Peirce's Logical Graphs for Boolean Algebras and Distributive Lattices

Peirce's logical graphs provide a completely new diagrammatic logical syntax, which in the present paper is contrasted with the standard linear syntax for classical propositional logic. Peirce's transformation rules of alpha graphs are shown to form a deep inference system for classical propositional logic. With the alpha system, Peirce was able to formulate a new diagrammatic approach to Boolean algebras. This is justified by the connection between the alpha system and Peirce's sequent system PC for Boolean algebras. The graphical methodology is extended to the variety of distributive lattices, which shows the flexibility of Peirce's logical graphs. Scrolls of finite arity are introduced to represent the join operation in distributive lattices. A graphical system that is a variant of Peirce's alpha system is introduced for distributive lattices.