I'm trying to get clear on Craig's new a posteriori argument for the beginning of the universe, and I'd be grateful for any constructive feedback. As far as I can make out, the core of his argument can be expressed as follows:
1. The Borde-Guth-Vilenkin Theorem (BGV) is true.
2. If the BGV is true, then any universe or multiverse which has, on average, been expanding throughout its history has a beginning to its expansion. (implication of BGV)
3. Therefore, every universe has, on average, been expanding throughout its history.
4. Any particular universe or multiverse that has a beginning to its expansion has a beginning of its existence.
5. If any particular universe or multiverse has a beginning of its existence, then there is a beginning to the existence of the aggregate composed of all physical objects (or the Aggregate for short) -- including all universes and multiverses there may be.
6. Therefore, the Aggregate has a beginning of its existence.
Some questions and comments:
(i) Does this look right, or am I leaving out something of substance?
(ii) I'm assuming (1) is the consensus view among the relevant scientists, and that (2) just falls out of (1).
(iii) I have no idea why we're supposed to accept (3).
(iv) I'm not entirely sure what the justification for (4) is supposed to be, but I think it's roughly as follows: (a) The initial state I of any given expanding universe U must be one of quiescence or activity. But (b) if I was a state of quiescence, then it would've remained in that state forever, as (c) there would be no way for an event to begin within U (which contradicts the assumption that it's expanding). On the other hand, (d) if the initial state of U was one of activity, then the expansion of U would have had an infinite amount of time to start its expansion phase I, in which case (e) it would've been expanding for eternity (which contradicts the assumption that U's expansion had a beginning). Therefore, (f) any universe that has a beginning of its expansion has a beginning of its existence.
As I said, I'm not sure if that's Craig's reasoning, but if it is, it seems pretty bad. One problem with it is that (c) doesn't entail (b). For even if the initial state of a universe were quiescent, and even if there is no way for an event to begin from within a quiescent universe, it doesn't follow that there's no way for a physical entity or event outside of a quiescent universe to initiate activity within it.
Furthermore, the best way I know of to support the inference from (d) to (e) ("if x can be completed in an infinite stretch of time S, then x must (or at least will) be completed within S") is Craig's a priori "immortal counter" argument, but that argument has an undercutting defeater.
Unfortunately, I don't know of another way to support (4), but perhaps there is one, and Craig has stated it somewhere (it may be that it's in his chapter in The Blackwell Companion to Natural Theology, and I missed it, though. It's been a while since I've read it, so it's quite possible that that's what's going on). If so, I'd be grateful for help on this.
(iv) Without independent justification, the inference from the antecedent to the consequent in (5) is an instance of a quantifier shift fallacy or the fallacy of composition. Thus, grant that each universe and multiverse is an inflating structure, and grant that each particular inflating universe or multiverse must have a beginning. It doesn't follow from these claims that the set of all such structures must have a beginning (let alone all physical reality. Note that the argument assumes the two are co-extensive). So, for example, it's epistemically possible that each inflating universe or multiverse has a beginning, but that there is a beginningless series of such universes or multiverses. It's also epistemically possible that there is some non-inflationary structure that's eternal (say, some sort of field structure), from which a finite or infinite number of inflating universes or multiverses arise.
Are such structures scientifically acceptable? Given Craig's ultimate argumentative aims, that's neither here nor there. For Craig's ultimate aim is to establish God as the originating cause of all other concrete entities that exist. As such, it's not enough for Craig to rule out all scientifically acceptable rival hypotheses of the origin of the universe besides theism; he must also rule out all metaphysically acceptable rival hypotheses of the origin of the universe besides theism. Perhaps, though, Craig has other arguments for (5) that can achieve this stricter aim. If so, I'd be grateful to learn what they are.
(v) Assume my reconstruction of Craig's argument adequately captures Craig's reasoning and that all the concerns raised above can be answered. The argument is nonetheless undercut by the following G.E. Moore Shift argument:
1. If all of premises 1-5 are true, then the Aggregate has a beginning of it's existence.
2. The Aggregate does not have a beginning of it's existence.
3. Therefore, not all of premises 1-5 are true.
The argument's valid, and (1) is true in virtue of the validity of my reconstruction of Craig's argument. Furthermore, (2) is justified in virtue of our a priori and a posteriori evidence for the following version of the principle of material causality (PMC): every concrete object (or aggregate thereof) that begins to exist has a material cause of its existence (i.e., it's made from pre-existing stuff, whether material or immaterial). So unless we allow that either (a) the Aggregate is made out of God's being, (b) the Aggregate is ultimately made out of some eternal stuff distinct from God, or (c) things that begin to exist but aren't preceded by other events don't need causes (all of which Craig is committed to rejecting), PMC should push one to accept (2).
Now here's the rub: if the G.E. Moore Shift argument is sound, then Craig's argument is unsound. And if Craig's argument is sound, then the G.E. Moore Shift argument is unsound. Unfortunately, The premises in the latter have at least as much going for them as those in Craig's argument. And if that's right, then the epistemic force of each cancels out that of the other, in which case Craig's argument is subject to an undercutting defeater.
 Of course it goes without saying that Craig's posited immaterial, tri-personal creator-out-of-nothing is scientifically unacceptable.
 Indeed, Craig also wants to argue that God is the cause of all abstract objects as well, but that's a topic for another day.