[FOM] Tangential to Slater and Numbers

[Neil Tennant]

On 8 Oct 2003, Peter Smith wrote:

> On Oct 7 2003, Hartley Slater wrote:
> 
> > If one expresses 'there are exactly two Fs' as
> >          (Ex1)(Ex2)(y)(Fy <-> y=x1 v y=x2)
> > then the counting of the variables is explicit, and so the number is 
> > referred to in the expression.
> 
> That claim seems crucial to him. But if I express the symmetry of identity 
> (for example) by
>              (Ax1)(Ax2)(x1 = x2 <-> x2 = x1) where the counting of the 
> variables is explicit, have I in fact referred to the number two? If I 
> write our old friend
>              (Ex1)((Ax2)(KFx2 <-> x1 = x2) & Bx1) have I failed to refer to 
> a bald king of france but managed to refer to the number two instead?? That 
> seems an extraordinary claim to me. But if there is not numerical reference 
> in these claims, why is there in Hartley's?

... to which one might add that Slater's formal sentence 
	(Ex1)(Ex2)(y)(Fy <-> y=x1 v y=x2)
is true in any world with exactly one F. (Choose that F as the value of x1
and as the value of x2. That assignment of values satisfies the open
formula (y)(Fy <-> y=x1 v y=x2).)

Upon the obvious emendation called for--namely,
	(Ex1)(Ex2)(~x1=x2 & (y)(Fy <-> (y=x1 v y=x2)))
---one can then expand Peter's criticism by asking "What about
re-lettering of bound variables?" Where is the explicit `counting of the
variables' in the logically equivalent formal sentence
	(Ex)(Ey)(-x=y & (z)(Fz <-> (z=x v z=y))) ?

Neil Tennant