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) = ———————-
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. Unless 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.

About Keith Parsons
  • Wes


    Interesting read.

    Paul Draper has been doing some really fascinating new work on prior probability (well, he doesn't describe it this way, but it fits the bill). He's developing a new theory of simplicity that touches on burden of proof arguments, which of course is relevant to prior probabilities.

    He sent me a copy of a paper he's been presenting at conferences, and I'm sure he'd forward along to you if you are interested.

    I personally think naturalism does well in these kinds of Bayesian comparisons. I'm not sure it is so easy for the theist to get to the 1's given many of their typical auxillary assumptions.

  • Keith Parsons


    Thanks for the tip. Draper is always interesting and informative. Fine philosopher. Wish we could get him to contribute here. I'll put a very heavy burden of proof on any attempt to give meaningful probabilities to ultimate metaphysical posits.

  • TaiChi

    Hi Keith, interesting post.

    "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."

    I think I can understand your reasons for the first and third sentences here, but I'm not sure why you think the second one is correct – can you explain how it follows from p(G) being undefined that P(E|G) is undefined?
    (Not that this is a criticism of your position – with undefined prior probabilities we cannot determine posterior probabilities, regardless of whether or not conditional probabilities are also undefined – I'm simply curious as to why you believe this additional bit is true).

    "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? .. We have absolutely no empirical basis for assigning probabilities to ultimate facts."

    I think it's quite clear that those who use fine-tuning arguments are appealing not to physical probabilities, but to logical probabilities instead, where logical probabilities are determined by truth value across the set of all possible worlds, and physical probabilities are determined by a subset of worlds, those in which our physical laws hold.
    If that's roughly correct, then what you would need in response to fine-tuning and other arguments of this sort is a defense of the view that there are no other probabilities than physical probabilities. Merely noting that physical probabilities are undefined for that which we define physical probabilities in terms of – i.e. physical laws – is going fail to engage with the argument being offered.

  • Bradley Bowen

    "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."

    There are many believers who will twist and bend over backwards to try to make the facts fit with their cherished beliefs. So, your objection applies to many people and many arguments. However, I'm not convinced that this objection is universally applicable and logically unavoidable.

    Swinburne attempts to derive the "wants" or purposes of God from a fairly thin and abstract conception of God as a perfect person, as a person who is perfectly powerful, perfectly knowing, and perfectly free. Granted, Swinburne may have various biases and religious assumptions that distort his thinking in favor of a desired outcome. So, your objection probably has some merit against his views about God. However, it is not clear to me that Swinburne's attempt to objectively determine what God, if he existed, would be likely to "want" or do, is a logically impossible task, or even a completely hopeless task.

    Can you explain why you think that no thinker, even someone who has less of an ax to grind than Swinburne, could ever possibly succeed at the task that he sets for himself?

    Also, doesn't your view of the hopelessness of such a task, completely undermine the problem of evil as an argument against the existence of God? How can the evils of this world count against the creator being a perfect person, if it is logically impossible to objectively determine what sort of purposes and values a perfect person would have, and what sort of world a perfect person would be likely or unlikely to create?

  • Dianelos Georgoudis


    I trust we agree that the butler’s fingerprints are evidence *for* the butler being the murderer, even given the fact that there is other much stronger evidence for the butler not being the murderer. Thus I trust we agree that you are not here offering a counterexample to Schlesinger’s principle.

    It is true that on theism if God wants X then X obtains, but I don’t see that as a “built-in” advantage for theism, given that, for example, the problem of evil is based precisely on this proposition. “If God wants X then X obtains“ represents a huge restriction on theism as long as we are confident about what an all-good and all-powerful being would want.

    The arguments from the apparent fine-tuning of the universe and from evil belong to a type of argument which centers on the internal coherence of worldviews. Its form in the case of the fine-tuning goes like this:

    1. If a worldview is more internally coherent than another, then the former worldview is more probably true than the latter. (By “internally coherent worldview” I mean a worldview in which all facts are ontologically accounted for, i.e. in which all facts are grounded. By “fact” I mean a belief one is certain about for all practical purposes.)

    2. Therefore, if a fact is more easily accounted for in a worldview than in another then this fact is evidence for the former and against the latter worldview.

    3. The fact of the apparent fine-tuning of the universe is easily accounted for by theism, but is not easily accounted for by naturalism (since, as it seems, it can only be accounted for by the multiverse for which no good evidence exists).

    4. Therefore the fine-tuning of the universe is evidence for theism and against naturalism.

    (That's the same form of the argument from evil. As long as there is not a theistic account for the existence of evil which works as well as the naturalistic account, the existence of evil is evidence for naturalism and against theism. Now, considering the above argumentation, I realize that an inherent epistemic problem is that such arguments depend on one’s subjective judgment about what is “easily accounted for”. For example in the case at hand the naturalist may concede that there is no good evidence for the multiverse, but argue that there is also no good evidence that God would want to create a universe in which intelligent life would naturalistically evolve. One way or the other ontology looks like an engineering project: To build a conceptual reality which better fits with all the facts we have.)

