On Mar 19, 2007, at 11:09 AM, andersen@xxxxxxxxxxxxxxxxx wrote:
>>> Subtype has a very clear and simple definition: X < Y
>>> means that every instance of X is an instance of Y.
>>
>> I would prefer to say that it implies this, but subtype is actually
>> somewhat stronger. Not every occassion of one category being included
>> in another need be viewed as a subtyping. BTW, this 'intensional'
>> perspective gives more efficient reasoning, as well as being more
>> natural.
>
> There are other versions as well. One stronger version involves
> modality,
> and corresponds to one view of classes as universals. Take
> (subclass A B)
> to read "A is a subclass of B" and (class A) to read "A is a class").
> Then we have (please excuse my extended CLIF - I introduce Nec as the
> necessity operator, and Pos for possibility, assuming S5):
>
> (forall (A B)
> (if
> (and (class A) (class B))
> (iff
> (subclass A B)
> (and
> (Nec (forall (x) (if (A x) (B x))))
> (Nec (exists (x) (and (A x) (not (B x)))))
> )))) (01)
You've got A and B turned around in that last line, Bill (and given
that you are generally averse to asserting contradictions, this is
obviously a hasty typo rather than a thinko); I take it what you
intended was: (02)
(Nec (exists (x) (and (B x) (not (A x))))) (03)
But that's too strong! Surely it's possible there are no dogs, but
Collie is a subclass of Dog all the same. This might suggest
replacing the last line instead with: (04)
(Nec (if (exists (x) (B x))
(exists (x) (and (B x) (not (A x))))) (05)
But that too entails that Collie is not a subclass of Dog, as there's
surely a logically possible world where Lassie and her kinfolk are
the only canines. :-) I think that, for A to be a genuine subclass
of B, you only want it to be *possible* that there be
Bs that aren't As: (06)
(Poss (exists (x) (and (B x) (not (A x))))) (07)
-chris (08)
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