I wonder if he’s familiar with Peirce’s alpha existential graphs. They are a complete propositional logic with a single axiom and, depending how you count them, 3-6 inference rules. They use only negation and conjunction.
They also permit shockingly short proofs compared to the customary notation. Which, incidentally was also devised by Peirce. Peano freely acknowledges all he did is change some of the symbols to avoid confusion (Peirce used capital sigma and pi for existential and universal quantification).
John Sowa is a good resource. Here is his annotation of Peirce's tutorial[1]. Another paper explores the influence of EG on Sowa's Conceptual Graphs[2]. I happen to find the juxtaposition of Frege's notation with Peirce's interesting. Sowa's commentary on yet another Peirce manuscript has some fun historical tidbits about the influence of Peirce on the design of SQL[3]. Here is another reference that mentions Peano's adoption of Peirce's notation[4].
That should be plenty to get you started! Digging through the references in those papers and the bibliography on Sowa's site will find you plenty more modern Peirce scholarship. I think Peirce would be pleased that his seemingly abstract approach to logic ended up inspiring one of the most pragmatically useful classes of software ever, the relational database.
They also permit shockingly short proofs compared to the customary notation. Which, incidentally was also devised by Peirce. Peano freely acknowledges all he did is change some of the symbols to avoid confusion (Peirce used capital sigma and pi for existential and universal quantification).