Saturday, September 29, 2012

Bayes’ Theorem and the LCA (p. 3)

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Bayes’ Theorem and the LCA
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Pr(E|¬H)

One could make this objection: there are infinitely many ways for there to be something, and only one way for there to be nothing. Pr(E|¬H) should therefore be extremely high, somewhere near 1.

I don’t think this objection is quite as good as it first appears. In some respects simplicity is evidence of truth, and we tend to have a special place in our intellectual hearts for the number zero. To illustrate, consider the case of “How many gods are there?”

Suppose one has no background information regarding the existence of “How many gods are there?” and little to no background information about what sorts of gods, if any exist, are like (e.g. whether they are friendly or hostile, whether they are beautiful or ugly, and whether any have six arms). Given the lack of such background information, what should one’s default view be about whether there are any gods? The default view, it seems to me, is to be truly agnostic, i.e. award “there are no gods” the probability of 50%. Some atheists would award “there are no gods” a default probability greater than 50%, so to broaden the agreement let’s say that the default probability of “there are no gods” is greater than or equal to 50%. But provided we would also award nonzero (even if low) probabilities for “there is exactly one god” and “there are exactly two gods,” we would be giving especially high probability status to there being zero gods, as in “the probability of there being zero gods is greater than or equal to 50%,” thereby awarding the “zero” value to be more probable than any other number of gods and it being no less probable than any other proposition that entails the existence of gods (e.g. “there is a god with six arms”). Notice how significant this is: we are awarding “there are zero gods” a probability value that is no less than all other possibilities combined even when there are infinitely many ways for gods to exist. This strongly suggests, I think, the value of ontological simplicity when coming up with prior probabilities especially as they relate to “there exist things of type X” when there is no background information.

The same principle holds, I believe, for “Why is there something rather than nothing?” Nothing is the ultimate zero, and abandoning all the background information we would otherwise have about things existing and focusing on just ¬H, it seems to me that Pr(E|¬H) is 0.5 or something close to it. The question, “Why is there something rather than nothing?” is interesting precisely because in the absence of a sufficient reason, there being nothing at all to exist seems like a very real possibility.

Prior Probabilities: Pr(H) and Pr(¬H)

Disputing the prior probabilities doesn’t actually affect my main point: a person who is truly agnostic (thinks “there is an explanation for the universe’s existence” is equally likely as “there is no reason for why the universe would exist”) but hadn’t yet considered the evidential force the universe’s existence has for H should re-assign a probability of about 67% for H in the absence of further arguments. That said, one could say that a person who didn’t consider the evidence for E has for H should have Pr(H) be less than Pr(¬H).

One could say that, but in that case the objector would have to argue that in this situation “there is an explanation for the universe’s existence” is less probable than “there is no reason for why the universe would exist.” In abandoning the agnostic’s “neutral probability” default position, the objector would need to give some argument for why the background information would favor ¬H over H. Furthermore, one could argue that if anything the opposite should be done. Hypothesis H has some inferential virtues (things that make an inference good) that go beyond merely entailing E, and thus there are inferential virtues that aren’t captured in Pr(E|H).

One reason to rank H higher than we would otherwise rank it is that it’s part of the nature of rational inquiry to look for explanations for why things exist. To quote what I said in a previous entry:
Here I’ll borrow a bit from philosopher Richard Taylor’s illustration of finding a translucent ball in the woods. “How did it get there?” you ask. I reply, “There is no explanation for it being in the woods; the ball just exists inexplicably.” My response seems less plausible than the idea that there is some explanation for the ball’s existence. What if we enlarged the ball to the size of a car? Same problem: some explanation seems to be needed. How about a city? Same problem. A planet? Same problem. A galaxy? Same problem; increasing the size does nothing to remove the need for an explanation. How about if the ball were as big as the universe? Same problem. All things considered, it seems intuitively plausible that if a contingent thing exists, there is some reason why it exists, since it could have failed to exist.
The universe is contingent, and our default rational preference should be to accept that there is an explanation for its existence. Regarding the translucent ball illustration, Maverick Atheism says:
True enough, increasing the size of the translucent ball does nothing to remove the need for an explanation (let us also assume arguendo that all translucent balls are contingent). Size doesn’t matter, but what does matter is whether the translucent ball existed eternally. It is quite conceivable that there are possible worlds where a translucent orb has existed for all eternity without an external cause. If we had no evidence that the translucent ball began to exist, it would seem at least premature to simply assume it had an external cause
This may be a situation where reasonable people can disagree, but I’m not quite convinced that knowing the translucent ball to be eternal is sufficient to remove the need for an explanation (recall that it wasn’t sufficient in the case of the eternal monument scenario). In any case, suppose also in our eternal translucent ball scenario we had an explanation for its existence that was readily available, fairly straightforward, is the only known viable explanation, and we have no reason to believe the explanation is false. In that case I think the prior probability of there being a sufficient reason for the translucent ball’s existence was fairly good, at least more than 50%.

Similarly, perhaps it is logically possible for the contingent universe to exist eternally and uncaused, and for there to be no explanation for there being something rather than nothing. But it seems more intellectually satisfying to accept that there is an explanation for why there is something rather than nothing, especially if we have an explanation readily available, is the only known viable explanation, and no reason to think the explanation is false. The atheist could argue that we do have reason to think that the theist’s explanation (a transcendent personal cause) is false, but again that would require some kind of argument. In the absence of such an argument, the background information (the nature of rational inquiry, a readily available explanation etc.) if anything favors H over ¬H.

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