How would propositional logic fit with regard to the Aristotelian Organon? By the Aristotelian Organon, I mean the extended (medieval) Organon, which numbers the books of logic as nine:
1) Isagoge (Porphyry's)
3) Peri Hermeneias (De Interpretatione)
4) Prior Analytics
5) Posterior Analytics
7) Sophistical Refutations
We can set aside 1,7, 8, and 9 as not relevant to the question. Now, of the rest, we can roughly say that logic qua Categories discusses terms; logic qua De Interpretatione discusses the structure of non-complex propositions; logic qua Prior Analytics discusses the formal structure of deduction; logic qua Posterior Analytics discusses demonstrative argument; and logic qua Topics discusses dialectical argument. In a philosophy of logic structured according to the Organon, where would propositional logic fit?
There's a sense, of course, in which all formal structures fit into Prior Analytics, but that's not really a very interesting thing. But there is more to logical argument than formal structure alone on this view (there is beyond this the actual application of the structure to subject-matter), so that's not the whole answer. Demonstrative reasoning according to this view proceeds terministically: that is, it is in virtue of the terms of the premises that the deduction goes through. Propositional logic, however, is (by definition) non-terministic. So it can't be demonstrative, which means it would have to be dialectical. The application of propositional logic to actual subject-matter is a matter of dialectical reasoning rather than demonstrative reasoning. And this is exactly the Aristotelian view. So how does this sort of thing work?
Let's take as true a single complex proposition like (p -> q), i.e., q if p. One can reasonably ask why, given p, q follows; but this cannot be answered in purely propositional terms, since the answer to this question would require looking at the terms of the proposition. To say why (p -> q) is true requires looking at the terministic and operational structure of p and q. So what good is q if p in argument? An Aristotelian would reply in this way. Without complex propositions like these, one would have to argue directly for q. If, however, we can take (p -> q) as a principle agreed upon by all relevant persons, we can then argue for q indirectly by arguing for p. This radically increases the flexibility of our argumentative resources. So suppose we were arguing that Something is caused. If we can take it as agreed upon by all relevant persons that If something begins to exist, something is caused, we can then argue for our conclusion by arguing for the conclusion Something begins to exist. On some views (e.g., Aristotelian, or Shepherd's), If something begins to exist, something is caused is a necessary proposition, since something's beginning to exist is just its being caused (under a different description). But this is not necessary for the propositional logic itself; all we need for that to work is to be able to posit the complex proposition (p -> q). If that taken as agreed upon, our dialectical resources have been expanded.
This suggests something about what is going on in the so-called 'special topics'. Every special science is held to have (in addition to the general topics that govern all sciences) its own 'special topics' peculiar to it. Our little meditation on the dialectical work of complex propositions suggests that what these special topics usually (always?) do is provide inferential guidelines to expand our dialectical resources in that particular field. In other words, because they draw connections between different propositional truths, they allow us to substitute one thing for another. For instance, one might have in physics a special topic devoted to saying how one gets a proposition expressed in terms of energy from a proposition expressed in terms of matter. If one needs to know or prove something about energy, this sort of complex proposition allows one to tackle this problem by way of what one knows or can prove about matter. Such a complex proposition has increased one's resources for concluding things about energy. Now, it's possible that this complex proposition expresses a necessary truth; but this requires terministic analysis that the special sciences do not in their normal progress usually require, unless one wishes strictly to demonstrate something in an Aristotelian sense. All the special sciences require is that the complex propositions we choose not be arbitrary, but chosen due to their aptness for rendering true conclusions. This is all dialectical reasoning: we don't need to show the premises to be strictly necessary; all we need is for the premises to be agreed upon, for expert reasons, by the relevant experts.
[UPDATE: In a revision slip I originally stated that on an Aristotelian view or on Shepherd's view 'beginning to exist' is equivalent to 'being caused'. This, of course, is blatantly false; so I've fixed the error.]