geese fly
> south for the winter." I don't think anything in this statement implies
> that in addition to there being some number of geese there is also a *set*
> of geese.
Actually i think it does. This shouldn't frighten us though, since the two
'entities' ('goose' on the one hand and 'set of geese' on the other) have, to put
it in a rather dangerous way, different *types* of existence. Geese do not exist
in the same way *sets of* geese do. Your phrasing ("...in addition to...")
insinuates somehow perversly that we should try to identify sets of geese
following the same criteria we use to identify geese themselves (much like
Boolos' bowl of Cheerios:
"One might doubt, for example, that there is such a thing as the set of Cheerios
in the (other) bowl on the table. There are, of course, quite a lot of Cheerios
in that bowl, well over two hundred of them. But is there, in addition to the
Cheerios, also a set of them all? And what about the >1060 subsets of that set?
(And don't forget the sets of sets of Cheerios in the bowl.) [
] [I]t doesn't
follow just from the fact that there are some Cheerios in the bowl that, as some
who theorize about the semantics of plurals would have it, there is also a set of
them all." (To be is to be a value...))
If one takes this "...in addition to..." as implying that sets of Cheerios (and
sets of sets of Cheerios and so forth) should respond to the same individuation
criteria as individual Cheerios themselves, then Boolos' fear is obviously
justified. However I honestly don't see any *other* reason of concern aside from
this one (...which, of course, is preposterous). What else could justify Boolos'
Tim Heap wrote:
> Pat Hayes writes:
> Pat> In contrast to Silver's intuition, it seems to me that saying
> Pat> "for all x..." implies that one has some notion of the extent
> Pat> over which that "all" is supposed to reach. If I can't (even
> Pat> conceptually) distinguish a Canada goose from a hooting swan,
> Pat> then for me to say "All Canada geese..." is impossibly
> Pat> vague. (How can we evaluate a proposed counterxample if we can't
> Pat> know whether or not is a goose?) The meaning is indeterminate
> Pat> until we decide what counts as a Canada goose. But that amounts
> Pat> to deciding what is in the set of Canada geese. The point is that
> Pat> the criteria which determine the truthvalue of "for all x..", and
> Pat> those which make all the x's into a set, are both purely
> Pat> extensional; and they coincide; so they are the same.
> ...but what happens to this argument when you replace "Canada Goose"
> with "set"?
The short answer would be: "...well you DON'T (...replace "canada goose" with
"set")!" At least not in this context.
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