# [FOM] Re: Unschooled grammatical intuition v. theoretical identities

Neil Tennant neilt at mercutio.cohums.ohio-state.edu
Thu Oct 16 22:33:59 EDT 2003

On Fri, 17 Oct 2003, Hartley Slater wrote:

> Now Neil Tennant is getting enraged (FOM Digest Vol 10 Issue 23):
>
> >Let's get clear ... Someone who cannot write down a correct formula of
> >first-order logic to capture "there are exactly two Fs" is now telling us
> >that his grammatical intuitions should prevail over the enormously
> >fruitful theoretical identification of the natural numbers with the finite
> >von Neumann ordinals ?
>
> I am not unschooled; I was well taught; my teacher was Steen.  So
> have another try, Neil, at interpreting the formula I gave for 'There
> are exactly two Fs'.  (Hint: suppose that numbering variables has
> implications).

"Enraged" is a mistaken exaggeration; let us just say "drolly amused"
instead. In your posting of Oct 7 2003 you wrote

> If one expresses 'there are exactly two Fs' as
>          (Ex1)(Ex2)(y)(Fy <-> y=x1 v y=x2) ...

and I subsequently pointed out that in standard(ly interpreted)
first-order notation, the claim that there are exactly two Fs would not be
expressed by the formula you gave, regardless of the happenstance of
numerical subscripts. You forgot the needed condition "~x1=x2". It is a
basic principle governing the use of bound variables that it does not
matter at all what symbols are used as bound variables, so long as their
pattern of occurrences establish the intended linkages between quantifiers
and argument places. Thus you could use coins, if you wished, stuck to the
page, and your formula could be written

(E dime)(E nickel)(penny)(F penny  <->  penny=dime v penny=nickel).

And it would still fail to capture the truth-conditions of "there are
exactly two Fs", for want of the condition "~dime=nickel", which even
logicians living in the land of George W. Bush would need to be told, when
their legal tender is being used as bound variables.  It is not for me to
"suppose that numbering variables has implications". That hint is so broad
that one could drive a covered wagon through it.

The most charitable construal I can come up with is that you intend that
your numerically subscripted variables should affect the interpretations
of the quantifiers binding them in such a way that once (E x_1) has been
instantiated, the individual assigned to x_1 is not eligible to be
assigned to x_2 when (E x_2) is instantiated. This deviant interpretation
had its brief day in a paper by Hintikka long ago in the JSL.

This is one of the mistaken interpretations of bound variables in standard
first-order logic that beginning students have to be *educated out of*. No
one uses it any more. Indeed, in the history of symbolic logic, it may
well be that only Hintikka ever used it (and then only briefly). it was
quickly abandoned, and for good reasons.

The standard interpretation is that each quantifier-occurrence *ranges
over the whole domain*. This is so fundamental a constraint that your
"hint" that I should "suppose" that numbering variables "has implications"
is little more than hand-waving evasion at being corrected on an obvious
and elementary point concerning standard logical notation. And I don't
think it is relevant who taught you your logic; they ought to be
embarrassed either by your inability to capture the truth-condition "There
are exactly two Fs", or by your cavalier abuse of standard notation.

> Faced with the fact that in, for example, '([n+1]x)Fx iff
> (Ex)(Fx.(ny)(Fy,-(y=x))', the 'n' is clearly a variable, people seem
> to have turned to some trivial pursuit, to avoid the anxiety which
> realisation of the dimension of the real issue would bring.

You are simply misunderstanding the role played by a metalinguistic
numerical variable in an inductive definition of a sequence of quantified
sentences in the object-language having the intended truth-conditions

There is at least one F
There are at least two Fs
There are at least three Fs
:
and so on

Your claim that "the 'n' is clearly a variable" is a comment on a schema
of definition provided in the metalanguage. The 'n' does not occur in
the defined object-language sentences themselves.

> And they are not just my grammatical intuitions, if they are
> 'intuitions' at all; they are also Neil's, see
> http://www.cs.nyu.edu/pipermail/fom/2003-June/006806.html :
>
> >In the singulary case, I proposed (in my 1987 book Anti-Realism and
> >Logic) an adequacy condition on a theory of number, using what I called
> >Schema N:
> >
> >Schema N        #xF(x) = _n_ iff there are exactly n Fs

Thank you for quoting Schema N to me; but I'm afraid I don't agree that it
shows that I share your grammatical intuitions at all. That would be
immediately apparent if you read the careful explanation in the book of
how to identify the sentences that are to go on the right-hand side of
Schema N.

Neil Tennant

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