The Paradox of Rational Believability

July 2, 2008

A while back I considered a weakening of Fitch’s paradox for knowability, to true belief instead of knowledge. I was thinking that factivity was essential to the proof, and so things couldn’t just be run with belief. But now I think you can do things with just rational belief. That is, letting Bp be ‘it’s rationally believed that p’, we get the principle of rational believability for any true proposition is false:

  • (p \rightarrow \Diamond Bp)

Suppose, like with knowledge, we have that belief distributes over conjunctions. Now grant further that we have positive introspection for belief, and belief consistency:

  • B(p \wedge q) \equiv (Bp \wedge Bq)
  • Bp \rightarrow BBp
  • \neg B(p \wedge \neg p)

Then we can show that the rational believability principle is false. Suppose for contradiction that a true Fitch sentence (p \wedge \neg Bp) could be rationally believed. B(p \wedge \neg Bp) gives us (Bp \wedge B\neg Bp) by distributing over the conjunction. We then get (BBp \wedge B\neg Bp) by introspecting on the first conjunct. Finally by the converse of distribution we get B(Bp \wedge \neg Bp) which contradicts belief consistency. Necessitating on that gets the desired result.

Update: this doesn’t just seem to be a problem for anti-realists. Suppose you think that you should only believe what’s true. Add to that ‘ought implies can’, and we’re in business.

  • p \rightarrow OBp
  • Op \rightarrow \Diamond p


  1. Mackie has a short paper in Analysis from 1980 arguing that factivity isn’t essential to the proof. You just need (what Linsky, in an Analysis paper six years later, calls) reflection:

    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.

  2. Ah! Thanks for that reference. Yeah, I was thinking it would be a little strange if no one had noticed this before.

    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.

  3. I’m wondering about distribution over conjunction.

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

    Bp \wedge B \neq p

    but notice

    \neg B (p \wedge \neg p)

    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.

  4. Sorry, that should have been Bp \wedge B \neg p.

  5. Sorry for not getting back quicker!!

    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 (K(p\wedge q) \rightarrow (Kp \wedge Kq)) 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.

  6. […] been thinking about to do with a version of the knowabality principle for rational belief. Back in this post I considered a version of Fitch’s paradox for rational belief, which shows the following […]

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