Frege’s principle of logical parsimony, the indispensability of “ξ = ζ” in Grundgesetze, and the nature of identity

Abstract

In Section 2, I analyze Frege’s principle of logical and notational parsimony in his opus magnum Grundgesetze der Arithmetik (vol I, 1893, vol. II, 1903). I argue inter alia that in order to carry out the proofs of the more important theorems of cardinal arithmetic and real analysis in Grundgesetze Frege’s identification of the truth-values the True and the False with their unit classes in Grundgesetze I, §10 need not be raised to the lofty status of an axiom. Frege refrains from doing this but does not provide any reason for his restraint. In Section 3, I argue that he considered the primitive function-name “ξ = ζ” indispensable in pursuit of his logicist project. I close with remarks on the nature of identity. I suggest that there is no need to interpret identity in a non-standard fashion in order to render it logically palatable and scientifically respectable.