The Principle of Indifference

Anyone who has been reading this blog for any length of time will know that I’m not a fan at all of Bayesian epistemology.  Richard Carrier is one of its biggest advocates, constantly casting all epistemological questions in its light and developing its components in his normal, inimitable style of insisting that it’s all just correct.  In this post, he talks about “The Principle of Indifference” and tries to defend it against all attacks.  As is normal for Carrier, he is as much if not more concerned about it as a practical principle as opposed to just being a theoretical or philosophical principle.  However, the problem is that in this article his own defenses actually destroy it as a practical principle and his use of it in the Bayesian context actually weakens it as a philosophical one as well.  In short, his defenses of it against the charges show that it can never actually be used in real life and that all of the problems with it that he needs to mount the defenses against are all caused by treating it as something expressing probabilities.

So let’s start with the basic idea of what the Principle of Indifference is:

In lay terms, the PoI means this: when you have no information making any logical possibility more or less likely than any other, they are, so far as you know, all equally likely. For example, when you say you don’t have any idea whether a claim’s truth is more likely than 50% or less likely than 50%, you therefore mean it has a probability of 50%. So far as you know, that is. In other words, this is a principle of epistemic probability, not “objective” probability. Because it is a statement about what you know, not about what’s the case apart from what you know. For any claim h, if all you know is that h is as likely true as false, then what you know is, by definition, that h is as likely true as false. And that translates into mathematical terms as “the probability of h is 50%.” That’s simply a description of your state of knowledge at that time.

So the base principle isn’t unreasonable, as it basically says that if you have no idea and no information that would lead you to thinking that one option is more likely than another, then to you they are all equally likely to be the case.  But note that then Carrier immediately starts translating that into direct probabilities, which he needs to do to maintain his idea that all epistemology is ultimately Bayesian.  This means that what he can’t advocate for is someone simply saying that this probability or likelihood is indeterminate, and so someone treating the situation as one where they really have no idea how likely any of these are.  Carrier would no doubt argue that what I just said is simply stating that the probability is 50% or that they are all equally likely, but this is false.  For a proposition that is, say, testable empirically, if the two possibilities are sufficiently distinct so that specific experiments would easily make it clear that one was much more likely than the other, then I’d know that one of them really is far more likely than the other, but that I wouldn’t know which one that is until I actually look.  Carrier can argue that I should treat the two as equally likely, but I could easily respond that if the likelihood is at all important to me I shouldn’t consider them equally likely, but should instead simply go and look to see which it is.  It wouldn’t be a smart move, say, to make decisions or plans on the basis that both were equally likely when I could indeed easily check to see which one is more likely.

The advantage of it being indeterminate over having to give it a probability is that this behaviour flows naturally.  If I don’t need to know how likely it is or need to act on which one I think is correct, then I can ignore and don’t have to consider it as a probability at all.  If someone asks which I think is more likely, I need not answer that they are equally likely, but instead can simply answer that I have no idea.  But if I need to act on either of the options or even on the basis of what I think the likelihood of each options is — for example, whether I should hedge my bets on which one is true — I immediately recognize the absolutely true fact, even for Carrier, that I really need to know more before I can act on that likelihood.  Even under the Bayesian model, we really need to be able to distinguish between the cases where the probability is 50% because that’s what I’ve come to after considering a lot of evidence and all the evidence I can reasonably consider and where the probability is 50% because I didn’t start with any evidence or information at all and didn’t bother to gather any additional information.  In the former case, it’s far more reasonable to act as if each option is equally likely whereas in the second case it’s far more reasonable to go and look to determine what the real likelihood is than to assume that they are all equally likely.

