From: Jeff Medina (analyticphilosophy@gmail.com)
Date: Sun Sep 11 2005 - 14:58:18 MDT
Eli wrote: "Your statement is certainly not logically correct. I can
easily generate situations in which observing a non-black non-raven
can generate evidence favoring the hypothesis "All ravens are black"
over its alternatives. For example, the set of objects includes 7
ravens and 1 non-raven."
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.
Interestingly, this assumption is also required if sampling a raven
and finding it to be black is to count as evidence for [all ravens are
black].
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.
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).
Best,
-- Jeff Medina http://www.painfullyclear.com/ Community Director Singularity Institute for Artificial Intelligence http://www.intelligence.org/ Relationships & Community Fellow Institute for Ethics & Emerging Technologies http://www.ieet.org/ School of Philosophy, Birkbeck, University of London http://www.bbk.ac.uk/phil/
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