The soul is propagated by generation, either from both parents or from one only; either as to its totality or only as to a part. But neither can be said. Not the former because thus two souls would coalesce into one and be mingled. Not the latter, for if from one (either the father or the mother only) no reason can be given why it should be propagated by the one rather than by the other (since both parents are equally the principle of generation). If the whole is propagated, then the parents will be without it and so will be deprived of life. If a part, it will be divisible and consequently material and mortal. Nor can it be reasonably replied here that neither the whole soul nor a part of it is propagated, but a certain substance born of the soul and (as it were) an immortal seed of the soul. For it is taken for granted that there is a seed of the soul by which it either generates or is generated; yet such a seed cannot be granted (which does not fall from the soul), and therefore proves it to be material and divisible.
One way to break down the argument is as follows.
1. Assume the soul to be propagated.
2. Either it is (2a) from both parents or (2b) from only one.
3. (2a) implies that two souls are coalesced or mingled into one.
4. Therefore not (2a).
5. If (2b), no reason can be given why it should come from one rather than the other.
6. Therefore not (2b).
7. The soul is propagated either (7a) as a whole or (7b) in part.
8. If (7a), conception would deprive the parent(s) of life.
9. Therefore not (7a).
10. If (7b), the soul would be divisible.
11. Therefore not (7b).
12. Therefore the soul is not propagated, with contradicts (1).
For the background on traducianism, see the Catholic Encyclopedia article on it.