mathematics, probability, statistics

The extended Hempel’s paradox

The absence of proof  isn’t yet proof of absence. However, the absence of X makes the existence of X less likely with every passing second in which X fails to present itself. This is also the essence of the Hempel’s Law, with which you can become familiar here  http://en.wikipedia.org/wiki/Raven_paradox

We won’t claim, that all ravens are black. Too many grey, almost white ravens live on Vancouver Island, instead we shall say:

All ravens are non-green.

Hempel assures us that:

When you see a non-green raven, the probability that all ravens are non-green, goes up!  (1)

Which is hardly a surprise. Our intuition happily agrees, but Hempel says more:

When you see a non-green cat, the probability that all ravens are non-green, goes up!  (2)

It gets stranger:

When you see a green cat, the probability that all ravens are non-green, goes up!  (3)

The absence of a green raven persists, so it is even more likely that there are no green ravens at all. And this is the usual conclusion of the story, an  extension follows.

When you only hear a raven call, the probability that all ravens are non-green, goes up!  (4)

Why is that? Nothing like a green raven sighting has happened. So it is even more likely that there are no green ravens in this world, at all! Whatever is not a proof of a green raven, makes it slightly less probable. Very well, but does it get even stranger? I guess, it sure does.

When you actually see a green raven, the probability that all ravens are non-green, goes up!  (5)

Why? Because if you see a green raven it’s more likely that it’s some good people’s joke or a hallucination caused by drug abuse or  a male duck and you are a lousy taxonomist. So something else must have happened and your green raven observation may be proof of the fact that all ravens are non-green.

ravens

DISCLAIMER:

We assumed that a green raven is highly improbable. For a brown one, it might be a different story, this extended principle maybe wouldn’t hold. It’s questionable if a green raven is odd enough. Maybe a pink raven with golden spots would be a better illustration of  (5). Well, we need something which is less likely than the “increase of the probability of non-existence by non observing it once again”. 

Generally speaking, an observation of X can sometimes be evidence AGAINST X!

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Image by msjr

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