A categorical interpretation of C.S. Peirceʹs propositional logic Alpha

C.S. Peirceʹs graphical system Alpha for propositional logic is given a geometric representation in terms of isotopy classes of planar diagrams, and surgery rules on these diagrams called illative transformations. An algebraic representation theorem is proven, stating that Alpha, as an equational theory, is isomorphic to the theory of Boolean algebras. The geometric and algebraic representations are extended to give multi-sorted analogues of Alpha. An interpretation of Alpha is given in terms of linear logic and the theory of closed categories.