The last 50 years have seen significant progress in clarifying the philosophical issues involved in the Glendower, Regress, and Taxicab problems. Indeed, several rigorous versions of the cosmological argument are available to overcome these. The Gap Problem has yet to see as much progress. Perhaps the reason is merely sociological. The typical philosophical atheist or agnostic not only does not believe in God but also does not believe in a necessarily existing First Cause. The typical philosopher who accepts a necessarily existing First Cause is also a theist. Thus, there is not much of an audience for arguments that the necessarily existing First Cause is God. Moreover, it makes sense to proceed in order - first, get clear on the argument for a necessarily existing First Cause, and only then on the argument that this is God.
Probably the most important part of the Gap Problem is the question whether the First Cause is an agent. After all, if the First Causes would have to be nonagentive necessarily existing substances that randomly spit out island universes, then the conclusion of the cosmological argument would be incompatible with theism.
In addition to the problem of personhood, there is the question of the other attributes that God has traditionally been believed to have: uniqueness, simplicity, omniscience, omnipotence, transcendence, and, crucially, perfect goodness. At the same time, it is quite reasonable for a defender of the cosmological argument to stop deriving attributes of the First Cause at some point, and say that the other attributes are to be accepted by a combination of faith and data from other arguments for the existence of God. In any case, rare is the Christian cosmological arguer who claims to be able to show that the First Cause is a Trinity, and indeed Christian theologians may say this is good, since that God is a Trinity is a matter of faith. Nor does the inability to show by reasoned arguments that the First Cause has some attribute provide much of an argument against the claim that the First Cause has that attribute.
There are two general approaches for bridging the gap between the First Cause and God: inductive and metaphysical. Inductive arguments may claim that supposing that the First Cause exemplifies some attribute is the best explanation of some feature of the First Cause's effects, and in doing so the arguments may reprise the considerations of design arguments. Typical metaphysical arguments, on the other hand, argue that a First Cause must have some special metaphysical feature, such as being simple or being pure actuality, from which feature a number of other attributes follow.
Considerations of space do not, however, allow a full discussion of these arguments, and of objections to them, so I shall confine the discussion to the barest sketches.
One might argue for agency in the causal activity of the First Cause in several ways. In order to not beg the question as to the number of First Causes, simply stipulatively define "the First Cause" as the aggregate of all First Causes - it might be a committee or a heap, but that is fine at this point.
If we got to the First Cause by means of the PSR, then the First Cause's activity must in some way explain everything contingent. If one accepts that all explanations of contingent states of affairs are either scientific, agential, or conceptual - at least these are all the kinds of explanations we know of, and since the concept of explanation is a concept of ours, we have some insight into what can and cannot yield an explanation - then one can argue that the First Cause is an agent. For the First Cause's activity does not provide a scientific explanation. As far as we can tell, science explains things in terms of contingent causes. Nor does the First Cause's activity conceptually explain everything contingent. In contingent reality we find substances, and the existence of a substance is not conceptually explained by the activity of something other than that substance - substances are self-standing. At worst, the existence of a substance is conceptually explained by the existence of constituent parts, but if so, then these constituent parts will themselves be substantial. In the end we shall have to give a nonconceptual explanation, or else to find parts that are necessary. But it is false that goats and people are made up of necessarily existing parts. So, in the end, a noncon-ceptual explanation must be given. Hence, the explanation cannot be entirely conceptual. Since the explanation is not scientific either, it follows that it is at least in part agential, and hence the First Cause either is or contains a necessarily existing agent, unless there is some fourth relevant kind of explanation.
Alternately, one might argue that the only way to resolve the van Inwagen problem is to posit agency in the explanation of the BCCF. Perhaps only agential explanations in terms of a necessary being combine the two crucial features: contingency of effect and the impossibility of asking for a further explanation of some further contingent fact. However, if one thinks that nonagential statistical explanations can also have this feature, then this argument will not impress.
Finally, one can bring to bear the full panoply of design arguments available. The First Cause is an entity that has produced a universe apparently fine-tuned for life, containing beauty and creatures attuned to beauty, containing moral obligations and creatures aware of them; a universe containing conscious beings with free will; and a universe some of whose contents have objective functions (eyes are for seeing and so on - these kinds of functional attributions arguably cannot be reduced to evolutionary claims, although there is a large literature on this controversial claim). We have shown, let us suppose, that there is a First Cause. The further supposition that the First Cause is a highly intelligent and very powerful person acting purposively is highly plausible given all this data.
Finally, one might argue for agency on metaphysical grounds. If the metaphysical arguments show that the First Cause has every positive property, then the First Cause will in particular have knowledge and will, and hence be an agent.
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