As mentioned previously, I run into very little opposition to the following argument:
1. The universe is either caused or uncaused. (Definition)
2. Complex things are most likely caused. (Premise)
3. The universe is complex. (Premise)
4. Therefore, the universe is most likely caused. (From 1 - 3)
In our experience, we all know why (2) is considered true. The only reservations are usually from radical Humeans. The earth, for example, is a complex thing, and we know that the earth was formed by (caused by) various cosmological and geological processes.
(3) is probably an understatement. The universe is astronomically complex (complexity being understood as the interrelationship between diverse objects), so given (1) - (3), it follows logically that the universe is most likely caused.
Interestingly, a number of atheistic philosophers agree with the conclusion that the universe is caused. Quentin Smith is one such atheistic philosopher, and he has argued that the universe is self-caused.
By way of introduction, we have two more disjunctions to consider. First:
5. The universe is either self-caused or externally caused. (Premise)
6. Nothing can be self-caused. (Premise)
7. Therefore, the universe is externally caused. (From 5 and 6)
Again, Smith would agree with both (4) and (5), but he disagrees with (6). He postulates that the universe, while being finite in the past, does not begin at a singularity. Rather, he says that t0 is an impossible state of affairs. His argument hinges on this and on the possibility that between any two moments immediately after the Big Bang, after t0 and before/at t1, there are infinitely-many points. So, t1 is caused by t1/2 and t1/2 is caused by t1/4. This process continues infinitely, such that every moment of time is explained by a previous one without t0 having been an actualized state of affairs.
This is much like treating 0 to 1 as an open interval, with 0 not being a point in the interval. This is problematic for at least two reasons. For one, Smith is assuming that the infinity of points between t0 and t1 are concrete and not merely abstract. Secondly, as Robin Collins points out in his response to Smith, if there were infinitely-many points between t0 and t1, this would result in all kinds of paradoxes.
Take the interval between t1/2 and t1. There are infinitely-many points between t1/2 and t1; but surely according to Smith's own argument, t1/2 is a point in the interval that has been actualized, e.g. t1/2 to t1 is not simply another open interval. Hence, an actual infinite is formed by successive addition, which is impossible, given that another member of the set can always be added before arriving at infinity.
Smith's argument for a self-caused universe, therefore, does not appear to be sound. Yet, are there arguments against the possibility of a self-caused universe, or against anything being self-caused? If there are, then we may consider the final disjunction:
8. The external cause of the universe is either personal or impersonal. (Premise)
The universe's external cause must already be timeless, changeless, immaterial, and enormously powerful. If a case can be made that the external cause of the universe is personal, then we have the icing on the cake of an argument for the existence of God. The focus for philosophers of religion, then, should be not only on the first disjunction (which we have already seen is almost certainly true), but on the second and third disjunctions.