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.


# on 08 July 2011, 06:58

A few disorganized thoughts:
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 lucky—suppose 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. 16–32), 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 Lindström, 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 plausible—how 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 Agustín 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 content—linguistic, mental, whatever—and 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.


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.

# on 08 July 2011, 08:05

Whoops, I realized I didn't actually give any references at all for what you'd explicitly asked about. Here are a few explicitly on expression forms of the Liar:

William Kneale, "Propositions and Truth in Natural Langauges," Mind 81, pp. 225–243, 1972.

Charles Parsons, "The Liar Paradox," Journal of Philosophical Logic 3, pp. 381–412, 1974.

Michael Glanzberg, "A Context-Hierarchical Approach to Truth and the Liar Paradox," Journal of Philosophical Logic 33, pp. 27–88, 2004.

As I said above, I'm not convinced that the resolutions proposed here will work without amendment for other paradoxes. Also, none of these authors is working with a possible-worlds analysis of propositions, but I don't think that makes much of a difference.

# on 08 July 2011, 08:26

Thanks Dustin, that's very helpful! I'll have to follow some of these references.

A few quick comments.

I'm thinking about paradoxes of propositional attitudes in the style of Montague 1963 and Thomason 1980/2011. These paradoxes seem to arise if one takes the objects of attitudes to be sentences, but not if one takes them to be propositions. Asher and Kamp 1986 argue that a propositional treatment can't escape the paradox as long as we can express the expression relation between sentences and propositions. That's correct, but it seems to me that this isn't much of an argument against propositional treatments of attitudes, given that adding such an expression relation all by itself already leads to paradox, even if we say nothing at all about attitudes. Hence I wondered about paradoxes generated directly by "expresses".

I've used "expresses the empty set" instead of "is (necessarily) false" in order to avoid having any sort of truth predicate. All other ingredients of the paradox apart from "expresses" should be completely harmless and easily available in a formal system that can prove the diagonal lemma.

About your general point, I'm inclined to say that many of the intensional paradoxes you mention do belong to a different class than paradoxes brought about by semantic vocabulary ("true", "refers", "expresses"). But of course taxonomising paradoxes is always a bit arbitrary and theory-laden.

The paradox with "I believe that I believe something false" is neat. Like the preface paradox, it looks like it might disappear if we move from a binary notion of belief to a graded notion.

# on 08 July 2011, 09:39

I think I was being too sloppy with my use of "proposition." I didn't mean to be talking about possible-worlds propositions in particular. I don't think this is a huge deal, but it might have led to some talking past. (And caution: Kneale, Parsons, and Glanzberg are speaking more generally, too.) I'll just use "content" to talk about whatever it is that we're related to by our beliefs, hopes, fears, desires, knowledge, speech, etc., and also whatever it is that sentences express, and so on, and hope that there's actually something there to talk about.

Anyway, I think I have a better sense of what you're getting at now. Here are two rather different points.

1. I'm inclined to think of the expression paradoxes and the attitude paradoxes as differing only in the unimportant respect that in some cases it's sentences that are related to content and in other cases it's people. The paradoxes are so similar, and admit of so many similar resolutions, that this seems like a useful way to proceed.

Also, my thought was that the move from talk about the empty set to talk about falsity wasn't very important. But note that I meant "false" to not apply to sentences at all; there is a truth predicate that applies to sentences and a separate one that applies to content, and while the two are certainly related, I think they're importantly different. I was thinking that crucial to your expression paradox was that the empty set was necessarily false (in the non-sentential sense), and that any resolution of the other paradoxes would be easily and naturally extended to yours.

With all of that said, I think perhaps at the heart of this is that I don't think of the expression paradoxes as being brought about by semantic vocabulary. Certainly some paradoxes are due to vocabulary, such as paradoxes involving satisfaction predicates (which are what I take truth predicates in the sentential sense to be), but I'm inclined to think of expression paradoxes as being brought about by vocabulary in the same way that attitude paradoxes are brought about by brain states—if the vocabulary is central, then brain states are going to be central in an analogous way, and at any rate we can make progress by abstracting a little bit away from those details.

To be clear, I don't mean to be arguing that any of this is right. I'm just trying to get a handle on the possibly non-common assumptions I was bringing to the table.

