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

**Date:** Mon Sep 12 2005 - 15:16:32 MDT

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*> Ben, just to be clear on this, do you mean that, under PTL, and under
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*> your own view of probability, sampling a random non-black object, and
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*> finding that it is not a raven, should count as no evidence in favor of
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*> the proposition that all ravens are black? Given, we shall say, that at
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*> least one raven exists, and that the *ratio* of ravens to nonravens is
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*> greater than zero. And again to be clear, by "evidence" I am trying to
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*> get at the Bayesian concept of evidence: After sampling a random
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*> nonblack object and finding it to not be a raven, would you/PTL
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*> increase, or not increase, the odds at which you would be willing to bet
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*> that "All ravens are black" is true of the sample space?
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Eli,

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

have

N_(non-black) = 499

and

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

population."

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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