This isn’t analogous to the situation with the PoE, however. There, the information you have is that ‘Bob is a vegetarian who buys meat (at least some, at least sometimes)’—this is what follows from the argumentation of the FWD. Since you have no idea how often and how much, you have no grounds on which to claim that the amount of meat bought in a given instance exceeds what you should expect—and hence, no grounds on which to find it in conflict with the idea that Bob is a vegetarian. Again, just ask yourself: given that information, what exactly is it that makes you doubt that Bob is a vegetarian as a result? It can’t be the mere fact that he is a vegetarian, since you already know that despite this, he buys (an unspecified quantity of) meat.
But you only have a single observation (a single world). Again, if you could compare across distinct observational instances, there would be no problem to establish a baseline—but you don’t have that option (except virtually, through assumptions and arguments).
Let me address this and @Irishman’s points with an explicit example. Suppose your gran has contributed a jar of cookies to a bake sale. You’re tasked with bringing her jar home, but unfortunately, you don’t know what the jar looks like. How can you tell which jar is the one containing your gran’s cookies? Without further information, evidently, you can’t.
Suppose you know your gran disapproves of raisin cookies. If you know that she’d never, ever even bake a single raisin cookie, your task becomes easy: any jar from which someone draws a raisin cookie is out, immediately. (This would be the situation if the logical problem of evil worked.)
But now suppose you know that despite her disapproval of raisin cookies, you knew that she sometimes makes them, perhaps because you, her favorite grandchild, really likes them. What does somebody pulling a raisin cookie from a jar now tell you? Well, nothing at all, evidently. Beyond the fact that there may be raisin cookies in your gran’s jar, you simply have no information. There might be none, or a single one, or the entire jar might be raisin cookies. Each of these cases is compatible with the information you have, and equally well so; and so are all the ones in between.
So how could you nevertheless make a decision? Well, what you need is further information: some idea of how many raisin cookies to expect, given that the jar is your gran’s. Then you can check whether the observed raisin fraction agrees with what you expect—or not. This is an application of Bayesian reasoning.
Bayes’ theorem allows you to assign a certain credence, or belief, to a hypothesis on the basis of the evidence you have obtained, and allows you to modify that belief when additional evidence comes to light. If P(A|B) is the likelihood that the hypothesis A is correct, given the observed evidence B, then we have:
P(A|B)=\frac{P(B|A)}{P(B)}\cdot P(A),
where P(A) is the base probability of the hypothesis being correct, P(B) is the base probability of observing the evidence, and P(B|A) is the probability that you observe a certain piece of evidence B, if the hypothesis A is correct. Here is a nice visual proof of this relation (taken from the wiki article in case there are problems with the image embedding):
This should look familiar: it’s just the illustration of the situation in the PoE I provided above. Here, the hypothesis A is G, i.e. the existence of God, and the evidence B is E, the observation of evil. Essentially, the question is: how likely are we to be in the overlap of both?
It should be clear that just observing the existence of evil does tell us essentially nothing, here. The overlap could be tiny, or envelop all of B—both are equally compatible with what we know. Anything further than that must thus necessarily appeal to additional information. I think this becomes clear by studying cases where we do have additional information, and hence, can say something more than this.
So let’s get back to your gran and her jar of cookies. Suppose that there are two jars, between which you have to decide. This means that the probability of selecting the correct jar, P(N) (N for ‘nanna’ to avoid confusion with G for God), is a priori 50%. Suppose also that you know that gran would never, under any circumstances, put more than 10% raisin cookies into her jar. Then suppose that the other jar comes from somebody S who doesn’t care whether they make raisin or non-raisin cookies, such that the probability of drawing a raisin cookie from their jar is P(R|S)=0.5. Then, you have enough information to update your probability that you have chosen the right jar, based on whether you draw a raisin cookie. Thus, you have the following items of knowledge:
- Likelihood of having selected gran’s jar (prior probability): P(N)=0.5
- Likelihood of drawing a raisin cookie R given that you have selected gran’s jar: P(R|N)=0.1
- Likelihood of drawing a raisin cookie from either jar: P(R) = P(R|N)\cdot P(N) + P(R|S)\cdot P(S) = 0.3
Now we can easily calculate the probability that you have selected gran’s jar, given that you have just pulled out a raisin cookie:
P(N|R)=\frac{P(R|N)}{P(R)}\cdot P(N)=\frac{0.1}{0.3}\cdot 0.5=0.1\bar{6},
so you only have a 17% chance that this is, in fact, granny’s jar. Drawing another cookie, and obtaining again a raisin one, lowers that probability to about 5.5%. This is the basic method how to properly take evidence into account in order to revise one’s belief in a certain hypothesis. Mutatis mutandis, this would be the way to revise one’s belief in a tri-omni God given that one observes evil.
Now, to pre-empt the complaint, nobody does this in such a formal manner in everyday situation. But just because one usually uses heuristics that in everyday situations are sufficient to approximate this sort of process, doesn’t mean that this isn’t the procedure one ought to follow if one wants to be strictly correct. Furthermore, the process is to a certain degree independent of one’s prior beliefs: if you were initially pretty sure that you had granny’s jar, you might need more evidence to convince you that this isn’t the case, but you’d eventually get there.
But what is absolutely the upshot of this discussion is that the mere information that granny doesn’t like raisin cookies, but may nevertheless include some, doesn’t suffice to tell you which way the needle of your belief should move upon finding a raisin cookie. For it is just as compatible with this information that granny’s jar contains, say, 60% raisin cookies—in which case your belief that you have granny’s jar will in fact increase upon finding a raisin cookie, and continue to increase upon finding more of them.
Hence, whether or not evidence (finding more and more evil in the world, say) should increase your confidence in a hypothesis depends on information that you simply do not have. Claiming that it always decreases your confidence thus is simply irrational.