Hartley Slater slaterbh at cyllene.uwa.edu.au
Mon Sep 29 21:29:54 EDT 2003

```Randall Holmes thinks I have not made my point (FOM Digest Vol 6 Issue 30):

>Slater asserts that he has "dealt with my point", but I note for the
>record that this is not the case.   He has asserted his views about the
>proper interpretation of the grammar of sentences about numbers and
>sets -- but not in such a way as to compel my agreement, which is not
>forthcoming.
>
>He says:
>
>"But I also think there is no choice about the matter: the place for
>'n' in '(nx)(x is in Y)' has a quite different syntax from the place
>for 'Y', and any formalisation which does not respect this difference
>is not going to mix up numbers and sets, since, for instance,
>iota-n(nx)(x isin {{},{{}}}) is identical to the number 2, but is not
>identical to {{}, {{}}})"
>
>It is quite true that if you set up your syntax correctly, you will
>not mix up numbers and sets (you will say about numbers only those
>things which are appropriate to numbers and about sets only those
>things which are appropriate to sets); that's what I meant by saying
>that numbers are an abstract data type.  This does not mean that
>numbers are not sets; it means that you have avoided the question as
>to whether they are or not (of course, it may be a very good idea to
>do so!).

There is a typo in my earlier remarks which does not seem to have
troubled Holmes ('is not going' should be 'is going'), but let us
take the whole matter more slowly.  Using standard predicate logic we
can construct a string of quantificational expressions which are
normally abbreviated as involving 'numerical quantifiers'.  Thus we
can formulate, for instance,
(Ex)(Ey)(-(x=y).(z)(Fz iff z=x v z=y)),
and shorten this to
(2x)Fx,
which is normally read 'There are exactly 2 Fs'.  The grammar of such
remarks requires the place of '2' to be filled by numerals, or
numerical variables, and quantifying over that place then enables us
to say, for instance,
(En)((nx)Fx.(m)((mx)Fx -> m=n).n=2),
which is equivalent to
iota-n(nx)Fx = 2,
and so we get, specifically
iota-n(nx)(x isin {{},{{}}})=2.
That would allow one to say, somewhat circuitously, if there are
exactly two Fs,
(iota-n(nx)(x isin {{},{{}}})y)Fy,
but if Holmes wants to put set abstracts in the place of numerals,
and such iota term descriptions, he will have to interpret instead,
for instance,
({{},{{}}}x)Fx.
A canonical set with two members, however, such as {{},{{}}}, merely
might have the same number of members as the set of Fs, {x|Fx}, and
so is not the number of members in either.  The number of even
numbers, for instance, is not {n| n>0}.

>The type-theoretic view described by Hazen is definitely also
>coherent.  From this viewpoint, remarks like Slater's could be made.
>I do not ascribe a view to Slater, because he has not clearly
>explained what his view is.

There were a good number of postings from me on FOM earlier this year
(in April and May), on related matters, and I referred again to
Mayo's JSL Dec 2002 paper recently.

>I can't imagine what Slater means by the following:
>
>"But premise 2 is false, since, following on from the above, the
>number two is not the set of things with two members, and so it is
>not the extension of the property of having two members - it is the
>extension of the property of being 2!"

There was another typo here (which I communicated to Holmes, with the
first): in place of 'it is the extension' there should be 'it is in
the extension'.  Both typos arose, I confess, from a quite
inexcusable lapse of concentration while I watched the murderous,
local state croquet championships.  Keep off the croquet!
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html

```