Is 'can' information-sensitive in an interesting way, like 'ought'?
An example of uninteresting information-sensitivity is (1):
(1) If you can lift this backpack, then you can also lift that bag.
Informally speaking, the if-clause takes wide scope in (1). The truth-value of the consequent 'you can lift that bag' varies from world to world, and the if-clause directs us to evaluate the statement at worlds where the antecedent is true.
Here's a more interesting case.
(2a) Fred can dance if nobody's watching.
(2a) Fred can't dance if people are watching.
(More simply: Fred can dance, but not if people are watching.)
On the wide-scope reading of the if-clause, as in (1), (2a) says that under the assumption that nobody is watching, Fred has the ability to dance. This is not the most natural reading. It becomes salient for example in the following scenario.
A group of people, including Fred, have been divided into those who can dance and those who can't. Those who can dance have been put into rooms where they are alone; those who can't dance have been put into rooms where they are being observed. Either way, they are instructed to stand still. Now (2a) and (2b) are true, and most naturally understood in the wide-scope sense. If people are watching Fred (as he stands still), then he's one of those who can dance; if nobody's watching him, then he can't dance.
But in normal contexts, that's not how (2a) and (2b) would be understood. Normally, the if-clause in (2a) and (2b) would be understood to specify a restriction on the kinds of situations in which Fred manifests his ability to dance, not as a restriction on the possible situations in which Fred has the ability. Note that on the wide-scope reading, (2a) and (2b) could be true even if Fred can only dance when he is watched by others. Intuitively, this is just what (2b) denies.
So the if-clause in (2a) and (2b) normally doesn't have wide scope. Does it have narrow scope inside the modal, so that (2a) says that Fred has the ability to dance-if-nobody's-watching? Arguably not.
Here's one reason. Suppose an evil scientist has rewired Fred's brain so that if Fred intends to dance, he stands still, and if he intends to stand still, he dances. Fred doesn't know about this. And he is alone. Now I think there's a good sense in which "Fred can dance" is false. The classical conditional analysis roughly captures this sense: it is not true that Fred would dance if he intended to dance. So, on at least one salient reading, ability statements require that in the relevant scenarios where the agent performs the act in question, she intends to perform this very act. On the narrow-scope reading of (2a), we would therefore require Fred to dance as a result of an intention to dance-if-nobody's-watching. But that seems wrong. It's enough if he intends to dance.
Moreover, what is it to dance-if-nobody's-watching? Do you automatically perform this act whenever somebody's watching? If so, the narrow-scope reading would falsely imply that (2a) is true even if Fred is paralysed and unable to dance under any conditions whatsoever. To get a halfway sensible reading, we'd have to understand dancing-if-nobody's-watching as equivalent to dancing-while-nobody's-watching. But it's not clear by what mechanism a narrow-scope if-clause could suddenly become equivalent to conjunction.
So on the most natural interpretation, the if-clause in (2a) and (2b) is neither narrow-scope nor wide-scope.
From a Kratzerian perspective, that's what you would expect. You would expect the if-clause to restrict the modal 'can'.
Let's assume that 'S can phi' is true iff S phis (with the right intention, perhaps) at some accessible worlds, where accessibility is a matter of holding fixed certain physiological features of S as well as certain features of her environment. (Which features? Depends on context). If-clauses primarily serve to restrict the space of accessible worlds.
This seems to give the right result in (2a) and (2b). If we consider a range of possible worlds that differ from the actual world with respect to Fred's motivation, his environment, and so forth, but not with respect to his general physiology and character traits, we find some worlds where Fred is dancing, but no worlds where he is dancing while he is being observed. So restricting to worlds where Fred is being observed, we no longer find any worlds where he is dancing.
We also get the prediction that if (2a) is true, then 'Fred can dance' is true as well. That might seem problematic, but I actually think it's right. Note that it would be very strange to affirm (2a) and go on to deny that Fred can dance. Moreover, there are good pragmatic reasons for uttering (2a) instead of the simpler 'Fred can dance' if the only relevant worlds at which Fred dances are worlds where he is unobserved. (In some contexts, we might also start out with a sphere of accessibility that does not contain any worlds where Fred is unobserved. 'Fred can dance' would then be false, but (2a) would be true, because the if-clause forces an extension of the accessibility sphere.)
Another case: Tim is travelling back in time to kill his grandfather (before the conception of his father). As Lewis pointed out, (3) then has a true reading and a false reading.
(3) Tim can kill his grandfather.
On the Kratzerian account I have outlined, we first have to assume that 'his grandfather' in (3) can scope out of the modal, to denote Tim's actual grandfather. The two readings then depend on whether the sphere of accessibility holds fixed the fact that the person Tim is aiming at is his grandfather. Among worlds where the person is Tim's grandfather, Tim never kills him; among worlds where he isn't, Tim often kill him. Now what about (4)?
(4) If the person Tim is aiming at is going to be Tim's grandfather, then Tim can't kill him.
Intuitively, I'd say this also has a true reading and a false reading. The restrictor account predicts that it is straightforwardly false. But arguably the if-clause has wide scope in the true reading.
In none of these cases do we see the more problematic kind of information dependence we find with 'ought' -- a kind that can't be handled by either a wide scope or a narrow scope or a restrictor interpretation.
It's hard to prove, but I don't think such a more problematic kind of information dependence exists. Whether an agent can do something, I think, depends just on the facts, not on any information we might have.
Imagine there's a button on the wall; you don't know what it's for. In fact, pressing it would stop climate change. Whether you ought to press the button is obviously sensitive to what you (or we) believe. But whether you can press the button is not. Of course you can press it.
Information might be relevant for the sense in which an ability to phi requires that in the relevant worlds, the agent phis with the intention to phi, because whether or not one can (or would) intend to phi might depend on the available information. But here what matters is the agent's own information in the relevant accessible worlds, which arguably can't be restricted or otherwise affected by an if-clause.