I have managed to write seven posts describing and explaining Swinburne’s case for God, but have not yet discussed a single specific argument for or against God. So, it is now time to examine an actual specific argument. (In my defense, the first 132 pages of EOG are introductory, and I have spared you many details from those first six chapters.)

In Chapter 7 of EOG, Swinburne gets around to presenting his cosmological argument. A cosmological argument is an argument “to the existence of God from the existence of some finite object or, more specifically, a complex physical universe.” (EOG, 2nd ed., p.135) Swinburne’s cosmological argument draws on Leibniz cosmological argument (EOG, p.136, and 143) , but Swinburne departs from Leibniz somewhat and does not base his cosmological argument on the Principle of Sufficient Reason (EOG, p.148-149).

Swinburne’s cosmological argument starts with an empirical fact:

Let e be the existence over time of a complex physical universe.
(EOG, p.149)

Although generally reluctant to provide specific probability estimates, Swinburne does quantify at least one conditional probability relating to his cosmological argument:

I argued [in Chapter 6] that it would be an equal best act to create or not to create such creatures [humanly free agents], and so we should suppose the logical probability that God would create such creatures to be 1/2. I argued that these creatures would need to have bodies, and so there would need to be a physical world. So for this reason alone the probability that a God will create a physical world will be no less than 1/2.

In terms of conditional probability, this claim by Swinburne can be stated as follows:

P (e I h & k) ≥ .50

In other words, that this evidence e (i.e. the existence over time of a complex physical universe) would occur, given the assumption that God exists (hypothesis h) and given our background knowledge k, is at least as probable as not.

For this first argument about the existence of God, the background knowledge k is limited to just a priori or tautological truths, such as the truths of logic and mathematics (EOG, p.17 and p.146).

Swinburne does not indicate an estimate for the probability of the existence of God, given the evidence that a complex physical universe has existed for a period of time and given our background knowledge (of a priori truths). So, it is unclear how big of a role the cosmological argument plays in supporting premise (1b) of Swinburnes’ case for God.

But if we assume that this cosmological argument has about the average force of the arguments that (supposedly) confirm the existence of God, then we can figure out roughly what that probability needs to be, in order for Swinburne to be able to establish the truth of premise (1b).

We also need to make some assumption about the prior probability of the existence of God and about the force of the disconfirming argument from evil in order to make get an idea of what sort of numbers are required to achieve the minimal target posterior probability of .21 (the probability based on consideration of all the arguments other than that of religious experience).

So, I make the following assumptions, not because they are known to be true, but in order to narrow down the range of possibilities, in order to get a feel for the sort of numbers needed for Swinburne to be successful in showing (1b) to be true:

(A1) The force of the problem of evil cancels out the force of one of the confirming arguments considered by Swinburne, leaving six confirming arguments to consider in determining the posterior probability of the existence of God.
(A2) The cosmological argument has the same force as the average force of the remaining six arguments for God (the arguments not cancelled out by the problem of evil).
(A3) The prior probability of the existence of God given only a priori (tautological) background knowledge is .03: P(h I k) = .03
(A4) The posterior probability given the evidence of the seven confirming arguments and the one disconfirming argument is .21, thus meeting the minimum threshold to show that premise (1b) is true.
(A5) There is a 50/50 chance that there will be a complex physical universe that exists for a period of time, given that God exists and given a priori (tautological) truths, as Swinburne claims: P (e I h & k) = .50
Based on (A1) and (A4), the posterior probability of God’s existence based on the evidence of the six confirming arguments (that remain after one of the seven confirming arguments is cancelled out by the problem of evil) is .21.

Based on this inference and (A3), the total increase or bump up of probability from the six remaining confirming arguments is .18, so that when added to the prior probability of God’s existence, we arrive at the posterior probability of .21.

Thus, the average force of each of the confirming arguments would be: .18 / 6 = .03.

Based on this inference and (A2), the bump up of probability from the evidence of the cosmological argument is .03.

We know that the ‘answer’ to the Bayesian probability calculation for the cosmological argument will be .06 (.03 prior probability + .03 increase from the evidence of the cosmological argument):

P (h I e & k) = .06
Bayes’ theorem (once again):

P(e I h & k) x P(h I k) / P(e I k) = P(h I e & k)
Lets fill in the numbers for the cosmological argument, based on the above assumptions and inferences.

Replace the ‘answer’ with the target posterior probability outcome for the cosmological argument:

P(e I h & k) x P(h I k) / P(e I k) = .06
Now replace the conditional probability expression ‘P(e I h & k)‘ with the probability estimate Swinburne gives for it:

.50 x P(h I k) / P(e I k) = .06
Multiply both sides of the equation by 2:

2 x .50 x P(h I k) / P(e I k) = .06 x 2
P(h I k) / P(e I k) = .12Add the assumption that the prior probability of God’s existence is .03:

.03 / P(e I k) = .12Now we can solve for the remaining conditional probability:

P(e I k) x .03 / P(e I k) = .12 x P(e I k)

.03 = .12 x P(e I k)
100 x .03 = 100 x .12 x P(e I k)
3 = 12 x P(e I k)

1/12 x 3 = 1/12 x 12 x P(e I k)

3/12 = P(e I k)

.25 = P(e I k)
Now we can fill in all the numbers for the Bayes’ theorem for the cosmological argument to illustrate the sort of numbers required for Swinburne to be successful in establishing premise (1b).

Bayes’ theorem (once again):

P(e I h & k) x P(h I k) / P(e I k) = P(h I e & k)
Conditional probability values for the cosmological argument that would help Swinburne to be successful in establishing premise (1b):

P(e I h & k) = .50

P(h I k) = .03

P(e I k) = .25

P(h I e & k) = .06
Bayes’ theorem, with the suggested probability values filled in:

(.50 x .03) / .25 = .06Obviously, these are not the only numbers that will allow Swinburne to be successful in establishing premise (1b), but this does give us a feel for the sorts of numbers required.

Because Swinburne does not spell out such specific probability estimates for the cosmological argument, it might be useful to work at evaluating this suggested probability calculation, to see if there are any issues with it. If there are significant problems with this probability calculation, those problems might well point to more general problems that would apply to a broad range of probability calculations that might be given for the cosmological argument and in support of premise (1b).