Proof Analysis of Peirce’s Alpha System of Graphs

Abstract

Charles Peirce's alpha system $$\mathfrak {S}_\alpha $$S? is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of $$\mathfrak {S}_\alpha $$S? is given in terms of two embedding theorems: the system $$\mathfrak {S}_\alpha $$S? and Brunnler's deep inference system for classical propositional logic can be embedded into each other; and the system $$\mathfrak {S}_\alpha $$S? and Gentzen sequent calculus $$\mathbf {G3c}^*$$G3c? can be embedded into each other.