Probabilistic reasoning and disjunctive Gettier cases

A disjunctive Gettier case looks like this. You have a justified belief in p, you have no reason to believe q, and you justifiedly believe the disjunction p or q. But it turns out that p is false and q is true. Then you have a justified true belief in p or q, but that belief doesn’t seem to be knowledge.

Some philosophers, like myself, accept Lottery Knowledge: we think that in a sufficiently large lottery with sufficiently few winning tickets, for any ticket n that in fact won’t win, one knows that n won’t win on the probabilistic grounds that it is very unlikely to win.

Interestingly, assuming Lottery Knowledge, in at least some disjunctive Gettier cases one has knowledge of the disjunction. For suppose that 99.8% is a sufficient probability for knowledge in lottery cases. Consider a lottery with 1000 tickets, numbered 1–1000, and one winner. I will then have a justified belief that the winning ticket is among tickets 1 through 998 (inclusive). Let this be p. Suppose that unbeknownst to me, p is false and the winning ticket is number 999. Let q be the proposition that the winning ticket is number 999.

Then I have the structure of a disjunctive Gettier case: I have a justified belief in p, I have no reason to believe q, and I justifiedly believe p or q.

Now given Lottery Knowledge, I know that ticket 1000 doesn’t win. But p or q is equivalent to the claim that ticket 1000 doesn’t win, so I know p or q.

Thus, given Lottery Knowledge, I can have a case with the structure of a disjunctive Gettier case and yet know.

Note that usually one thinks in disjunctive Gettier cases that one’s belief in the true disjunction is inferred from one’s belief in the false disjunct p. But that’s not actually how I would think about such a lottery. My credence in the false disjunct p is 0.998. But my credence in the disjunction is higher: it’s 0.999. So I didn’t actually derive the disjunction from the disjunct.

So, someone who thinks probabilistically can have knowledge in at least some disjunctive Gettier cases.

Even more interestingly, the point seems to carry over to more typical Gettier cases that are not probabilistic in nature. Consider, for instance, the standard disjunctive Gettier case. I have good evidence that Jones owns a Ford. You have no idea where Brown is. But since I accept that Jones owns a Ford, I accept that Jones owns a Ford or Brown is in Barcelona. It turns out that Jones doesn’t own a Ford, but Brown is in Barcelona. So I have a justified true belief that Jones owns a Ford or Brown is in Barcelona, but it’s not knowledge.

However, if I think about things probabilistically, my belief in the disjunction is not simply derived from my belief that Jones owns a Ford. For my credence in the disjunction is higher than my credence that Jones own a Ford: after all, no matter how unlikely it is that Brown is in Barcelona, it is still more likely that Jones owns a Ford or Brown is in Barcelona than that Jones owns a Ford.

So it seems that I have a good inference that Jones owns a Ford or Brown is in Barcelona from the high probability of the disjunction. Of course, a good deal of the probability of the disjunction comes from the probability of the false disjunct. However, that doesn’t rule out knowledge if there is Lottery Knowledge: after all, a good deal of the probability of the disjunction in our lottery case could have been seen as coming from the false disjunct that the the winning number is between 1 and 998.

Perhaps the difference is this. In the lottery case, there were alternate paths to the high probability of the true disjunction. As I told the story, it seemed like most of the probability that the winning ticket was either from 1 to 998 (p) or equal to 999 (q) came from the first disjunct. But the disjunction is equivalent to many other similar disjunctions, such as that the ticket is in the set {2, 3, ..., 999} or is equal to 1, and in the case of the latter disjunction, the high probability disjunct is true. But in the Ford/Barcelona case, there doesn’t seem to be an alternate path to the high probability of the disjunction that doesn’t depend on the high probability of the false disjunct.

But it’s not clear to me that this difference makes for a difference between knowledge and lack of knowledge.

And it’s not clear that one can’t rework the Ford/Barcelona case to make it more like the lottery case. Let’s consider one way to fill out the story about how my mistake in thinking Jones owns a Ford came about. I’ve seen Jones driving a Ford F-150 at a few minutes past midnight yesterday, and I knew that he owned that Ford because I drove him to the car dealership when he bought it five years ago. Unbeknownst to me, Fred sold the Ford yesterday and bought a Mazda. Now, it is standard practice that when people buy cars, they eventually sell them: few people keep owning the same car for life.

So, my belief that Jones owned a Ford came from my knowledge that Jones owned a Ford early in the morning yesteray and my false belief that he didn’t sell it later yesterday or today. But now we are in the realm of a lottery case. For from my point of view, the day on which Fred sells the car is something random. It’s unlikely that that day was yesterday, because there are so many other days on which he could sell the car: tomorrow, the day after tomorrow, and so on, as well as the low probability option of his never selling it.

Now consider this giant exclusive disjunction, which I know to be true in light of my knowledge that Jones hadn’t yet sold the Ford as of early morning yesterday.

1. Jones sold the Ford yesterday and Brown is not in Barcelona, or Jones sold the Ford today and Brown is not in Barcelona, or Jones is now selling the Ford and Brown is not in Barcelona, or Jones will sell the Ford later today and Brown is not in Barcelona, or Jones will sell the Ford tomorrow and Brown is not in Barcelona, or … (ad infinitum), or Jones will never sell the Ford and Brown is not in Barcelona, or Brown is in Barcelona.

Each disjunct in (1) is of low probability, but I know some disjunct is true. This is now very much like a lottery case. Its being a lottery case means that I should—assuming the probabilities are good enough—be able to know that one of the disjuncts other than the first two is true. But if I can know that that one of the disjuncts other than the first two is true, then I should be able to know—again, assuming the probabilities are good enough—that Jones hasn’t sold the Ford yet or Brown is in Barcelona. And if I can know that, then there should be no problem about my knowing that Jones owns a Ford or Brown is in Barcelona.

So, it’s looking like I can have knowledge in typical disjunctive Gettier cases if I reason probabilistically.