Swinburne and the Probability of TheismAnother modern attempt to revive the moribund state of philosophical theism is that of Richard Swinburne. His arguments are presented in his book The Existence of God (1979). One of his most well known argument in that book is his inductive version of the cosmological argument. Rather than trying to prove without a doubt that God exists, Swinburne, in a method he claims is analogous to science, intends to merely show that God's existence is more probable than not.
Swinburne's ArgumentSwinburne's argument is based on the field of inductive logic known as "confirmation theory". In particular the Bayesian confirmation theory. This theory is used in assessing the strengths and weaknesses of hypothesis in relation to the evidence. The relevant formula is as follows:
h = hypothesis to be evaluated
e = evidence
k = background knowledge
P(x/y.z) = Probability of the occurrence of x given conditions y and z.
Let us look at one example first for this. Suppose that a detective is investigating a murder and his hypothesis (h) is that "the butler did it". The background knowledge (k) is the general information pertaining to the case, for instance that the butler lives in a small house next to the manor where the owner was murdered. Now let us suppose that a piece of evidence (e), a set of footprints matching the shoes belonging to the butler, leading from the butler's house made during a late night rain storm the night of the murder, was found.
Let us look at the middle terms in the Bayesian formula above. The denominator [p(e/k)] gives the probability that the evidence holds given the background information alone. While the numerator [p(e/h.k)] gives the probability that the evidence holds given the hypothesis and the background knowledge. Obviously the footprints, if it fits the size of the butler, and the time of the murder, as an evidence, is made more probable with the hypothesis that the butler did it.
Thus the numerator has a higher probability than the denominator, thus making this number more than 1. Which means we now have an inequality: p(h/e.k) > p(h/k); provided p(h/k) is not equal to zero. This, Swinburne calls a "C-Inductive argument", i.e. the addition of the evidence makes the probability that the hypothesis is true more likely than without the evidence. Note the limitation here, it does not say that the hypothesis is more likely to be true than not; i.e. it does not say the probability is above 0.5 (or 50%), simply that it is made more likely than before with the presence of the evidence.
One of the consequences of Bayesian theory is that p(h/e.k) > p(h/k) is true if and only if p(e/h.k) > p(e/~h.k). [~ is the symbol for "not"]. This means that for the evidence to support the hypothesis, the evidence must be more likely to occur if the hypothesis is true (h) than if the hypothesis is false (~h).
Finally, a "P-Inductive argument" makes the hypothesis more probable than its negation: p(h/e.k) > p(~h/e.k). In this argument the p(h/e.k) is actually more likely that it's negation, in other words, the probability is more than 0.5 (or 50%).
So how does all this apply to the existence of God? Here "e" is the evidence, in this case, the whole universe, the fact that it exists; "h" is the hypothesis that God exists and "k" is the background knowledge. Note that since "e" already covers the whole universe what is left as background knowledge is simply the tautological knowledge of mathematics and logic.
According to Swinburne, p(e/h.k) (the probability that the universe exists given that God exists) is not very probable. However he claimed that p(e/k) (the probability that the universe exists as an unexplained brute fact) is even less probable. He claims that p(e/k) is very improbable as it is "strange and puzzling". However the existence of God can explain the universe and that it was more likely that God "would exist uncaused". The reason for this, according to Swinburne, is that God's attributes are infinite, thus they are "simpler" and hence more likely to exist than the universe which has finite values. In his own words: "A finite limitation cries out for explanation of why there is just that particular limit, in a way that limitlessness does not." Thus p(e/h.k) > p(e/k) and we get,as before, p(h/e.k) > p(h/k). The addition of the existence of the universe as an evidence increases the probability that God exists.
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However even his modest aims fail. Let us see why.
Simplicity, Infinity and the Need for ExplanationFirst we must state that Swinburne finds the universe "strange and puzzling" by itself, not bacause there can be uncaused things (as God is) but that its very finititude of attributes makes it unlikely that it could have existed uncaused. God, however, having the "simpler" infinite attributes does not require an explanation, thus could be uncaused.
However why should infinity be "simpler" than finiteness? In fact, postulating infinite attributes such as omnipotence, completely free will and omniscience for God leads to contradictions which have not been resolved. And postulates leading to contradictions is anything but "simple".
It is unclear, and Swinburne does not offer any help here apart from asserting that it is, why infinite attributes do not cry out for an explanation. While it is true that zero power or attribute does not require an explanation-since it does not exists-this is not so for infinite attributes. Let us take as an example "omnipotence"-the power to do anything. This seems to require an explanation. For the power to do anything, move mountains at will etc., certainly cries out for an explanation.
More importantly it does not follow that just because a hypothesis is "simpler" it is a priori more probable than a less simpler one. "All heavenly bodies travel in straight lines" sounds simpler than "All heavenly bodies travel in elliptical orbits", yet the latter is the correct observation.
It should further be noted that any analogy between Swinburne's principle of simplicity to Ockham's Razor is false. Ockham's principle, named after William of Ockham (c1280-1349), applies to theory formulation and states that if two hypothesis explain the same phenomenon equally well, the one with the fewer assumptions is to be preferred. In fact applied here, it seems to show that the hypothesis that the universe exists uncaused seems simpler since it has one less assumption (it does not assume God's existence).
Estimates of ProbabilityAlthough Swinburne claims that p(e/k) is small he gives no clue as to exactly how small it is nor does he provide any suggestion of a methodology of acquiring such an estimate. Yet even a cursory look at the issue will suffice to show how uncertain this is. What is the a priori probability that the universe exists? Is it a choice between two possibilities: i.e. that there is a universe and there isn't? And if we are ignorant about everything else (principle of ignorance), then we can assume that both possibilities are equally likely, thus p(e/k) is 0.5. But what if the alternatives are between non-existence of the universe and various sorts of universes. Say there are 999 different types of complex universes, all equally likely. Then p(e/k) is 99.9%. Or very likely. Thus depending on arbitrary assumptions, we get differing values for p(e/k). Thus a priori probability of the universe's existence is something which is purely subjective. 
ConclusionsSwinburne's C-Inductive argument for God's existence fail because:
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