On Chwistek's criticism of Principia
Stephen Boyce
TL;DR
The paper investigates whether Chwistek's claim that Richard's paradox can be formulated within Principia's class definitions is valid. It reconstructs Chwistek's argument, identifying two main errors: an invalid elimination of a defined term and a faulty substitution rule not sanctioned by Principia. The author argues that, when Principia's scope conventions and axiom of reducibility are correctly applied, the problematic equation is a theorem and does not produce a paradox, attributing the confusion to misinterpretation rather than foundational flaws. The discussion places the critique in the context of historical debates, including contributions by Gödel and the Russell–Whitehead framework, and contends that misrepresentations, not Principia itself, gave rise to the criticized claims. In sum, Chwistek's claim is deemed false, and the paper underscores the importance of correct substitution and scope rules in evaluating Principia's definitions of class.
Abstract
This paper examines Chwistek's claim that with Principia's definition of a class "Richard's paradox can be formulated". It is shown that the demonstration fails since it requires an incorrect elimination of a defined term and use of a faulty substitution rule, neither of which form part of the system of Principia.