Fitch’s Paradox and True Believability

April 20, 2008

Fitch’s paradox shows us that if all truths are knowable, then all truths must be known. (Proof sketch. Show the contrapositive: if there is an unknown truth, there is an unknowable truth. Let p be an unknown truth. Claim: that p is an unknown truth is unknowable. Suppose that it was known – then it must both be known and known to be unknown, for to know than both p and that p is unknown you must know each seperately. But of course by factivity, if it’s known to be unknown it must be unknown. Therefore its known and unknown. The supposition that that p is an unknown truth is known leads to contradiction, so knowing it is impossible. I.e. it’s unknowable.) The above can be made precise by formalising the knowability principle as the following schema for p:

  • WVER: (p \rightarrow \Diamond Kp)

Of course, this argument does not rely on knowledge in particular, it is true of any operator that is both factive, and distributes over conjunctions

  • Op \rightarrow p
  • O(p \wedge q) \rightarrow (Op \wedge Oq)

For example, if all truths are possibly necessary, then all truths are necessary. Indeed, it holds for any operator whose accessibility relation is reflexive.

Clearly not all substitutions for K are interesting. E.g. it yields results for truth, actuality, determinateness (in the context of vagueness), necessity, provability and the operator true only of eternal truths, but not all these results are particularly philosophically interesting.

The strongest version of Fitch’s paradox I know of results from substituting O for true belief (I shall write BTp for (Bp \wedge p).) And by strongest, I mean has the weakest premise a verificationist would accept, and has an unsavoury Fitch like consequence: if it is possible to truly believe any truth, then all truths are in fact believed by someone or other. Whatever notion of verifiability the anti-realist uses, if the possibility of X-ing p entails the possibility of truly believing p we are in trouble. For example, the possibility of knowing p entails the possibility of truly believing p, since knowledge entails true belief.1

One tactic an anti-realist might take is to reject the analysis of knowability in terms of possibility and knowledge (or: some modal notion plus some epistemic notion) and take knowabality as a primitive epistemic notion, \mathcal{E}, over a set of epistemic states or worlds. The knowability principle would then be

  • (p \rightarrow \mathcal{E}p)

Of course, we must then be careful that \mathcal{E}p doesn’t entail \Diamond BTp. For example we might take \mathcal{E} to be \neg \neg where \neg is intuitionistic negation (note: negation is an epistemic operator in intuitionist logic.) This gives us some insight into what the accessibility relation for \mathcal{E} might look like, taken over a set of situations (states of potential knowledge):

  • w \Vdash \mathcal{E}p \Leftrightarrow \forall u \geq w , \exists v \geq u, v \Vdash p

This is only one way to go, others have suggested \Diamond K@p, and no doubt there are many more. I just feel that most of these approaches are wrong headed. It seems to me that an anti-realist would already have a notion of “verifiability” that is conceptually prior to metaphysical possibility and knowledge. And I see no reason why that notion of verifiability should be definable in terms of more familiar operators. It seems to me the best approach is to start with a notion of verifiability, see what principles hold, and work backwards from there. E.g. we’d plausibly want

  • p \leftrightarrow \mathcal{E}p
  • If \vdash p, infer \vdash \neg\mathcal{E}\neg p

and so on. We’d then need to think about how this would interact with knowledge and true belief. If truth just is knowability, then the claim that knowability entails the possibility of true belief should be harmless. Anyway, this requires some careful thought, and its all a little unclear in my head right now – but I hope to post on this stuff again (so stay tuned!)

1 I guess we could make the paradox even stronger if we interpreted \Diamond as consistency (or, if we’re not treating K as a logical constant, consistency with a minimal logic of knowledge/true belief.) Then the knowability principle is: if p is true, then it is consistent that it be truly believed. Note also that we don’t need full necessitation for the proof, only: if \vdash \phi and \phi does not contain box or diamond, then \vdash \Box \phi. So it doesn’t matter whether we treat consistency as logical.



  1. That BT version of the argument is nice! The response that I’ve considered (but never been quite sure how to phrase) looks like it’s just the \Diamond K@ p version.

    I wonder though about taking a primitive notion of verifiability – could this really do the work that’s wanted here? Letting verifiability be primitive seems just as bad as letting truth be primitive – perhaps even worse, because the epistemic facts that it appeals to seem much less natural to countenance than a realist world.

  2. Hmm, I just meant that to be the code for this formula that you used above:

  3. Hi Kenny – thanks for the comment!

    I think I agree with you that making epistemic facts metaphysically primitive is bad. But I’m not sure you need to go that far – I just want to argue that the notion of verifiability is, as it where, a ‘semantic atom’ much like knowledge is, in that an analysis which decomposes it into more primitive notions is not possible, or particularly illuminating.

    For example, the analysis from which Fitch’s paradox arose was possible knowledge. But as Salerno has argued, the expressions used in English to express knowability are factive (‘could have known’ etc.) which possible knowledge isn’t – contingent falsehoods are possibly known.

    Another reason I don’t like the Fitch analysis is that according to it: ‘that there are an even (odd) number of books on my shelf is an unknown truth’ is unknowable according to the possible knowledge analysis (just run Fitch’s argument.) However, I could just come by tomorrow, count the books on my shelf, and since I know no-one else would be bothered to count them, I can infer ‘there was an even (odd) number of books on my shelf and nobody knew this’. On the pretheoretic notion of knowability, some Fitch sentences *are* knowable.

