Chris  You wrote... In general, however, the problem of determining when one thing follows from
another is undecidable (hence intractable). So a serious challenge of
AI is to figure out how to do useful reasoning in the face of this
theoretical limitation.
One approach to this is of course to choose a subset of formal logic that is (a) useful for AI, (b) decidable, and (c) has tractable complexity on current computers. The paper [1] and the system [2] represent one set of choices in this area. No doubt there are many others.
Cheers,  Adrian [1] Backchain Iteration: Towards a Practical Inference Method that is Simple Enough to be Proved Terminating, Sound and Complete, A. Walker. Journal of Automated Reasoning, 11:122.
[2] Internet Business Logic A Wiki and SOA Endpoint for Executable Open Vocabulary English over SQL and RDF. Online at www.reengineeringllc.com Shared use is free
Adrian Walker Reengineering On Sun, Sep 28, 2008 at 2:22 PM, Christopher Menzel <cmenzel@xxxxxxxx> wrote:
On Sep 27, 2008, at 1:24 AM, Rob Freeman wrote:
> If you are really confused by my reference to the "pitfalls of
> formal logic" I apologize.
No apology needed, unless it is for what appears to be an attempt to
deflect attention away from your refusal to answer my simple question
about what you had in mind by the "pitfalls of formal logic" by
depicting me as confused.
> My comment was most directly in the context of the artificial
> intelligence work of Marcus Hutter and Juergen Schmidhuber
> immediately above in my original message.
There is nothing whatever in the work of Hutter and Schmidhuber that
suggests there are any "pitfalls of formal logic". As proponents of
approaches to AI that are alternatives to the more traditional logic
based approach, they no doubt argue that pitfalls await those who
attempt to *base AI* on formal logic alone. The problem I saw with
your original post, and which seems to be persisting, is that you
appear to be attributing the existence of pitfalls that can arise in
an *application* of formal logic to formal logic itself. That is is
simply a confusion about the nature of logic (and, in this case AI)
that I think it is quite important to point out.
> Traditionally AI has been approached as a problem of formal logic.
> This has proven intractable for reasons which I believe closely
> resemble those which prevented a basis in formal logic for
> mathematics, broadly "Gödelian incompleteness".
There are several errors and unclarities here. First, AI has never
been "approached as a problem of formal logic". Problems of formal
logic have to do with such matters as the axiomatizability of validity
relative to a certain semantics, the decidability of validity, the
expressive power of a certain semantics, the consistency of a certain
theory, etc. AI, by contrast, is the problem of implementing
something approaching human intelligence on a computer. That is an
engineering problem, one that, depending the particulars of your view
of AI, also bleeds over into a variety of other disciplines including
cognitive science and linguistics. And there are, of course, many
points at which one might *apply* formal logic to solve particular
problems in AI. And here, of course, pitfalls await. The wellknown
problems of intractability in pure logic of course have implications
for AI  indeed, these purely theoretical facts lead to perhaps the
most fundamental challenge to (and, hence potential pitfalls for) a
logicbased approach to AI. One of the things human do quite well is
reasoning, drawing valid conclusions from given premises. In general,
however, the problem of determining when one thing follows from
another is undecidable (hence intractable). So a serious challenge of
AI is to figure out how to do useful reasoning in the face of this
theoretical limitations. But, again, the limitation itself are no
more a "pitfall" of formal logic than, say, the infinitude of the
primes is a pitfall of arithmetic.
As for your remark about finding a basis for mathematics in formal
logic, I'm not sure which of two projects you are alluding to. The
idea of finding a basis for mathematics in logic sounds very much like
the logicist project of Frege and (subsequently) Russell. But the
reasons for the failure of this project had nothing to do with the
issue of tractability and indeed was largely dead in the water long
before 1931 when Gödel published his famous paper on incompleteness.
So I'm thinking that, if anything, what you have in mind here is the
formalist project of Hilbert, whose failure can indeed be attributed
to Gödel's work, from which the general intractability of deduction
noted above follows. Historically interesting, for sure, but I'm not
seeing any clear point.
> Hence by my lights formal logic has "pitfalls" as a basis for AI
> (and mathematics.)
Once again, that there may be is exactly the point I made (more
generally) in my original reply: there may well be pitfalls in any
attempt to *apply* formal logic to solve a problem. It might simply
be the wrong framework for the problem at hand.
> The work of Schmidhuber and Hutter avoid these "pitfalls", by
> ignoring formal logic and seeking a basis for meaning in prediction/
> probability.
I doubt very much that they are "ignoring" formal logic, as they still
have to implement certain forms and patterns of reasoning.
> It is interesting to compare this with, e.g. Greg Chaitin's work on
> mathematical randomness...As I said in the "Axiomatic ontology"
> thread I personally prefer to look for a basis for this randomness
> in something like Vitiello's "manybody system" approach (or Chris
> Anderson's "Google"/Robert Laughlin's "bottom down", crystalline
> structure inspired, physics.) But the different approaches don't
> conflict (no less the success of "transformation invariance", as a
> fundamental parameter for understanding the world, in "geometric"
> models.).
Sounds Very Deep.
Chris Menzel
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