Pat, (01)
On Mon, Oct 13, 2008 at 12:21 AM, Pat Hayes <phayes@xxxxxxx> wrote:
>
> On Oct 12, 2008, at 6:49 AM, Rob Freeman wrote:
>
> > ... What matters to me is that, even for finite
> > sets, the number of possibly "meaningful" relations could be very
> > large (unless we impose an artificial limitation in some way.)
>
> But do you have any reason to suppose that it WILL be large, in fact? (02)
I think we have reason to suppose this, yes. (03)
At the very least the possibility should be studied carefully. I don't
think it has been. (04)
Articles like the one by Chris Anderson I referenced earlier in this
thread are one indication the number is beyond expression in any one
model. Other people are beginning to say similar things. (05)
> I'm talking here about FO semantics as opposed to higher-order semantics.
> Were you then arguing for the use of classical higher-order logic? (06)
I was trying not to assume any ab initio semantic model at all. (07)
> Set theory does not suggest anything about 'meaningful' sets. It simply
> talks about sets. Yes, there are a lot of sets. Nothing at all follows about
> how many 'meaningful' sets there are. The fact that human beings do not seem
> to be involved in combinatorial explosions when they think or speak
> rationally, suggests that the number of meaningful sets might be a lot less
> than the number of combinatorially possible sets. And one can give many
> other reasons to suggest that there are far fewer 'meaningful' sets than
> sets altogether: for example, the fact that people can do induction quite
> successfully, or that one-shot learning often works very well. You may hold
> other opinions, but it falls to you to come up with some reasons why this
> might be an issue. Simply re-iterating elementary mathematical facts doesn't
> cut it. (08)
I agree there must be "far fewer 'meaningful' sets than sets
altogether". I'm not arguing that all sets are meaningful. (09)
On the other hand I don't see that we have any justification for
simply throwing away all sets which do not fulfill some arbitrary
condition, e.g. conformance with one or other logic or model. (010)
Rather than defining "meaningful" as conformance with a model, why not
take it from the other direction? Sets can be limited not by a model,
but by say, a distance measure (like John's analogies.) Let that limit
us. Then let the sets which result from clustering on a distance
measure determine the model relevant to a particular case. (011)
What happens? Has anyone looked into it? (012)
> You have just summed up one the central foundational notions
> of computer science. (013)
Great. Yes. Thanks for making that association. (014)
It is somewhat my point that we do not use the full expressive power
of these "foundational notions of computer science", a power beyond
any single logic, when we attempt to represent knowledge. (015)
Why? (016)
-Rob (017)
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