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Byron Jennings | TRIUMF | Canada

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The Raven Paradox and the Flaw in Verification

One of the more interesting little conundrums in understanding science is the raven paradox. It was proposed by Carl Hempel (1905 –1997) in the 1940s. Consider the statement: All ravens are black. In strict logical terms, this statement is equivalent to: Everything that is not black is not a raven. To verify the first we look for ravens that are black. To verify the latter we look for coloured objects that are not ravens.  Thus finding a red (not black) apple (not raven) confirms that: Everything that is not black is not a raven, and hence that: all ravens are black. Seems strange: to learn about the colour of birds, we study a basket of fruit.

While the two statements may be equivalent for ravens, they are not equivalent for snarks.  The statement: Everything that is not black is not a snark, is trivially true since snarks do not exist, except in Lewis Carroll’s imagination. However, the statement: All snarks are black, is rather meaningless since snarks of any colour do not exist (boojums are another matter). Hence, the equivalence of the two statements in the first paragraph relies on the hypothesis that ravens do exist.

One resolution of the paradox is referred to as the Bayesian solution.  The ratio of ravens to non-black objects is as near to zero as makes no difference.  Thus finding 20 black ravens is more significant than find 20 non-black, non-ravens. You have sampled a much larger fraction of the objects of interest. While it is not possible to check a significant fraction of non-black objects in the universe, it may be possible to check a significant faction of ravens, at least those which are currently alive.

But the real solution to the problem seems to me to lie in different direction. Finding a red apple confirms not only that all ravens are black but also that all ravens are green, or chartreuse, or even my daughter’s favorite colour, pink.  The problem is that a given observation can confirm or support many different, and possibly contradictory, models.  What we do in science is compare models and see which is better. We grade on a relative, not absolute scale.  To quote Sir Carl Popper:

And we have learnt not to be disappointed any longer if our scientific theories are overthrown; for we can, in most cases, determine with great confidence which of any two theories is the better one. We can therefore know that we are making progress; and it is this knowledge that to most of us atones for the loss of the illusion of finality and certainty.

We do not want to know if: All ravens are black is true but rather if the statement all ravens are black is more accurate than the statement all ravens are green. A red apple confirms both statements, while a green apple confirms one and is neutral about the other. Thus the relative validity of the two statements cannot be checked by studying apples, but only by studying ravens to see what colour they are.  Thus, the idea of comparing models leads to the intuitive result. Whereas, thinking in terms of absolute validity, leads to nonsense:  Here, check this stone to see if ravens are black. Crack, tinkle (sound of broken glass as stone misses raven and goes through neighbor’s window)

We can go farther. Consider the two statements: All ravens are black, and Some ravens are not black. The relative validity of these two statements cannot be checked by studying apples or even black ravens. Rather what is needed is a non-black raven. This is just the idea of falsification. Hence, falsification is just a special case of comparing models: A is correct, A is not correct.

In practice, all ravens are not black. There are purported instances of white ravens. Google says so and Google is never wrong. Right? Thus, we have the statement: Most ravens are black. This statement does not imply anything about non-black objects; they may or may not be ravens.  Curious… this whole raven paradox was based on a false statement and as with: All ravens are black, most absolute statements are false, or at least, not known for certain.

Even non-absolute statements can lead to trouble. Consider: Most ravens are black, and: Most raven are green. So we merrily check ravens to see which is correct. But is it not possible that the green ravens blend in so well with the green foliage that we are not aware that they are there? Rather like the elephants in the kid’s joke that paint their toe nails red so they can hide in cherry trees. Works like charm. Who has seen an elephant in a cherry tree?  We are back to the Duhem-Quine thesis that no idea can be checked in isolation. Ugh. So, why do we dismiss the idea of perfectly camouflaged green ravens and red-nailed elephants? Like any good conspiracy theory, they can only be eliminated by an appeal to simplicity. We eliminate the perfectly camouflaged green raven by parsimony, and as for the red apple, I ate it for lunch.

Additional posts in this series will appear most Friday afternoons at 3:30 pm Vancouver time. To receive a reminder follow me on Twitter: @musquod.

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