Occasionally one meets with the idea that, granted, bringing in eternal life really does help a lot with the problem of evil or with hiddenness, nonetheless bringing in eternal life is a cheat because it begs the question or something like that.
I can see how one can object to the eternal life move by saying that some things are so horrendous that God shouldn't allow them even if he compensates those to whom they happened. Or that an omnipotent God shouldn't need to compensate. Or that God has some kind of a duty never to be hidden (but: surely a lover is permitted to hide for a while, since otherwise it would be wrong to play peekaboo with infants who don't understand about object persistence). But the "it's a cheat to bring in eternal life" move is not this move. Rather, it grants, at least for the sake of argument, that if there is eternal life, then God can have a justification for allowing the evil or being hidden.
I am having a hard time seeing how this "it's a cheat" move is supposed to work. Let T = theism, L = eternal life and E = the atheological arguer's favorite evil/hiddenness evidence. Then: T is equivalent to T&L or T&~L. Now to grant that eternal life would solve the problem would be to grant that P(T&L|E) is not significantly less than P(T&L). Now let the theist grant, in a spirit of mutual accommodation and simplification, that E is conclusive evidence against T&~L: P(T&L|E)=0. But now:
P(T|E) = P(T&L|E) + P(T&~L|E) = P(T&L|E).
But P(T&L|E) is not significantly less than P(T&L), it was granted. So, basically, the atheological evidence E lowered the probability of T to around the probability of T&L before that evidence.
Now, if in our background there is the fact that there are person, then P(L|T) is quite high. If God made persons, it is very likely that they (or at least those who do not deserve to not have it--there might be room for tweaking of what exactly L says), it is very likely that he made them to have eternal life. But if P(L|T) is quite high, then P(T&L) is pretty close to P(T). Since P(T|E) is not much smaller than P(T&L), it follows that P(T|E) is not much smaller than P(T).
So I just don't see how the "it's a cheat" move is supposed to work. Once one grants that the probability of T&L does not go down very much given E, then given the very plausible claim that most of the probability of T is contributed by T&L portion, it simply follows that the probability of T does not go down very much given E.
Note that it doesn't matter for the above whether there is any independent evidence for L. All that the argument needs is that before E is brought to the table, L is very probable given T.
What if there were strong independent evidence E* against L? Well, then E* would provide a pretty powerful argument against theism. P(L|T) is high, but P(L|E*) is low, and so P(T|E*) is low. But notice that here were did not bring in E at all. This is a new atheological argument: the atheological argument from the non-existence of an afterlife. It would be a pretty good argument if, contrary to fact, there were strong independent evidence against L.
Here's another way to put the point. If it's granted that eternal life would completely solve problem that E posed for theism, then the only thing that E could do to the theist would be to eliminate the possibility T&~L. But the typical working theist believes both T and L, and what she cares about is T&L, not just T by itself. So a refutation of T&~L is not something she need particularly mind.
I don't claim any originality here. I know that Trent has independently had the same thoughts. I haven't heard the following point, though.
Suppose the atheological arguer (aa) denies that P(L|T) is high. If one says that P(L|T) is not high, then I think one needs to say that if God existed, God wouldn't be very likely to care that much about the persons he created. But if that's true, that should significantly damage the aa's intuition that P(E|T) is low. For if God wouldn't care that much about the persons he created, then maybe he would let them suffer terrible evils.
Or let's put it this way. If there is no eternal life, then created persons perish forever. But few if any of the evils in E are worse than that! Let's say that E is some instance of torture. Well, perishing forever is worse than E, since if suffering torture in this life were the price of eternal life, it would surely be worth it (as many martyrs of have quite rationally concluded).
So the position that P(E|T) is low but P(~L|T) is not low is untenable. In fact, it is very plausible that P(E|T) > P(~L|T). (One might be able to get some nice theorem using this inequality combined with P(T&L|E) = P(T&L), but that's enough for the day.)