Overview of Categorical Logic

Categorical logic is the oldest logic in Western thought. It was first advanced by Aristotle and was believed by many to be the heart, if not the whole, of logic for centuries. Traditionally its subject matter included standard syllogisms, such as

together with extended forms, or sorites (so-RI-tes), such as

also it could handle such nonstandard form syllogisms and sorites as

although these required more effort. (These are nonstandard form in that they contain both A and nonA, B and nonB, etc.)

This logic was a regular part of the medieval and early modern curriculum but in the 20^th century it was subsumed as a small and peripheral portion of the more comprehensive logic developed by Gottlob Frege, Bertrand Russell, and Alfred North Whitehead. One unfortunate consequence of this development is that although Aristotle's small portion seems most relevant to everyday life, it turns out that fewer students are exposed to it now than before, and many of those who do get access to it have to wait until their studies are sufficiently advanced to master the entire, comprehensive system of logic.

(In the late 20^th century Fred Sommers, followed by George Englebretsen, developed an alternative approach to the comprehensive logic which begins with categorical logic and in which categorical logic remains central. Moreover, Lorne Szabolcsi also developed a numerically expanded logic from that perspective which may surpass the numerical expansion I have proposed in important ways. However, the logic of Frege, Russell, and Whitehead continues to be entrenched as the dominant approach.)

The schematics of ReasonLines provide a quick and intuitive exposition of this important part of logic whose introduction is now often delayed, if not neglected altogether. I contend that these schematics display the logical relationships more adequately than do the medieval system of rules and mnemonics or the circular diagrams advanced in the 18^th and 19^th centuries and this, in turn, makes the study appropriate for much younger students than ever before. Also, unlike the earlier approaches, the schematics accommodate sorites (i.e., long chains of reasoning) as easily and clearly as they do the syllogisms (i.e., the basic units of reasoning), and they also accommodate nonstandard form syllogisms and sorites as straightforwardly as they do the ones in standard form.

Finally, the schematics provide the easy exposition of the numerically expanded logic for which they were initially designed. That is, syllogisms and sorites with numerically specific quantifiers, such as

lie completely beyond the purview of the traditional approach to categorical logic, but are plainly displayed by the schematics.

Unlike calculators for arithmetic, when users become familiar with ReasonLines they will no longer have to have access to the app; rather, they will be able to do all the same work by sketching the schematics with pencil on paper.