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

Antonielly Garcia Rodrigues; Eduardo Mario Dias

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arXiv · 2308.12973

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

Antonielly Garcia Rodrigues·Eduardo Mario Dias

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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.

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Antonielly Garcia Rodrigues, Eduardo Mario Dias. 2023-08-17. Proofs of valid categorical syllogisms in one diagrammatic and two symbolic axiomatic systems. https://arxiv.org/abs/2308.12973

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