In the Stanford Encyclopedia's page on Russell's Paradox, we get the following anecdote about an additional, lesser-known paradox from Russell:

...in Appendix B Russell also presents another paradox which he thinks cannot be resolved by means of the simple theory of types. This new paradox concerns propositions, not classes, and it, together with the semantic paradoxes, led Russell to formulate his ramified version of the theory of types.

The new, propositional version of the paradox has not figured prominently in the subsequent development of logic and set theory, but it sorely puzzled Russell. For one thing, it seems to contradict Cantor’s theorem. Russell writes: “We cannot admit that there are more ranges [classes of propositions] than propositions” (1903, 527). The reason is that there seem to be easy, one to one correlations between classes of propositions and propositions. For example, the class m of propositions can be correlated with the proposition that every proposition in m is true. This, together with a fine-grained principle of individuation for propositions (asserting, for one thing, that if the classes m and n of propositions differ, then any proposition about m will differ from any proposition about n) leads to contradiction.

There has been relatively little discussion of this paradox, although it played a key role in the development of Church’s logic of sense and denotation. While we have several set theories to choose from, we do not have anything like a well-developed theory of Russellian propositions, although such propositions are central to the views of Millians and direct-reference theorists. One would think that such a theory would be required for the foundations of semantics, if not for the foundations of mathematics. Thus, while one of Russell’s paradoxes has led to the fruitful development of the foundations of mathematics, his “other” paradox has yet to lead to anything remotely similar in the foundations of semantics.

I have never heard of this but it seems extremely interesting. If I get the gist correctly above, one can look at the "collection" of all propositions, whether it is a set, class, or otherwise. Then one can look at the collection of unary predicates that take a proposition as argument and return true or false.

On the one hand, a naive "Cantor's theorem"-style argument says there should be more predicates than propositions; the collection of predicates is, naively, the "power collection" of the collection of propositions. However, to each such predicate we can associate a proposition, which is simply the conjunction of all propositions for which the predicate is true, so there would seem to be an injection from these predicates to propositions.

The above is, of course, hopelessly informal and "naive," but it seems to be the basic paradox Russell is getting at. I would like, however, a less naive, more formal treatment of it. Is there a name for this paradox? What are some different ways to rectify it? Is there any direct expositional writing about it?


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It seems to me that everything's going to resolve itself once we make precise "proposition" and "class of propositions." For example, consider the line:

the class m of propositions can be correlated with the proposition that every proposition in m is true.

Why should "every proposition in m is true" be a proposition? In order for that to be true, we need to be restricting attention to "definable" classes. But at that point Russell's argument can essentially be phrased much more familiarly as:

  • How can there be no more definable sets than formulas?

The resolution, of course, being that the attempted diagonalization runs up against Tarski's undefinability theorem. E.g. in arithmetic we might want to define the set of $n$ such that $\neg\varphi_n(n)$ holds, where $(\varphi_n)_{n\in\omega}$ is an appropriate enumeration of the one-variable formulas in arithmetic, but we would need a truth predicate to do this, which isn't in fact definable.


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