BTW, what would you consider evidence for a genuine attempt to lull the government into a false sense of security (in an analagous situation)?

It is impossible for A and ~A to both be evidence for B. If a lack of sabotage is evidence for a fifth column, then an actual sabotage event must be evidence against a fifth column.

This is not correct.

One explanation (call it A) for why there fails to be sabotage is that the Fifth Column is trying to be sneaky and inflict maximum damage later on when no one expects it. The probability of that is greater than 0, so it is a legitimate potential explanation for the apparent absence of sabotage. But, on further thought, there is this other possible explanation (call it B): the absence of a Fifth Column will produce an absence of sabotage. The probability of this is also greater than 0.

So here we have the event (Fifth Column exists) constituting evidence for (absence of sabotage) (perhaps the probability is low, but not zero). Surely it is fair to take it for granted that ~(Fifth Column exists) also constitutes evidence for (absence of sabotage). So that's an example where an event and its negation can potentially both be evidence for something.

I think what you really mean to say is that since P(no sabotage) = P(no sabotage | Fifth Column) * P(Fifth Column) + P(no sabotage | no Fifth Column) * P(no Fifth Column), and since no sabotage has been observed, making P(no sabotage) = 1, this must imply that P(no sabotage | Fifth Column) * P(Fifth Column) = 1 - P(no sabotage | no Fifth Column) * P(no Fifth Column).

If we then make the (perhaps unwarranted) assumption that the prior probabilities are equal, i.e. P(Fifth Column) = P(no Fifth Column), then when deciding via a maximum a posterior decision rule which hypothesis to believe, we wind up with P(no sabotage | Fifth Column) = 1 - P(no sabotage | no Fifth Column), and thus we simply select the hypothesis corresponding to whichever conditional probability is larger... and from this, our intuitions about basic logic would say it doesn't make sense to assign probabilities in such a way that (no Fifth Column) is less likely to cause no sabotage than (Fifth Column), and this is what creates the effect you are noting that some event A and its negation ~A shouldn't both be evidence for the same thing.

Compactly, it's only fair to claim that A and ~A cannot both be evidence for B in some very special situations. In general, though, A and ~A definitely can both serve as supporting evidence for B, it's just that they will corroborate B to different degrees and this may or may not be further offset by the prior distributions of A and ~A.

But it is important not to assert the incorrect generalization that "It is impossible for A and ~A to both be evidence for B."

As I understand it, there were at least three hypotheses under consideration: a) No members (or a negligibly small fraction) of the ethnic group in question will make any attempt at sabotage. b) There will be attempts at sabotage by members of the ethnic group in question, but without any particular organization or coordination. c) There is a well-disciplined covert organization which is capable of making strategic decisions about when and where to commit acts of sabotage.

The prior for A was very low, and any attempt by the Japanese government to communicate with saboteurs in the States could be considered evidence against it. Lack of sabotage is evidence for C over B.

The reply that it "is impossible for A and ~A to both be evidence for B" is to ignore what Frank said in favor of insisting on the very overgeneralization I think he was trying to point out. It's not impossible at all when we are being imprecise enough about the prior expectations involved, such as when we lump all moments in a sustained effort together.

Here's an example to illustrate what I'm saying: Say you are a parent of a 10 year old boy who generally wants to stay up past his bedtime. His protests vary from occasional temper tantrums to the usual slumped-shoulders expression of disappointment that bedtime has finally arrived. Under normal circumstances, the expectation is that he would give at least some evidence of wanting to stay up later. We'll call this resistance "A," and A is evidence for "B": his desire and motivation to stay up later. What shall we say when ~A happens? That is, what shall we say when the boy one day suddenly goes enthusiastically to bed? That he has given up his desire? That it is impossible for this to be evidence of his continued desire and motivation? Of course not. It is exactly what we would expect a motivated and reasonably intelligent person to do: try different and probably more effective strategies. If we generalize the ongoing experience of the little boy's quest to say up later, A and ~A are both evidence of B. "It is impossible for A and ~A to both be evidence for B" is simply not narrow enough to be a true statement, and using it in that way can easily amount to a bad counterargument.

Rather, we need to be specific about each situation. What I think we should pay attention to here is the prior expectation of B. With a high enough prior, A and ~A could either (but not both) be evidence of B. But if we are not being specific to each precise situation, the generalization "it is impossible for A and ~A to both be evidence of B" can be a very subtle straw man, because the person being argued against may not be relying on the assumption that A and ~A are equal evidence for B at the same time and in the same situation.

Returning to the Japanese Fifth Column argument, unlike the little boy in my example the Japanese (and, in general, descendants from countries that go to war with their current country of citizenship) do not have a consistent track record of wartime sabotage. Also, there isn't any reason to think they would not generally be more loyal to their country of citizenship than the country of their parents, grandparents, or even of their own childhood. So there is no particularly strong expectation that they would commit sabotage... and thus no such expectation that some mysterious lack of sabotage is itself a sign of a new strategic attempt as part of a sustained effort. The argument should come down to the prior expectation of Japanese sabotage. That seems to be the crux of it to me.

It seems to me the weakness in Frank's argument also lies in the basic premise that we should expect the Japanese to commit sabotage. And I believe the governor would need to rely on that premise, or a similar one, in order to sustain his argument beyond what Eliezer presented.

But. The inside information premise seems nearly undefeatable to me. We can't comment on information we don't have. I think that is always a possibility with controversial official responses that most people would prefer to deny. If the person whose claims you are evaluating has secret but pertinent information you don't have access to, then it can be very difficult to offer a fair analysis. For one, you will have very different yet subjectively valid prior expectations.

I think you're right, but there's an adjustment (an update, isn't it called?) warranted in two directions.

The absence of sabotage decreases the likelihood of the fifth column existing at all.

But if there is a fifth column, it could be reasonably predicted that there would be evidence of sabotage unless there was an attempt to keep a low profile. If they were to favor this hypothesis for other reasons, as in the classified data mentioned by Frank, then the lack of apparent sabotage would also increase the probability that if the unlikely fifth column DID exist, it would be one which is keeping a low profile. I grant, of course, at the same time, the decreased probability of there being any kind of fifth column in the first place.