I’ve recently been learning the logic of philosophy which I’ve wanted to learn since taking a mathematical logic and proofs class while getting my Computer Science degree. Using logic to understand, analyze and create arguments expands my understanding of logic because I primarily use logic in software engineering as well as math. To learn how philosophy uses logic I’ve been reading Introduction to Logic 13th Ed. By Copi, Cohen, and McMahon. In chapter 5, while learning about categorical propositions and their immediate inferences, I was struck with why the converse immediate inference works only for the E and I propositions. After some thought I realized that this was a result of both the predicate and subject classes being either distributed or undistributed. I thought it would be a fun data visualization exercise, another interest of mine, to expand on the square of opposition to illustrate the distribution of classes in each proposition. This led me down the path of significantly expanding the square of opposition to show not only the distribution of classes, but also adding other immediate inferences and contextual information to help understand the propositions and their inferences. The resulting diagrams are below:

You can see in both diagrams that converse is only possible when both the subject and predicate classes are either distributed or undistributed. Furthermore, the contrapositive inference is only possible when the distribution is not the same across the subject and predicate classes. This is because the contrapositive can be expanded to a series of three immediate inferences: obverse → converse → obverse. The initial obverse changes the quantity of the initial proposition thus making the subject and predicate classes have the same distribution allowing the subsequent converse inference.

As a reminder, in the classical interpretation it is presupposed that all classes are non-empty, in the modern version the classes are only presupposed non-empty when the quantity is particular.

This was a fun data visualization and learning exercise for me. I tried my best to keep the diagram readable while still containing a lot of information to the standard square of opposition diagram. I hope you will find it helpful!