#845 PPI and ECREE

Dr. craig’s response

What a great question!  This really illustrates the value of sharing ideas publicly to get feedback from others. (It’s nice to know that so many unbelievers are reading our Monthly Reports, too!) I also think that there is a broader lesson to be learned here, to which I’ll return at the end.

The short answer to your question is that “extraordinary” and “overwhelming” are not, in their respective contexts of use, synonymous terms. The long answer involves explaining how each word is used in its respective context. The appearance of inconsistency can be dissolved by examining more closely exactly what I was saying in each case.

In the case of the claim “extraordinary events require extraordinary evidence” the clue to my meaning is found in my assertion that the claim is “demonstrably false.” That is because the claim as I understand it violates the probability calculus expressed as Bayes’ Theorem. So it is demonstrably wrong. As I point out in Reasonable Faith,

there arose a discussion among probability theorists from Condorcet to John Stuart Mill over how much evidence it takes in order to establish the occurrence of highly improbable events.[1]  It was soon realized that if one simply weighed the probability of the event against the reliability of the witness to the event, then we should be led into denying the occurrence of events which, though highly improbable, we reasonably know to have happened.  For example, if on the morning news you hear reported that the pick in last night’s lottery was 7492871, this is a report of an extraordinarily improbable event, one out of several million, and even if the morning news’ accuracy is known to be 99.99%, the improbability of the event reported will swamp the probability of the witness’s reliability, so that we should never believe such reports. . . .

            Probability theorists saw that what also needs to be considered is the probability that if the reported event has not occurred, then the witness’s testimony is just as it is. . . .Thus, to return to our example, the probability that the morning news would announce the pick as 7492871 if some other number had been chosen is incredibly small, given that the newscasters had no preference for the announced number.  On the other hand, the announcement is much more probable if 7492871 were the actual number chosen. This comparative likelihood easily counterbalances the high prior improbability of the event reported.

So Bayes’ Theorem requires us to consider not only the ratio of the prior probabilities of the competing hypotheses on the background information B:

Pr (H|B)

__________

  Pr (not-H|B)

We must also consider the ratio of the posterior probabilities of the evidence E on the respective hypotheses and the background information:

Pr (E|H&B)

______________

   Pr (E|not-H&B)

In other words, you cannot judge the probability of H based simply on its probability relative to the background information; you have to also consider how probable it is that the evidence would be just as it is if H were not true. The second factor can easily overwhelm the first factor, so that something that seems extraordinarily improbable actually turns out to be quite probable on the total evidence: Pr (H|E&B) >> Pr (not-H|E&B). The evidence E doesn’t need to be extraordinary; what matters is that E is much more probable given H than not-H. It’s the ratio that counts.

Now what about the Principle of Personal Incredulity? It implies that we should accept an improbable claim only if we have evidence that overwhelms the improbability of that claim. Here “overwhelming evidence” does not mean “extraordinary evidence,” as you suggest, Agustin. Rather it refers to evidence that overcomes or outbalances the initial improbability. So far from being inconsistent, (PPI) and my denial of (ECREE) are actually saying the same thing! Namely, you needn’t believe a highly improbable claim unless you have evidence that outweighs that improbability (which needn’t be extraordinary if the second ratio is lopsided).

Now what’s interesting about (PPI) is that it also seems to apply to cases where the hypothesis is not improbable but just unbelievable. As my illustrations of the Boltzmann Brain Hypothesis and the Omphalos Theory reveal, I’m talking about a case in which the total evidence is equally well explained by either hypothesis Pr (H|E&B) = Pr (not-H|E&B). Neither the Boltzmann Brain Hypothesis nor the Omphalos Theory is improbable on the background information. You can’t refute the illusion of an external world or the appearance of age by appeal to sensory evidence! Neither is the evidence more probable on the contradictories of these claims than on the claims themselves, for precisely the same reason.

Thus, (PPI) seems to be a sort of mirror image of the widely accepted Principle of Phenomenal Conservatism:

PPC. If it seems to me that p, then, in the absence of defeaters, I am thereby justified in believing that p.

So if it seems to you that an external world exists and that the past was real, then, in the absence of overwhelming, i.e., outweighing, evidence to the contrary, you are justified in believing that you are not a Boltzmann Brain and that the world is not only a few thousand years old.

So suppose the atheist says it’s incredible to him that God exists. We should suggest to him that his intuitions may be mistaken, since he has no good reasons against God’s existence and God’s existence does not strike most people as incredible. Moreover, we can share with him evidence for God’s existence, as well as insist on the proper basicality of belief in God, which constitute defeaters for his atheistic belief. Given these defeaters, his atheism is unjustified.

This leads us to the broader lesson I mentioned. There is a difference between reading superficially and reading with understanding. The superficial reader just looks at the surface grammar of the sentences he reads and so jumps to conclusions; the careful reader tries to dig beneath the surface to understand the reasoning behind the claims being made. By reading with understanding we can avoid being misled by impressions. We may come to see that what appear to be inconsistent claims are really not at all incompatible.