As I understand it, the objective of chapters 3 and 4 is to test out the idea that the methodology of philosophy consists in something like searching for analytic truths and that the linguistic/conceptual-turners can take refuge in this fact on the grounds that analytic truths are linguistic or conceptual in some sense.
I was with Williamson in chapter 4 right up until the end where he argues against KUt’ (i.e. whoever knows …insert epistemic-analytic true thought here… in the normal way does so on the basis of their grasp of the thought).
Here is my formulation of Williamson’s argument:
(1) Either ‘their grasp of the thought’ in KUt’ can be taken in a thick way or in a thin way.
(2) If we take ‘grasping’ in the thick way (i.e. on the basis of the facts underlying the grasping), then grasping depends on knowledge that is not conceptual.
(3) If KUt’ is taken in a way that doesn’t depend on conceptual knowledge, then it will not help the conceptual-turner any.
(4) If we take ‘grasping’ in the thin way (i.e. on the mere fact that the thought in question has been grasped), then there is no way in which grasping is a basis for knowing in any useful sense.
(5) If KUt’ is taken in a way that makes it epistemologically useless, then it will not help the conceptual-turner any.
(6) If (1)-(5), then (7).
(7) KUt’ will not help the conceptual-turner any.
Ignoring the sloppy grammar in my formulation of Williamson’s argument, I think that (4) is false. There is still an important way in which ‘grasping’ in the thin sense is a useful basis for knowing. Namely, such grasping is a useful basis for knowing in cases where the truths are so thin, fundamental and basic that there can be no other way to explain our knowledge of them. These cases include truths of logic and truths about truth. To be sure, these are truths about the way the world is. But I’m sure Williamson would agree that we do not come to know about them by coming to know about the world – we come to know them merely by having grasped them.
Despite this point, I think that to restrict the role of philosophy only to the study of these “thin” truths is to place too strong a constraint on philosophy, so the conceptual-turner is not saved by resorting to this line of reasoning.