(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?

But if I do it with temporal operators throughout, not quantifiers I get:
.
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.

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: . (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 .)

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.

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.)

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.