>>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. (01)
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): (02)
(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)))))
)))) (03)
and, for all classes A (04)
(if (class A) (Pos (exists (x) (A x)))) (05)
>From these it follows, for example, that (06)
(forall (A B)
(if (subclass A B)
(not (subclass B A)))) (07)
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