I'm not sure how the graded notion of belief will help, though I'm also not at all sure that it won't. Here are a couple cases I'd want to know more about: Will it help with every attitude? One of Prior's paradoxes involves thinking at 6 o'clock that everything thought in Room 7 at 6 o'clock is false while being the only person in Room 7 at thinking nothing else. I would also worry about things like saying that everything said (in the indirect sense) in Room 7 at 6 is false, but maybe there one could retreat to drawing a sharp distinction between cases involving sentences and the other cases.

2. You're right, of course, that none of these paradoxes can even arise on a possible-worlds treatment of propositions. But I don't think that can be the end of the story. Paradoxes arise when intuitively consistent assumptions allow us to derive contradictions. One response, which has at least been very well represented in the Liar literature, is to explain where the derivations have gone astray. (Truth-value gaps, for instance, say that they've gone astray by assuming excluded middle or something like that.) The other response is to bite the bullet and say that the assumptions really are inconsistent. (The other other response is to be paraconsistent, but I'll set that aside.)

It sounds like you're taking the second tack—this is, at any rate, what a possible-worlds theory of propositions is committed to, I think, since we know full well that there are models. There's nothing wrong with this approach in principle, but I think that it ought not be the end of the story: We still want an explanation of why our intuitions have gone wrong. Why is it that if I have all true beliefs, I cannot come to believe that something I believe is false? What happens if I believe that something I believe is false, but then come to lose all my other beliefs that were false? Or even all my other beliefs all together? Replace belief with any attitude, of course.

Worse, as Prior discovered when he took this route to address the paradoxes, we can get into bizarre trouble when we involve multiple people. He imagined saying that what he is saying is false if and only if what Tarski says immediately after is true, and then realized that Tarski couldn't say anything true, on pain of contradiction.

Now, this sort of first-come, first-served approach is definitely wrong, because if I think to myself that /what I am thinking right now is false if and only if someone else thinks something at some point in the future/, we have a contradiction so long as anybody else ever thinks something; according to Prior, I am now the only thinking being. (Actually, according to Prior, I've been the only being capable of bearing a propositional attitude toward anything for the last four years or so, due to a thought I had in 2007, unless someone beat me to it.)

But what is the right story? Why is it that we can't have the attitudes we think we can have, and in the multi-agent cases, how can we ensure that the right assumptions in the paradoxical set turn out false? (I've suggested one way to address the second question, but don't pretend to have anything to say about the first.)

Anyway, this is why I think there's still work (and what seems to me to be all the interesting and important work, as far as the paradoxes are concerned) to be done if we have a theory of content according to which the paradoxes cannot even get off the ground, such as a possible-worlds theory.

# on 08 July 2011, 11:59

I think Dustin already mentioned Prior's paradox, but that's a pretty important one.

Paraphrasing it into singular quantification, the theorem says: if someone thinks at t that everything he thinks at t is false, then he thinks something true and he thinks some thing false at t. The only assumptions are: (1) classical propositional logic and (2) propositional universal instantiation: \forall p\phi \rightarrow \phi[\psi/p].

The surprising thing is that you can't currently think at t that everything your thinking at t is false and nothing else. You can substitute "thinks at t" for says that, writes that, fears that, makes it the case that, etc.

Now if you accept the disquotational schema: "phi" says that phi, and that every sentence says at most one thing (formalised in propositional quantification, you can weaken this premise to everything a sentence says has the same truth value) you can derive a contradiction in the presence of self referential sentences. This is just by substitutiting "L says that" in Prior's theorem, where L identical to the sentence "everything L says is false".

Now given that the the premises of Prior's theorem are so weak I'm inclined to accept it, but draw a sharp distinction between saying that phi and uttering the sentence "phi", and similarly between thinking that phi and being in a particular intrinsic mental state. Assuming this there are a number of options:

1) that L doesn't say anything
2) L says two or more things
3) L="everything L says is false" says exactly one thing but it's not the proposition that everything L says is false.

My preferred solution is (3). However it does not seem to be particularly well explored, but it's a very natural position about "expressing" to take if you're someone who thinks every sentence is true or false (such as McGee, etc.)

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