# Probabilities and Ultimate Posits

Victor Reppert has a short essay on his Dangerous Idea blog on the use of probability arguments in the philosophy of religion. Here I would like to offer my own take on the question.

A principle often invoked by theists making probability arguments is what Robin Collins calls the “prime principle of confirmation (PPC),” which he defines as follows:
Simply put, the principle says that whenever we are considering two competing hypotheses, an observation counts as evidence in favor of the hypothesis under which the observation has the highest probability (or is the least improbable). (Or, put slightly differently, the principle says that whenever we are considering two competing hypotheses, H1 and H2, an observation, O, counts as evidence in favor of H1 over H2 if O is more probable under H1 than it is under H2.) Moreover, the degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which the observation is more probably under the one hypothesis than the other.
George Schlesinger offers a more succinct version of this principle, which he calls “Principle E”:
Principle E: when a given piece of evidence E is more probable on H than on H’ then E confirms H more than H’.
He holds that E can be quite easily justified:
This I believe should appear very reasonable to everyone and can be shown to be so in various ways. After all, when the probability of E on H is zero, that is when p(E/H)=0, then E falsifies H, thus the further p(E/H) is from zero the less E falsifies H or the more it confirms it. Or, one may say, the hypothesis we wish to adopt is to account for E, and it is clear, a hypothesis on which E is highly improbable does not account for E; in fact the very meaning of the expression “account for” implies that the more an hypothesis makes E probable the more it may be said to account for it.
The technical term for what Collins and Schlesinger are talking about is the likelihood of evidence E given hypothesis H. “Likelihood” is the rather unfortunate name given to such quantities by Sir Ronald Fisher, stellar statistician and one of the founders of population genetics. To see what a “likelihood” is, consider the simplest form of Bayes’ Theorem:
p(e/h) x p(h)
p(h/e) = ———————-
p(e)
What Bayes’ Theorem tells us is that if we want to know the probability of a hypothesis h given evidence e, we figure it by multiplying the likelihood of e given h, p(e/h)—how likely the evidence is given the hypothesis—and multiply that times the prior probability of h (more on this below), and divide the result by the total probability of the evidence e, that is, the probability of e and h plus the probability of e and not-h.

