RE: Hempel's Paradox -- OOPS!

From: Ben Goertzel (
Date: Tue Sep 13 2005 - 19:56:29 MDT

Eli wrote:
> If PTL's
> two numbers lead it to bracket probabilities by two numbers, as in
> Dempster-Shafer theory, so that Novamente will buy gambles priced under
> one number and will sell gambles priced above a higher number, then
> dutch book may not be made against Novamente so long as every bracket
> contains the Bayesian value.

Yeah, that is basically correct, via the mapping between PTL's
truth values and Walley's interval probabilities (which are derived
based on reasoning about betting, as it happens...)

> However, I can still arrange
> a set of gambles which Novamente will refuse and which would be a
> deductively guaranteed gain, unless all lower and upper brackets
> coincide with the exact Bayesian price. That's the cost of saving on
> computing power, even if your approximation doesn't give absurd answers.

That is also correct, and is quite unproblematic from an AGI perspective,
since given finite resources we are not trying to make an all-knowing

> But what you *needed* to
> do was sit bolt upright and say "Wait, something's wrong with my math!"
> as soon as you thought PTL gave an answer substantially different from
> the Bayesian one. This would have saved some time, even if you didn't
> know whether you'd messed up the Bayesian math, the PTL math, or both.

Well, actually, it wasn't my MATH that was wrong per se, it was the way I
was feeding the information about the conceptual problem into the math

What I was thinking was that the mapping of the conceptual
Hempel paradox into PTL math somehow led to a different formalization than
mapping of the Hempel paradox into conventional probabilistic math. In
that case, the answers could have been different because the formalizations
of the question were somehow different.

But anyway, this turned out not to be the case...

The discussion improved my understanding of PTL a little ... and
gave me another nice, simple example for the PTL book...

Basically, the "different formalization of the Hempel paradox" I was
using turned out to be a formalization that assumed the knowledge of the
reasoner was restricted to a "local context" consisting only of his own
recent observations. But this local context didn't take into account
extra assumptions like "there is at least one raven" and therefore
wasn't really the right one to assume the reasoner was using. PTL can
do reasoning using various different contexts; restricting the context
can improve efficiency at cost of decreasing accuracy. In this
particular case restricting the context as I was doing wouldn't
decrease accuracy much but it would make it miss the small effect
noted in the Bayesian analysis of Hempel's paradox.

> If you
> find a case where you think PTL gives a different result from
> probability theory, and you see nothing alarming about this, but instead
> conclude offhandedly that probability theory must be wrong, you will end
> up not noticing when you divide by zero.

Well, truth be told, I end up not noticing when I divide by zero all
the time ;)

Generally speaking, I tend to have a pretty high error rate but also a
pretty high rate of generation of interesting and high-quality ideas ...
and I'm ultimately careful enough to check my ideas and remove errors ...

> The problem is that you're regarding Bayesian probability theory as a
> computational library, a tool for solving problems. It's not.
> Probability theory is the underlying mathematics of the thing you're
> trying to do.

Probability theory is PART OF the underlying mathematics of intelligence,
it's not a complete mathematics of intelligence...

And, it does of course provide a set of tools for solving problems.

It also provides a "way of thinking" which is valuable in solving many

My statement about the limitations of probabilistic thinking as regards
AGI was not intended to mean that there are aspects of AGI to which
probability theory is inapplicable. Rather, it was intended to mean
that there are crucial aspects of AGI for which the probabilistic way
of thinking is not all that overwhelmingly useful, and for which other
quite different ways of thinking are much more useful (even though
these other ways of thinking are ultimately *consistent* with probability

-- Ben

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