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.

翻译：戈特弗里德·莱布尼茨启动了一项研究计划，旨在通过图解和代数方法证明所有亚里士多德式的直言三段论。他成功使用欧拉图证明了它们，但未留下包含代数证明的手稿。我们展示了散见于莱布尼茨各类逻辑学草稿中的关键片段已包含足够的要素，能够通过一种代数方法（我们称之为莱布尼茨-凯莱系统）来证明这些三段论，而无需求助于更具表现力和复杂的一阶量词逻辑机制。此外，我们使用一种关系方法（我们称之为麦科尔-拉德系统）再次证明了经典的直言三段论，该方法运用了休·麦科尔和克里斯汀·拉德研究过的直言关系。最后，我们展示了ML和LC与布尔代数的联系，证明了ML是LC的推论，而LC是布尔格公理的推论，从而确立了莱布尼茨在刻画并应用（一个充分的）布尔代数片段以有效处理直言三段论方面优先于乔治·布尔的历史地位。