# [FOM] Godel numbers, use, and mention

Richard Heck heck at fas.harvard.edu
Sat Jun 7 02:28:30 EDT 2003

```> [Someone] 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?

There are two quite different questions here. (i) Do we have any reason
to suppose that Leibniz's Law is unrestrictedly valid? (ii) How are we
to distinguish the contexts in which it is valid from those in which it
is not? The answers to these questions are: (i) Not much, so far as I
can see; and (ii) Uhh...that's quite a hard question, one I really wish
that I could answer, as then I could write a really, really good paper.
But, while no-one, so far as I know, has a convincing answer to (ii), I
myself see no reason to despair. Plenty of interesting work has been and
is being done on this very question, both in logic and in semantics.

It is not an unreasonable proposal, for example, that Leibniz's Law
always fails to be logically valid when the term in question falls
within a finite clause that occurs as an argument (that is, roughly,
within that-clauses). Note the phrasing: The suggestion is that, in
these cases, Leibniz's Law *fails to be logically valid*. There may be
some special cases where all inferences made in accord with it would be
truth-preserving, but not logically so. For example, in:
(1)	It is true that a is F.
Assuming that "a is F" is not itself intensional,
(2)	It is true that b is F
will be true if (1) is. But, or so it seems to me, (2) does not follow
logically from (1), in any reasonable sense of 'logically'.

Note that small clauses, such as in
do not seem to be obstacles to Leibniz's Law. And famoulsly, infinite
clauses, such as in
(4)	Ralph believes [Ortcutt to be a spy]
are not always barriers, either. On the other hand, if we consider
(5)	Ralph wants [Ortcutt to eat dirt],
then it is not clear to me whether, if Ortcutt is Smith, it follows,
logically, that Ralph wants Smith to eat dirt. Perhaps not. But there
may be an ambiguity here. The ambiguity in question is obvious with
quantificational constructions like
(6)	Ralph wants someone to love him,
which can mean either that Ralph wants to be loved or that he's got a
special someone in mind. So it is possible that (4) and (5) can exhibit
the same ambiguity, so that (4) has one reading on which it just means
(4')	Ralph believes that Ortcutt is a spy
and another on which it means
(4'')	Ortcutt is such that Ralph believes that he is a spy,
depending upon where "Ortcutt" actually occurs in the underlying
syntactic structures (i.e., whether it raises). It's also possible that
(5) can exhibit this ambiguity, but that (4) cannot. If so, that would
presumably trace to some difference between the verbs in (4) and (5):
perhaps to the fact that "believes" can take a finite complement, but
"want" cannot.

So there are all kinds of complications here. But again, I see no reason
to despair. Careful attention to issues of logical form can be expected
to pay dividends here, as elsewhere.

Richard Heck

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