    (I fixed your formula BTW – I can’t work out how to give posters edit rights over their own posts yet. Sorry.)

  4. Thanks for fixing the code!

    I’m not quite sure that you’re right about the Fitch sentences being knowable – after all, the verificationist just means it’s possible for someone at some point to know any given sentence, and not that it’s possible for someone now to know it. Thus, Kp must mean “someone, at some time knows p” – and thus, if you ever count your books, then “that there are an even (odd) number of books on your shelf” is not an unknown truth, and so no one could know that it is.

  5. Right, so it’s knowable because it seems I’m gaining future knowledge of now p but nobody now knows it. (I’m inclined to think the present tense implicitly does something like this anyway, but that’s a side issue.)

    So I guess you’re wondering where the ‘now’ came from if you’ve stipulated what p and K are. I want to say that it’s a sufficient condition for a truth, p, to be knowable, that it is possible to know that Np (i.e. that p is now true.)

    In this case it’s not that the quantifiers in K are restricted to currently existing knowers as you suggest, but rather that they are unrestrictedly quantifying over all (future) knowers saying they don’t currently know p. That is: \neg \exists x \exists t NKp. (Where did that come from? Well it’s sufficient for p to be knowable that it’s possible to know Np – so in this case N(p \wedge \neg \exists xt Kp) \leftrightarrow Np \wedge \neg \exists x \exists t NKp.)

  6. Hmm, actually I’m getting a bit worried. That last formula comes out true on a fixed domain kripke semantics which is why I wrote it out, but it doesn’t make too much sense because I’m mixing tense operators and quantifiers over times.

    But if I do it with temporal operators throughout, not quantifiers I get:
    N(p \wedge \neg \exists x FPKp) \leftrightarrow (Np \wedge \neg \exists x NFPKp).
    Here FP is ‘at some time or other.’ But I can’t commute NFPKp to FPNKp I don’t think, which is what we need if its to come out knowable. If I remember correctly, Edgington’s paper glossed over this in her discussion of the temporal version of Fitch’s paradox – she (incorrectly it seems) claimed that future knowers could know that Fitch sentences currently obtain.

  7. […] Bacon, a Philosophy student at Oxford University, presents ‘Fitch’s Paradox and True Believability‘, postulating whether truth is truly knowable; (or conversely, if unknown truths are […]

  8. Hello Andrew and Kenny (and others). Thanks for this post and discussion. I’ve been thinking about Fitch’s Paradox recently. I’ve got a comment about the original formulation of the paradox at SEP (you linked to it in the post). By way of introduction, I believe there’s a bit of conflation on the ‘K’ operator that causes confusion. Let me try to spell out what I think is the problem. Brogaard and Salerno write that Kp iff it is known by someone at some time that p. I suggest that instead of working with K, we work instead with what I call ‘K3’ whose semantics follow: K3(p, k, t) iff k knows that p at t. To relate this to the operator K in the entry, we can say that Kp iff (Ey)(Ez)K(p, y, z). Using K3 we can spell out principle (KP) and (NonO) like so:

    (KP): (x)(x -> (poss.)(Ey)(Ez)K(x, y, z)

    (NonO): (Eq)(q & ~(Eu)(Ev)K(q, u, v)

    An instance of NonO is:

    (p & ~(Eu)(Ev)K(p, u, v))

    [in other words “p is true, but it’s not the case that there is a knower u and time v, such that at v, u knows p”]

    From this and KP we have:

    (p & ~(Eu)(Ev)K(p, u, v)) & (poss.)(Ey)(Ez)K((p & ~(Eu)(Ev)K(p, u, v)), y, z)

    informal quantifier exchange to simplify:

    (p & (u)(v)~K(p, u, v)) & (poss.)(Ey)(Ez)K(p & (u)(v)~K(p, u, v)), y, z)

    Now if we’re serious about possible world semantics for the ‘poss.’ operator, would we say the following? (Curioser and curioser…)

    In the actual world, p is true, but at no time does anybody in the actual world know p, but there is a possible world in which someone at some time knows that p is true in the actual world and knows that at no time, in the actual world, does anyone, in the actual world, know that p is true.

    Here’s a thought experiment to show that this last English translation of the supposedly inconsistent result from Fitch is not, in fact, inconsistent. This is a bit far fetched, but this is after all philosophy…

    Suppose the actual world (W@) had had no knowers, but was otherwise exactly as it is, so specifically the W@ has a certain physical feature F. Now suppose, that in possible world WP, there exists a modal clairvoyant who comes to know about W@, in particular that it has feature F and that there are no knowers in it. If p is the proposition expressed by the sentence ‘W@ has feature F’ then p is true and nobody in W@ knows it, as there are no knowers in W@, but the clairvoyant knows that world W@ has feature F and that nobody in W@ knows this, again as there are no knowers in W@.

    Is this a case a sort of model for K3 version of the supposedly inconsistent sentence (3) from the SEP entry?

  9. […] « Logic for Philosophy 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 […]

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