I will present one of Williamson’s arguments, and then give a critic of that argument, and finally I will refute the critic.
Williamson argues for the conclusion that “[w]e should adopt no conception of philosophy that on methodological grounds excludes elusive objects.”
1. If it is possible that reality contains elusive objects then it is possible that not everything is thinkable.
2. If it is possible that not everything is thinkable then “[w]e should adopt no conception of philosophy that on methodological grounds excludes elusive objects.”
3. It is possible that reality contains elusive objects.
4. Therefore it is possible that not everything is thinkable.
5. Therefore “[w]e should adopt no conception of philosophy that on methodological grounds excludes elusive objects.”
A criticism one could give to this argument is to deny the 2nd premise, on the grounds that there is no evidence that there are elusive objects and therefore it is more likely that there are no elusive objects, and everything is thinkable. Thus, just because it is possible that not everything is thinkable does not mean “[w]e should adopt no conception of philosophy that on methodological grounds excludes elusive objects”, since there is no evidence that elusive objects exist.
6. If there is no evidence that elusive objects exist then we should not abandon conceptions of philosophy that on methodological grounds excludes elusive objects.
7. There is no evidence that elusive objects exist.
8. Therefore we should not abandon conceptions of philosophy that on methodological grounds excludes elusive objects.
This criticism does not work because premise 6 is false. McDowell’s argument requires the premise that everything is thinkable. Just there being a logical possibility, (no matter how unlikely) is enough so that McDowell can not claim the premise that everything is thinkable.