A well known argument by Fitch proves that a plausible claim, namely that all truths can be known, collapses into a much less plausible claim: all truths will be known. Since truth entails possibility, Fitch’s proof licenses the following equivalence :

where the diamond expresses possibility and the epistemic operator ‘K’ reads ‘it is known by someone at some time that’. This is the knowability paradox. According to Jonathan Kvanvig, the two notions of truth universally knowable and truth universally known are semantically different, at least according to Frege’s criterion of identity for senses. Therefore, he argues, they should not be equivalent. If we rely on our semantic intuitions, Kvanvig concludes, Fitch presents us with a genuine modal paradox.
   Is Kvanvig’s diagnosis correct? Or does Fitch’s proof just constitute a counterexample to anti-realism, as Timothy Williamson has recently claimed? The paper argues that both Kvanvig and Williamson are wrong: (F) is not paradoxical and Fitch’s proof does not undermine the thesis that all truths are knowable.
Different solutions of the paradox have been proposed so far. Among the most discussed are the so-called restriction strategies. Eminent anti-realists, such as Neil Tennant and Michael Dummett, have recently proposed to restrict the left hand-side of (F) to some class of non-problematic truths. Once knowability is restricted, those authors argue, Fitch’s paradox is blocked. Such proposals, however, look desperately ad hoc. Moreover, new Fitch-like paradoxes undermining Tennant’s and Dummett’s restrictions have been recently proposed in the literature. How to solve Fitch’s puzzle?
The thesis I defend is that source of the paradox lies in the standard formalization of the knowability principle “All truths are knowable” (henceforth KP). The principle carries an equivocation between two notions of knowability: “in w, knowledge that j” and “knowledge that, in w, j”. That is, with respect to actual truths: possible knowledge of a possible truth and possible knowledge of an actual truth. The distinction is subtle and standard modal quantified logics do not have the resources to accommodate it. That’s why, as Dorothy Edgington pointed out, we need a richer logic for a better formalization of (KP). Edgington suggests to add the actuality operator A to S5 to get a new knowability principle :

which does not entail Fitch’s conclusion. Her proposal, however, faces serious objections. Helge Rückert has recently suggested to frame KPA into Kai Wehmeier’s new modal logic S5*. The upshot is a new knowability principle KP*, semantically equivalent to Edgington’s, but syntactically different :

As modal contexts are concerned, in S5*, expressions followed by stars must be evaluated with respect to the possible worlds at stake; otherwise, they must be evaluated with respect to the actual world.

The paper supplies an independent argument from our everyday uses of the expression ‘it possible to know that’ to the effect that our common notion of knowability is factive, and defends it from criticisms. As a result, it is argued, Fitch’s paradox does not undermine our concept of knowability. Rückert’s non-factive interpretation of (KP*) is criticized and an alternative reading is offered. A new formalization of (KP) is provided within an extension of Wehmeier’s framework :

The principle has it that knowability is a necessary and sufficient condition for both actual and possible truths. Among its virtues, the proposed formalization does not entail the unwelcome right hand side of (F) and blocks other knowability paradoxes recently discovered in the literature: once knowability is given an explicit logical interpretation, the new arguments can be seen to be all provably invalid.

Edgington-type conceptions of knowability face at least two powerful objections. First, they appear to restrict knowability to necessary truths. Second, they require transworld knowability, i. e., in our case, possible knowledge of actual truths. The objection from necessary truths, I claim, can be easily addressed. For one thing is to be true at every world, and another thing is to be evaluated as true from every world: the first notion, but not the latter, is the one of being a necessary truth. The paper also provides an answer to the second problem: how can merely possible knowers refer to the actual world? According to Edgington, counterfactual knowledge can do the trick. The paper defends Edgington’s account of transworld knowability against Williamson’s well-known charge of trivialization. Williamson’s trivialization argument, I argue, is harmless, provided that counterfactuals can be informative. Since Williamson’s argument was the main obstacle faced by Edgington-like treatments of Fitch’s paradox, I conclude that the knowability paradox is either S5*-invalid or metaphysically uninteresting.

References:

Brogaard, B. and Salerno, J., 2002. “Clues to the paradox of knowability: reply to Dummett and Tennant”, Analysis 62, pp. 143-50.

-----, 2006. “Knowability, Possibility and Paradox”, forthcoming in D. Pritchard and V. Hendricks (eds.) New Waves in Epistemology, Ashgate.

Edgington, D., 1985. “The Paradox of Knowability” Mind 94, 557-568.

Fitch, F., 1963. “A Logical Analysis of Some Value Concepts” The Journal of Symbolic Logic 28, 135-142.

Kvanvig, J. 2006. The Knowability Paradox, Oxford University Press.

Rückert, H., 2003. “A Solution to Fitch's Paradox of Knowability” in Gabbay, Rahman, Symons, Van Bendegem (eds.) Logic, Epistemology and the Unity of Tennant, N., 1997. The Taming of the True. Oxford, Chapter 8.

Wehmeier, K. F., 2004. “In the Mood”, Journal of Philosophical Logic 33, pp. 607-630.

Williamson, T., 2000. Knowledge and its Limits. Oxford, Chapter 12.