But as it turns out all of the issues raised against the Principle of Indifference precisely follow from it being considered as a probability of 50%, and that Carrier’s defenses against those charges end up killing it as a practical principle.  Let me just look at one of the more indicative ones, which is the Partition Problem:

For instance, if you assume the absence of all pertinent information and then demarcate h and ~h as “God exists or God does not exist” you might assume the PoI entails P(h), in this case P(God exists), is 50%. But on another day you might demarcate h and ~h as “the Christian God exists or the Christian God does not exist” and assume the PoI entails the same conclusion. But it cannot be the case that the probability that just any god exists is exactly the same as that the specifically Christian God exists. Since there is a nonzero probability that a non-Christian God exists instead, the probability “that God exists” must necessarily be higher than the probability “that the Christian God exists,” as the former includes the latter—and all other gods. The “and” here is additive: all their probabilities add together. But you can’t add any positive number to x and still have x; you will always have some number larger than x.

This problem arises not only there. Because “God exists” contains a large number of assumptions—each of which must demarcate the probability space. For example, “God exists” entails “the supernatural exists,” but that the supernatural exists cannot be as likely as “the supernatural does not exist” and at the same time “God exists” be as likely as “God does not exist.” Because it is logically possible the supernatural exists and God does not. So P(supernatural) must be higher than P(God). It can’t be the same. And here the PoI should give us a different result: if we assume no information exists to render any of these possibilities more likely than its converse, then so far as we know P(supernatural) must be 50% and so far as we know P(God|supernatural) must be 50%. And that actually gets us the conclusion that P(God|supernatural) is 25%, not 50% (as 50% x 50% = 25%).

The basic idea is this:  there will always be a number of related propositions to pretty much any proposition that we are considering.  So if we are to consider a proposition for which we have no knowledge or evidence as having a probability of 50%, then what do we do with those related propositions if we don’t have any knowledge or evidence for them either?  Well, we’d be adding or multiplying them, which would give them a probability that isn’t 50%.   But the Principle of Indifference says that for any proposition where we have no information as to how likely each option is we must assume that they are equally likely, which means in those cases where there are only two options — which is generally true for propositions since they can only be true or false — if we do treat them as probabilities then by the PoI we would have to give them a probability of 50% and yet the probability calculation would insist that we not assign that probability to them, thus contradicting the PoI.

Carrier’s defense is to argue that in those cases we actually do have information as to the probability:

And that’s generally the solution to the Partition Problem: to properly account for dependent probability in any hierarchy of assumptions. Since P(God|~supernatural), i.e. the probability that God exists if the supernatural doesn’t, is zero (unless we change what we mean by “God,” but then we would be talking about a different thing—more on that shortly), then necessarily the PoI first operates on the demarcation between the supernatural and the natural, and then operates on God only inside the domain of the supernatural. In other words, we are talking about P(God|supernatural) and not just P(God). Discovering this fact—which is a logical fact, about the definition of God (and thus what we are “actually” seeking the probability of)—is information that changes the probability.

So in a broad sense, the PoI doesn’t apply: because you know it is not “as likely as not” that God exists, but rather that it’s “as likely as not” that the supernatural exists, and then “as likely as not that God exists if the supernatural exists” (again, assuming we have no other information bearing on either question). But the PoI still narrowly applies: within the probability space where we have no distinguishing information. Within the set of all logically possible worlds in which “the supernatural exists,” we have no information indicating that “God exists” is any more likely than not. But that entails P(God) = 0.25, not 0.50, as just noted. So once we do the math correctly, the Partition Problem solves itself.

But note that there and throughout the post any challenge to the PoI based on these probability calculations is solved by saying that in those cases we actually do know something about the proposition and so the probability calculations are actually correct and giving us the right probability.  We should immediately be able to see that any move like this weakens the PoI as a practical principle, because it limits its applicability.  Without doing all the work for every proposition in existence, we can wonder how many propositions have no such dependencies.  For example, while Carrier exempts atheism from being dependent on the supernaturalism proposition, could it be dependent on the naturalism proposition?  Or a host of other propositions?  So if we didn’t know anything about atheism itself, could we have a number of other propositions that would reduce its probability in the same way that Carrier has reduced theism’s and Christianity’s?  It seems pretty likely, so it seems that in most cases we’d always have additional information that we could get just from knowing the propositions exist and logically analyzing them without having any specific evidence or information at all.  Given that as per Carrier the PoI doesn’t apply in such cases, it looks like the PoI is going to legitimately apply very much at all in real life.

