Re: Hempel's paradox redux

From: Eliezer S. Yudkowsky (
Date: Thu Sep 15 2005 - 22:37:59 MDT

Ben Goertzel wrote:
> Just one more thing...
> I started out this whole silly thread by saying that:
>>>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.
> I admit that in my followup discussions, after making this statement,
> I manifestly failed to demonstrate its truth...

Ben, standard probability theory says no such thing. You have to
provide a number of additional assumptions before probability theory,
the logic of science, makes any logical statements about the
probabilities involved. Observe a green lampshade one way, it means one
thing, observe it with different prior information, it means something else.

> Instead, I made some careless and silly errors, both with the standard
> formulation of probability theory and with my own PTL formulation. I
> apologize for this -- I'm not usually quite *that* error-prone even
> when badly overworked, but what can I say, it happens from time to
> time....
> However, after all that, I *still* hold the same intuition that I had
> originally. And this is with the probabilistic arguments regarding the
> Hempel paradox quite fresh in my mind and quite fully understood both
> conceptually and arithmetically.
> I don't doubt the math of probability theory, but I still have a nagging
> intuitive suspicion that the way the math is being applied to this situation
> is not conceptually right. Furthermore, I still have the same suspicion
> that this conceptual wrongness is related to other problematic issues
> with standard AI deployments of probability theory such as Bayes nets.

Ben, did I wish to justify your intuition, I would use the following
reasonable-sounding assumptions.

1) Random sampling of objects (not random sampling of a non-black object).

2) No correlation between your belief in the color mixture of ravens
and your belief in the number of ravens. That is, whether some ravens
are non-black does not, so far as you know, have anything to do with the
total number of ravens of all colors, nor the proportion of non-ravens
to ravens. (For all you know there might be a correlation, but if so,
you have no idea what direction it's in.) I.e., if you think that half
of all ravens are white, this doesn't change your estimation of how many
ravens there are in the world.

3) The only thing you know about the result is that it was not a raven.

Now if you observe an object randomly selected from the set of all
objects, and you know only that it is not a raven, this says nothing
about the color mix of ravens, because your color mix hypothesis classes
all assign equal likelihood to your seeing a non-raven (of whatever color).

But that's not what probability theory says, as such.

Probability theory just says to further specify your assumptions.

Probability theory has plenty of room to encompass structure of the sort
Russell Wallace refers to, if you know more about the object than that
it is not a raven.

And if you randomly sample a non-black object, and the only thing you
know about the result is that the result is not "Raven", you are
essentially stuck with this increasing the probability of the hypothesis
"All ravens are black" under any reasonable prior assumptions. If you
sampled a nonblack object and it was a raven, the probability of "All
ravens are black" would go (way) down. So if you get the converse
result of nonraven, the probability "All ravens are black" goes up.
Unless you are a priori certain that the result will not be a raven
because ravens are too rare. p(A) = p(A|B)p(B) + p(A|~B)p(~B). If
p(A|B) > p(A) then p(A|~B) < p(A) if both probabilities are defined.

Eliezer S. Yudkowsky                
Research Fellow, Singularity Institute for Artificial Intelligence

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