The premises of the Thomistic Cosmological Argument (TCA) need not be supported by deduction, even though the syllogism is itself deductive. Consider the following:
1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
Obviously, (2) is supported by induction, since we have to observe that there is a man named Socrates. However, we may conclude that (1) is true by universalizing what we generally observe. Even if we were to theoretically come across an immortal man, (3) would still be more likely true than its negation, given the mortality of the vast majority of human beings (men and women) we have observed.
Likewise, we may appeal to purely inductive inferences in our defense of the TCA. We might state the argument like this:
1. Evident to the senses is motion.
2. Everything in motion is moved by another.
3. If there is no First Mover, there is no motion.
4. Therefore, a First Mover exists.
(1) is obviously based on induction. I have spent some time deflecting objections to (2) that are based on quantum mechanics, but I don't think that is even necessary. For, even if there are instances in which a thing is in motion but is not moved by another, still the majority of things in motion we observe actually are moved by another. If we take a look at the motion of the celestial bodies, we realize that each body's gravity affects the others, so it is reasonable to conclude that one body's motion is caused by (re: "moved by") another.
(3) can also be supported by induction. Just as the motion of the gears of a watch are best explained by the cause of motion found in the spring (ontologically first member of the watch), so too can the motion of the celestial bodies be best explained by the cause of motion found in the First Mover. The removal of the spring would result in the removal of motion among the gears; so by analogy, the removal of the First Mover would result in the removal of motion among the celestial bodies.
Given the truth of (1)-(3), we may conclude that (4) is more likely true than its negation; and we may conclude this without having to appeal to deduction. However, we may infer even more than this. Just as the uniformity* of motion found in the gears is reducible to a single spring, so too is the uniformity of motion found in the celestial bodies reducible to a single First Mover. Hence, we have a legitimate starting point for an inductive argument for monotheism. After all, uniformity is best explained by the causation of a single agent than it is by many agents.
*Uniformity refers to 1) patterns of regularity; and 2) the exemplification of a system in which each member affects the others. The gears of a watch fit this description, and more importantly, the celestial bodies match this qualification.