[FOM] Leibniz's Law (Was: Goedel numbers, use, and mention)

Richard Heck heck at fas.harvard.edu
Thu Jun 5 23:51:41 EDT 2003

On Thu, 2003-06-05 at 14:30, Dean Buckner wrote:
> The point is,
> (A) It is improbable that Bacon wrote Macbeth
> (B) It is probable that Shakespeare wrote Macbeth
> does not imply
> (C) Shakespeare <> Bacon
> whereas by Leibniz it should.  

No, Leibniz's Law does NOT imply that it should, not unless you assume
that "It is probable that" is an extensional (transparent) context. The
example you mention gives us reason to suppose, however, that it is not,
at least in this use. That's because, in this use, "probable" expresses
an epistemic (sometimes called "subjective") conception of probability.
If you're operating with an "objective" conception, then Leibniz's Law
will apply as usual.

It's worth being careful here about how one formulates Leibniz's Law.
(The following is essentially stolen from a paper of Dick Cartwright's.)
Leibniz's Law is not schematic. It does not say that, for every
sentential context ...x..., if "...a..." is true and "a=b" is true, then
"...b..." is also true. Rather, at least as Leibniz himself formulated
it, and as Frege and Russell did, too, more or less, Leibniz's Law says
that, for every property of objects P, if a has P and a=b, then b has P.
The schematic version would then follow if we knew that every sentential
context ...x... defined a property of objects. But familiar examples
give us reason to doubt that every context does define such a property.

Richard Heck

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