    Now, at the very least there is a hypothetical naturalistic account, namely the multiverse, which accounts for the apparent fine-tuning of the universe. There are other facts we know about from the physical sciences for which only highly implausible naturalistic accounts exist. The physical fact which I think represents the greatest problem for naturalism is the deeply mathematical nature of the universe. The deeply mathematical nature of our universe is not accounted for by the multiverse. We know that such a deeply mathematical nature is not needed for natural evolution (including for the natural evolution of intelligent beings), nor is it needed for the success of the physical sciences. But it is there, a remarkable fact which precisely physicists (Taner, are you reading this?) should be keenly aware and wonder about. My point is that the deeply mathematical nature of the universe is the strongest evidence for theism and against naturalism to have come out of the physical sciences yet.

  • Keith Parsons


    Thanks for your perceptive comments and stimulating questions.

    First, since p(E/G) =

    p(E & G)

    if p(G) is undefined, I would think p(E & G) and p(E/G) would have to be also.

    Second, the kind of probability Collins invokes is epistemic probability which he defines as follows:

    "Roughly, the epistemic probability of a proposition can be thought of as the degree of confidence or belief we rationally should have in the proposition. Further, the conditional epistemic probability of a proposition R on another proposition S—written as P(R/S)—can be defined as the degree to which the proposition S of itself should rationally lead us to expect that R is true. Under the epistemic conception of probability, therefore, the statement that the fine tuning of the cosmos is very improbable under…[a hypothesis]is to be understood as making a statement about the degree to which the [hypothesis] would or should, of itself, lead us to expect cosmic fine-tuning."

    Collins then argues that the epistemic probability of fine tuning on the hypothesis of theism is higher than the epistemic probability of fine tuning given the Atheist Single Universe Hypothesis (i.e., the hypothesis I defend which is that there is one universe that exists as a brute fact).

    My response is that in our ordinary assessments of epistemic probability, we base our expectations on a body of empirical or theoretical information that we take as background knowledge. For instance, we expect a tossed coin to turn up heads about fifty percent of the time given that it is a fair coin. We think that there is a 40% chance of rain tomorrow given a body of meteorological knowledge. Ultimately, we base our rational expectations about physical phenomena upon our knowledge of the laws of physics. But against what background or given what information are we supposed to judge, as Collins would have us do, that the laws of physics themselves are highly unexpected?

    My question to Collins, then, is, given the necessary lack of all empirical or theoretical background information, how do we arrive at rational expectations for the actualization of possible universes? On what possible basis do we say that the postulated single, brute-fact universe is or is not likely to be fine-tuned? Rational expectations need a rational basis, and I don't see what that could be here.

    Collins answers that the Principle of Indifference can give us rational expectations here. Graham Oppy corrosively addresses this suggestion at length in his terrific book Arguing About Gods.

  • Keith Parsons


    I just had a terrific meal at my favorite Greek restaurant, so I have to say "Opa!"

    No, I consider Schlesinger's principle to be false, at least in the unrestricted form he offers it. I think it should hold only for admissible hypotheses. What is an admissible hypothesis? Consider an example:

    E = Patient P is exhibiting various forms of bizarre and demented behavior.

    H1 = Patient P is suffering from mental degeneration due to syphilitic infection.

    H2 = Patient P is possessed by a blue demon.

    Now, we have to say that p(E/H1)is quite low. However, such behavior is precisely what we expect when people are possessed by blue demons. Therefore, p(E/H2) is very high. I take it, though, that no rational person would say that evidence E confirms H2 over H1. H2 is not an admissible hypothesis. We just throw it out and do not even consider any evidence for it. If someone seriously suggests such a hypothesis, we do not argue with them; we just raise our eyebrows and try to change the subject.

    Well, maybe some would say that the evidence does confirm H2 over H1, but that the prior probability of blue demons is so low, that the overall probability of the syphilis hypothesis is higher despite the much higher likelihood of E given H2.

    I am dubious about that kind of a response. On what basis do we say that the demons, or angels, or ancient astronauts, or poltergeists, or leprechauns, or witches have an extremely low prior probability? I can't think of any really convincing reasons. Have we disproven the existence of such putative paranormal entities? I would be at a loss to disprove the existence of Zeus, Thor, or Quetzalcoatl.

    No, I think that the reason we reject paranormal hypotheses is not that we have found them to be improbable, but that centuries of bitter experience have taught us their enormous obscurantist potential. The obscurantist potential of paranormal hypotheses comes from their untestability and the automatic advantage they usually have with respect to likelihoods (ghosts, gods, and demons are simply attributed with occult powers to do whatever they are supposed to do). Since paranormal hypotheses have a pesky way of hanging around and getting in the way when real explanations are discovered, we have learned simply to rule them out.

