On Linear Existential Graphs

Peirce's linear versions of the language of his Existential Graphs (EGs), presented in 1902, are examined. Differences between linear and non-linear languages are explained by permutational invariance and type- vs. occurrence-referentiality: Standard EGs are permutationally invariant with respect to linear EGs, while the Beta part of the system, which corresponds to first-order quantificational theory with identity, is occurrence-referential. However, occurrence-referentiality of Beta graphs constitutes a defect of expressivity: since the meaning of a quantifier is inextricably connected to the meaning of the sign of identity, certain complex assertions cannot be expressed in the language of Beta graphs without a new extension of its standard notation.