Yes, I didn’t think about that. That’s interesting because the principle doesn’t hold for K either, when K is read as ‘it is known by someone or other’. That said, only the other direction is needed for the Fitch proof () and that direction is much less controversial.

Either way, the application I had in mind in the update required only reading Bp as ‘one rationally believes that p’. For the original verificationist principle, you could give the quantifier over believers wide scope: “there is some possible individual, x, such that, if p is true, then it is possible for x to rationally believe p.” The proof acts as a reductio of this principle, which also sounds like the kind of thing the verificationist would be committed to.

I think most people (except dialetheists) won’t accept it for the following reason:

but notice

I’d imagine most won’t accept that it is rational to believe a contradiction, even though it may be rationally believed that p and also rationally believed that not-p by someone else (at some other time) etc.

For my own part, I’ve got no problems with contradictions being rationally believed.

It seems reflection is weaker than introspection + consistency, which I guess says something for going about it via that route. I’ll definitely check it out.

Reflection: O~Op -> ~Op

Now, if O is factive, we obviously get reflection for free too. But Mackie explores some arguments for thinking some non-factive operators obey reflection. For example, he argues that ‘it is justifiably believed at t’ supports reflection, even though it’s not factive. Definitely worth a look, especially since it’s such a short read.