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If you don't think there is any reason to want to discriminate against
non-symbols as names, then why call the function SYMBOL-CLASS?  Why not
name it NAME-CLASS? Then X is a proper name if (CLASS-NAME <class>) = X and
(NAME-CLASS X) = <class>.

If you do this, isn't there some other functionailty you'd like to specify
about names, like associating a name with an object, like registering
names (default, intern), mapping over them (default, do-symbols),
destroying them (default, ???)...?

If you go beyond function specs into the realm of class names, what about
the other naming contexts in Common Lisp: types, variables...?  

If you eliminate SYMBOL-CLASS and use NAME-CLASS, are implementations free
to use SYMBOL-VALUE to implement NAME-CLASS? Or is there also the thing
meant by the old SYMBOL-CLASS that is used when symbols are specified as
the names?  Does this mean that NAME-CLASS must be a generic function?

Chapters 1 and 2 specify a possible world as implementable by chapter 3.
mechanisms. By selecting that world we eliminate possibilities from it.
This doesn't mean that an adjacent world cannot be different.  I don't
feel that we have considered enough of the can of worms opened by general
naming to want to casually introduce it into our nice little world.