The Principle of Insufficient Reason

I was reminded of the Principle of Insufficient Reason (PIR) while finishing the book by Bouveresse on Wittgenstein vs. Freud (and, among others, vs. Schopenhauer). The PIR states that if one is considering a number n of alternative hypotheses, if these are mutually exclusive, and if there is no reason to prefer one over any of the others, then it is sensible to assign an a priori probability of 1/n to each hypothesis. If this smells a lot like Bayesian analysis to you, your sense of smell is keen indeed.

Though the principle is rather intuitive, it has plenty of detractors. For one thing, if one adopts the chief non-Bayesian approach to probability and statistics, the so-called "frequentist" interpretation (where probabilities are not measures of belief in a hypotheses, but of frequencies that a certain event will occur) then the PIR makes no sense.

Nevertheless, the PIR has a fascinating history, going back to Jacob Bernoulli and Pierre Simon Laplace. Apparently the name PIR was given to it by later authors, in possible reference to Leibnitz's Principle of Sufficient Reason (more on this in a moment). It was the economist John Maynard Keynes who renamed the PIR the principle of indifference and stressed that it is valid only in the rather special case when there is no knowledge indicating unequal probabilities.

The philosophical use of the PIR can perhaps be best understood precisely in contrast to Leibnitz's principle mentioned above. The principle of sufficient reason basically says that every fact has a sufficient reason for why it is the way it is and not otherwise (Leibnitz was the guy who said that we live in the best of all possible worlds, a philosophical statement mercilessly caricatured by Voltaire in his Dr. Pangloss character in Candide). The principle of sufficient reason has often been used to argue for the existence of God, basically as underpinning some version of the cosmological argument, and some philosophers have seen it as a generalization of the dictum ex nihilo nihil fit, "nothing comes from nothing."

The PIR, on the other hand, would suggest that God is just one among many possibilities, and that the God hypothesis should be granted equal probability against its alternatives. Obviously, the question immediately arises about how many mutually exclusive hypotheses there are here. At minimum, we have "God exists" and "God doesn't exist," which would make Leibnitz rather unhappy because the probability of God would drop from 1 (certainty) to 0.5. But of course one could argue that many different kinds of gods have been proposed, and that each one of them should count as a hypothesis. This would make the likelihood of the existence of a particular God (say, the Christian one -- of which, of course, there are also several sub-species) pretty low. Indeed, from there it's a short step to suggest that there is an infinite number of possible gods, and that therefore the probability of existence of any one of them is infinitely small. Leibnitz is getting more and more unhappy! Ironically, then, the Principle of Insufficient Reason becomes a fairly strong anti-theistic (and anti-deistic) argument.

Of course, things aren't that simple, because theists can argue that there are in fact other facts known about the universe that make the PIR invalid (e.g., the exquisite apparent "design" of living organisms). Equally reasonably, tough, atheists can argue that the facts actually point even more strongly against the existence of gods (e.g., we have naturalistic explanations of the complexity of living beings, the problem of evil, etc.).

But these latter arguments go beyond the scope of the PIR, and therefore it is no longer useful in this context. However, the PIR is a simple and yet powerful tool to begin any such discussion, because it at least forces the defenders of one position or the other to state clearly why they think the PIR is not applicable to the issue at hand, i.e. why one thinks that the a priori probability of one hypothesis is significantly higher than the probabilities of all alternatives. And that, of course, is where things become really interesting.