The Fitch-Church Paradox and First Order Modal Logic

Reformulation strategies for solving Fitch's paradox of knowability date back to Edgington (Mind 94: 557-568, 1985). Their core assumption is that the formula p -> lozenge Kp, from which the paradox originates, does not correctly express the intended meaning of the verification thesis (VT), which should concern possible knowledge of actual truths, and therefore the contradiction does not represent a logical refutation of verificationism. Supporters of these solutions claim that (VT) can be reformulated in a way that blocks the derivation of the paradox. Unfortunately, these reformulation proposals come with other problems, on both the logical and the philosophical side (see Percival in Aust J Philos 69: 82-97, 1991; Williamson in Knowledge and its limits, Oxford University Press, Oxford, 2000; Wright in Realism, meaning and truth, Blackwell, Oxford, 1987). We claim that in order to make the reformulation idea consistent and adequate one should analyze the paradox from the point of view of a quantified modal language. An approach in this line was proposed by, among others, Kvanvig (Nous 29: 481-499, 1995; The knowability paradox, Oxford University Press, Oxford, 2006) but was not fully developed in its technical details. Here we approach the paradox by means of a first order hybrid modal logic (FHL), a tool that strikes us as more adequate to express transworld reference and the rigidification needed to consistently express this idea. The outcome of our analysis is ambivalent. Given a first order formula we are able to express the fact that it is knowable in a way which is both consistent and adequate. However, one must give up the possibility of formulating (VT) as a substitution free schema of the kind p -> lozenge Kp. We propose that one may instead formulate (VT) by means of a recursive translation of the initial formula, being aware that many alternative translations are possible.