Knowing-Who in Quantified Epistemic Logic

Abstract

This article proposes an account of knowing-who constructions within a generalisation of Hintikka’s (Knowledge and belief. Cornell UP, Ithaca, MA, [10]) quantified epistemic logic employing the notion of a conceptual cover Aloni PhD thesis [1]. The proposed logical system captures the inherent context-sensitivity of knowing-wh constructions Boër and Lycan (Knowing Who. MIT Press, Cambridge, MA, [5]), as well as expresses non-trivial cases of so-called concealed questions Heim (Semantics from different points of view. Springer, Berlin, [9]). Assuming that quantifying into epistemic contexts and knowing-who are linked in the way Hintikka had proposed, the context dependence of the latter will translate into a context dependence of de re attitude ascriptions and this will result in a ready account of a number of traditionally problematic cases including Quine’s well-known double vision puzzles Quine (The ways of Paradox and other essays. Random House, New York, [16]).

Appendix

Let $M=\langle W, R, D, I, C \rangle$ be a CC-model and $M'=\langle W, R, D, I \rangle$ be the corresponding classical QEL-model. And given a CC-assignment g and a world $w \in W$, let $h_{g,w}$ be an element of $D^{{\mathscr {V}}_N}$ such that for all $v \in {\mathscr {V}}_N: h(v) = g(v)(w)$. We prove the following theorem for any closed $\phi$ in ${\mathscr {L}}_{CC}$:

Theorem 4.3

$$M, w, g \models _{CC} \phi \hbox { iff } M', w, h_{g,w}\models _{QEL} \phi . $$

Proof

The proof is by induction on the construction of $\phi$. We start by showing that the following holds for all terms t:

Suppose t is a variable. Then $[\![t ]\!]_{M,w,g}= g(t)(w)$. By definition of $h_{g,w}$, $g(t)(w)=h_{g,w}(t)$, which means that $[\![t ]\!]_{M,w,g} = [\![t ]\!]_{M', w, h_{g,w}}$. Suppose now t is a constant. Then $[\![t ]\!]_{M,w,g}= I(t)(w)=[\![t ]\!]_{M', w, h_{g,w}}$. We can now prove the theorem for atomic formulae.

Suppose $\phi$ is $Pt_1,...,t_n$. Now $M , w, g\models _{CC} Pt_1,...,t_n$ holds iff (a) holds:

By (A), (a) holds iff (b) holds:

which means that $M', w, h_{g,w} \models _{QEL} Pt_1,...,t_n$.

Suppose now $\phi$ is $t_1=t_2$. $M , w, g\models _{CC} t_1=t_2$ holds iff (c) holds:

By (A) above, (c) holds iff (d) holds:

which means that $M', w, h_{g,w}\models _{QEL} t_1=t_2$.

Suppose now $\phi$ is $\Box \psi$. $M , w, g\models _{CC} \Box \psi$ holds iff (e) holds:

By induction hypothesis, (e) holds iff (f) holds:

And (f) holds iff (g) holds:

Since $\psi$ does not contain any free variable, (g) is equivalent to $M' , w, h_{g,w} \models _{QEL} \Box \psi$.

Suppose now $\phi$ is $\exists x_n \psi$. $M , w, g\models _{CC} \exists x_n \psi$ holds iff (h) holds:

By induction hypothesis, (h) holds iff (i) holds:

By definition $h_{g[x_n/c],w}= h_{g,w}[x_n/c(w)]$, and $c(w)\in D$. But then (i) holds iff (j) holds:

which means $M' , w, h_{g,w} \models _{QEL} \exists x_n \psi$. The induction for $\lnot$ and $\wedge$ is immediate. $\square$