[FOM] Re: 176:Count Arithmetic

Neil Tennant neilt at mercutio.cohums.ohio-state.edu
Thu Jun 12 15:58:35 EDT 2003

If I understand Harvey's proposal correctly, the following would be
well-formed, and have the interpretations indicated:

#(F(x);0)	The number of Fs = 0
#(F(x);s0)	The number of Fs = s0
#(F(x);_n_)	The number of Fs = _n_

Call these the singulary cases, since there is a single variable x being
bound in the formula F before the semi-colon.

Likewise, we would have the binary cases:

#(G(x1,x2);_n_)	The number of ordered pairs <x1,x2> such that G = _n_ ;

and then the ternary cases:

#(H(x1,x2,x3),_n_) 	The number of ordered triples <x1,x2,x3> 
			such that H = _n_ ;

and so on.

While Harvey is of course correct in saying that we can use a single
symbol # (called "sharp") for any such k-ary binding of variables
x1,...,xk in the formula before the semi-colon, it is nevertheless the
case that the *conceptual* operation, for each k, is distinct. The Fregean
syntactic category of # is different in the singulary case, for example,
from what it is in the binary case. In the singulary case, sharp takes one
one-place predicate [F(x) above] and one expression of category Name [_n_
above] to form an expression of category Sentence. Since a one-place
predicate is an expression that takes one Name to form a Sentence (i.e.,
an expression of category 1SN) we can categorize the singulary sharp as


The binary sharp would be categorized as


i.e. as an expression that takes a two-place predicate and a name to form
a sentence.

Hence my suggestion that what is involved in Harvey's axiomatization of
Count Arithmetic is in fact a countable infinity of "sharps", each with
its own distinctive syntactic category. This conceptual point can easily
be lost sight of if one uses just the same symbol # in every case, and a
generalized formation rule.

Harvey asked for information about connections with current neo-Fregean
analyses of arithmetic. I have one suggestion (or observation) to make.
Neo-Fregeans thus far have dealt only with the singulary sharp, and indeed
have used it as a term-forming operator rather than as a quantifier the
way Harvey does. Thus a neo-Fregean would write an explicit identity

	#xF(x) = _n_

where Harvey writes


(In the latter, one has to understand that free occurrences of x in F have
now become bound.)

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

The adequacy condition is that every instance of this Schema should be
derivable in one's neo-logicist theory of natural number. This condition
is analogous to Tarski's well-known adequacy condition on a theory of
truth, which requires that one's theory of truth allow one to derive every
instance of the T-Schema:

T-Schema	s is true iff p

(where p is a translation, into the metalanguage, of the object-linguistic
sentence s). Just as Tarski's theory is "disquotational" by virtue of
satisfying this adequacy condition, so too a logicist theory of natural
number satisfying the condition that I proposed would be "disnumerical".
For, on the right-hand side of Schema N, the condition "there are exactly
n Fs" can be spelled out explicitly (for any given choice of n) without
any reference to, of quantification over, natural numbers (unless, of
course, F itself were a numerical predicate, or had at least one
numerical argument). By way of illustration, one
instance of Schema N is

	#xF(x) = ss0 iff EyEz(-y=z & Fy & Fz & (w)(Fw -> (w=y v w=z)))

The point of all this in the case at hand is as follows. Given that
Harvey's treatment now has sharp "binding" any finite number of variables
simultaneously, the Adequacy Condition involving Schema N is going to have
to be generalized. Thus (sticking with our choice of 2, i.e. ss0, by way
of illustration) we shall need to be able to derive the likes of

	#(G(x1,x2);ss0) iff there are exactly two ordered pairs <y,z> 
			    such that G(y,z),

where the right-hand side has to be spelled out without use of sharp, and
without any apparent reference to numbers, or quantification over numbers,
except in the case where G itself is a numerical predicate (or one that
involves at least one numerical argument).

Now, Harvey intended the domain of interpretation for Count Arithmetic to
be just the natural numbers themselves. This, unfortunately, involves a
loss of generality crucial to the neo-Fregean project. That project
regards it as essential to show how numbers are *applied* in the course of
talk about even non-numerical subject matters. That is why one so often
resorts to examples involving apples in a basket, or salad forks and
plates. The neo-Fregean wants to be able to prove, for example,

	#x(x is an apple) = ss0 iff (Ey)(Ez)(-y=z & y is an apple
				    & z is an apple & (w)(w is an apple ->
				    (w=y v w=z)))

So let's imagine, for the time being, that Harvey would allow one to
generalize the intended range of application of his primitive predicates,
so as possibly to include non-numbers in their extensions. Let us call a
"one-way affair" an instance of one person loving another person. If the
loving is reciprocated, then it's a two-way thing, i.e. we have two
one-way affairs involving the same two people. We need, then, to be able
to derive instances such as

	#(x loves y;ss0) iff there are exactly two one-way affairs,
			     i.e. there are exactly two ordered pairs
				  <z,w> such that z loves w

The right-hand side would be satisfied in a situation where the following
is the only loving taking place:

	Bill loves Monica;
	Hillary loves Monica.

It would also be satisfied in a situation where the following is the only
loving taking place:

	Monica loves Bill;
	Monica loves Hillary.

Likewise in this situation: 

	Bill loves Bill;
	Monica loves Monica.

And so on.

So, how do we capture this with an appropriately quantified sentence on
the right-hand side of the above instance of Schema N ? We have to say
something like

	(Ex1)(Ex2)(Ey1)(Ey2)(x1 loves y1 & x2 loves y2 & (-x1=x2 v -y1=y2)
	& (x)(y)(x loves y -> [(x=x1 v x=x2) & (y=y1 v y=y2)]))

It's a nice teaser to try to give an inductive definition of the
quantified formulae that would express "there are exactly n k-tuples
<x1,...,xk> such that F(x1,...,xk)". Assuming one can do this, one would
then have the general instance of a generalized Schema N for the k-ary

	#(F(x1,...,xk);_n_) iff there are exactly n k-tuples 
				<x1,...,xk> such that F(x1,...,xk)

Count Arithmetic would be all the more welcome if it could be developed so
as to allow predicates F that have non-numerical arguments, for which
adequacy via generalized Schema N could be established. 

So my question to Harvey is: can one do this?

Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science


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