The following is the first draft of the opening to the final section of the imaginative resistance paper. Don’t think that because I’m drafting the final section it’s close to being done. The drafting is not exactly chronological. But it is turning out. It’s all about the in virtue of relation that does 110% of the philosophical work in the paper, my lame attempts at comic short-stories easily accounting for the other -10%. There aren’t many footnotes in other parts of the paper (yet) but somehow we get four in three paragraphs here.

Since Greek times
we have called numbers that equal the sum of their proper factors perfect.
Euclid knew that a certain class of numbers were all perfect. These are numbers
of the form 2^p-1(2^p ‑ 1),
where 2^p ‑ 1 is an odd prime. Call numbers of
this form flawless. Euler proved that all even perfect numbers are
flawless. It is not known whether there are any odd perfect numbers, but
suspicion seems to be that there are not. Let us imagine it is proven in a few
years that there are not. The proof, we can be confident, will not be short,
else it probably would have been discovered. And imagine that sometime after
that, an imaginative underemployed philosopher writes a fictional work in which
a philosopher becomes famous for in fact discovering an odd perfect number, despite
the existence of this magisterial proof of its non-existence.

This looks a lot like a story in which an impossibility, indeed several salient
a priori impossibilities are true in the fiction. One such impossibility is
that there is a perfect number that is not flawless.

Indeed, such stories are sometimes taken to be evidence against the thesis that
only possible truths are true in fictions. But note what kind of story we could
not have. The young philosopher could not discover an odd number that, although
obviously not flawless, does equal the sum of its proper factors and
thus conclude that, since all and only flawless numbers are perfect, that some
numbers that are the sum of their proper factors are not perfect. That, I
think, could not fail to be perfect in the story.

There is something odd about all
that. I said, truly, that (3) could be true in a story even though (4) could
not be.

(3) Some number that is the sum of its
proper factors is not flawless.
(4) Some number that is the sum of its proper
factors is not perfect.

The oddity is that
this distinction could hold even if it is known that flawless is
co-intensional with perfect. There is a natural enough explanation of
this in terms of virtue. A number is not flawless (or flawed) in virtue of
being the sum of its proper factors (or not). It is flawless in virtue of being
of the form 2^p-1(2^p ‑ 1),
where 2^p ‑ 1 is an odd prime. But a number is
perfect (or not) in virtue of being the sum of its proper factors (or not). Any
instantiation of (4) would be an asymmetric compound impossibility, and hence
(4) is impossible.

I think this
is quite good news for the theory that which impossibilities are ruled out
should be determined by which have conjuncts that are false in virtue of their
other conjuncts.

On
the other hand, this is is pretty bad news for an analysis of ‘in virtue of’.
For it suggests that this relation is hyper-intensional. If it were an
intensional relation, then a number would be flawless (or not) in virtue of the
same things in virtue of which it is perfect (or not). So we won’t be able to
analyse ‘in virtue of’ in terms of modal locutions, such as counterfactuals.
Nor will we be able to analyse it in purely epistemic terms, since in the
circumstance imagined it is known that perfect and flawless are
co‑intensional.