[FOM] Tangential to Slater and Numbers

[A.P. Hazen]

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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