[FOM] Tangential to Slater and Numbers
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Wed Oct 8 15:38:39 EDT 2003
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
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