Re: Hempel's Paradox

From: Eliezer S. Yudkowsky (
Date: Mon Sep 12 2005 - 10:49:17 MDT

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

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                
Research Fellow, Singularity Institute for Artificial Intelligence

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