bookmark_borderSwinburne’s Case for God – Part 8

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).

bookmark_borderSwinburne’s Case for God – Part 7


The first premise of Swinburne’s case for God makes a fairly modest claim:
1. Based on evidence other than religious experience, the existence of God is not very improbable.
Because the expression “not very improbable” is a bit vague, I argued for the following clarification of premise (1), in my last post:
1b. Where e is the specific evidence (considered by Swinburne in EOG) for and against the existence of God, excluding the evidence of religious experience, and where h is the hypothesis that God exists, and where k is our background knowledge: P (h I e & k) > .20

In Swinburne’s estimation, the relevant evidence (other than the evidence from religious experience) makes the probability of God’s existence significantly
…all that my conclusion so far amounts to is that it is something like as probable as not that theism is true, on the evidence so far considered. However, so far in this chapter I have ignored one crucial piece of evidence, the evidence from religious experience. (EOG, 2nd ed., p.341)
In other words, according to Swinburne:
P(h I e & k) ≈ .50
This is where e is the relevant evidence other than religious experience.
Since this is only an approximation, we could consider Swinburne to be correct if the actual probability were somewhere from .40 to .60:
(4) Where e is the specific evidence (considered by Swinburne in EOG) for and against the existence of God, excluding the evidence of religious experience, and where h is the hypothesis that God exists, and where k is our background knowledge: .40 ≤ P(h I e & k) ≤ .60
Clearly, (4) implies (1b), becuase if the probability of h is between .40 and .60, then the probability of h is greater than .20.
In EOG, Swinburne considers eleven arguments on the question of the existence of God. The first premise of Swinburne’s case for God is concerned with ten of the arguments, and excludes the final argument that is based on religious experience. Two of the ten arguments are set aside as having no significant force, leaving eight arguments to use as the basis for showing that the probability of God’s existence is between .40 and .60.
Seven of the eight remaining arguments are supposed to provide some confirmation of the existence of God, while one argument (the problem of evil) is supposed to provide some disconfirmation of the existence of God. Since each of the arguments that confirm the existence of God is supposed to bump up the probability a bit, so that the cumulative force of the arguments is greater than that of any one particular argument, the claim that Swinburne has to make for each of the confirming arguments is very modest.
Suppose that the problem of evil is only strong enough to cancel out the weight of one of the seven confirming arguments. That would leave six confirming arguments to support the cumulative probability of between .40 and .60.
If we assume that the prior probability of the existence of God was equal to one chance in a hundred, i.e. P(h I k) = .01, and if we assume that each of the six remaining confirming arguments bumps up the probability by the same amount (on average), then the probability of h would seven times .01:
P (h I e & k) = 7 x .01 = .07
On this scenario, Swinburne’s claim that the probability of h (given the relevant evidence other than religious experience) was between .40 and .60 would clearly be false, as would the weaker claim made in premise (1b), namely that h was greater than (or equal to) .20.
However, suppose that the prior probability of the existence of God was a bit higher: .05, and suppose that each of the six confirming arguments (remaining after cancelling out one of the confirming arguments with the problem of evil) added .05 to the probability (on average). In that case, the probability of h would be seven times .05:
P (h I e & k) = 7 x .05 = .35
On this scenario, Swinburne’s stronger claim (4) would be false, but this would still be enough to establish his first premise (1b), because a probability of .35 is greater than a probability of .20.
If we were to assume that the prior probability of h was .06, and that each of the six confirming arguments (remaining after the problem of evil cancelled out one confirming argument) bumped up the probability by .06, then not only would premise (1b) be true, but so would Swinburne’s stronger claim:
P (h I e & k) = 7 x .06 = .42
These different scenarios make two important points. First, Swinburne only needs to establish fairly weak claims about the force of his confirming arguments. If each confirming argument bumps up the probability of the existence of God just a bit (say .04 or .05) that may well be sufficient to establish the first premise of his case for God (if force of the problem of evil is only about the same as one of the confirming arguments, and if there is at least a tiny prior probability of the existence of God).
Second, the difference between success and failure is small, now that we have clarified the probability required to make the first premise true. If the average bump up of probability by the confirming arguments (and the prior probability of God’s existence) is .02 (i.e., two chances in a hundred), then premise (1b) is likely to be false, but if the average bump up of probability by the confirming arguments (and the prior probability of God’s existence) is .03 (i.e. three chances in a hundred), then premise (1b) is likely to be true (depending on the force of the problem of evil in disconfirming God’s existence).
greater than .20:

bookmark_borderSwinburne’s Case for God – Part 6

Swinburne’s case for God (in The Existence of God, 2nd ed.) can be summed up this way:

1. Based on evidence other than religious experience, the existence of God is not very improbable.
2. If based on evidence other than religious experience, the existence of God is not very improbable, then the evidence from religious experience (in combination with other relevant evidence) makes the existence of God more probable than not.
Therefore:
3. The evidence from religious experience (in combination with other relevant evidence) makes the existence of God more probable than not.



The expression ‘not very improbable’ in premise (1) is a bit vague, making it difficult to determine whether or not Swinburne has succeeded in establishing this claim, so we should try to clarify this expression.

Let’s say that h is the hypothesis that God exists. Clearly Swinburne means that the probability of h (given the evidence he considers and given our background knowledge) is greater than some low probability at the upper border of a range of probabilities that we would consider to be ‘very improbable’.

One chance in a million or a probability of .000001 is obviously in the ‘very improbable’ range, but is presumably not at the upper border of that range. What we need to clarify is where (approximately) that upper border falls.

