RE: Fundamental problems

From: H C (
Date: Fri Feb 17 2006 - 16:12:58 MST


'Probability Theory : The Logic of Science' by E. T. Jaynes



>From: Marc Geddes <>
>Subject: Fundamental problems
>Date: Fri, 17 Feb 2006 18:04:00 +1100 (EST)
> >Eliezer has posted a job notice at the SIAI website,
> >looking for research partners to tackle the problem
> >of rigorously ensuring AI goal stability under
>self->enhancement transformations.
> >I would like to see this problem (or perhaps a more
> >refined one) stated in the rigorous terms of
> >theoretical computer science; and
> >I'd like to see this list try to generate such a
> >formulation.
>Fascinating, fascinating.
>I thought as a final post I'd better try to say
>something actually intelligable, so I shall take one
>crack at actually pointing to a solution :D
>Is there such a thing as
>'a probability of a probability' ?
>See a new paper by Robin Hanson arguing for a new
>Bayesian framework wherein probabilities can be
>assigned to priors:
>Also see blog entry by Ben Goertzel:
>*The management of uncertainty in the human brain: new
>experimental insights*
>"In other words, some of us maverick AI theorists have
>been saying for a while that using just ONE number
>(typically probability) to measure uncertainty is not
>enough. Two numbers -- e.g. a probability and another
>number measuring the "weight of evidence" in favor of
>this probability (or to put it differently, the
>"confidence" one has in the probability) -- are needed
>to make a cognitively meaningful algebra of
>I assume Bayesian probability theory could be
>reformulated in terms of some kind of fuzzy set
>theory. Then the notion of 'a probability of a
>probability' would be referring to fuzzy sets
>containing other fuzzy sets. The problem of sets
>containing other sets has never been fully solved.
>According to Roger Penrose:
>'In fact, the way that mathematicians have come to
>terms with this apparently paradoxical situation is to
>imagine that some kind of distinction has been made
>between 'sets' and 'classes'...Roughly speaking, any
>collection of sets whatever could be allowed to be
>considered as a whole, and such a collection would be
>called a *class*. Some classes are respectable enough
>to be considered as sets themselves, but other classes
>would be considered to be 'too big' or 'too untidy' to
>be counted as sets. We are not neccessarily allowed
>to collect *classes* together, on the other hand, to
>form larger entities. Thus 'the set of all sets' is
>not allowed...but the 'class of all sets' is
>considered to be legitimate...
>There is something unsatisfactory about all
>this...This procedure might be reasponable if there
>were a clear-cut criterion telling us when a class
>actually qualifies as being a set. However the
>'distinction' appears often to be made in a very
>circular way."
>-Roger Penrose. 'The Road To Reality' , Page 373
>(Hard-back version)
>Paper on possible extensions to set theory:
>"Abstract: We discuss the problems of incompleteness
>and inexpressibility. We introduce almost
>self-referential formulas, use them to extend set
>theory, and relate their expressive power to that of
>infinitary logic. We discuss the nature of proper
>classes. Finally, we introduce and axiomatize a
>powerful extension to set theory."
>O.K, so... did I solve it? Are any of these ideas of
>"Till shade is gone, till water is gone, into the shadow with teeth bared,
>screaming defiance with the last breath, to spit in Sightblinder’s eye on
>the last day”
>Do you Yahoo!?
>The New Yahoo! Movies: Check out the Latest Trailers, Premiere Photos and
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