Russell on Contextual Definition and the Elimination of Sets
This chapter explores Russell’s “no class theory,” originally expressed by his contextual definition of classes in Principia Mathematica. In recent years, some Russell scholars have trumpeted the virtues of the interpretation of Russell’s quantification as substitutional, among which is the sense it makes of the “no-class theory.” Such an interpretation does make some sense of Russell’s philosophical remarks about that theory, about the significance of his logicist reduction, and about the ability of the reduction to serve as a model for similar reductions outside the philosophy of mathematics. However this substitutional interpretation is not sufficient, since it is inconsistent with important aspects of Russell’s philosophical logic and is technically inadequate to support his logicist reduction. In short, if substitutional quantification is the source of the “no class theory,” then the theory is not vindicated, but refuted.
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