The following is from William Lane Craig's "The Existence of God and the Beginning of the Universe". It is part of the defense of premise 2.11 of his version of the kalam cosmological argument, which says that "an actual infinite cannot exist:"
Perhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.
Over the course of this summer, I've been forming an argument for belief in the Bible. Part 2 of my argument was critically depndent on the claim that "if we accept ... [the] postulate ... that a very powerful being is trying to get our attention, then the most coherent ... [explanation] ... is that ... Jesus did in fact rise from the dead." This is, of course, a fairly modest claim, as it explicitly presupposes a pretty substantial chunk of theology (there is a God, he actually cares what we think/do, he wants us to know about him, etc.). However, some would still dispute it. Some time ago, after reading this post on The Prosblogion, I downloaded the transcript of the debate between William Lane Craig and Bart D. Ehrman on the historicity of the resurrection, and I've just now got around to reading it. Not all of the debate is that good or that interesting (I got irritated with Ehrman repeatedly, and I didn't think Craig always addressed my irritations in his responses - also, while it started out very friendly, the participants became noticeably irritated with each other later on and both got into some repetitive rhetorical attacks). However, Craig's first speech gives an excellent and concise outline of the type of argument needed to establish this claim (though, disappointingly, Craig is not as straightfoward about the presuppositions required for the argument to work as I would have liked). In short, if you found my argument unsatisfactory (or even if you were utterly convinced by my argument), I highly recommend that you read at least Craig's first speech in the debate, though I cannot unequivocally endorse everything he says throughout.
Middle knowledge is a problem that has been bothering me for quite some time now. It goes like this: middle knowledge is knowledge of the truth or falsity of counterfactuals of freedom, where a counterfactual of freedom (sometimes called a counterfactual of creaturely freedom) is a statement about what some agent having libertarian free will would do in a purely hypothetical situation, i.e. one that never has and never will occur. Libertarian free will means that one is free because one could do otherwise than one actually does. So, for instance, if human beings (including me) have libertarian free will (as I believe they do), then the statement "if Kenny was offered $1 million to kill someone yesterday, he wouldn't have done it" is a counterfactual of freedom: yesterday has come and gone, and I can assure you that no such situation arose yesterday. It seems that middle knowledge ought to exist, because it seems that God ought to have it. First, there are a few places in Scripture that seemingly make claims about counterfactuals of freedom, but I find this to be unimportant to the discussion, as the Bible is presumably speaking in a 'loose and popular' sense and not in 'metaphysical rigor.' What I find more problematic is this: assuming that human beings have libertarian free will and that God has foreknowledge of their actions, but there is no such thing as middle knowledge, it would be the case that God wouldn't have known that Adam would sin if he hadn't created this world, the one in which Adam did sin. That is, because God created this world, and it is a world in which Adam sins, God knew from eternity that Adam would sin. However, if God had created a different world - say, for instance, one that didn't have any beings with libertarian free will (other than, I suppose, God himself) - God would never have known what would have happened if he had created Adam and Eve and the garden. Furthermore, if God has foreknowledge but not middle knowledge, then God does not now know with absolute certainty whether Adam would still have sinned if he had placed the Tree of the Knowledge of Good and Evil three inches east of where he in fact placed it. This seems more than a little problematic!
So suppose that there is middle knowledge (and God, being omniscient, has it). In addition to allowing God to make more informed decisions, it is useful for theodicy, in that it allows us to take the step that William Lane Craig does in explaining the justice of the condemnation of those who are under non-ideal conditions with regard to Christian salvation in terms of what Craig calls "transworld damnation" (William Lane Craig, "Is 'Craig's Contentious Suggestion' Really So Implausible?," Faith and Philosophy 22 (2005): 358-361). Craig's interlocutor, Raymond Van Arragon, gave one definition of "transworld damnation" as follows, and Craig seems to accept his formulation: "The property of being such that in every feasible world in which one exists, one does not accept Christ" (ibid. 359, emphasis original). What this means is that we might say that for everyone who rejects Christ and is condemned to hell, there was absolutely nothing that could have been done, no evidence that could have been offered, such that he would accept Christ. Leibniz offers a similar defense from a more fatalistic perspective: he thinks that all properties are essential, and therefore person A's spending eternity in hell is an essential characteristic of person A, and if A's eternal fate were changed, A would no longer be A. In other words, according to Leibniz, it is logically necessary that (A exists)->(A will spend eternity in hell). Not a very comforting thought, but the idea, again, is that if A went to heaven, A would simply not be A.
