If They Had No Tails...
In Chapter 5 of Williamson's manu The Philosophy of Philosophy we find a proposed answer. To know that it is metaphysically necessarily that A, one needs no more than whatever it takes to know that not-A counterfactually implies a contradiction. And this requires nothing over and above the cognitive faculties employed to acquire ordinary knowledge about the empirical world. Therefore, if (metaphysical) modal knowledge---indeed, if philosophy---is possible at all (and we suppose it is), then it requires no special cognitive faculties over and above what is required for ordinary knowledge of the world. Let's focus on the purported connection between modal and counterfactual knowledge.
Williamson demonstrates that modal claims are logically equivalent to some counterfactual claims. A necessary proposition is one whose negation counterfactual implies a contradiction, and a possible proposition is one that does not counterfactual imply a contradiction. Respectively,
(1) □A (¬A □→ ⊥)
(2) A ¬(A □→ ⊥)
The lessons, according to Williamson, are these: (i) Having what it takes to understand □→ and ¬, implies having what it takes to understand and □, and (ii) modal thinking is a special case of counterfactual thinking.
I take it that (ii) is meant to entail (i)---or is equivalent to it. So there is a natural objection here. Understanding A (or grasping its meaning) is not closed under logical consequence. If A is logically equivalent to B, s may grasp A without grasping B, because B embeds concepts that s doesn't understand---e.g., 'A' is equivalent to 'A or A', and though s understands the former, he doesn't understand the latter because he doesn't grasp the meaning of 'or'. Hence, (i), and so (ii), is false, and we lose the argument for thinking that an epistemology of counterfactuals will give us everything needed for an epistemology of modality. Carrie Jenkins expresses a version of the concern over at Long Words.
I don't think the objection hits Williamson. We need to read him as offering an account of how modal knowledge is possible for beings like us, and not as defending the empirical thesis that our modal and counterfactual knowledge covary. Whether or not our knowledge of necessity and possibility does (or must) take a detour through counterfactual knowledge, is not the question. The epistemological claim is that beings like us, who are in a position to know the right-hand side of the equivalence, are also in a position to know the modality on the left. After all, we might infer the left from the right.
Here's a different objection. It is one that I think is more serious than Williamson suggests. Williamson's proof of the first equivalence requires a certain theory about how to handle counterpossibles (i.e., counterfactuals with impossible antecedents). He sides with Lewis that we should treat them as vacuously true. If this is right, then necessary implications entail counterfactual implications:
(3) □(A → B) (A □→ B)
(3) underwrites Williamson's proof of (1). If this choice about how to handle counterpossibles is a mistake, then things unravel for Williamson's epistemology of modality. For a genuine counterexample to (3) can be turned into a counterexample to (1).
Here's one way to do it:
Let "LEM" be the universal formulation of the law of excluded middle, "∀A(A v ¬A)". And let "DNE" be the classical theorem "∀A(¬¬ A → A)". Suppose classical logic is unrestrictedly valid, so that LEM is metaphysically necessary (since logically necessary). Then the strict conditional, "□(¬ LEM → DNE)", is true. It's vacuously true. However, the corresponding counterfactual appears to be false: "¬LEM □→ DNE". This is false because if LEM were false then intuitionistic logic (and not classical logic) would be unrestrictedly valid. And in intuitionistic logic, ¬LEM does not imply DNE (but rather entails ¬DNE).
If this is right, then we also have a counterexample to (1). Although LEM is in fact metaphysically necessary, it is not equivalent to "¬LEM □→ ⊥". For in a counterfactual circumstance where ¬LEM is true, intuitionistic logic (but not classical logic) obtains. And, ¬LEM does not enjoin an intuitionistic contradiction.
Williamson evaluates other purported counterexamples to (3). I don't see that his replies touch the above example. Williamson attributes a confusion akin to the confusion made by philosophers for centuries prior to the realization that a generalization, "All S are P", is vacuously true when nothing is an S. But I'm not guilty of this confusion. If our counterfactual supposition is the denial of LEM, then not all of our knowledge of classical logical truth is available in the description of the counterfactual circumstances from which we are to develop a path to the consequent. In particular, any logical truth intuitionistically equivalent to LEM is not available in the description of the counterfactual circumstances. But then in such counterfactual (intuitionistic) circumstances, ¬LEM does not materially imply DNE. The reason is that, in those circumstances, ¬LEM is not a logical falsehood. Vacuity vanishes!
[UPDATE: Berit locates a vacuous assertion of Williamson's that he did not intend to be vacuous, and draws a dark lesson for the prospects of doing metaphysics.]