Reference by Occurrence in Quantification

This paper investigates occurrence-referentiality at the quantificational level and argues that it clarifies distinctive features of Peirce’s Beta Existential Graphs. Extending the distinction between type-referential and occurrence-referential systems to quantification, it contrasts standard first-order logic with identity (FOL=) with Beta Graphs. In FOL=, co-reference depends on reuse of the same variable type (“x”), while different types (“x” and “y”) allow distinct reference; reference is thus determined at the level of type. In Peirce’s Beta graphs, by contrast, reference depends on symbol occurrences: existential quantification is represented by Lines of Identity (LIs), where continuity expresses co-reference and disconnection distinctness. The system is therefore occurrence-referential at the quantificational level. Two consequences follow. First, Beta graphs display an equivalential observational advantage: logically equivalent FOL= formulas collapse into a single graph type and are directly observable as equivalent. Second, Existential Graphs achieve the expressive power of FOL= while using only one representational device—the LI—to express both variables and identity.