From: Ben Goertzel (ben@goertzel.org)
Date: Sun Sep 11 2005 - 20:08:56 MDT
Hi Jeff,
> Ben replied, "Yes, an observation of a non-black raven TOGETHER WITH
> SPECIAL ASSUMPTIONS ABOUT THE SITUATION can yield evidence toward the
> hypothesis that all
> ravens are black.
> But in the absence of explicitly stated special assumptions, an
> observation
> of a non-black non-raven should provide NO evidence toward the hypothesis
> that all ravens are black."
>
> The only assumption required is that our sample space is finite.
This isn't true.
Suppose you have a population of 10 birds of different colors, and no other
knowledge about the population.
If you sample one of the birds and find that it's a purple goose, why does
this count as information that all the RAVENS in the population are black?
This would be true only if there were additional information of some sort...
For instance, if it's believed that geese are generally similar to ravens,
then a purple goose provides a limited amount of speculative evidence that
purple ravens exist; and, given a finite population, this is speculative
evidence against the existence of black ravens ... it decrements the
estimated probability of ravens being black.
But, this inference requires an additional assumption, it's not implicit in
the original problem.
> We can modify Eli's earlier example to demonstrate:
>
> Given a set of M ravens and N non-ravens, we randomly sample a raven
> and find that it is black. As the number of times we repeat this
> increases, it becomes asymptotically certain that all ravens are
> black, as compared to the hypothesis that a positive non-zero integer
> number of ravens are non-black. Taking the limit as the number of
> ravens goes to infinity, the evidence provided by sampling a raven and
> finding it is black for [all ravens are black] approaches zero.
All this is correct, of course. But it has nothing to do with Hempel's
paradox of confirmation...
> Considering you don't take this as a proper criticism of my
> informal/colloquial assertion that [observing a black raven should be
> evidence for the claim that all ravens are black], it cannot be
> deployed in defense of your claim that [observing a non-black
> non-raven should provide NO evidence toward the hypothesis that all
> ravens are black]. It does provide evidence, and it should. Hempel's
> ravens peck no holes in Bayesian philosophy of science.
>
> Hempel's paradox is not a paradox at all; it was a confusion based on
> a lack of knowledge on the part of humans. The evidential situation
> proposed was counterintuitive, yes. But that's just yet another reason
> for researchers to stop letting their intuitions overrule the math (or
> science, as the case may be).
All math and science paradoxes are paradoxical only in the context of overly
limiting formal or conceptual systems.
For instance, Epiminides' paradox ("This sentence is false") is problematic
in some formal systems but is fine in G. Spencer-Brown's "imaginary logic"
or in logic based on Aczel-style non-foundational set theory.
Hempel's paradox is problematic in some formal models of evidence and not
others.
I agree that "Bayesian philosophy of science" deals with Hempel's ravens in
an internally consistent way, but not that it deals with them in a
conceptually adequate way.
-- Ben
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