How Would You Describe Your Evidence? Strong or Weak?

Roger White has recently argued that we should use the logically strongest formulation of our evidence when evaluating competing claims [1]. To see what he means, think about the following case. We know that our universe is finely-tuned for the existence of life. Had various physical constants and quantities differed slightly in their respective values, our universe wouldn't be life-permitting. Some have argued that this discovery is evidence for theism. But there are at least two ways of describing the relevant evidence:

• (E1) This universe is finely-tuned.
• (E2) Some universe is finely-tuned.

E1 entails E2 and, hence, is logically stronger than E2. White concedes that E2 may support the multiverse hypothesis as much as (or more than) theism. If there were billions of universes being spawn by some mechanism for a long period of time, it isn't unlikely that some universe would turn out finely-tuned. However, White argues that a principle of total evidence should require that we use our logically strongest evidence -- E1, not E2. Here's the rule:

• (Principle of Total Evidence) When evaluating competing hypotheses, use the logically strongest formulation of your evidence. 

Following the principle of total evidence, E1 is our evidence. And relative to E1, the multiverse hypothesis is not confirmed. Compare: you learn that a roulette wheel lands on the winning option 5 times in a row. Relative to chance alone, that seems wildly implausible. Intuitively, you're rational in suspecting that someone is tinkering with the outcome. But suppose an attendant at the casino pulls back a curtain and reveals that billions of roulette wheels are being spun at the same time (it's a rather big casino). Given the vast number of roulette wheels spinning, it is less improbable that some roulette wheel will get lucky and land on the winning slot 4 times in a row. Even so, the probability that this (or any particular) roulette wheel lands on the winning slot 4 times in a row is entirely unaffected by the outcomes of the other wheels. The wheel spins are independent. So, you should still think that it is just as improbable that this roulette wheel will land on the winning slot 4 times in a row, regardless of the fact that a vast number of other roulette wheels. Likewise, the likelihood that this universe is finely-tuned is unaffected by the number of other universes being "spun." The likelihood will be just as low as on the original chance hypothesis.

Recently, however, Peter Epstein has argued that you shouldn't always use the logically strongest formulation of your evidence. Doing so will sometimes give the intuitively wrong results. Consider:

• Pond Case. You’re fishing in a pond that contains large and small fish. However, you don’t know the distribution of large to small fish in the pond. You throw in your net and catch a large fish -- call this evidence “E.” 

Intuitively, you’ve just gained evidence that there are more large fish in the pond than small fish. Were the pond comprised of more large fish than small fish, you'd be more likely to catch a large one on the first cast (all else being equal). Probabilistically, Pr(E|More Large Fish) > Pr(E|More Small Fish). But suppose you name your fish “Bob." Following this ostensive designation, you choose to abide by White's principle of total evidence and describe your evidence as follows:

• (E*) I caught Bob.

This is stronger than:

• (E) I caught a large fish.

Now ask, How likely is it that you would catch this fish -- i.e., that you would catch Bob? Is it more likely on the first hypothesis than on the second? Intuitively, no. On this way of carving the evidence, it’s just as likely that you’ll catch Bob as it is that you’ll catch any other fish. Pr(I catch Bob|More Large Fish) = Pr(I catch Bob|More Small Fish) = 1/n, where n is the number of fish in the pond. 

Hence, if you treat your evidence according to White's principle above -- by taking the more specific evidence -- you shouldn’t think you’ve gained evidence for More Large Fish than for More Small Fish [3]. But, intuitively, you have. So White's principle is false. 

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[1] White, "Fine-Tuning and Multiple Universes", in God and Design (2003).

[2] Elliott Sober defends the same principle. See his forthcoming book, The Design Argument. 

[3] The reason is that, according to a standard way of viewing evidential support -- one used by White and Epstein -- E is evidence for H just in case Pr(E|H) > Pr(E|~H). Alternatively, E favors H over H* just in case Pr(E|H) > Pr(E|H*). And since neither inequality is satisfied in pond case when you go with E* over E, your catch is not evidence (or does not favor) thinking there are more large fish in the pond.