Re: SUO: Set/Class Distinction
Ian wrote:
> A while back John Sowa recommended that we eliminate the concept of
> 'Class' in the SUMO, because it is not clearly differentiated from 'Set' and
> because it does not seem to have any utility beyond the concept of 'Set'.
> If no one objects, I'll go ahead and replace all of the occurrences of
> 'Class' in the SUMO with 'Set'.
Wow, that strikes me as a really seriously bad idea that runs into
problems similar to the ones John noted in regard to "Class". The term
"set" is deeply entrenched in mathematics, logic, linguistics, theoretical
computer science, and philosophy. There are different set theories, of
course, but, with the quirky exception of Quine's NF, which no one uses,
in *all* of them sets differ markedly from SUMO classes in a number of
critically important ways. Notably:
1. Sets are extensional. SUMO classes are not assumed to be.
2. SUMO classes have complements. Sets do not.
3. There is a universal SUMO class, i.e., a class (viz., Entity) of which
everything is an instance. There is (on pain of contradiction) no
universal set.
4. In most set theories, sets are well-founded; in particular they cannot
be self-membered. Some SUMO classes (notably, Entity) are instances of
themselves.
5. SUMO appears to have a general comprehension principle -- made possible
through the KappaFn operator -- that yields the existence of a SUMO class
corresponding to every statable condition of the language. There is no
such principle for sets, again on pain of contradiction.*
It therefore strikes me as rather perverse, to say the least, to ignore
these profound differences and appropriate the term "Set" in the manner
you propose.
*The SUMO teeters on the brink of contradiction here, since it appears to
allow the existence of the Russell class of all classes that are not
instances of themselves: (KappaFn ?x (not (instance ?x ?x))). The only
reason SUMO seems to avoid contradiction is that it does not have a
general conversion principle for classes: it is not provable in general
that x is in the class of A's if and only x is an A, i.e., formally, one
can't in general prove, for any given condition A of the language
containing the variable ?x, (iff A (instance ?x (KappaFn A))). This
principle is essential for deriving the Russell paradox, and its omission
is intuitively unwarranted if you're gonna have KappaFn around. But
KappaFn is nothing but trouble. You keep ignoring me, but I'll say it
again: you ought to clobber it.
-chris
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