Proofs of valid categorical syllogisms in one diagrammatic and two symbolic axiomatic systems

TL;DR

This paper demonstrates algebraic and relational methods to prove categorical syllogisms, connecting Leibniz's early work with Boolean algebra and establishing historical precedence over Boole.

Contribution

It introduces the Leibniz-Cayley and McColl-Ladd systems for proving syllogisms, linking them to Boolean algebra and highlighting Leibniz's priority in algebraic logic.

Findings

Leibniz's drafts contain sufficient ingredients for algebraic proofs.

The ML system employs categorical relations for syllogism proofs.

ML is derived from LC, which in turn is derived from Boolean algebra.

Abstract

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn't produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz's drafts on logic contained sufficient ingredients to prove them by an algebraic method -- which we call the Leibniz-Cayley (LC) system -- without having to make use of the more expressive and complex machinery of first-order quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method -- which we call the McColl-Ladd (ML) system -- employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz's historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic.