On Feb 13, 2007, at 2:18 PM, Pat Hayes wrote:
> ...For example, adding the power to make
> definitions to IKL would make the entire logic paradoxical, and
> re-create the Russell paradox in CL. (01)
Pat, I agree strongly with your general point, but I think what you
say here is not true about CL and moreover reflects an incorrect
concept of definitions (which puzzles me because I know you know what
a good definition of "definition" is!). What you say above seems to
identify "the power to make definitions" with the ability to call
things into existence ex nihilo. But that is *exactly* what one
cannot do in giving a definition. A critical condition on a genuine
definition is that it be *non-creative*: A definition (within a
theory) cannot entail the existence of anything that was not already
entailed by the theory. Hence, any purported definition in IKL of a
Russell set (property, class, type, whatever), or any other
paradoxical entity, would be illegitimate, for the same reason that
the Russell set {x | x not in x} is illegitimate in ZF set theory.
You can't prove the existence of a set of all non-self-membered sets
in ZF, hence, you can't legitimately introduce the name "{x | x not
in x}", as it violates the non-creative condition on definitions.
Same for CL. (02)
-chris (03)
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