On Philosophy

September 4, 2007

The Paradoxical Nature Of Probability And Justification

Filed under: Epistemology — Peter @ 12:00 am

In many situations we have the ability to estimate the probability that a particular proposition is true given the evidence at hand. With that knowledge may realize that one particular proposition is highly likely. And so it seems natural to tie our estimations of probability to justification; if a proposition is likely it is justified, and if it is unlikely it is unjustified. There are some definitional problems, regarding how likely a proposition has to be before it is justified, but these seem like they might be swept under the rug without too many problems. However, we also expect that we can use the rules of logic to move from one justified proposition to another. For example, if we are justified in believing P then we expect that since P and ~P are exclusive options we are then justified in disbelieving ~P. But when we allow probability to be the basis of justification that simply isn’t the case; the laws of deduction we expect to hold can contradict what we know about probability.

Previously I used probability to deduce how likely various hypotheses regarding frequency were given evidence in the form of the event occurring or failing to occur. Here I am going to use those calculations to illustrate my claims concretely, specifically those outlined in the last paragraph, which allow us to estimate the probability that the frequency falls within a certain range. Suppose then that there is an eighty percept probability that the frequency is between 60% and 70%. Since this is more likely than not we can conclude that J([60%,70%)), meaning that the proposition that the frequency is between those two values is justified. Which in turn means that J([60%,61%) ∨ [61%,62%) ∨ … ∨ [69%,70%)), which indicates that we are justified in believing that the frequency lies in one of the 1% ranges between 60% and 70%. Again, if we are justified in believing the first it seems like we must be justified in believing the second, assuming this is an ordinary kind of event if it has a frequency between 60% and 70% it must lie in one of the 1% ranges between the two. However, we are clearly justified in claiming that J(~[60%,61%)). Assuming that probabilities are distributed roughly evenly the probability that the frequency lies in that 1% range is roughly 8%, which justifies the opposite claim, since there is a 92% probability that it lies in some other range. And, by similar reasoning we can make the claim that J(~[61%,62%)), and so on. And, if we can logically reason involving propositions that are justified, then we can conclude that J(~[60%,61%) ∧ ~[61%,62%) ∧ … ∧ ~[69%,70%)), and thus that J(~([60%,61%) ∨ [61%,62%) ∨ … ∨ [69%,70%))), and thus that ~ J([60%,61%) ∨ [61%,62%) ∨ … ∨ [69%,70%)), and thus ~J([60%,70%)). Which is a contradiction; we have derived from the probabilities both the conclusion that we are justified in believing that the frequency lies in the 60% to 70% range and that we aren’t justified in believing that claim.

Obviously one of the steps that was used to derive this contradiction must be rejected. There are quite a few options. One is to deny that we justified in believing that the frequency lies outside a small range because of its low probability. While promising, accepting that solution raises inconsistencies of its own, because accepting that it lies outside that small range is equivalent to accepting that it lies in one of the two ranges on either side of that small range. And it certainly has a high probability of doing that. So, to be consistent, we would have to also endorse the idea that a high probability can’t justify a claim, and that means we would be throwing out probability as a guide to justification altogether. Another possibility is to disallow drawing the conclusion that ~J(P) from J(~P). But to disallow that seems rather absurd, because it means that we could be perfectly justified in believing that something isn’t the case, for example, that it isn’t Christmas, but that from that we can’t draw any conclusions regarding our justification regarding the negation of the claim, which would be equivalent to reasoning as follows: “I am justified in believing that it isn’t Christmas, but I can’t rule out the possibility that I am justified in believing that it is Christmas”, which seems absurd since we are not reasoning about situations which have the possibility of both being and not being the case.

One, less absurd, alternative is to argue that we cannot have negative claims, in other words that we can’t claim that a frequency doesn’t lie within a certain range. Or, alternately, we could deny that we can reason from statements that are justified to a conclusion that is justified by that reasoning, and that instead we must always reason directly from the probabilities to the desired conclusion. Neither leads to any obvious contradictions, but neither is entirely satisfactory either. The first prevents us from making a great many claims. Even if the probability of a hypothesis being true is 0% (for example, the hypothesis that an event never occurs when it has been witnessed to occur) we can never say that we are justified in believing that not to be the case, all we can say is that we lack justification for making that claim. This seems to collapse cases where we just don’t have enough evidence to decide in favor of one possibility or the other (perhaps they are equally likely) and cases where we seem to have definitive evidence that something is not the case, and that is not a perfect solution. On the other hand, denying that we move from justified propositions to propositions that are justified by logical entailment is equally unpalatable, since such entailment justifies the probability calculus in the first place, although we can perhaps soften the blow by disallowing justification by entailment only in certain cases.

At their root both problems stem from the fact that claiming that a proposition is justified or unjustified loses a great deal of information. In a perfect world when we combined the propositions that the frequency wasn’t in any of the 1% ranges in our example earlier we would not have derived the conclusion that the frequency wasn’t in the entire range, but rather that the frequency probably was in that range. But to achieve that we would have to move to a conception of justification that allows it to come, at the very least, in degrees. And that in turn may prompt a reevaluation of knowledge as justification plus other conditions. But that is a topic for another day.

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