Teoría de Relaciones Humanas

Having argued that exceeding a probability-threshold is not necessary for acceptabil- ity, I now want to argue that it is not sufficient either. My argument is closely related to the Preface Paradox’s “sister paradox”, theLottery Paradox(first proposed by Kyburg (1961)). This paradox, appropriately modified to fit the topic of acceptability (as opposed to the related but distinct topics of knowledge and rational belief), goes as follows: Suppose you are almost certain that there was a lottery drawing yesterday, and that this lottery was a fair one with 1000 tickets. Let Ti be the proposition that ticket number i lost. Being almost certain that the lottery was fair, you assignp(Ti) = .999 for all1 ≤ i ≤ 1000. Yet, again because you are almost certain that the lottery was fair, you assign probability close to 1, say .999, to the propositionTsthat thatsometicket won, i.e. that not all tickets were losing tickets. So if a hypothesis is acceptable if its probability exceeds some threshold (and if the threshold is lower than 0.999) thenT1,T2,. . .,T1000,andTsare all acceptable.

The problem, of course, is that these propositions are inconsistent, so the Threshold View entails that a set of propositions that are jointly inconsistent would be acceptable. This goes against the following principle:

(C) If eachHi in{H1, . . . , Hn}is acceptable, then{H1, . . . , Hn}is consistent.

That is bad enough, but things get worse: Given (&) (or (D)), it follows from each of the claims being acceptable that their conjunction – a contradiction – is acceptable. Further- more, note that since a contradiction necessarily has probability 0, this conflicts with the necessity claim of the Threshold View (assuming of course that the threshold is not 0). Now, one might doubt that (&) (or (D)) hold true in all cases. Even without (&) (or (D)), however, it is quite implausible to say that a set of propositions could all be acceptable even if they are jointly inconsistent. Yet that is what the sufficiency claim of the Threshold View entails.

One might think that the Lottery Paradox can be resolved by making a minor mod- ification to the Threshold View. This thought starts by pointing out that the probability assigned to each hypothesisTiis based on a known objective chance (greater than zero but less then one), namely the objective chance that any particular ticket wins in a fair lottery with 1000 tickets. This suggests that perhaps both the Threshold View can be rescued from the Lottery Paradox by adding the qualification that hypotheses are not acceptable if they are based on known objective chances in this way.16

This modification won’t do the trick, however, because there are variations of the Lottery Paradox that have nothing to do with objective chances. It’s easy to see what such variations would look like: Suppose you have a set of 1000 propositions,{H1, ..., H1000}, none of which are based on known objective chances. Suppose further that you are almost certain that{H1, ..., H1000}contains at least one false claim, and thus assign a probability close to 1, say .999, to the proposition that the conjunction(H1&...&H1000)is false. And

16_{In this vein, Nelkin (2000) suggests that propositions likeT}

icannot be rationally believed because they

are based on what she calls “P-inferences”, where a P-inference is an inference to a propositionpfromp

having a high “statistical probability”. Now, I argue below that this won’t do as a solution to the Lottery Paradox, but some have found this requirement to be independently plausible. That is, one might think that whatever else is true of acceptability, a theory cannot be acceptable if its probability is merely “statistical” in Nelkin’s sense. While I do not find this restriction plausible, I don’t want to alienate the reader who does find it plausible. Fortunately, we can set this restriction aside in what follows since any account of acceptability that appeals to probabilities (including the one I present below) may simply adopt this restriction in addition to whatever else the account requires of acceptable theories.

suppose you have no reason to think that one of the claims in{H1, ..., H1000}is more likely to be false than any other, but you do have very, very good reasons to think that each claim is true. So let’s suppose that p(Hi) = .999 for all 1 ≤ i ≤ 1000. Again we have that, if a hypothesis is acceptable just in case its probability exceeds a threshold (and if that threshold is below 0.999), an inconsistent set of hypotheses could be such that all of the hypotheses are acceptable. This violates (C) of course. Moreover, (&) (or (D)) again entails that a contradiction would be explanatorily acceptable.

In the next subsection, I will address a worry about (C) having to do with the fact that scientists in fact often accept inconsistent theories. For now, note that as with the Preface Paradox, the Lottery Paradox applies even if the threshold is nonrigid. For we can simply fix the context and stakes so that the same threshold applies to all of the relevant propositionsT1,T2, ...,T1000, andTs). The same goes for a version of the Threshold View on which the threshold is vague: If so, i.e. if the threshold is an interval as opposed to a number, then one can still suppose that all the hypothesis involved have probabilities above the upper bound of that threshold-interval. So nonrigid and/or vague thresholds do not allow the Threshold View to avoid the Lottery Paradox.