From: Ashley Thomas (Ashley.A.Thomas@dartmouth.edu)
Date: Mon Sep 12 2005 - 14:55:48 MDT

If you test a hypothesis in a way which *could* falsify it, and *fail*
to falsify it, then your confidence in the hypothesis should increase.
The amount by which your confidence should increase should depend on
how likely it was that you would falsify your hypothesis.

Suppose we have a bag full of objects, which can be ravens or
not-ravens, and black or not-black. We can reach into the bag with
manipulators which can select an object based on whether it's a raven
or not, or whether it's black or not, but not both.

We hypothesize that all ravens in the bag are black. We can falsify
this hypothesis by finding a not-black raven. There are two possible
experiments we can perform: we can pull out a raven and see if it's
not-black, or we can pull out a not-black object and see if it's a
raven. Without knowing the relative numbers of ravens to not-black
objects, these two searches have an equal chance of finding a not-black
raven. Every time we fail to falsify our hypothesis, our confidence in
the hypothesis increases. If we find a black object when searching
through ravens, our confidence increases because we have failed to
falsify the hypothesis. If we find a not-raven when searching through
not-black objects, our confidence increases because we have failed to
falsify the hypothesis.

How much our confidence increases for either of the two searches
depends on how many of the objects we thought were ravens, and how many
of the objects we thought were not-black. If we think that there are
many fewer objects which are ravens than objects which are not-black,
then our confidence will increase faster by searching through the
ravens (alternate pov: the same number of searches gets us closer to an
exhaustive search faster checking the few ravens than checking the many
not-black objects), but our confidence will still increase a smaller
amount searching through the not-black objects. Even a single result of
a not-raven found when searching the not-black objects (even a purple
goose) increases our confidence, because we tested the hypothesis in a
way which could have falsified it, but got a result which didn't.

If the number of not-black objects in the bag goes to infinity while
the number of ravens in the bag stays finite, then the amount our
confidence in our hypothesis should increase goes to zero when
searching not-black objects for a raven, because it becomes
increasingly unlikely that we'll be able to falsify our hypothesis by
finding one of the finite number of ravens among the infinite number of
not-black objects.

Ashley "ASE" Thomas

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