Aristotle’s whole system of logic starts from two premises:
Logic is the direction of the act of reason
The direction of reason is from what is more universal in predication to what is less so.
Aristotle says the major premise everywhere, and at the slightest provocation; the minor is from St. Thomas, and Aristotle simply assumes it everywhere. The conclusion is that logic is the right order from what is more universal to what is less so. This is why Aristotle starts his logic with a study of most universal things (the Categories) Then shows all the ways that one universal thing can relate to another (On Interpretation) and then goes on to speak of arguments as the motion from what is major (or most universal) to what is minor (least universal) through a term of middle universality. The middle only has a middle universality when we speak in a way that follows what is called “the first figure syllogism” and so Aristotle rightly insists that this is the pre-eminent tool for ordering reason, and that all other tools of reasoning are correct so far as they can be reduced to it. Aristotle insists that this is even true of non-categorical reasoning, as Yvan Pellitier proves here (download “PelletierStrategy.pdf”). This is not to say it is the only way that reason can go from one thign to another: there is an ocean of dialectical tools that are used to bring us to the point where we can actually form a valid universal.
We more tend to look at terms like checkers that can be arranged in certain correct ways, and in doing so we can reach conclusions that cannot be reached by the mere categorical method. This is all fine, but it is not logic in the same sense of the term, and it is not clear how we can make one thing called logic relate to the other thing called logic.