Munich Center for Mathematical Philosophy (MCMP)

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Talk: Robert Carlen May (UC Davis)

Location: Ludwigstr. 31, ground floor, Room 021

19.05.2022 16:15  – 17:45 


Definition and Proof of Referentiality

(Rachel Boddy & Robert May)


In Grundgesetze, Frege undertook an attempt to demonstrate that his logical language, the Begriffsschrift, is a fully referential language, that is, that every term of the language, simple or complex, is referential. This demonstration – the Proof of Referentiality – is shown by Russell’s Paradox to fail. Yet the question remains why Frege undertook the proof; what was his purpose? While the meta-logical content of the proof is plain – if successful it would be a consistency proof – this was not Frege’s primary intent. Rather, Frege’s core purpose was to legitimize definitions, and accordingly the proof must be considered in the context of Frege’s broader concern with canons of proper definitions designed to be scientifically useful, or as he called it, fruitful. Our discussion begins with examining Frege’s placement of the Proof of Referentiality in the text of Grundgesetze. We then turn to the role the Proof plays in justifying definitions, and a distinction is drawn between analytic definitions and proper definitions. Fruitful definitions are those that are simultaneously analytic and proper; Definition Z, the definition of the number of a concept, is the prime example. We turn to Frege’s approach to the definition of concepts, which is indirect, accomplished by their relation to their value-ranges. This discussion brings “On Concept and Object” into the main dialectic of Grundgesetze. Critical to this reduction is Definition A, which is intended to capture the predicational aspect of concepts. But this leads directly to Russell’s Paradox; to Frege, the consequence is that it undermines the definition of number, and hence the scientific content of the logicist project.