# [FOM] Tangential to Slater and Numbers

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Mon Oct 6 03:44:28 EDT 2003

```To clarify a few points about what I MEANT to say in my earlier post
(and say a bit about why I thought it was of philosophical
importance)...

(1) I meant to say that, for any given natural number, the
"numerically definite quantification" saying that there are that
number of x such that Fx is formulable in plain, unaugmented,
First-Order Logic with Identity (FOL with I). The numerically
definite quantification when so formulated does NOT contain a numeral
or other term referring to the number in question. In making this
claim, I used the numerical variable, n: this was part of my
(informal) metalanguage, NOT part of the FOL-with-I construction I
was describing.

(2) Slater has since pointed out that the FOL-with-I way of saying
there are n F-things contains n quantified individual variables. This
is true, and I suspect vitally important at some deep level of
philosophical semantics. At a more superficial level, however, I
would insist that this syntactic fact has no direct semantic
significance. If I wrote out the formula, the token I produced would
contain ink, but would not thereby contain a term REFERRING TO ink.
Similarly, I claim, it would contain n variables, but no expression
REFERRING TO the number n.  (In other words, I deny, what Slater has
asserted in his latest (6.x.03) posting, that "with numbered
variables there is a reference to numbers in the standard
re-expression of numerical quantifiers.")

(3) 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.'  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
AxAyAzEw(-w=x&-w=y&-w=z&Fw)
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.)

(4) Slater's 6.x.03 posting says
>  (a) Hazen himself proposed a logic in which the numerical place
>  in '(nx)Fx'  can be quantified over,
which I did: this was proposed as a possibly interesting EXTENSION of
FOL-with-I.  It is perhaps easiest to think of extending FOL-with-I
in two steps.  FIRST STEP: introduce the numerical quantifiers,
(2x)Fx, (3x)Fx and so on, as abbreviations for the purely FOL-with-I
statements that there are at least 2 (at least 3, etc) x such that
Fx.  This simply extends the language by (infinitely many)
definitions, and so does NOT increase its  expressive power (in
principle: in practice it might be useful).  SECOND STEP: Treat the
numerals appearing in the defined quantifiers as TERMS, by allowing
quantifiable variables to be substituted for them.  I apologize if I
didn't make it clear enough that I was talking about two DIFFERENT
formalisms, one properly stronger in its expressive power than the
other.

-----

Allen Hazen
Philosophy Department
University of Melbourne
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