Now, what Collins and Schlesinger are telling us is that if we consider two hypotheses h1 and h2 a given body of evidence e confirms h1 more than it confirms h2 if that evidence is more likely given h1 than h2, that is, if p(e/h1) > p(e/h2). Now we have to be careful here. To say that e “confirms” one hypothesis more than another does not mean that e shows that hypothesis to be more probable than the other. It could well be that a given piece of evidence is far more likely on a given hypothesis than on another, but the overall hypothesis of that other hypothesis can be far greater. How can this be? Consider an example:
Colonel Blimp has been found murdered in the library, with an ornamental dagger thrust into his back. Suspicion immediately falls upon the butler and the chamber maid. The butler’s fingerprints were found on the dagger. This evidence strongly implicates the butler (i.e. strongly confirms the hypothesis that he was the murderer) because the likelihood of the fingerprints being on the dagger is much higher given the hypothesis that the butler did it than the hypothesis that the chamber maid did it. So, do we conclude that the butler did it? Not necessarily. As Bayes’ Theorem shows us, likelihoods are only part of the story. Another factor we must consider is the prior probability of the hypothesis; in this case it would be how probable, apart from the fingerprint evidence, it is that the butler did it. Suppose that the butler has an ironclad alibi. Suppose that at the time of the murder the butler was recorded on the security camera, quietly imbibing at his favorite bar at the time of the murder. Now, even with the apparently damning fingerprint evidence, the hypothesis that the butler did it appears highly unlikely.
The upshot is that to say that a piece of evidence confirms one hypothesis more than it does another is only to make a comment about the evidential import of that bit of evidence; it can conclude nothing about the overall probability of one hypothesis over another.
Defenders of theistic hypotheses like to talk about likelihoods, and it is pretty clear why. If your hypothesis is the existence of an all-powerful being who wants x, then the existence of x is certain, i.e., the likelihood of x is one. For instance, if the evidence e is all the “finely tuned” features of the universe, and our hypothesis is that the theistic God exists, then, since the God is conceived as all-powerful and as wanting a universe fit for such splendid beings as ourselves, then the likelihood of a finely-tuned universe given the existence of God is one. That is, given that the God exists, we are certain to have a finely-tuned universe. Unl
ess a naturalistic hypothesis also entails the existence of the finely-tuned features of the universe then the existence of those finely tuned features of the universe confirm the hypothesis that God exists over the competing naturalistic hypothesis. That is, those finely-tuned features are strong evidence for the existence of God. Winning the contest of likelihoods is therefore a cinch for defenders of theistic hypotheses. It is a given (indeed, a necessary truth) that if an all-powerful being wants x, you will have x!
How can a competing naturalistic hypothesis, even if it is true, have any chance at all of winning since the theistic hypothesis has such an automatic advantage vis-à-vis likelihoods? Well, I guess a many-universes hypothesis could be invoked to make the likelihood of finely-tuned features in our universe one or close to one. Then theists would point to other features of the world, such as the existence of morality or consciousness or beauty which, they will assert, God would want (and so are guaranteed to exist) but are not certain on any naturalistic hypothesis. Really, it is a pretty neat trick. There are turtles in the world, and if God wanted turtles, we were sure to get turtles. So, the existence of turtles confirms a turtle-wanting God! Whatever you find in the world you hypothesize an all-powerful being who wants exactly that, and you have a likelihood of one that you will have it! You can even do this with the rotten stuff in the world by claiming, as theodicists always have, that a world with evil is overall better than one without.
One way for naturalists to try to level the playing field is to put the prior probability of theism so low that it more than cancels out the automatic advantage of the theistic hypothesis with respect to likelihoods. If I place the prior probability that God exists near zero (And we are dealing with personal probabilities here, so why not?) then the higher likelihood of theism is insufficient to make that hypothesis probable overall.
My take is somewhat different. It seems most reasonable to me to regard ultimate metaphysical posits as having no definable prior probabilities at all. That is, if U = “The universe exists as an ultimate, uncaused fact,” and G = “God exists as an ultimate, uncaused fact,” then it seems to me that p(U) and p(G) ought to be regarded as undefined. I gave my reasons for this conviction in an essay I wrote some years back:
The assignment of meaningful probabilities upon the hypothesis of atheism is…difficult. If atheism is correct, if the universe and its laws are all that is or ever has been, how can it be said that the universe, with all of its “finely tuned” features, is in any relevant sense probable or improbable? Ex hypothesi there are no antecedent conditions that could determine such a probability. Hence, if the universe is the ultimate brute fact, it is neither likely nor unlikely, probable or improbable; it simply is…If we were in a position to witness the birth of many worlds—some designed, some undesigned—then we might be in a position to say of any particular world that it had such-and-such a probability of existing undesigned. But we simply are not in such a position. We have absolutely no empirical basis for assigning probabilities to ultimate facts.
Put another way, we can make all sorts of probability assessments given the fundamental laws of physics, but what can we say about the probability of those fundamental laws themselves? As Robin Le Poidevin pointedly asks in his terrific book Arguing for Atheism, against what possible background could we judge, say, that it was extremely improbable (as Collins and other defenders of the FTA assert) that the charge on the electron would be 1.602 x 10-19 coulomb? The laws of physics cannot constitute the background since they will either be irrelevant to the charge on the proton or will entail precisely the charge it has.
Since, therefore, we necessarily have no empirical basis for the assignment of probabilities to ultimate facts, my personal probability for ultimate posits is nothing at all—not zero; I just don’t have any. If p(U) and p(G) are thus undefined, then so must be p(E/U) and p(E/G) for any E whatsoever. If there are no likelihoods for evidence given ultimate posits like U and G, then the principles of confirmation cited by Collins and Schlesinger do not apply to such hypotheses.
The upshot is that I am extremely skeptical about the application of probability arguments to ultimate metaphysical posits. This does not mean that I think that arguments between theists and atheists are pointless. I think that we can argue the evidence for or against theism and naturalism. However, I think that those arguments should take the form of inferences to the best explanation, not probabilities.

Let me be plain—unphilosophically plain. Arguments such as Collins’ appeal to “fine tuning” rest on a cheap, shoddy trick, the trick of using a sure-fire, one-size-fits all hypothesis that can work for any e whatsoever. We just say that God wanted e, and, Like Lola, the sexy demoness in Damn Yankees, what God wants God gets. Science does not permit such hypotheses. Lyell rejected catastrophism precisely because catastrophists could “explain” any peculiar geological feature by invoking an ad hoc catastrophe which they would invest with whatever powers were needed to account for that geological peculiarity. Such cheap tricks were tossed out of science long ago, and we need to do the same in philosophy.