https://www.umsu.de/blog/ Musings in Analytic Philosophy https://www.umsu.de/blog/2026/830#c2477 https://www.umsu.de/blog/2026/830#c2477 Tue, 09 Jun 2026 00:26:45 +0000 https://www.umsu.de/blog/2025/815#c2476 https://www.umsu.de/blog/2025/815#c2476 Sun, 31 May 2026 06:01:16 +0000 arxiv.org/abs/1310.0017


There's a lot of quantum stuff you can ignore. The "dF without symmetry" theorem is self contained.]]> https://www.umsu.de/blog/2026/829#c2475 https://www.umsu.de/blog/2026/829#c2475 Mon, 11 May 2026 14:54:54 +0000 https://www.umsu.de/blog/2026/829#c2474 https://www.umsu.de/blog/2026/829#c2474 Sun, 10 May 2026 18:28:21 +0000 https://www.umsu.de/blog/2025/823#c2473 https://www.umsu.de/blog/2025/823#c2473 Mon, 20 Apr 2026 14:47:08 +0000
I think it's a bit harder to say this one piggy-backs on the Liar. ]]> https://www.umsu.de/blog/2025/823#c2472 https://www.umsu.de/blog/2025/823#c2472 Mon, 20 Apr 2026 10:49:27 +0000
More generally, the fact that knowledge entails truth makes it not too surprising that sentential accounts of knowledge (knowledge predicates that apply to sentences, rather than e.g. sets of worlds) run into paradox. My interest in paradoxes for sentential belief stems partly from trying to show that the paradoxes for sentential attitudes don't simply piggy-back on the Liar.
]]> https://www.umsu.de/blog/2025/823#c2471 https://www.umsu.de/blog/2025/823#c2471 Mon, 20 Apr 2026 01:01:05 +0000
Now, just put something in for M(p) that has nothing to do with knowledge or belief and is such that M(⌜G⌝) is contingently true. E.g., M(p) could be "There is no inscription of p on the far side of the moon in giant neon letters" or "The smallest mammal in my household doesn't know p" (we have a cat) or just "The sky is blue." We then have an argument for the contingent absurdity ¬M(⌜G⌝).

Another fun variant. Let I(p) say that x is ignorant of the fact numbered p. Then I is factive, just like knowledge. So, we get to construct a G analogous to yours and get ¬I(⌜G⌝) and G. But now let x be a baby. :-)

I agree that although it's unsurprising that B can't contain ¬BC(⌜0=1⌝), it is surprising that B* can't either. ]]> https://www.umsu.de/blog/2025/823#c2470 https://www.umsu.de/blog/2025/823#c2470 Sun, 19 Apr 2026 05:53:36 +0000
Maybe you're right that we can't fully grasp the meaning of G, although (1) by the MRDP theorem there's a version of G that's "just" a diophantine equation, and (2) I'm not sure how much is required to grasp the meaning of a mathematical statement in general. But more importantly (as you mention) the paradox returns if we reason about the knowledge of smarter agents, so I still think it cautions against assuming that we have a clear grip on what mathematical knowledge involves.

You're right that there's a gap in my argument involving 'B': thanks for bringing this out! I still suspect there's a real paradox here, but I'm not sure how best to develop it. The fact that B* can't contain ¬BC(⌜0=1⌝) already seems odd.]]>