Nice eye-grabbing headline.

There is a Wittgenstein quote I've been thinking about ever since going through AI to Zombies around a decade ago:

If there were a verb meaning 'to believe falsely', it would not have any significant first person, present indicative.

As a statement of boolean logic, that's fine, it's p ∧ ¬p being false — claiming that “I falsely believe p” cannot be asserted, because the moment you avow that a belief is false, you withdraw your belief — and given Wittgenstein was famously a logician this is natural…

… but that's not how humans actually do their thinking. There's a lot of ways we just plain don't work like that.

We don't actually fully disavow beliefs when we first recognise them false, and we (fail to) do so in more than just one way:

1. People routinely hold beliefs they themselves distrust or judge unreliable, tracking (consciously or subconsciously) the probabilities and meta-beliefs separately from first-order beliefs. In a Bayesian sense, consider what someone is saying by "I believe the weather a year from today will be today's temperature, with a standard deviation of 7°C": there's important differences between Bayesian reasoning and formal logic, that's fine, but both are consistent with natural use of words like "believe", and therefore this use is entirely consistent with "I believe p and I believe that my belief p is false".
2. People also have non-unified minds, with one part of themselves disagreeing with another. This can show up as, e.g. saying about a superstition they held from childhood “I know this is probably nonsense, but I can’t shake the feeling it’s true.”; another example would be phobias, as someone who goes for therapy to eliminate their arachnophobia does actually have part of their mind that believes spiders are not dangerous (or they wouldn't want the therapy), while also another part of their mind believes spiders are dangerous (otherwise they wouldn't need the therapy). One may object that awareness of subsystems within our own mind makes this more like “part of me says p; another part of me says ¬p”, but again, normal people don't do that: precisely because it is natural for us to speak of ourselves as a single unified being, to not introspect that deeply, this is also a way in which we are saying as a first person present indicative "I falsely believe this". They're not withdrawing from the false belief, they're (perhaps even literally) of two minds about it.
3. There's also the sense in which most people don't have a sufficiently good world model to fully understand the implications of what they believe, so we also have p ∧ ¬p type issues within our minds whenever any of us demonstrates hypocrisy, and sometimes people notice they're hypocritical without needing someone else to point it out, so anyone who says "huh, I'm being a hypocrite here…" is, in that way, using "hypocrite" as a first person present indicative of 'to believe falsely'. They may or may not withdraw from either belief, we can be all over the place on this, those who do not withdraw from at least one of those beliefs are an example of non-Wittgenstein behaviour.

So, linguistically, we do sometimes say things equivalent to "I currently falsely believe this specific thing".

I have of course asked everyone's favourite Shoggoth to critique this and tell me everything wrong with this post before posting it; the AI is now trying to simultaneously give me two mutually incompatible directions — telling me Wittgenstein was actually talking about language instead of logic, and also trying to say that I'm focusing on language when I should be looking at logic — I clearly can't get more useful feedback from it. I will be amused if there's a similar split in the comments.