Possible Worlds and the Probability of God

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In a previous article I claimed that theistic arguments are of little value in providing undeniable proofs of God’s existence. Denying the reality of God is, I believe, more a matter of the will and passions than of reason and intellect. This is one of the reasons that ontological arguments, which rely on reason and intuition alone, are almost completely unpersuasive to those with agnostic inclinations.

Such arguments, though, are similar to other types of theistic arguments in that their usefulness is independent of their value as convincing proofs. The acclaimed philosopher Alvin Plantinga came to a similar conclusion after initially failing to believe that his ontological argument was a “successful piece of natural theology.” He later realized that he had set the standard for success unreasonably high:

In God and Other Minds, I argued first that the theistic proofs or arguments do not succeed. In evaluating these arguments I employed a traditional but wholly improper standard: I took it that these arguments are successful only if they start from propositions that compel assent from every honest and intelligent person and proceed majestically to their conclusion by way of forms of argument that can be rejected only on pain of insincerity or irrationality. Naturally enough, I joined the contemporary chorus in holding that none of the traditional arguments was successful. (I failed to note that no philosophical arguments of any consequence meets that standard; hence the fact that theistic arguments do not is of less significance than I thought.) I then argued that the objections to theistic belief are equally unimpressive; in particular, the deductive argument from evil (the argument that there is a contradiction between the existence of God and the existence of evil), I said, is entirely unsuccessful. So I saw, as I thought, that neither the arguments for the existence of God nor the arguments against it are conclusive. (Warranted Christian Belief, p. 62.)

Although Plantinga’s formulation of the ontological argument is inconclusive, it provides another plank in the defense of the claim that belief in the existence of God is more probable, more plausible, more reasonable, and more rational than its denial.

Part I — God and Possible Worlds (or Why Ontological Arguments are Really Complicated)

An ontological argument is any one of a category of philosophical arguments for the existence of God the rely on the nature of being. Ontological arguments are a bit complicated and aren’t always easy to follow. Also, because Plantinga’s version relies on modal logic we’ll need to understand the term possible worlds and its related concepts. As Wikipedia explains:

Those who use the concept of possible worlds consider the actual world to be one of the many possible worlds. For each distinct way the world could have been, there is said to be a distinct possible world; the actual world is the one we in fact live in. The modal status of a proposition is understood in terms of the worlds in which it is true; thus:

  • True propositions are those which are true in the actual world (for example: “Richard Nixon became President in 1969.”)
  • Possible propositions are those which are true in at least one possible world (for example: “Hubert Humphrey became President in 1969.”)
  • Contingent propositions are those which are true in some possible worlds and false in others (for example: “Richard Nixon became President in 1969”, which is contingently true, and “Hubert Humphrey became President in 1969”, which is contingently false.)
  • Necessary propositions are those which are true in all possible worlds (for example: “all bachelors are unmarried.”)
  • Impossible propositions (or necessarily false propositions) are those which are true in no possible worlds (for example: “Melissa and Toby are taller than each other at the same time.”)

Plantinga uses the concept of possible worlds in his case for the existence of a “maximally great being” (i.e., a being who has such qualities as omnipotence, omniscience, and moral perfection). One version of his argument is as follows:

  1. By definition a maximally great being is one that exists necessarily and necessarily is omniscient, omnipotent, and perfectly good.
  2. It is possible that a maximally great being exists.
  3.  If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
  4.  If a maximally great being exists in some possible world, then it exists in the actual world.
  5. If a maximally great being exists in the actual world, then a maximally great being exists.

Note the key premise (“It is possible that a maximally great being exists”) is a metaphysical claim (i.e., relating to the fundamental nature of being) rather than an epistemic claim (i.e., relating to what can be known). As philosopher William Lane Craig explains, it is tempting to say, “It’s possible that God exists, and it’s possible that he doesn’t exist!” But if God is a maximally great being, then his existence is either necessary or impossible regardless of our epistemic uncertainty. Craig gives an example of a mathematical formula that is beyond our ability to grasp. We may believe that it is either true or false but it is either necessarily true or necessarily false.

