Williamson’s argument for rejecting UAt and UDAt:

(1) Every vixen is a vixen.

UAt: Necessarily, whoever grasps the thought every vixen is a vixen assents to it.

UDAt: Necessarily, whoever grasps the thought every vixen is a vixen has a disposition to assent to it.

Williamson wants to know if we can maintain UAt or UDAt while acknowledging UA1 and UDA1 fails by the Stephen and Peter counterexample.

(2) Stephen assents to (1) if and only if he assents to the though every vixen is a vixen.

(3) He doesn’t assent to (1) so he doesn’t assent to the thought every vixen is a vixen.

(4) It follows that if t is the thought every vixen is a vixen then Stephen doesn’t assent to the thought every vixen is a vixen and is also a counterexample to UAt.

(5) If the thought that he associates with the sentence ‘Every vixen is a vixen’ is not the thought every vixen is a vixen then he is not a counterexample

(6) Using the word ‘thought’ indicates that inferential differences between Peter and Stephen and us represent differences between the thoughts we associate with (1). Peter and Stephen associate different thoughts then our own when assenting to (1).

(7) If (6), then we should translate their idiolects non-homophonically into ours.

(8) A translation scheme would invoke refusal to acknowledge the full challenge which Peter and Stephen have issued to (1).

(9) To claim (6) then the disagreement between Peter and Stephen and us sounds less threatening because it bundles together logical and semantic differences without unification.

(10) So, the links from understanding to assent and dispositions to assent fail for thought like they do for language.

For reasons other than (8) I think (6) fails because it separates thought from language. If this should be the case then the thought someone associates with a sentence isn’t just the thought of that sentence. This seems kind of strange to accept. I think Williamsons argument for failing UAt and UDAt is a good one.