In Chapter 3 of Knowledge and Its Limits, Timothy Williamson offers recombination arguments for the primeness of knowledge (and other mental states). To say that knowledge is prime is to deny that it is a composite of narrow (internal) and broad (external) conditions. Each argument begins with two cases of knowledge that are like with respect to their internal condition but different with respect to their external condition. If knowledge is composite, then recombining the internal condition of one case with the external condition of the other case will produce a third case of knowledge. Williamson needs one such recombination that fails to produce a case of knowledge to undermine the thesis that knowledge is composite. Here's Williamson:
...in [case 1] there is water on the right and gin (which looks just like water) on the left, and a brain lesion causes one visually to register only what is on the right. In [case 2] there is gin on the right and water on the left, and a brain lesion causes one visually to register only what is on the left; in the [case 3] internally like [case 1] and externally like [case 2], there is gin on the right and water on the left (as in ), and the brain lesion causes one visually to register only what is on the right (as in ). Thus, given appropriate background conditions one sees water in  and  but not in . (70)
My criticism of Williamson's argument is this. It is all but clear that cases 1 and 2 are in fact cases of seeing. After all, for Williamson seeing entails knowing (Chapter 1), and he is sympathetic to the thesis that one can know only if she could not easily have gotten it wrong (Chapter 4). But it would seem that in cases 1 and 2 the subject could very easily have gotten it wrong, since she could very easily have looked at the gin when forming her water-belief. The worries here are the same for barn-beliefs in Barn County. Just as it seems strange to say that I know that there is a barn when there are fake barns in the vicinity, it seems strange to say that the subject in cases 1 and 2 knows that there is water when there is gin (indistinguishable from water) in the vicinity. Additionally, the subject in each of the two cases has a brain lesion blocking half of her visual information! One would not be remiss to pause and question the respectability of the partially disabled visual process. In sum, two worries arise. There are Ginet-Goldman style barn considerations to worry about, and there are Bon Jour-Plantinga clairvoyance-brain lesion considerations about the epistemic respectability of strange but reliable belief-forming processes. Both worries go against a claim to knowledge in cases 1 and 2. And so, if seeing entails knowing, then arguably the subjects in cases 1 and 2 fail to see. Related worries surround Williamson's other arguments. Here is Williamson arguing more directly for the primeness of knowledge.
Let [case 1] be a case in which one knows by testimony that the election was rigged; Smith tells one that the election was rigged, he is trustworthy, and one trusts him; Brown also tells one that the election was rigged, but he is not trustworthy, and one does not trust him. Let  be a case which differs from  by reversing the roles of Smith and Brown.... Now consider a case  internally like  and externally like . In , one does not trust Brown, because one does not trust him in , and  is internally like . Equally in , Smith is not trustworthy, because he is not trustworthy in , and  is externally like . Thus, in , neither Smith nor Brown is both trustworthy and trusted. Consequently, in , one does not know that the election was rigged. Thus the condition that one knows that the election was rigged is prime. (72)
My criticism here is this. In recombining the internal condition from case 1 and the external condition from case 2, Williamson fails to include the entire external condition from case 2. Part of the external story in case 2 is that the belief in question was produced by reliable testimony. In case 3, the belief was not produced by reliable testimony. But if in case 3 the belief was not produced by reliable testimony, then Williamson has not properly recombined the cases. He has not included in case 3 the full external condition from case 2. Therefore, the recombination is incomplete.
A general criticism that I am making is that Williamson, in all of his recombination arguments, fails to include some causal or counterfactual conditions as part of the broad (external) condition of the subject in cases 1 and 2. The result is a mistaken attribution of knowledge in the first argument and an incomplete recombination in the second argument. Therefore, the question about the primeness of knowledge remains open.