I think it makes sense to first clarify the notion ‘very probable’ and then use that as the basis for clarifying the notion of something being ‘very improbable’. My linguistic intuitions seem firmer when dealing with the positive concept of probability, as opposed to the negative concept of improbability.

Six chances in ten or a probability of .60 would qualify an hypothesis as being ‘probable’ in the sense of being more probable than not. But a probability of .60 would not ordinarily count as being ‘very probable’. So, the lower border of the range of probabilities categorized as ‘very probable’ must be some number above .60. I would say the same of seven chances in ten or a probability of .70.

Would eight chances in ten or a probability of .80 count as being ‘very probable’ ? Perhaps, but my linguistic intuition is that .90 is closer to being at the lower edge of the range of probabilities that we would ordinarily consider to be ‘very probable’. If this is correct, then the claim that ‘hypothesis h is very probable’ means something like this:

P (h I e & k) ≥ .90

If this clarification of ‘very probable’ is correct, then it is reasonable to infer that the upper border of ‘very improbable’ would be one chance in ten or a probability of .10. In that case, the claim that ‘hypothesis h is very improbable’ means something like this:

P (h I e & k) ≤ .10

If this is a proper way to understand the expression ‘very improbable’, then Swinburne’s premise (1) which speaks of God’s existence as being “not very improbable” can be restated with a bit more precision:

1a. Where e is the specific evidence (considered by Swinburne in EOG) for and against the existence of God, excluding the evidence of religious experience, and where h is the hypothesis that God exists, and where k is our background knowledge:
P (h I e & k) > .10

However, Swinburne probably intends something slightly stronger than this.
Suppose that Swinburne agrees that a probability of .10 is at the upper boundary of the range of probabilities in the category ‘very improbable’. Suppose that Swinburne’s case for this first premise only proved this:

P (h I e & k) = .11

This would be enough to prove that (1a) was true, but the probability in this case is so close to the border between what is considered ‘very improbable’ and what is considered ‘not very improbable’ that the truth of Swinburne’s original claim would still be in doubt. So, I think a slightly higher number needs to be used as the cutoff in order to ensure that the probability indicated by the evidence is clearly beyond what could reasonably be considered ‘very improbable’. Thus, I would suggest that the cutoff be bumped up from .10 to .20, to remove any doubts introduced by the fuzzy boundary here:
1b. Where e is the specific evidence (considered by Swinburne in EOG) for and against the existence of God, excluding the evidence of religious experience, and where h is the hypothesis that God exists, and where k is our background knowledge:
P (h
I e & k) > .20


Therefore, if Swinburne can show that the evidence relevant to the hypothesis that God exists (specifically, the evidence that he considers in EOG), excluding the evidence of religious experience, and given our background knowledge, makes the probability of this hypothesis greater than .20, then I think we should accept the first premise of his argument, namely the claim that this evidence makes the existence of God “not very improbable”.

Most of the arguments presented in EOG are part of a cumulative case to establish premise (1). For this part of his case, Swinburne is making a fairly modest claim. He claims to have shown that the evidence for and against the existence of God taken together, and setting aside the evidence of religious experience, provides some degree of confirmation of the hypothesis that God exists, making this hypothesis “not very improbable”.

bookmark_borderSwinburne’s Case for God – Part 5

Swinburne makes use of Bayes’ Theorem in presenting most of the a posteriori arguments for and against God in The Existence of God (EOG), and he makes significant use of it in summing up his case for God.


Bayes’ Theorem:

P (h I e & k) = P(e I h & k) x P(h I k) / P(e I k)

By the symmetry of equality we can restate Bayes’ Theorem with the “answer” on the right hand side of the equation:

P(e I h & k) x P(h I k) / P(e I k) = P (h I e & k)

We have previously discussed the conditional probability that constitutes the answer that we seek:

P (h I e & k)

This is the posterior probability of the hypothesis h, or the probability of the hypothesis, given the specific evidence e and our background knowledge k.

Bayes’ Theorm (with the answer on the right-hand side) has the following simple form:

A x B / C = X

If one of the factors in the numerator (either A or B) increases, then X also increases.
If the denominator (C) increases, then X will decrease. The same relationships hold in
Bayes’ Theorem. If either P(e I h & k) increases or P(h I k) increases, then P(h I e & k) will also increase. If P(e I k) increases, then P(h I e & k) will decrease.

This makes sense if you think about what each of these conditional probability expressions means.

P(e I h & k)

This conditional probability in the numerator is the probability that the evidence would occur given that the hypothesis was true and given our background knowledge. If the evidence is likely to occur given the truth of the hypothesis, then the occurence of the evidence is favorable towards the truth of the hypothesis. So an increase in this probability should mean an increase in the probability of the hypothesis, i.e. P(h I e & k).

P(h I k)

This factor in the numerator is the probability of the hypothesis given only our background knowledge. If the hypothesis is already very probable, prior to considering evidence in support of the hypothesis, then the probability of the hypothesis after consideration of the evidence will also tend to be high, at least higher than it would be if the hypothesis was improbable to begin with. So, it makes sense that the higher this prior probability of the hypothesis h is, the higher the posterior probability of the hypothesis will be.

P(e I k)

This conditional probability is in the denominator. It means the probability of the evidence occurring given only our background knowledge. If the evidence was likely to occur whether or not the hypothesis is true, then the evidence is a poor indicator of the probability of the hypothesis. Thus it makes sense that an increase in this prior probability of the evidence e would decrease the posterior probability of the hypothesis, i.e P(h I e & k).