Sunday, January 14, 2007

Past Events and Probability

I once read an article that tried to invalidate a favorite premise of people who argue in favor of Intelligent Design, specifically the extreme improbability of life spontaneously arising from non-life. The article started with the assertion that the probability of any past event is necessarily 1, therefore Intelligent Design partisans were committing a dark and dangerous fallacy by insisting that of life existence of life was highly improbable if it arose through purely natural means. I won't deny that Intelligent Design partisans can and do commit dark and dangerous fallacies, but I don't think that believing the stated premise is one of them. Instead, I doubt the validity of the notion that we should always consider the probability of a known past event to be 1 when reasoning about past events. I do so because such an assumption to deduce some apparently false conclusions. Since we should not be able to deduce false conclusions from any true premise, we should not make it a rule that the probability of any known past event is 1, modus tollens.

Here goes my argument. Let's call the assertion under dispute "Proposition 1" and state it as follows:
  1. If an event occurred in the past, then the probability that the event occurred is 1.
At first glance, this seems reasonable, even intuitively true. It seems internally consistent, and can I think of a way to derive a contradiction from it (not that I've thought that much about it). But that's not what I'm arguing.

Now for Propositions 2 and 3:
  1. 1 is greater than 1/2
  2. If an event has a probability of 1, then the probability that the event occurred is greater than 1/2.
Proposition 2 should be axiomatic enough for anyone, and Proposition 3 is just an instance of Proposition 2. Sorry to bore you with such triviality, but I need Proposition 3 so I can use it in a Hypothetical Syllogism with Proposition 1 to infer Proposition 4:
  1. If an event occurred in the past, then the probability that the event occurred is greater than 1/2.
To state it more simply, if it happened, then it probably happened. This does not seem to pose a problem; if George Washington crossed the Delaware, then we wouldn't be amiss to say that George Washington probably crossed the Delaware, right?

Things only become interesting when we consider a particular case of the contrapositive of Proposition 4:
  1. If the probability of a past event occurring is less than or equal to 1/2, then the event did not occur.
  2. If the probability of a past event occurring is less than 1/2, then the event did not occur.
(Proposition 6 is just Proposition 5 limited to the particular case where the probability of the event is not equal to 1/2). Proposition 6 is my basis for rejecting Proposition 1. To state it simply, if something probably didn't happen, then it didn't happen, or, in other words, improbable events never occur. In a sense, this is believable. If something happened, then it can't not have happened, right? Maybe so. I can't disprove the law of cause and effect. But I can say that trouble ensues when we try to apply Proposition 6 to a situation where something "probably would have" or "probably shouldn't have" happened.

Take an example: Let's say I find out that my brother the college student failed an exam. I know he's a smart guy, and the class he failed it for wasn't anything difficult like thermodynamics, and most of the professors at the college he goes to are competent teachers and test writers. Therefore, he probably didn't study for the test. If I accept Proposition 6, then I must conclude that he simply did not study for the test.

Here is the problem: I should not be able to draw this conclusion based on the information I have. It is still possible that he did study, but he got a bad professor or he just wasn't understanding the material or perhaps something else. However, Proposition 6 follows necessarily from Proposition 1, Proposition 1 makes this conclusion not only possible, but logically necessary!

Therefore, it would be an obvious fallacy to apply Proposition 1 to reasoning about the probabilities of past events. I am not saying that I've reduced Proposition 1 to absurdity. I'm just saying that it is not a valid rule to be used for reasoning about the probability of past events, which is exactly what was attempted in the aforementioned article regarding intelligent design.

1 comment:

highlandhistory said...

Dang, boy!

Nice, mathematical approach.