<i>Deinonychus antirrhopus</i>: Trusting the Mind

April 04, 2006

Trusting the Mind

Over at the Evangelical Outpost there is an old post of Joe Carter's that has been annoying me, but I've never really had the time to respond...until now. The gist of the post is based on Alvin Plantinga's supposed defeat of methodological naturalism. The argument is a probabilistic one and can be summarized thusly,

E represents evolution,
N represents naturalism,
E & N is the intersection of these two events,
R represents that our brain and senses are reliable.

Plantinga essentially argues that Prob(R) pretty close to 1, that Prob(R | E & N) is going to be small and that Prob(E & N) is "comparable" to Prob(TT) where TT is traditional theism. With these assumptions Plantinga points out that Prob(E & N | R) is going to be small no matter what value is chosen for Prob(E & N). Go ahead, try it out, set Prob(R) to something like 0.99 and randomly pick Prob(E & N) both high and low values, and set a low value for Prob(R | E & N) (say .02). If you are really serious about this select a low Prob(R | E & N), say 0.02, and set Prob(R) = 0.99 and then graph Prob(E & N | R) for different values of Prob(E & N). The values will all be very low.

That looks pretty darned convincing, eh? No matter what we observe for Prob(E & N) it means we should reject R, which means we should end up rejecting E & N since that is based on R. Neat trick. But there are some problems and these are very neatly described by Branden Fitelson and Elliot Sober (pdf).

First off, there is something interesting with Plantinga's initial assumption that Prob(R) is close to 1. In Bayesian statistics the researcher likes to use the following relationship:

H represents an hypothesis we are interested in,
O is an observation--i.e. data,

Now we can re-write the above so that we have,

Prob(H |O)/Prob(H) = Prob(O | H)/Prob(O).

With Prob(O) = 1, the above can now be written as,

Prob(H | O) = Prob(O | H) * Prob(H).

But it is also true that 0 < Prob(H) < 1, hence it is the case that observation O, does not confirm hypothesis H. Another way of thinking about this is that observing something that is common no matter what hypothesis we are considering does nothing to help us validate our hypothesis. Think of it this way. Suppose we could observe O₁ and O₂ where Prob(O₁) = 1 and Prob(O₂) is pretty small say 0.05. Observing O₁ is no big deal since we expected to observe it whether H is true or not, hence its value in evaluating H is minimal or non-existent.

While this problem is as acute for Plantinga's argument it is still a problem when one sets Prob(R) close to 1. What it means is that R has little or no bearing on the hypothesis of E & N in terms of confirmation. This is a pretty serious problem for Plantinga in that it does have serious repercussions for his entire argument.

However, that isn't the only problem. Recall that Plantinga argues that Prob(R) is close to 1, Prob(R | E & N) is low, and that Prob(TT) and Prob(E & N) are "comparable". Also, Plantinga argues that Prob(R | TT) is also high. Now using the theorem of total probability we have,

Prob(R) = Prob(R | E &N) Prob(E & N) + Prob(R | TT) Prob(TT).

Now, if Prob(E & N) and Prob(TT) being comparably means that they are equal then we have a problem. It follows that we have,

1 aprox. low*(0.5) + high*(0.5).

But this violates the axioms of probability. To see this, pick any number really close to 1, say 0.99999999, and pick a low number say 0.01, now add them together and multiply by 0.5. You should get a number close to 0.05 which last I checked is not approximately 1. No matter how close you pick a number to 1...Hell, go ahead and even pick 1, the result is still the same, a very small number that is nowhere near 1.

Fitelson and Sober note that there is one way out, but that it isn't particularly palatable to Plantinga nor his argument. That is there is a third possiblity, X. Thus, using our friend the theorem of total probability we get,

Prob(R) = Prob(R | E &N) Prob(E & N) + Prob(R | TT) Prob(TT) + Prob(R | X) Prob(X).

But recall that Prob(R) is close to 1, and that Prob(E & N) is comparable to Prob(TT) and P(R | E & N) then it has to be the case that Plantinga assigns both Prob(E & N) and Prob(TT) negligible probabilities. If this is the case, then it turns out that Prob(X) must be close to one and we can reject both E & N and TT in favor of whatever X is (Raelians must be loving this kind of stuff, eh?).

Fitelson and Sober point to a third way out, which Plantinga also seems to endorse (via private communication) which holds that "comparable" means that Prob(E & N) and Prob(TT) are "not too far apart". The probelm with this is that it also removes undermines Plantinga's argument. To see why going back to our first expansion of Prob(R) via the theorem of total probability,

1 aprox. low*(?) + high*(?).

While the above is no longer in violation of the axioms or probability, it still remains that we must set Prob(E & N) very low and Prob(TT) very high. The problem here though is that this is basically stacking the deck against E & N. No matter what value Prob(R | E & N) takes, Prob(E & N | R) is always going to be smaller than Prob(E & N). Plantinga has basically come up with a system that is self-reinforcing and impervious to any and all data by assumption.

So what does this mean for Joe Carter's post? Well since it is largely based on the writings of Alvin Plantinga and these problems are pretty devastating to Plantinga's argument, by extension Joe's post is pretty devestated. The same goes for Joe's other posts that rely on Plantinga's argument as well. For those of you interested in this kind of thing, I also strongly recommend the article by Fitelson and Sober, it discusses additional problems with Plantinga's leaving it in pretty bad shape. They also point out that despite its serious problems Plantinga's argument does pose a serious question for evolutionary theory and its adherents, one that is pretty much unanswer (AFAIK). The question is, if evolutionary theory suggests that our cognitive skills are in some contexts unreliable, how is this factored into our own theoretical beliefs?

Posted by Steve at April 4, 2006 10:53 PM

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