FW: Hempel's Paradox -- OOPS!

From: Ben Goertzel (ben@goertzel.org)
Date: Mon Sep 12 2005 - 15:16:32 MDT

> 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?


First a general contextualizing statement: I am pretty confident in the PTL
mathematics, but less so in my interpretation and application of that
mathematics "by hand." PTL is new and the art of applying it to various
situations is still a young one.

Now, to answer your question. My current thinking is that, given only the
information you've provided, I WOULD increase the odds, just like you would.
In other words, it seems on reflection that the PTL approach is more
consistent with the standard Bayesian approach than I'd thought, but that in
my prior e-mail I was not applying the PTL math carefully enough.

In fact, this seems to be a case where the Novamente PTL implementation
would come up with a different answer than I originally did, via applying
the rules in a way that took into account all information ;-) Maybe we'll
try the experiment, it wouldn't be hard to feed in the data...

Specifically, in my reply to Jeff Medina, I was not taking into account his
"special assumption" that at least one raven exists in the assumed-finite
sample space.

The PTL "amount of evidence" inference equation I gave in my prior email is

N_raven = N_(non-black) P(raven|non-black)

What I said there was

So if we've observed 499 non-black entities and none of them are ravens
(they're all purple geese, or orange orgasmotrons, or whatever), then we

N_(non-black) = 499


P(raven|non-black) = 0

thus an inferred

N_raven = 0

So, the amount of evidence about P(non-black|raven) [or P(black|raven)]
obtained from P(non-raven|black) is zero.

However, if we assume that there is at least one raven in the finite sample
space, then it now seems to me this is not correct. Under this assumption,
we have a situation where one of the two possibilities hold:

-- we have not yet looked at all the non-black items in the sample space,
and so there are some non-black items where it's still unknown whether
they're ravens or not. But since we know there is at least one raven and
have no prior certain knowledge that this raven is black, then our estimate
of P(raven|non-black) should be > 0 based on our total knowledge.

-- we have looked at all the non-black items in the sample space (because
there was as it happened only one), and have observed that there are no
non-black items there. But in this case there must be a black raven, since
there is a raven, and so in this branch we know P(raven|non-black)>0

Thus in either of these branches P(raven|non-black) > 0, and

N_raven > 0

according to the PTL evidence formula, unlike what I said before.

So, in conclusion, according to PTL applied correctly, we may say:

"The observation of a nonblack nonraven in a population known to be finite
and *known to contain at least one raven* may be considered as a small
amount of evidence in favor of the existence of a black raven in that

Without the assumption that the population is known to contain at least one
raven, then the argument I've given above fails.

However, it can be made to succeed if one adds various other special
assumptions, of course. For instance, if one assumes that P(raven) >0 and
that raven-ness and blackness are independent, then a variant of the above
argument obviously works.

I don't, at the moment, see any way to make the above argument work without
making SOME special assumption in addition to the finiteness of the
population. This is consistent with my original statement about Hempel's
paradox, but I erred by overlooking the way the introduction of the one
black raven changes the situation, from a PTL perspective.

Anyway, this has been an interesting thread. In fact it's pleasing for me
to see that PTL's evidence algebra is agreeable with traditionally-applied
probability theory in this instance. (However, I do sorta wish I could get
myself to think harder before replying to emails on technical topics ;-)

-- Ben

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