Now, I don't know that I would want to accept Craig's claim, even if the problem of middle knowledge were solved, but there are clearly many reasons why a Christian would want to say that God has middle knowledge besides just this. So what is the problem of middle knowledge? It is the question of "in virtue of what are conterfactuals of freedom true?" That is, in general, when we say that a proposition is true, we mean something like the classic correspondence theory of truth (no, I haven't read the long article I just linked to) - we claim that it corresponds to some "way things are" out in the world. But what is the "way things are" out in the world such that counterfactuals of freedom are true? If it is facts about my character, I don't have libertarian free will (I act always according to my character, as in compatibilist free will). If it is divine will, I don't have libertarian free will (God compels my hypothetical actions). It seems that if I can actually do otherwise in actual situations, I should be able to hypothetically do otherwise in hypothetical situations! But since the hypothetical situation never happens (the proposition is counterfactual), I never have the chance to act one way or another.
This is known as the 'grounding objection,' and is discussed in another paper by William Lane Craig, which I just finished reading online, entitled "Middle Knowledge, Truth-Makers, and the 'Grounding Objection.'" I was hoping for a fantastic solution from Craig that would solve all of my problems but, alas, I was quite disappointed. Craig's paper actually only argues that oponents of middle knowledge have not demonstrated that counterfactuals of freedom need truth-makers, and that the whole theory of truth-makers is convoluted, controversial, and quite possibly false. Furthermore, even those who embrace the theory often hold that not all propositions need truth-makers. This is all well and good, but what, I ask, about truth-conditions?! Surely propositions must have truth-conditions! If a sentence is meaningful (i.e., actually expresses a proposition), there must be some condition in virtue of which it is true (or false). I'm not talking about verificationism here - that is, I'm not claiming that in order for a sentence to be meaningful there must be some process by which we can determine its truth. Rather, I am claiming that in order for a sentence to be meaningful there must be some way that existing things in the universe are such that the sentence is true or false. Craig doesn't seem to address this in either of papers I read, but perhaps he could do so with a kind of limited modal realism:
I'm not sure what is meant by a "feasible world" in the transworld damnation discussion, but suppose it means a world that would have occurred had God acted differently than he actually did. That is, God acts differently, and then leaves the other beings with libertarian free will to act as they wish. Now suppose that all feasible worlds exist - perhaps on a lower ontological plane than the actual world, or perhaps we should say like David Lewis that 'actual' is simply an indexical term, picking out one of the feasible worlds and they are all on the same ontological level. Now we will need to claim that you and I are in fact 'transworld entities' - that is, we are made up not just of our selves in the actual world but of our counterparts across all feasible worlds. If this is the case then we actually do choose whether counterfactuals of freedom about us will be true or false, and middle knowledge is saved. However, the whole transworld damnation thing only works if God picks out certain feasible worlds to create. So perhaps we should say that only the people who exist in the actual world are 'real' and God has created all of the feasible worlds which contain one or more 'real' people, and that he does so specifically so that counterfactuals of freedom will have meaning.
At this point, I think we are worse off than we started. (1) It's still not clear that God really has middle knowledge, because he only knows about what free beings do in worlds he creates (he just created a lot of worlds). (2) What happens to the merely 'feasible' people (those who do not exist in the actual world)? Are there 'feasible' heavens and hells? Or will those people simply cease to exist at the end of the actual world? (3) Are merely feasible people actually people with moral value and so forth? (4) If the feasible worlds are real, how do I know I'm in the actual world and not one of those? Furthermore, how do I know I'm a 'real' person (transworld entity) and not merely feasible? If I'm merely feasible, doesn't that mean that God still doesn't have (complete) middle knowledge about me?
In short, problem not solved, by me or by the Craig papers I read. Does anyone have better suggestions? Craig seems to have written a lot on the subject. Perhaps one of his other papers solves this problem?