On Feb 13, 2007, at 4:38 PM, Pat Hayes wrote:
>> 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.
>>
>> 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.
>
> I wont argue with what you are saying, but you are here using a
> very sophisticated notion of what a "definition" is. I don't think
> this notion (which is informed by a century of post-Russellian
> thought about how to deal with paradoxes) is what people usually
> mean by "definition". You are, to use a philosophical semi-joke
> term, assuming that all definitions come pre-Quined; but that is
> not how they are usually understood. (01)
I won't dispute your point about what people usually mean by
"definition", but your claim that it is "informed by a century is
post-Russellian thought about how to deal with paradoxes" is not
historically accurate. Russell sent his fateful letter to Frege
informing him of the paradox in the Grundgesetze der Arithmetik in
1902; Padoa published his first paper on the theory of definitions
(which included a notion of non-creativity) in 1901. (02)
-chris (03)
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