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.
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Notes
Informally speaking, a subgraph of G is any collection of parts of G that can be inscribed within a sep which does not intersect any other sep.
A literal is either an atomic variable or a negated atomic variable.
In the sequence below, for the sake of brevity the two iterations for P and Q have been collapsed to a single one.
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Caterina, G., Gangle, R. & Tohmé, F. Native diagrammatic soundness and completeness proofs for Peirce’s Existential Graphs (Alpha). Synthese 200, 471 (2022). https://doi.org/10.1007/s11229-022-03903-8
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DOI: https://doi.org/10.1007/s11229-022-03903-8