On Linear Existential Graphs

Peirce’s linear version of the language of his Existential Graphs (EGs), which he presented in 1902, is examined. The difference is explained by introducing the notion of permutational invariance and the type- vs. occurrence-referentiality: Standard EGs are permutationally invariant with respect to linear EGs, while the Beta part of the system, which corresponds to the 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 identity sign, certain complex assertions arguably cannot be expressed in standard Beta graphs without a modification to its received notation.