**From:** Ben Goertzel (*ben@goertzel.org*)

**Date:** Fri Sep 09 2005 - 10:17:00 MDT

**Next message:**Ben Goertzel: "RE: Inference control"**Previous message:**Phil Goetz: "RE: Inference control"**In reply to:**Eliezer S. Yudkowsky: "Re: The Relevance of Complex Systems [was: Re: Retrenchment]"**Next in thread:**Eliezer S. Yudkowsky: "Re: Hempel's Paradox"**Reply:**Eliezer S. Yudkowsky: "Re: Hempel's Paradox"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

Eli,

*> Googling on Hempel + Bayes turns up some standard responses, for example
*

*> in http://plato.stanford.edu/entries/epistemology-bayesian/
*

*>
*

*> "Hempel first pointed out that we typically expect the hypothesis that
*

*> all ravens are black to be confirmed to some degree by the observation
*

*> of a black raven, but not by the observation of a non-black, non-raven.
*

*> Let H be the hypothesis that all ravens are black. Let E1 describe the
*

*> observation of a non-black, non-raven. Let E2 describe the observation
*

*> of a black raven. Bayesian Confirmation Theory actually holds that both
*

*> E1 and E2 may provide some confirmation for H. Recall that E1 supports H
*

*> just in case Pi(E1/H)/Pi(E1) > 1. It is plausible to think that this
*

*> ratio is ever so slightly greater than one. On the other hand, E2 would
*

*> seem to provide much greater confirmation to H, because, in this
*

*> example, it would be expected that Pi(E2/H)/Pi(E2) >> Pi(E1/H)/Pi(E1)."
*

*>
*

*> A fine answer so far as it goes, though really it is only half the
*

*> solution.
*

IMO this is *not* a fine answer, it's a dodge of the issue.

The problem is that what is logically correct is that an observation of

a non-black non-raven should provide NO evidence toward the hypothesis

that all ravens are black.

If probability theory as standardly deployed states that an observation

of a non-black non-raven provides a NON-ZERO amount of evidence toward

the hypothesis that all ravens are black, then this shows there is

something wrong with probability theory as standardly deployed.

Of cousre, an approach that yields small errors may still be valuable

for practical AI purposes.

However, what frustrates me about the quote you cite, and your attitude,

is that you seem to be denying that probability theory as standardly

deployed is conceptually and logically erroneous in this case -- albeit

the magnitude of its error is generally small.

Your followup comments are intelligent and well-thought-out, but, they

don't really solve the problems with the attempted solution given in

the paragraph you cite above.

I believe the Hempel problem is handled more artfully in Novamente.

In PTL, we can estimate the truth value of

ForAll x { is_raven(x) ==> is_black(x) }

as a transform of the truth value of

P( is_black(x) | is_raven(x) )

[the simplest rule for this is, e.g. s^N estimates the probability

of the former statement where s is an estimate of the probability

of the latter statement and N defines the amount of (explicit or

implicit) evidence used to arrive at s].

The definition of evidence in PTL makes clear that the only evidence

that counts for

P( is_black(x) | is_raven(x) )

is the set of x for which is_raven(x) has a nonzero truth value,

and therefore Hempel's paradox does not exist.

The key point is that PTL explicitly defines the concept of evidence

and keeps track of the evidence in favor of each statement. Evidence

is defined as something separate from probability, though related

to probability. Basically, the evidence in favor of an assertion is

the "number of observations" made to estimate the probability of

the assertion.

I realize these comments are only evocative rather than convincing,

which is pretty much inevitable given the constraints of expression

in a brief and semi-technical email.

Pei Wang and I have recently written and submitted for publication

a paper arguing that, to be adequate, an uncertain logic system must

use at least two numbers to quantify truth value. Examples of

uncertain logic systems using two numbers to quantify truth values

are:

* Novamente's PTL framework

* Pei Wang's NARS framework (which I have some serious issues with)

* Walley's theory of interval probabilities (which I haven't explored

that fully, though it has some nice algebraic similarities to PTL

and NARS)

I continue to believe that a purely probabilistic approach is not

adequate, but that if one augments probability theory by considering

truth values with more than one component (e.g. a "weight of evidence"

as well as a probability), then things work out more adequately

(though one still needs to introduce a bunch of heuristic approximations

to tractably handle real-world inferencing).

*> I shall now demonstrate the folly of adulterating Bayes with lesser wares.
*

*>
*

*> Suppose that I know that, in a certain sample, there is at least one
*

*> black raven, and at least one blue teapot, and some number of other
*

*> ravens of unknown color. I now observe an item from the group that is
*

*> produced by the following sampling method: Someone looks over the
*

*> group, and if there are no non-black ravens, he tosses out a blue
*

*> teapot. If there are non-black ravens, he tosses out a black raven.
*

*> Now observing a black raven definitely shows that not all ravens
*

*> are black.
*

*>
*

*> How would Novamente's "augmented" probability theory handle that case, I
*

*> wonder?
*

Given the constraints you've introduced, the only way Novamente has to

handle this problem is to use "higher-order inference", which means

to explicitly represent the definition of the problem in terms of

variables and quantifiers, in a manner similar to predicate logic.

The difference is that, unlike standard predicate logic, Novamente has

formulas for managing uncertain truth values attached to quantified

logical formulae.

I could write out the details of this example in Novamente formalism,

and may do so later as it's a moderately amusing exercise, but I don't

have time at the moment.

-- Ben G.

**Next message:**Ben Goertzel: "RE: Inference control"**Previous message:**Phil Goetz: "RE: Inference control"**In reply to:**Eliezer S. Yudkowsky: "Re: The Relevance of Complex Systems [was: Re: Retrenchment]"**Next in thread:**Eliezer S. Yudkowsky: "Re: Hempel's Paradox"**Reply:**Eliezer S. Yudkowsky: "Re: Hempel's Paradox"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

*
This archive was generated by hypermail 2.1.5
: Wed Jul 17 2013 - 04:00:52 MDT
*