On Wed, Jan 20, 2010 at 4:57 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
> ...
> RF> Is it not true that a lot of what we might regard as
> > "meaningful" about some programs, such as whether they
> > will ever halt, will escape any such description?
>
> ...
>
> For some pairs of (M,T), the predicate Will_Halt can be
> determined by a proof in FOL. But for others, the theorem
> prover will loop forever. But any (M,T) that is undecidable
> in FOL will be just as undecidable in English or any other
> language, formal or informal. (01)
It may be enough to emphasize that point, i.e. that for some systems
the only meaningful arbiter is the system itself (viewed as a
computable process?) There is no more succinct "description". (02)
What I am always trying to draw people's attention to are the
implications of this, or equivalent logical results, for meaning
representation. (03)
I fully believe it will eventually prove possible to represent any
meaning, completely, in the steps of a computable process. And I
accept it will be possible for the steps of that computable process to
be described in FOL, or any other formalism. What I think will not
prove possible, what is demonstrated to be impossible in general, and
what I think your use of the word "described" implied, for some,
perhaps leading to confusion, is for the sum of the steps to be
summarized in FOL. For that there may be no more succinct
representation than the computable process itself. (04)
Your use of "described" is fair. I can see how you came to it. I just
think it tended to hide this point: that FOL, any logic, anything,
cannot in general summarize the totality of steps in a computable
process. (05)
>From a knowledge representation point of view, if it turns out
"meaning" can only be fully represented by computable processes, then
there will indeed be something (Paola's "new logic"?) which cannot be
captured by FOL: the sum of steps in a computable process. (06)
Rob (07)
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