Plantinga concludes there is neither evidence for nor against the truth of premise #2 (I disagree, see Addendum #1 below). He concedes that since either option is possibly (though not necessarily) true, the outcome is even and the result is that it could go either way — 50/50. On first glance this may appear to be a useless result. Neither option is considerably more provable than the other, according to Plantinga, though it makes believing in the existence of God at least as rational as believing the opposite.

We can go further, though, and use this argument as a useful starting point for a Bayesian proof of the probability of God.

Part II — Calculating the Probability of God

[Note: This part isn’t necessarily part of the ontological argument. It’s just an interesting exercise that helps us take the argument one step further. Much of this section is adapted from Stephen Unwin’s The Probability of God.]

British philosopher Betrand Russell, famous for his agnostic views as much as for his theories on logic, was once asked how he would answer if he turned out to be wrong about God. Russell was delighted with the question and answered, “Why, I should say, ‘God, you gave us insufficient evidence.’”

I suspect that upon their meeting, God corrected the ol’ Brit, showing how the evidence was there and that Russell had simply chosen to ignore it. But it does raise the question of why different people when presented with much the same evidence, come to such varying conclusions about the existence of God.

Presumably, rational people weigh the evidence of God’s existence or non-existence in order to determine the probability of one being more likely than another and proceed from there. After all, since we can’t know the answer with absolute certainty, we have to base it on our best probabilistic assumption.

But how can we determine what is more likely when applied to an issue such as the ontological status of God? By applying Bayesian probabilities, a statistical method devised by 18th-century Presbyterian minister and mathematician Thomas Bayes. Using Bayes theorem we can calculate for ourselves the probability of God’s existence (a typically prideful thing for humans to do, but an interesting apologetic exercise nevertheless). This method won’t tell us whehter God exist or not, but it can tell us based on the evidence we have whether we should consider the probability of the belief being true to be more or less likely.

The best (or worst) case scenario for the ontological argument is that it gets us to the probability for or against God’s existence is 50-50. Since 50-50 represents “maximum ignorance” (the outcome of Plantinga’s argument), we can begin with a 50 percent probability that God exists (it’s the same as 50 percent that God doesn’t exist) and then apply it to the following modified Bayesian theorem. Fifty percent will be the P(before):

The probability of God’s existence after the evidence is considered a function of the probability before multiplied by D, a function we’ll call the “Divine Indicator Scale.” We can create the scale as follows:

10 indicates the evidence is 10 times as likely to be produced if God exists
2 is two times as likely if God exists
1 is neutral
0.5 is moderately more likely if God does not exist
0.1 is much more likely if God does not exist

Let’s say we decided to plug in a factor of D for each of six categories: recognition of goodness; existence of moral evil, existence of natural evil, intra-natural miracles (e.g., a friend recovers from an illness after you have prayed for him); extra-natural miracles (e.g., someone who is dead is brought back to life); and religious experiences. We can use the following lines of evidence and applies our own, admittedly subjective, figures for their likelihood.

Recognition of goodness (D = 10)
Existence of moral evil (D = 0.5)
Existence of natural evil (D = 0.1)
Intra-natural miracles  (D = 2)
Extra-natural miracles (D = 1)
Religious experiences (D = 2)

If we plug these figures in one and a time and recalcuate each time into the above formula (in sequence, where the P after figure for the first computation is used for the P before figure in the second computation, and so on for all six Ds), we’d arrive at the conclusion that the probability that God exists, based on our weighting of the evidence, is 67 percent.

My own calculations would be slightly different;

Recognition of goodness (D = 10) — I agree that the existence of goodness is more probable in a world in which God exists.

Existence of moral evil (D = 1) —I believe that the existence of evil is a neutral factor in regards to God’s existence. As Alvin Plantinga shows with his “free will defense” argument, moral evil is just as probable in a world in which God exists as it would be in a world in which He didn’t.

