## Paradoxes for "expresses the proposition"

There are familiar semantic paradoxes for "truth" and "reference", such as the Liar paradox and Berry's paradox. I would have thought that there should be similar paradoxes for "expression", i.e. for the relation between a sentence S and the proposition expressed by S. A quick duckduckgo search didn't come up with anything. Pointers?

Here is a Liar-style one I came up with myself. Assume propositions are sets of worlds (which is the case I'm interested in). Consider the sentence

E: E expresses the empty set.

If E is true, then the proposition it expresses contains the actual world, in which case E doesn't express the empty set. So E can't be true. Since we've just proved not-E from no empirical assumptions, ~E expresses the set of all worlds. Hence E expresses the empty set. So E is true. Contradiction.

A few disorganized thoughts:

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1. Insofar as "expresses" refers to the relation between a sentence and the proposition it expresses (so that we could not truthfully say using the same word, "'the empty set' expresses the empty set"), which you seemed to stipulate earlier, I don't think that this paradox differs in an important way from one involving the sentence "This sentence expresses a [the] proposition that is necessarily false." Or something without self-reference, if we want. And then I think we're in the realm of paradoxical sentences like "This sentence expresses a false proposition" and "This sentence does not express a true proposition," which have been the focus of some of the Liar literature.

2. This isn't to say the paradox is unimportant. But I have a horse in this race, so I'll get up on a soapbox for a bit and preemptively apologize if this is going off on a tangent you're not as interested in. I think that the stories told for paradoxes like this one by most authors (such as a recent story from Michael Glanzberg) are not very helpful, at least without substantial amendment, when we consider a variety of other paradoxes involving propositions. My favorite, though perhaps not the most important, one of these other paradoxes is this: I believe that something I believe is false. I really do have this belief, because I think it would be a massive stroke of luck if all my beliefs turned out to be true. But suppose that I've gotten luckyâsuppose that everything /else/ I believe is true. Then we can prove that it is both true and false that something I believe is false, and that's no good.

There are lots of these paradoxes. They're kind of like intensional versions of the paradoxes that Kripke is concerned with in "Outline of a Theory of Truth," but they were discussed at great length nearly 15 years earlier by Arthur Prior in "On A Family of Paradoxes" (1961, NDJFL 2, pp. 16â32), so that's a slightly misleading characterization. (But the resolution Prior proposes is pretty hopeless without at least some amendment.)

Several other authors have discussed various forms of these paradoxes, too, including Anthony Anderson, George Bealer, Tyler Burge, Alonzo Church, David Kaplan, Kevin Klement, Sten LindstrÃ¶m, and Richmond Thomason.

The trouble with most existing resolutions of the Liar (including expression forms of the Liar, like the one you came up with in the original post, if I'm right about what's going on there) is that they rely on syntactic features of sentences at some point or other, and there's no guarantee that we'll have any analogues of that structure in something like the belief paradox. If there's a language of thought, or if mental attitudes have structure some other way, or if propositions themselves have rich syntactic structure, then we're off to the races, but that's a high price to pay to just resolve some paradoxes. (And structured propositions, at least, are of course not without their detractors.)

3. Other propositional (or "intensional" paradoxes), which might or might not require similar resolutions, go back to Appendix B of Russell's _Principles of Mathematics_. A modern variant of that paradox is to imagine that for each set S there exists the unique proposition /that S is my favorite set/. This seems plausibleâhow could it be that there are distinct sets S and S' such that I cannot even believe that one is my favorite without thereby believing that the other is my favorite as well (even if I know that they are distinct)? But of course we have a violation of Cantor's theorem if there's a set of all propositions. This is sometimes known as the Russell-Myhill antinomy. It also seems to have been rediscovered by Vann McGee and AgustÃn Rayo in terms of pluralities instead of sets; similar paradoxes arise for properties and propositional functions (and I'm sure plenty of other things, too).

4. Yet another class of intensional paradoxes is raised by Barwise and Etchemendy's _The Liar_: they consider the problems posed by a proposition that is identical to its own negation. We can eliminate the circularity by following Stephen Yablo and imagining instead an infinite sequence of propositions, each of which says that all the later propositions in the sequence are false.

5. I don't mean to say that all these paradoxes are equally important or definite problems. Of course, different theories will be more or less concerned with different paradoxes; this is just a rough taxonomy. For my part, I think that one important lesson here is that we should not focus on sentences and semantical relationships in order to resolve these paradoxes, because the deeper problem is one involving contentâlinguistic, mental, whateverâand not attending to that can sweep important issues under the rug. This is not to say that paradoxes involving satisfaction predicates are unimportant, of course. But it is to say that they and their ilk are not exhaustive of important, non-set-theoretical paradoxes.

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I'm not sure what part of this, if any, you're actually interested in, and this comment is long enough, so I'll stop there and refrain from filling out all the references. These paradoxes, especially the ones not involving expression, have been somewhat neglected, but there's still some good stuff out there on them.