But wait, it gets worse.  The PoI is supposed to be applied when we don’t have additional information telling us which option is more likely.  However, we always have some information telling us how likely a proposition is, since propositions do not simply spring fully formed to our minds.  No, propositions always have a source.  Someone tells us about it.  We read it in a book.  We sit in our armchair and imagine it up.  And these sources all have a reliability that we can determine and that we have all the information we need to determine as soon as the source provides a proposition for us to ponder.  So to not consider that information that we clearly have seems to be epistemologically negligent.  So we shouldn’t be relying on the PoI here and pretending that we have no information about the proposition, but should instead be using the information that we have to determine a more reasonable probability.  So if the PoI does not apply when we have additional information about the relative likelihoods, then the PoI will almost never apply.  And that’s not taking into account the fact that for almost all propositions we will encounter and certainly for all propositions where we care about those likelihoods we will have background knowledge that is relevant to it, or at least will have a way forward to gather information and evidence to determine what those likelihoods are.  So as a practical epistemological principle, we will or at least should be gathering the information to determine “real” likelihoods, and not use the PoI at all.

And here is where we get into the philosophical problems.  The first is the one I just noted:  epistemologically speaking, when confronted with the PoI what a good epistemology will advocate is that you don’t assume that all of the options are equally likely, but instead that you go out and figure out what the real likelihoods are.  After all, it is premised on the idea that you don’t actually have any real evidence or information about them, but if you care about which of them are correct surely you should go out and look for such evidence or information, no?  The only time you wouldn’t need to is for propositions where you really don’t care what the likelihood is, almost certainly because the proposition is irrelevant to your life.  But then what’s the point of assigning any probability to it at all?  If you don’t care, then you will never use the fact that it has a probability of 50% and so have no need to assign it one, and if you come to care later considering the probability indeterminate should prompt you to do the epistemically superior thing and go try to get evidence to get a better set of probabilities.  So one should, in fact, never use the PoI, and should instead use any situation where they think they might want or be able to use the PoI as an indication that they need to go out and gather more information.  So a good epistemology should never stop with the PoI.

The second one is that all of the issues that Carrier runs into are spawned from the fact that he translates the “equally likely” there to a specific probability, which then allows for probability operations, which then move the probabilities away from the probability that the PoI would try to assign, which then forces Carrier to claim that the PoI doesn’t really apply in those cases.  So then at a philosophical level we can ask, as I did above, why we’re even bothering to assign a probability to it at all.  Why not just leave it indeterminate?  If we did that, then we wouldn’t have any of these problems, and it would allow us to easily distinguish between propositions where we have no information and propositions where based on significant information we have determined that each option really is equally likely based on the information we have.  Saying that the former case is an example of the latter is indeed quite misleading since in the latter case we are clearly more justified in saying that each option is equally likely because we went out and gathered the information and did the mathematical work to determine that, whereas in the former case we have no information and so are merely assuming it on the basis of a complete lack of information.  So leaving the PoI case indeterminate makes it abundantly clear that we are working from a position of a complete lack of information instead of working from a position where we may have lots of information that supports all the options equally well.  Carrier’s use of the PoI cannot make this distinction on its own, and runs him into issues with probability calculations forcing him to abandon the PoI anyway.

The PoI doesn’t work as a practical principle because in real life we will almost always have at least some additional information that will let us calculate proper probabilities.  It also doesn’t work as a philosophical principle because in any case where it would entail if we cared about those likelihoods we really should be going out and gathering information before assigning any likelihood to the options.  And Carrier’s translation of it to a probability in line with Bayesian epistemology causes the issues that he needs to address, and ends up forcing him to abandon the PoI to avoid contradictions and paradoxes.  If Bayesian epistemology really depends on the PoI, so much the worse for Bayesian epistemology.