[FOM] Tangential to Holmes/Slater exchange

Hartley Slater slaterbh at cyllene.uwa.edu.au
Thu Oct 2 23:01:17 EDT 2003

Allen Hazen has tried to be 'tangential' to the Holmes/Slater debate, 
but despite himself has hit the bullseye!  He says (FOM Digest Vol 10 
Issue 3):

>         (A) Numerically definite quantifiers.  For any fixed n, FOL with
>Identity can express "There are n things such that ...".  (Aside: the most
>efficient way of doing this is apparently due to David Lewis.  To say
>"There are at least n x such that Fx," start with (n-1) UNIVERSAL
>         AyAz...Aw
>then put in one EXISTENTIAL
>                  Ex
>then a conjunction of (n-1) negated identities saying that x is not one of
>                    (~x=y&...~x=w
>and finally say that x is an F thing:
>                                  &Fx).
>The length-increase of these formulas is linear as n increases, which is a
>lot more user-friendly than other methods.)

Here, although Hazen tries to think he is dealing with 'any fixed n', 
he has produced a formula in which 'n' is a variable, and which 
therefore indicates a place which can be quantified over.  Yes, 
indeed, *for any n* start with ...n-1 universal quantifiers then 
...then a conjunction of n-1...'  If he numbered his individual 
variables, so that y,z,...w were x1, x2,...x[n-1], the point would be 
even plainer.

In a representation of 'x has n elements' as I said before (FOM 
Digest Vol 10 Issue 2), one cannot get rid of the 'n'.  Even if, for 
any fixed 'n', the numeral 'n' does not occur in the formula (because 
numbered variables are not used), still the number it denotes must be 
previously known, to fix the number of various elements, and so 'the 
set of those sets with n elements' cannot define the number n, since 
one must know what that number is beforehand.

So (a) Holmes' 'Frege numerals' no more define numbers than von 
Neumann ordinals like {{}, {{}}}, and (b) quantification over the 
numeral place in expressions like (nx)Fx is entirely possible.  Q.E.D.

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

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