For the past two years I have, in my copious free time, been studying Richard Swinburne’s case for God. Recently my focus has been on his evaluation of the cosmological argument (hereafter TCA) in his book The Existence of God, 2nd edition (hereafter EOG). His version of TCA is quite simple:

e: A complex physical universe exists.
Therefore
g: God exists.

But Swinburne’s argument about TCA is not so simple.

Swinburne does not present TCA as a proof of the existence of God, nor does he claim that TCA makes God’s existence probable. Rather, he claims that e makes g more probable that it would be otherwise:

P(gle & k) > P(glk)
Where k is tautological (a priori) background knowledge. In other words, e provides relevant evidence in support of of the hypothesis g, increasing the probability that g is the case.

I understand most of Swinburne’s argument in support of his claim about TCA, but there is a part of his reasoning that I just don’t get, and I’m hoping that someone can help me to get it.

There are two key premises in Swinburne’s argument about TCA:
(TCA8) The probability that there will be a complex physical universe given that God exists is at least ½. (EOG, p.151)

(TCA9) The probability that there will be a complex physical universe given that God does not exist is low. (EOG, p.151)

I fully understand Swinburne’s argument for (TCA8), and I think I understand his core argument for (TCA9), but I’m having difficulty figuring out his reasoning in support of a premise used to support (TCA9). Here is how I would translate (TCA9) into a conditional probability statement:

.2 ≤ P(el~g&k) < .4

(I’m interpreting ‘low probability’ as meaning ‘greater than or equal to .2, and less than .4’).Here is what I believe to be the core argument for (TCA9):
(TCA14) The probability that a complex physical universe exists without an explanation is very low. (EOG, p.152)
(TCA15) The probability that a complex physical universe exists given that God does not exist is approximately equal to the probability that a complex physical universe exists without an explanation. (EOG, p.149)Therefore:
(TCA13) The probability that a complex physical universe exists given that God does not exist is approximately a very low probability.Therefore:
(TCA9) The probability that there will be a complex physical universe given that God does not exist is low (at most). (EOG, p.151)

I think I understand Swinburne’s reasoning in support of (TCA14), but I cannot figure out his reasoning in support of (TCA15), even after reading and re-reading what he says in support of this claim.

I’m also uncertain about how to represent (TCA14) in terms of a conditional probability statement. Here is my attempt to do so:

0 < P(el~y&k) < .2

y: e has an explanation.

If I knew how to correctly represent (TCA14) in a conditional probability statement, perhaps that would help me to understand Swinburne’s thinking.

Swinburne eliminates the possibility that science might explain e, which leaves him with just two possibilities: either e has a personal explanation (e being the result of a choice of some person for some purpose) or else e has no explanation at all, and is just a brute fact.

Traditionally, cosmological arguments eliminate the possibility that the universe is simply a brute fact by an appeal to the Principle of Sufficient Reason (hereafter: PSR). But Swinburne rejects the strong version of PSR, and sees no good reason for accepting a weaker version of PSR. So, what he does is argue that the a priori probability of e being the case but having no explanation is very low, (TCA14).