 In a later paper, Church turns his attention to a paradox that Russell formulates in Appendix B of Principles of Mathematics (1903). Once again, Church’s aim is first to show how this contradiction arises in STT, and then to appeal to RTT as a means of solving the problem. This time, however, STT is not supplemented directly by any overt semantic principles concerning denotation or predication, but only by a predicate for identity of propositions and axioms supporting Russell’s view as of 1903, that if a pair of propositions differ by a constituent, then they are different propositions. For example, if \(X\) and \(Y\) are distinct classes of propositions, then the propositions expressed by “Every proposition in \(X\) is true” differs from that expressed by “Every proposition in \(Y\) is true”. Given this principle—call it “Structure”—if such propositions are correlated with the classes they mention, then it is possible to diagonalize and obtain a contradiction. For let \(W\) be the class of propositions that are not in the classes with which they are correlated, and consider the proposition expressed by “Every proposition in \(W\) is true”. Then this proposition is in \(W\) if and only if it’s not. (Note that this argument involves yet another violation of  CL,  since in effect it defines a function \(f\) from propositions to sets of them, and then assumes that \(D_f\) is in the range of \(f\). ) Church gives a careful proof theoretical analysis of this argument showing exactly what formal principles must be assumed for it to be formally derivable. (See  supplement F  for a sketch of some details.)