Hope for Hempel's Paradox

Try induction for wellness.

Posted Oct 30, 2012

I don't particularly care about the usual. ~ Nassim Taleb

I finally read The Black Swan. It is a wonderful and infuriating book. It is wonderful because it is written with passion, it makes great use of stories, and it pillories the hubris of the mechanistic, linear approach to science. The book is infuriating because the passion is overblown, most of the stories are self-indulgent, and not all Gaussian science is the kind of devil’s work that Dr. Taleb suggests it is.

The book is a commentary of David Hume’s critique of induction, or rather Hume’s critique of a primitive form of induction that seeks to extract general truths from a finite number of observations. Hume noted that no matter how many white swans you observe, you cannot be sure that the next swan will not be black. This seems right because there is no law in nature that forbids the existence of black swans. But what if there were such a law? My stock example is that pigs do not fly (unaided) because of gravity. Every non-flying pig adds to the confirmatory count, but the next pig might be a flying one (would it still be a pig, though?). What is this law of gravity? If it is another way of saying that no flying pig has ever been observed, we are back to Hume’s swans and their whiteness. If, however, the law is something larger, beyond the enumeration of confirming instances, then we can say that pigs will not fly unaided, as long as gravity prevails.

Given his skepticism regarding the logic of induction, Hume had to be creative when commenting on the validity of alleged miracles. Take reports of resurrection. Everyone who has died so far has remained dead. Yet, Hume would caution that we cannot infer from this unbroken record that the next corpse will not return to life. If the dead was reported to have risen, society would treat the event as a miracle. Hume was committed to not rejecting never-before-seens out of hand (he could not, lest he betrayed his critique of induction). So he found another way. Hume pointed out that the probability of the presumed witness of the resurrection being mistaken or lying was greater than the probability of a true resurrection having taken place. Notice how this distinction is impossible to make without taking into account the relative frequencies with which the events in question (self-deception or lying vs. resurrection) were observed in the past. Induction is back and going strong, even in Hume’s own hands. It is just not designed to yield absolute knowledge – but who needs that anyway?

Back to Dr. Taleb. Chapter 5 of his book is entitled Confirmation Shmonfirmation. Taleb is not above sarcasm, cynicism, or ridicule. The ad hominems fly. Most scientists sicken him (yet, he appears to seek their company at parties). In chapter 5, he hammers home Hume’s message that induction never provides certainties. Yet, Hume himself never left home without it because it helped him get on with life. Taleb’s correct and important point is that induction fails to predict when a black swan (or flying pig, or a hurricane hitting New England) will occur. But why throw out the baby with the bathwater?

One of the grenades Taleb lobs at induction is Hempel’s paradox. He writes (pp. 59-60) “If you believe that witnessing an additional white swan will bring confirmation that there are no black swans, then you should also accept the statement, on purely logical grounds, that the sighting of a red Mini Cooper should confirm that there are no black swans” (emphasis his). Taleb does not elaborate. He trusts, it seems, that this presumed confirmation is so ridiculous that the reader will take it as a confirmation for the view that confirmation is ridiculous.

Hempel’s paradox is actually no true paradox. It is treated as a paradox only because its logical validity offends our intuition. In the abstract, Hempel asserts that “All X are Y.” If X, then Y. This is modus ponens, and it is a valid inference. Another valid inference is that If not-Y, then not-X. This is modus tollens. All this is basic deductivie logic. Induction comes into play when we strengthen the belief that X implies Y when observing an X that is Y or a non-X that is non-Y. Hempel’s black raven example and Taleb’s white swan example set us up for doubt because there are far more non-ravens and non-swans in the world than there are ravens or swans. It’s a base rate trap. The rarity of X guarantees that observing a non-X that is non-Y only counts for little – but it does not count for nothing (Good, 1960).

Here's a numerical example. There are 2 ravens and 8 other entities (swans, Mini Coopers, peanut butter sandwiches, whatever). Three of the entities are black and 7 are not. Without having observed anything, we estimate the probability of both ravens to be black as .09 (3/10 x 3/10). Having observed 1 black raven, the probability that both ravens are black goes up to .22 (2/9). Surprisingly – but not paradoxically – the probability that both ravens are black goes up from .09 to .11 if 1 non-black non-raven is observed. This is so because this observation removes 1 entity from the world (so that p = 3/9 x 3/9). When the world gets large, this effect becomes negligible and this is what our intuition says.

If Taleb believes that his black swan (white raven) will come, what makes him so sure, if it is not induction?

Good, I. J. (1960). The paradox of confirmation. The British Journal for the Philosophy of Science, 11, 145-149.

Taleb, N. N. (2007/2010). The black swan. New York: Random House.

For an excellent review of The Black Swan see:

Blyth, M. (2009). Coping with the black swan: The unsettling world of Nassim Taleb. Critical Review, 21, 447-465.