[FOM] Godel numbers, use, and mention

Dean Buckner Dean.Buckner at btopenworld.com
Fri Jun 6 14:43:57 EDT 2003

Hartley Slater argues

>One needs to have uniqueness before an identification can be made,
> 1.  One needs K(Ex)(y)(My <-> y=x) with Ma.
>The former entails
>(Ex)(y)(My <-> y=x),

Only one person ("that person") committed the murders.  Agree

> and so (y)(My <-> y=ex(y)(My <-> y=x))

So any one who committed the murders = that person.  Agree.

> Ma <-> a=ex(y)(My <-> y=x), giving a=ex(y)(My <-> y=x),

So if A committed the murders, A is indeed that person, yes yes.

> and since KMex(y)(My <-> y=x)
> we get KMa.

So, since we know that person committed the murders, it follows that A
committed the murders.  Why?

Certainly it follows by Leibniz.  So, following this reasoning, since we
know that only one person ("Jack") committed the Whitechapel murders, if the
Duke of Clarence was Jack, we know that Duke of Clarence committed the
Whitechapel murders.  But we don't know that Duke of Clarence committed the
Whitechapel murders (do we?) so that rules out one suspect.

Hartley adds:

>because K(y)(My <-> y=ex(y)(My <-> y=x)),
> and so K[Mex(y)(My <-> y=x) <-> ex(y)(My <-> y=x)=ex(y)(My <-> y=x)],
> and automatically K[ex(y)(My <-> y=x)=ex(y)(My <-> y=x)]},


(a) we know that if any y was the murderer, y = that person.  Yes
(b) therefore we know ( that person was the murderer iff that person = that
(c) and therefore we know (that person = that person).

Fantastic, agree up to here.  But we still need to infer from (c) that we
know that A committed the murder, i.e. that we know that A = that person.
Which of course we can, so long as we have Leibniz' principle.  But then if
Leibniz' principle is unrestrictedly true, from

(A) it is not widely believed that Bacon wrote Macbeth
(B) it is widely believed that Shakespeare wrote Macbeth

we can infer (by Leibniz)

(C) Shakespeare <> Bacon (<> = does not equal)

which was the problem we started with, since clearly both (A) and (B) are
compatible with Shakespeare = Bacon.  As Woody Allen really did say,
"Reality is what refuses to go away when I stop believing in it."

I just got an offline email from a minor celebrity  saying we must be
careful from unrestricted application of Leibniz.  Clearly.  He argues
(following Cartwright) that not every sentential context "...x..." defines a
property of objects.  Yeah but why not?  So Leibniz' law fails in just those
cases where it fails?  Some law.  And what is a property of an object A?
Surely what I assert of A when I assert that "...x..." applies to A.  Or if
in certain cases not, why not?

Dean Buckner

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