In an earlier post, I discussed the merits of an Aristotelian logic as term logic: we begin by simply noting that things exist, and we know them. Only then can we move on to construct propositions and then chains of reasoning.
Aristotelian logic is also content logic. Content logic may be distinguished from formal logic by its dependence on the terms within it. Formal logic analyzes propositions solely by their form, its primary tool is the truth table. For a content logic, on the other hand, it makes all the difference in the world what we are actually talking about, because we are often interested in arguments that are not true by necessity. Formal logic can tell us whether an argument must always be true, may sometimes be true, or can never be true. A content logic is most useful in the middle case, because we may need to know just how often, or how much, sometimes really is. Especially in matters of natural science, we often deal with contingent being, things that could be otherwise.
One modern attempt at reintroducing the full range of logic was David Stove's Rationality of Induction. A brief teaser of this is available from William Briggs. Most of the examples provided therein are textbook cases of an undistributed middle. I have not read Stove's work, but a more robust logic like that of Aristotle can make use of formal logic while avoiding the kinds of mistakes highlighted in the teaser.