The knower paradox concerns a sentence such as, "This sentence is unknown." Now liar sentences can be paired with honest sentences, e.g. "This sentence is true." So suppose there was a sentence that goes, "This sentence is known."

Curry's paradox can involve conditional sentences whose antecedents resemble honest sentences. Is it possible to fuse an honest-knower sentence with the informal structure of a Curry sentence to yield an additional enigmatic sentence? What I mean is, can a paradox be derived from, "If this sentence is known, then φ," where φ is an arbitrary sentence/logical explosion?

Or consider liar cycles such as

A. Sentence B is true.

B. Sentence A is false.

What happens if we switch out "is true/false" for "is known/unknown"? Do we end up with another species of knower paradox?

I also wonder about Yablo's paradox, here. Generally, can we expand upon the knower paradox by combining it with other paradoxes?

  • To engage the mechanism of Curry's paradox you need something weaker than "true", like "meaningful", not stronger like "known". The secret is that x="if x is meaningful, then φ" together with the conditional entails "x is true" by the convention T. Since "x is true" implies "x is meaningful" (presumably) you then get φ by modus ponens. But "x is true" implying "x is known" is not very plausible except to radical anti-realists. – Conifold Apr 6 at 5:21
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    On the relation between the Yablo paradox and the knower paradox, there is a nice paper by Roy Sorensen, 'Yablo's paradox and kindred infinite liars'. – sequitur Apr 6 at 22:23

Fix your favorite sentence φ and let x be the sentence "If x is known, then φ." In order for x to be false, its hypothesis "x is known" would have to be true, which means a fortiori that x would have to be true. This constitutes a proof of x, and so we know x - from which we get φ.

This is exactly the same "shape" as the argument around the original knower paradox, so if we make the same assumptions we also get a paradox in this case. However, for what it's worth I've never found the relevant assumptions convincing at all, so I don't find the knower paradox (or this Curry-knower paradox) particularly compelling.

An interesting sub-question is whether we could get a more "parsimonious" paradox this way - that is, a paradox which doesn't require us to assume as much about knowability. I don't think that happens, though, since we need a reason why "true-but-unknown" should be problematic. (Keep in mind that in the setting of Godel's theorem "true-but-unprovable" is completely unproblematic, so this is a serious option.) To get any mileage out of this we need a rather strong connection between "is known" and "is true," and this is exactly what (PK)/(KF) give us with the classical knower paradox.

  • So we basically just end up with the same paradoxes (however compelling or not) that we started out with, just conjoined? – Kristian Berry Apr 6 at 0:24
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    @KristianBerry As far as I can tell, yes. To be fair it's hard to rigorously prove a result like that. That said, the Godel point I raise tries to do this: unless we make assumptions on knowability which distinguish it from provability, we're guaranteed (under mild assumptions) to never hit a contradiction since arithmetic itself doesn't break. This shows that in order to get any kind of knowability paradox we need strong rules connecting knowability and truth ... at which point a "truth-to-knowledge shift" doesn't seem particularly exciting anymore. (But that's all heuristic of course.) – Noah Schweber Apr 6 at 0:26

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