Gwiazda on Swinburne: Complexity Quotients and God's Simplicity

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This post is the second in a series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God. For an introduction to Swinburne's argument see here.

We saw in the first part that Swinburne's basic argument is Bayesian in form. He claims that the probability of theism (h) given certain empirical evidence (e) and background tautological evidence (k) is high. We derive this probability using Bayes theorem, as follows:

• Pr(h|e&k) = Pr(e|h&k) Pr(h|k) / Pr(e|k)

Swinburne estimates that Pr(e|h&k) is 0.5. He then argues that the intrinsic probability of h, or Pr(h|k), is much higher than the intrinsic probability of e, or Pr(e|k). He does so on the grounds that h is much simpler than e and that the simpler hypothesis must have the higher probability.

Gwiazda thinks that this line of reasoning gets Swinburne into probability theory's version of hot water. We are about to see why.

1. Complexity Quotients

To appreciate the flaws we need to introduce the concept of a complexity quotient. This is a figure that arises from the division of one intrinsic probability by another. It gives some indication of how complex one theory is when compared to another.

In the case at hand, the relevant quotient is derived when we divide the intrinsic probability of theism by the intrinsic probability of the evidence. Before we consider that case, let's consider an instructive example.

Let w be the hypothesis "exactly one wooden block exists". W must have some intrinsic probability based on the background tautological evidence. But when you think about, w is ambiguous: wooden blocks come in a variety of sizes and shapes.

For illustrative purposes we can divide w into four more specific hypotheses. Since we are going purely on background tautological evidence, we can assume that these more specific hypotheses are equiprobable. The hypotheses are as follows:

• w1 = "one small light wooden block exists"
• w2 = "one large light wooden block exists"
• w3 = "one small dark wooden block exists"
• w4 = "one large dark wooden block exists"

Because these are more specific versions of w, and because they are equiprobable, it follows that:

• Pr (w|k) = 4*Pr(w1|k)

Which implies the following complexity quotient:

• Pr (w|k) / Pr(w1|k) = 4

Which means that w is four times more probable than w1. This makes sense since w is the more general, non-specific hypothesis. 

So good, so far.

2. Swinburne's Error

Swinburne's error becomes apparent once we bring the concept of the complexity quotient to bear on his original equation.

• Pr (h|e&k) = Pr(e|h&k) Pr(h|k) / Pr (e|k)

Recall that all probabilities must be greater than 0 but less than 1. This means that the maximum value for each side of this equation is 1. Recall also that Swinburne put a figure of 1/2 or 0.5 on Pr(e|h&k). This would give us:

• 1 ≥ 1/2 * Pr(h|k) / Pr(e|k)

Which we can multiply through by 2 to give us:

• Pr(h|k) / Pr(e|k) ≤ 2

This is an odd result. Whereas the intrinsic probability of one wooden block existing was 4 times greater than the probability of one small light wooden block, we are now forced to conclude that the intrinsic probability of God is only 2 times greater than the intrinsic probability of all the evidence that Swinburne wants to explain.

In other words, despite his claims that the evidence was incredibly complex and so in need of an explanation, Swinburne's own figure for Pr(e|h&k) would force him to accept that the evidence is not that complex when compared with God.

3. Where did he go wrong?

Gwiazda suggests that there is a rational explanation for Swinburne's error. You see, Swinburne wants the posterior probability of God's existence -- that is, Pr(h|e&k) -- to be reasonably high (> 0.5). He wants this because he wants his argument to have some persuasive force. After all, if you were told that the probability that God explains the observable universe was, say, 0.1 you would be relatively unimpressed.

Because he wants the figure to be relatively high, he needs h to confer a relatively high probability on the evidence. This is why he comes up with the figure of 1/2 for Pr(e|h&k). 

But at the same time he is committed to: (i) the idea that simple things are more probable than complex things; and (ii) the idea that God is simple whereas the observable evidence is complex. This commitment implies that a simple thing can never confer a high probability on a complex thing.

Consequently, he would be better off abandoning his claim that God is simple (or that the evidence is complex).

That's it for now. In the next part we will consider an additional mathematical problem that arises when Swinburne addresses alternative explanations for e.