[FOM] Tangential to Slater and Numbers

Dean Buckner Dean.Buckner at btopenworld.com
Thu Oct 9 16:49:26 EDT 2003

Both Smith and Tennant appear to have exposed a problem with Slater's
original argument, that the number of variables in a FOL formula expressing
number, is somehow relevant to the number expressed.  For

    ExEyEz(x <> y & y <> z & x = z)

says there are TWO things, using THREE variables.

Does this matter?  Define a predicate N, such N(x,y) iff x <> y, N(x,y,z)
iff N(x,y) & N(x,z)& N(x,z) "and so".  Then, clearly, the number of
variables DOES indicate the number of things that are said to satisfy the
predicate "N".  It is false, for example, that

             N(Cicero, Tully)

Hazen writes:

"An idiot savant who had learned the notation of FOL-with-I without having
learned to count (and without having learned the number words) could
demonstrate an understanding of


by finding three F things (maybe F means 'is a toy in the box') and
concluding "There's got to be another one."  It is important that this
demonstration involves three successive Universal Quantifier Elimination
inferences, but the ability to make three inferences doesn't presuppose that
one has the concept of "THREE."  (The common use of numerical subscripts on
variables in writing FOL is a red herring, as any other way of
distinguishing variables would serve as well.)"

 I agree with the point about "common use of numerical subscripts".  But
think how an idiot savant would learn the use of "N" as defined above?
Suppose I successfully explain the use of N, with two variables and three
variables.  But then it turns out the person has not grasped that N(a,b,c,d)
is also true, even though he has grasped that a,b,c,d are all different.  I
explain patiently that "N" is also satisfied in this case, then he still
fails to grasp that N(a,b,c,d,e), even though he understands that a,b,c,d,e
are all different things.  How do I explain the general applicability of
"N", where any number of variables are concerned.  It depends on whether the
person can grasp that, for N to be satisfied, it must be satisfied *by
different things*.  One who can, understands the concept of number.

Hazen writes:
"As a matter of (logically contingent) fact, most of us learn how to count
before we learn the formalism of FOL-with-I, but that psychological fact
does not show that the FOL-with-I way of saying that there are at least n
things such that... PRESUPPOSES an understanding of numbers and counting, in
any logically interesting sense of 'presuppose.'  "

I think it DOES presuppose an understanding of numbers and counting.
Suppose the person points to a single toy soldier and says "that's the first
soldier".  Then he points to the same soldier and says "and that is the
second".  You point out the mistake.  "The second soldier has to be a
different one from the first".  So he points to the first soldier "that's
the first soldier", points to a second and says "and that is the second",
then points to the second saying "and that is the third".  No, you say, "the
third soldier has to be different from the first two".  And so it goes on.
Clearly the person has failed to understand the concept of counting - that
any object counted has to be different from any of the objects already
counted - in the same way and for the same reason that he failed initially
to understand "N".

In summary, grasp of FOL presupposes unthinking reliance of sameness or
difference of reference, which underpins our whole system of thought and
reasoning. To retiterate a point made by Sainsbury (departing from Frege p.

a = b
:. Fb

In accepting the validity of this argument, we assume that expressions that
are the same make the same contribution to validity.  The same conceptions
(but of numerical difference this time) are presupposed in counting.

Dean Buckner

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