1. Only series having a first member are series ordered as to prior and posterior.
Therefore, all series ordered as to prior and posterior are series having a first member.
2. So, all series ordered as to prior and posterior are series having a first member.
But all series of producing causes are series ordered as to prior and posterior.
Therefore, all series of producing causes are series having a first member.
3. But no series having a first member are infinite series.
And all series of producing causes are series having a first member.
Therefore, no series of producing causes are infinite series.
I don't think this is quite right as a statement of the formal structure of Turretin's argument. One of the problems, I think, is that the regimentation treats "Only series having a first member are series ordered as to prior and posterior" as an initial premise when it is clearly a conclusion. And I think the last half of the passage, which is omitted, is doing the bulk of the work. The beginnings of a better reconstruction would be something like:
1) A causal series must be ordered according to prior and posterior.
2) In causal series that infinitely regresses there is no first cause.
3) Therefore in such a series every cause is a middle cause, with a cause prior to it. (from 1 and 2)
4) But in causal series consisting of producing causes, any collection of serial producing causes in the series is itself a producing cause.
5) Therefore the collection of all the superior producing causes in the series is a producing cause. (from 4)
6) Therefore the collection of all the superior producing causes in the series is a middle cause, with a superior cause that is prior to it. (from 3 and 5)
7) Therefore in an infinitely regressing series of producing causes there is a cause that both has and does not have a prior cause. (from 6)
8) Therefore a causal series of producing causes that infinitely regresses both is and is not ordered according to prior and posterior (from 1 and 7)
9) Therefore no causal series of producing causes infinitely regresses. (from 8)
That's somewhat loose and on-the-fly; it could be tightened up by filling in the implied premises. The point is that it is not enough to respond to the argument by saying that we can identify order according to prior and posterior in a causal series without regard to a first cause (for which we'd need an account of how to do so, in any case); this is not an assumption of the argument but its very nerve. And the part about the collection of causes is actually important: it's an argument that a series of causes consisting only of middle causes is impossible.
The most important premise here is actually (4). Since Turretin is only summarizing a form of scholastic argument that was common in various versions, he doesn't justify (4); traditionally it would be justified by an analysis of the sort of causation involved. And, indeed, this is the major issue as to the soundness of the argument: whether there are any real causes that work in such a way that they can be called producing causes in exactly Turretin's sense. If there are, there is a first producing cause. If there aren't, then maybe there are first causes in some other sense of the term 'cause' or maybe there aren't.
