**From:** Eliezer S. Yudkowsky (*sentience@pobox.com*)

**Date:** Mon Sep 12 2005 - 10:49:17 MDT

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Ben Goertzel wrote:

*> Hi Jeff,
*

*>>
*

*>> Because it is evidence that [all non-black objects are non-ravens].
*

*>> If we know at least one raven exists, and sampling a non-black
*

*>> object produces a non-raven on each of N sampling events, then with
*

*>> increasing N comes increasing certainty that no non-black object
*

*>> is a raven.
*

*>
*

*> I agree so far...
*

*>
*

*>> And [no non-black object is a raven] is, of course, logically and
*

*>> conceptually equivalent to [all ravens are black], given the tiny
*

*>> extra assumption I left out earlier that at least one raven exists.
*

*>
*

*> This is the controversial part.
*

Controversial indeed: In conventional mathematics, "All ravens are

black" is true if no ravens exist. But let that aside.

*> To get from
*

*>
*

*> NOT(black) ==> NOT(raven)
*

*>
*

*> to
*

*>
*

*> raven ==> black
*

*>
*

*> requires a logical transformation that does not preserve "amount of
*

*> evidence", at least not according to PTL's theory of evidence. And
*

*> when you look at the algebra of evidence transformation that comes
*

*> along with this transformation, you find that in fact the amount of
*

*> evidence about raven==>black ensuing from NOT(black) ==> NOT(raven)
*

*> comes out to zero...
*

Ben, just to be clear on this, do you mean that, under PTL, and under

your own view of probability, sampling a random non-black object, and

finding that it is not a raven, should count as no evidence in favor of

the proposition that all ravens are black? Given, we shall say, that at

least one raven exists, and that the *ratio* of ravens to nonravens is

greater than zero. And again to be clear, by "evidence" I am trying to

get at the Bayesian concept of evidence: After sampling a random

nonblack object and finding it to not be a raven, would you/PTL

increase, or not increase, the odds at which you would be willing to bet

that "All ravens are black" is true of the sample space?

*> Elegant, huh? Hempel's paradox disappears when you move to
*

*> two-component truth values and tabulate evidence separately from
*

*> probability. It doesn't just quasi-disappear like in standard
*

*> Bayesian semantics, it *really* disappears.
*

No matter what hidden assumptions you require of your prior

probabilities, under the rubric of "no assumptions" or "no other data",

I cannot see any possible way - short of the ratio of ravens to

nonravens approaching zero as a limit - for sampling a non-black object

and finding it to not be a raven, to provide NO evidence about whether

all ravens are black. Simply no way to do it under Bayesian probability

theory.

This being the case, I am willing to bet real money against you, or

against Novamente, with odds set on the basis of our respective views of

probability theory, and we will see who has more money after 100 bets.

-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence

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