Dear Matthew, (01)
I agree with your point: (02)
> 1. Stop thinking of inheritance and specialisation as being synonymous.
> Inheritance of properties can happen through other relationships as well. (03)
> 2. Think in set theoretic terms. So each member of a subset is a member of
> the superset. If a method belongs to the superset, it is not "inherited" by
> the subset, but it applies to each member of the subset because it is also a
> member of the superset. (04)
But please note that inheritance is not applicable to an arbitrary
set, but only to a set S that satisfies two conditions: (05)
1. There is some predicate P(x) that is true of every element x of S. (06)
2. Any method, property, or whatever that can be inherited by every
element of S must be characterized by some predicate m(x) that is
implied by P(x): (07)
For all x, if P(x), then m(x). (08)
This predicate P(x) is not necessarily the defining predicate for S
because there might be some larger set {x | P(x)}, which includes
S as a proper subset. (09)
The critical point is that there must exist some such predicate P(x)
that meets the two conditions above. (010)
I would call that predicate P *intensional* information about the set S,
but you don't have to use the word 'intensional' if you don't like it. (011)
John (012)
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