    The God hypothesis is, of course, the ultimate paranormal postulation. Like ancient astronauts, poltergeists, demons, etc., it is an untestable hypothesis and has the same sort of automatic advantage in likelihoods. God is invested with a paranormal, occult power (omnipotence) that gives a likelihood of one that we have whatever God wants. Why, then, would it be unreasonable for me to classify theistic hypotheses with those postulating demons, ancient astronauts, poltergeists, etc., and view them as deserving the same short shrift? In short, why not be a methodological naturalist in philosophy just as in science?

  • Victor Reppert

    So, Keith, I take it that you are abandoning the probabilistic argument from evil.

  • Keith Parsons


    I think not; at least not if the argument requires us to conditionalize on ultimate posits. My little essay "A Simple Statement of the Problem of Evil" now available at Secular Web (and which I am here shamelessly plugging) gives my latest thoughts on the problem of evil.

    The "skeptical theists" say that no matter how gross and pointless an evil appears to us, omnipotence working over unlimited time might someday, somewhere, somehow, bring about a good that necessitated that evil and which is good enough to justify God's permission of that evil. Imposing this sort of burden of proof makes any sort of evidential argument impossible. An 88 year old Nazi nut walks into the Washington D.C. Holocaust Museum and kills two security guards. Given everything humans know about good and bad, this appears an utterly pointless evil. Yet to satisfy the skeptical theists, we would have to show that sub specie aeternitatis it will still be pointless. I have to admit that this stumps me. I don’t see how I can make any judgments about what an omnipotent being can or cannot do sub specie aeternitatis. In other words, it seems to me that William Lane Craig was right when he said to me once that (paraphrasing and adding emphasis) we are not in an epistemological position to judge with any confidence whether it is probable or improbable that God has (sub specie aeternitatis) a morally sufficient reason for permitting evils.

  • Keith Parsons


    OK, but, obviously, such an argument cuts both ways. If Parsons cannot have (given the evil evident in the world) any confidence that God does not have morally sufficient reasons for permitting evil, then Craig, pari passu, can have no confidence that he does. God might, so far as we can tell, have a reason for permitting the Holocaust Museum shooter. Then again, maybe not. Maybe even sub specie aeternitatis nothing justifies that evil, or God could have achieved great enough goods without that evil. We just cannot say. Indeed, the believer seems to be in a much worse epistemological situation. Recall that, in the whole long history of human and animal suffering, only one instance of all that suffering has to have been gratuitous, and God’s existence is impossible. If one Diplodocus once suffered needlessly in the Jurassic, then God does not—indeed, cannot—exist. The theist has to be confident that, over geological time, no instance of suffering has been gratuitous, and, given the imponderables thrown up by the skeptical theists, how could they possibly have such confidence? In short, skeptical theism is hoist with its own petard, and, as far as I can tell, simply reduces to agnosticism.

  • Keith Parsons

    Clarification: The first sentence of my reply to Victor is misleading. I meant to say that I AM abandoning the evidential argument from evil, at least if it requires us to conditionalize on ultimate posits (like God).

  • TaiChi


    Regarding your response to my first question, I'm still not sure that's right. That we can describe p(U|G) as mathematically equivalent to another description which employs undefined variables doesn't seem to show that p(U|G) is itself undefined unless p(U|G) is a purely formal device, whose meaning would be exhausted in that description taken as a stipulative definition. But I doubt that we have that here, since p(U|G) does seem to have its own sense: your remarks on how easily the theist ascribes to (U|G) the probability of 1 indicates that p(U|G) is intelligible independently of estimates of p(G), p(U), and of the formula for p(U|G).
    But suppose I'm wrong, that the presence of an undefined variable in the equation for p(U|G) does show that it is also undefined. In that case, I believe that for every p(x), p(x) would have to be undefined, since it is trivially easy to fit p(x) into an equation which contains an undefined term. Let p(R) be the probability of rain tomorrow. There is an equation for p(U|R), which employs the undefined p(U): p(U|R) = p(U&R;) / p(R). The equation p(R) = p(U&R;) / p(U|R) can be derived from this, and so we have an equation which equates p(R) to a mathematical description involving undefined variables. By the reasoning above, p(R) would therefore also be undefined, and since p(R) was chosen arbitrarily, any p(x) would likewise be undefined.

    "Collins answers that the Principle of Indifference can give us rational expectations here. Graham Oppy corrosively addresses this suggestion at length in his terrific book Arguing About Gods."

    That's how I'd answer it too, so obviously I'll need to read Oppy's stuff. Thanks for pointer.

  • Keith Parsons


    Excellent points! Being able to get feedback of this quality is one of the main reasons I blog. I don't have time for a response right now since I will not be doing anything on SO for the next couple of weeks. Please be patient, though, I would very much like to hear more from you on these issues.

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