Consider the following argument:
(1) If empiricism is true, then any synthetic belief is open to revision.
(2) At least one synthetic belief is not open to revision.
(3) Therefore, empiricism is false.
The argument is valid because it follows a standard modus tollens. Thus, we must look at the premises.
As to (1), if a belief is synthetic then it can in theory be falsified. For instance, Newton’s law of universal gravitation had a ton of support for it, but there were phenomena that showed Newton’s law to be incorrect and so it was abandoned. The same is true of any other belief under empiricism since a fact that falsifies the belief would lead one to revise the belief. Now this doesn’t mean that any anomaly must cause belief revision since Kuhn has pointed out that every theory faces anomalies, but that fact does not falsify (1).
As to (2), consider this argument for it:
(4) At least one synthetic belief is of such a nature that it has an intrinsic defeater-defeater.
(5) If a belief has an intrinsic defeater-defeater, then it is not open to revision.
(2) Therefore, at least one synthetic belief is not open to revision.
Again, the argument is logically valid by means of modus ponens so we must look at the premises to see whether they are true.
As to (4), examples seem to abound of this sort. If Quine is right that there are no such things as analytic truths, then something like “there are no married bachelors” or “if p, then p” would be great examples. However, if we think Quine is wrong on that matter, then there are other examples still. Consider Moore’s famous argument about his two hands: “here is one hand, and here is another.” Now what could defeat my knowledge that I have two hands? If I seem to wake up and find myself with only one hand, then I would suppose that a brief bit of my memory was about a dream within a dream and so I am still dreaming and once I wake up from my dream where I have one hand, I will find myself with two hands. Moreover, if it seems to me that all of the sudden I only have one hand and I think that I was having a delusion the whole time, I would consider that present state a delusion because I have my whole life experience and immediacy of knowledge as evidence. The same goes for skeptical arguments, etc. This type of Moorean shift is common and seems totally appropriate. For more examples, see Moore’s papers on the topic or consider beliefs that are usually taken as foundational in nature.
With regards to (5), if the belief that I have two hands is of such a nature that I cannot have a fact that will defeat my belief due to its intrinsic defeater-defeater, then that belief is clearly not open to revision. For if it were open to revision, that means it could be defeated. Since that is not the case, then (5) holds.
Therefore, (2) follows which means that (3) follows in the main argument. Empiricism is therefore defeated; that is, if it is open to revision.