Native diagrammatic soundness and completeness proofs for Peirce’s Existential Graphs (Alpha)

Abstract

Peirce’s diagrammatic system of Existential Graphs (\(EG_{\alpha })\) is a logical proof system corresponding to the Propositional Calculus (PL). Most known proofs of soundness and completeness for \(EG_{\alpha }\) depend upon a translation of Peirce’s diagrammatic syntax into that of a suitable Frege-style system. In this paper, drawing upon standard results but using the native diagrammatic notational framework of the graphs, we present a purely syntactic proof of soundness, and hence consistency, for \(EG_{\alpha }\), along with two separate completeness proofs that are constructive in the sense that we provide an algorithm in each case to construct an \(EG_{\alpha }\) formal proof starting from the empty Sheet of Assertion, given any expression that is in fact a tautology according to the standard semantics of the system.