Existence of natural evil (D = 1) — Since “transworld depravity” could apply to nonhuman moral agents, Plantinga’s argument still applies.

Intra-natural miracles (D = 2) — I’ve personally had a sufficient number of these types of experiences to believe that it is two times as likely that they are divine actions rather than coincidences. Since the mathematical probability of them being coincidences are impossibly high, though not outside the realm of possibility, I can only rate this a 2 rather than a 10.

Extra-natural miracles (D = 2) —I would put these in the same category as the ones above.

Religious experiences (D = 2) — I give my personal religious experiences the same weight as I did the intra-natural miracles.

When I plug these number into the formula I come up with the probability that God exist to be 99 percent. Even if I had started with the presumption that there was a 98 percent chance that God does not exist, when I factor in the evidence it still puts the probability at 62 percent. Clearly, based on my interpretation of the evidence, it would be more reasonable for me to believe that God exists than to doubt it.

Again, none of this proves beyong the point of rational dispute that God exists. But what it does show is that when we add this link to the chain of other theistic arguments, the rational reasons for believing in God far outweigh the rational reasons for atheism.

Addendum #1: On this point I disagree with Plantinga and agree with Ed Feser. Feser’s objection is that, “Unless someone has actually given evidence to think that Plantinga’s concept of a maximally great being entails a contradiction or is otherwise incoherent, the rational position (again, at least if we buy the whole “possible worlds” framework in the first place) would be to accept his key premise rather than the key premise of the “no-maximality” argument, and rather than suspending judgment.”

Addendum #2: Personally, I find the ontological arguments, taken on their own, to be the least useful in apologetic discussions. Indeed, I’ve never personally discussed them with an atheist who was even able to fully comprehend some of the most obvious points about ontological arguments. (I’m not saying atheists who understand what ontological arguments are about don’t exist; I’m just saying I’ve never encoutered them in person. Look around at atheistic-apologists’ websites that discuss the argument and you’ll see what I mean.) I think the reason is because to truly understand the argument requires admitting the possibility that there could be a “maximally great being.” Once you do that, the probability for atheism drops below the case for theism (or at least deism). Of course, if an atheist were willing to follow the obvious chain of logic and admit that the possibility of God’s existence was more probable than his non-existence they’d have to admit that atheism isn’t intellectually tenable. And than their passions and emotions will allow them to concede.

Addendum #3: A primary objection to ontological arguments:

If you’ve followed the discussion so far you may think that the argument suffers from what is called the “overload objection”— that the form of the argument would overload the world with an almost infinite number of items such as perfect islands, perfect lions, and perfect blogs.

This objection was first raised against Anselm’s version of the argument by his fellow monk, Gaunilo. In a response titled On Behalf of the Fool,

Gaunilo invited his readers to think of the greatest, or most perfect, conceivable island. As a matter of fact, it is likely that no such island actually exists. However, his argument would then say that we aren’t thinking of the greatest conceivable island, because the greatest conceivable island would exist, as well as having all those other desirable properties. Since we can conceive of this greatest or most perfect conceivable island, then it must exist. While this argument seems absurd, Gaunilo claims that it is no more so than Anselm’s. (Wikipedia)

The problem with this line of thinking is that:

Such objections always depend upon the accuracy of the analogy. That is, we must be able to show that the objector’s argument is sufficiently like the ontological argument for us to be able to conclude that if one works so must the other. There are at least two problems with Gaunilo’s version, though. First, what exactly is the concept of the perfect island—the island than which no greater can be conceived? In any group of people, there will be disagreements as to what makes an island perfect; there will be different preferences concerning size, climate, inhabitants, food-availability, etc. There is no single concept of a perfect island, because perfection here can only mean what is perfect for us, rather than perfect in itself. The notion of the perfect being, however, isn’t relativised to any individual; it’s the notion of a being that is maximally great—not for me or for you, but great, full stop.