There are many Cosmological Arguments. It is perhaps more surprising that there are many variations of the individual versions of the Cosmological Argument. One, expressed similarly by Kreeft and Tacelli, goes like this:
1. Everything in motion is moved by another.
2. The universe is in motion.
3. Therefore, the universe is moved by another.
In this case, "motion" just means "change." A pile of bricks is a brick wall in potentiality, but it requires someone or something to build it into a wall in order from the pile to move from potentiality to actuality qua wall.
The second premise is sometimes alleged to commit the fallacy of composition. This fallacy is committed by applying something to the whole, when it only really has application to its individual parts. Sticking with our brick wall analogy, even though all of the individual bricks may be small, the wall as a whole may very well be large. This is known as an incidental composition.
However, sometimes it does make sense to apply to the whole what also belongs essentially to its parts. Hence, this is known as an essential composition. For instance, given that each part of the wall is made of brick, it logically follows that the entire wall is itself made of brick.
A proponent of this version of the Cosmological Argument will contend that (2) makes use of an essential composition. In fact, it is difficult conceptually to think how if every part of the universe is in motion, that the universe as a whole would not itself be in motion. Now, if the universe really is in motion (and it appears more than reasonable to think it is), then we can either: a) accept (3) as a sound inference; or b) make an exception for the universe and say that it is not moved by another. The problem is that (b) must be supported by some line of reasoning, or else it commits the fallacy of special pleading. Barring any reason for this exception, then, it seems that the argument from motion is a sound one.
Our job is then to show what attributes we can infer about the universe's mover, the first mover, so to speak. We have looked at these attributes in some detail in other posts, especially in the one dealing with the Inductive Cosmological Argument.