Patricia Blanchette

Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent to which his understanding of logic differs importantly from that of the model-theoretic tradition that grows out of Hilbert's work.

This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of existence claims in mathematics. It is argued that Frege's strict requirements on existential proofs would rule out the attempt to ground arithmetic in. It is hoped that this discussion will help to clarify the ways in which Frege's position is both coherent and significantly different from the neo-logicist position on the issues of: what's required for proofs of existence; the connection between models, consistency, and existence; and the prospects for a logical grounding of arithmetic in the wake of the paradox.