# [FOM] Cantor's argument

Dean Buckner Dean.Buckner at btopenworld.com
Sat Feb 1 06:16:35 EST 2003

```Alasdair Urquhart wrote

> Dean Buckner presents a "non-mathematical"
> application of Cantor's diagonal argument.
> He seems to think it shows that there is
> a problem with the diagonal method.

No, I do not think it shows that there is a problem with the diagonal
method.  The statement

(E m)    f(m) = {x: x not in f(x)}

is false, and Cantor's argument shows that it is false, and to that extent
there is no problem with his argument.

My point is that the statement could be false for two reasons.  (a) Because
there is some object designated by "{x: x not in f(x)}" but this object is
not within the range of f.   (b) Because there isn't such an object .  Don't
we all agree that

(E m)    f(m) = Pegasus

is also false?  And that it is not false because some object named by
Pegasus is not within the range of any f?  That it is false because there is
no such thing as Pegasus?

Of course it follows from the assumptions of set theory that there is
something designated by "{x: x not in f(x)}".  Obviously.  My point was, it
we translate Cantor's argument into an argument about the semantics of
natural language, and thus outside the narrow confines of set theory, then
the argument looks decidedly dubious.  It is dubious because of the
assumption that the expression "is bald" in

Socrates is bald

names an object.  I drew a parallel with Russell's paradox because (it could
be argued) the paradox arises out of similar assumptions.

Alasdair writes

> It is unproductive to discuss such arguments
> in the early 21st century as if nothing had been done
> in foundations of mathematics in the last 100 years.

Plenty has been done in the period when two ideas represented in Frege's
philosophy diverged into (a) the subject of mathematical logic, which is
amply represented in this discussion group (b) the philosophy of language,
which is unfortunately not so well represented.

It is equally unproductive discuss such arguments in the early 21st century
as if nothing had been done in the philosophy of language in the last 100
years.  For an excellent discussion of the "Socrates is bald" fallacy, see
for example the Crispin Wright paper below.  See also two excellent papers
on Hartley Slater's U.W.A. website ("The Uniform Solution" and "Concept and
Object in Frege"), which discuss the same fallacy in detail.

http://www.arts.uwa.edu.au/PhilosWWW/Staff

"Why Frege Does Not Deserve His Grain of Salt: a Note on the Paradox of "The
Concept Horse" and the Ascription of Bedeutungen to Predicates", in Grazer
Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett ,
eds J.Brandl and P.Sullivan (Vienna: Rodopi) pp. 239-63

```