About Rationally Speaking

Rationally Speaking is a blog maintained by Prof. Massimo Pigliucci, a philosopher at the City University of New York. The blog reflects the Enlightenment figure Marquis de Condorcet's idea of what a public intellectual (yes, we know, that's such a bad word) ought to be: someone who devotes himself to "the tracking down of prejudices in the hiding places where priests, the schools, the government, and all long-established institutions had gathered and protected them." You're welcome. Please notice that the contents of this blog can be reprinted under the standard Creative Commons license.

Monday, October 03, 2005

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.


  1. i hope this comment will make sense as i am not nearly as well-read in philosophy - it seems to me that the problem with the probability of god is that god cannot be defined. therefore it seems that one would always default to the position of infinite possibilities for what god is and therefore infinitely small probability of god.

  2. The god probability model however ignores aggregation. If you take a god description, like the one in the bible, it will probably match a large number of the possible gods outcomes. So, in actual fact, you have to aggregate those, creating a common subset with a much larger probability, though still small. :-)

  3. Talking about "god's probability": Michael Schermer wrote about this a while ago in his Sci, Am. column (Skeptic). It was titled "God's number is up - among a heap of books claiming that science proves god's existence emerges one that computes a probability of 67%". Unfortunately it's not available online, but it's on the July 2004 issue in case you want to dig it up.

    He goes on to quickly describe a kind of Bayesian probabilities, which were used in the "67%" calculation from the book "The probability of god". Indidentally, they started with the PIR (a flat prior), a 50/50 chance of god's being there.

    Making a (not so) long story short, he shows that the number is not specially interesting, since plugging different likelihood ratios (assumptions about the relative merit of observations) in the formula yielded a probability of 2% for god's existence. Which simply means you can get any number you want, as long as you choose the right conditions.

    He then goes on to the obvious conclusion that this kind of thing is scientifically insoluble anyway and the religious would do better staying on their own turf instead of weakening themselves with this kind of approach.

    As Shermer retorically asks: "If faith is tethered to science, what happens when the science changes?"

    Our criationists in attendance will probably say that the science is clearly wrong, then... :-)

    Another interesting related "probability" article is this one: http://quasar.as.utexas.edu/anthropic.html

    Here the authors demonstrate mathematically (at least as far as my limited math capabilities could grasp) that the anthropic principle, even if it weren't slightly ridiculous to start with. DOES NOT support supernaturalism, quite the opposite actually.


  4. a comment from ...Reims (France): the PIR seems to be quite unknown in France. There are no reference for it in the french version of Wikipedia and very few in Google. May be Frenchs are reticent to say "I dont know"
    Perharps, I dont know !


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