>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. (01)
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. (02)
<I know you know all this>
For example, take ANSI KIF. That had a provision
for stating definitions which clearly and
explicitly separated them from assertions, and
had no provision for distinguishing what you are
here calling legitimate from illegitimate
'definitions'. If one defines, in this sense, R
to be the relation (lambda (x)(not (x x))), then
this is indeed what R *must* be; from which it is
easy to generate the Russell paradox. The
illegitimacy of this definition arises from the
fact that if stated as an assertion, it would be
contradictory: it implicitly assumes the
existence of something (the Russell "set") that
cannot exist. By your modern criteria, this means
that it cannot be regarded as a 'genuine'
definition: but it looks like a definition, and
it smells like a definition, it can be used to
draw conclusions like any other definition, and I
suspect that to most people it simply IS a
definition: and so the problem it gives rise to
must be solved elsewhere, for example by
restricting the syntax of the logic or invoking
the majestic structure of Zermelo-Fraenkel set
theory. For example, if you were to add a kind of
lambda-conversion to CL and call it a definition
feature (a common (ab?)use of terminology), this
is what you would get. And look how awful the
practical consequences of your sophistication
are: to even use a definition in the most
elementary way - to replace the defined term by
its definition - I must first formally prove that
the entity described by this instance of the
definition exists. I am virtually certain that
nobody but a professional set theorist would even
consider this a useful notion of "definition".
Most mathematicians would quaver at this
condition if they took it seriously, let alone
the medics and practical men that Barry is
concerned about.
</I know you know all this> (03)
What I meant by the remark about adding
definitions to CL was adding a simple definition
feature which is not protected by existence
criteria in this way, like the ANSII KIF
definitions. The problem is exactly that this
does in fact add a power to "call things into
existence" by defining them: or if you prefer,
put that last 'defining' inside scare quotes. (04)
Pat (05)
>-chris
>
>
>
>
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