EAVESDROPPER: Someone is on the way to the grocery store. I hear her say: 'Susan might be at the store. I could run to her.' No party to the conversation that I am listening in on knows that Susan is on vacation. But I know that she is. Despite the fact that it is compatible with what the conversants know that Susan is in the store and that the speaker will run into her, I am inclined to judge the speaker's modal judgements to be incorrect. (Hawthorne 2007: 92)
For the contextualist, this is awkward. It seems that the person who utters 'Susan might be at the store" has said something true. After all, Susan's being at the store is entirely compatible with the speaker's body of knowledge (and the body of knowledge possessed by those in the conversation I'm overhearing). However, having overheard this conversation, I am inclined to judge the speaker's claim to be incorrect.
But the picture that emerges is not a friendly one for the contextualist. It looks as though we have a case of disagreement: I reject what the speaker I overhear accepts. It also looks like neither the person I overhear nor I myself am mistaken: "Susan might be at the store" thus seems true relative to the body of knowledge operative in the conversation I overhear and false relative to what I know. But this sort of 'faultless disagreement' implies a sort of truth-relativism about epistemic modals. In order to avoid this result, the contextualist has got to explain away why the eavesdropper case only appears to be a disagreement (or at least a case where my denying what the speaker I overhear asserts is felicitious) when it's actually not, or alternatively, why the case is one that only appears to be faultless, when in fact either I or the person I overhear believes something false.
No matter how the contextualist tries to get out of this puzzle that appears to motivate truth-relativism about epistemic modals, one thing is clear: the contextualist is going to have a hard time explaining just whose body of knowledge is supposed to be the relevant one in this context.
Reference:
Hawthorne, John (2007). "Eavesdroppers and Epistemic Modals," in Philosophical Issues, 17, The Metaphysics of Epistemology, 2007.
MacFarlane, John (2011). "Epistemic Modals are Context Sensitive," in Epistemic Modality, eds. Egan, A. & Weatherson, B. (Oxford: OUP).