Sometimes, when we say that someone can (or cannot, or must, or must not) do P, we really mean that they can (cannot, must, must not) do Q, where Q is logically stronger than P. By what linguistic mechanism does this strengthening come about?
Example 1. My left arm is paralysed. 'I can't lift my (left) arm any more', I tell my doctor. In fact, though, I can lift the arm, in the way I can lift a cup: by grabbing it with the other arm. When I say that I can't lift my left arm, I mean that I can't lift the arm actively, using the muscles in the arm. I said that I can't do P, but what I meant is that I can't do Q, where Q is logically stronger than P.
Example 2. I have bought a piano and just took my first lesson. So I can play a few basic tunes. Can I play the piano? In most contexts, the answer would be no. Normally, when we say that someone can play the piano, we mean that they can play reasonably well, which is logically stronger than play.
(Without context, the standards vary widely, as this forum thread illustrates, where someone asked, 'at what point can you say that you can play the piano?'. Answers range from 'when you can play all of Chopin's etudes in one sitting' to 'anyone can play the piano'.)
Example 3. I'm standing in front of a safe, but I don't know the combination. I only have a minute. 'I can't open the safe', I say. But whatever the right combination is, I can dial that combination. So I can open the safe, but only by luck. What I can't do is open the safe deliberately or at will, which is logically stronger than open the safe.
Example 4. You don't know the way to the train station. I could walk you there, but due to a disability I can only walk very slowly, so that you would miss your train. 'I can't walk you to the station', I say, meaning that I can't walk you there in time.
It is not obvious that these are examples of a single phenomenon. But they all have in common that a 'can P' statement is interpreted as 'can Q', where Q is logically stronger than P.
The effects arguably also arise for deontic 'can' and 'must'. For example, 'you must (or must not) raise your arm' is naturally understood as conveying an obligation to actively raise the arm.
What might explain these effects? I can think of five explanations, none of them very good.
Explanation 1. The appearance of a strengthened prejacent comes about through a contextual restriction on the domain over which the modal quantifies. I don't see how this could work, if we want to retain the idea that the domain of the modal is given by contextually salient worlds compatible with relevant circumstances (or the most "ideal" of these worlds relative to some salient ordering). Consider example 1. The circumstances surely allow me to lift my left arm with my right arm. We might try to say that worlds in which I don't "actively" perform an act are ignored. But in worlds where I lift my left arm with my right arm, I am actively performing that act. And it is unclear what ideal the act might float. (The best candidate is perhaps an ideal of normalcy, but that doesn't generalise.)
Explanation 2. The prejacents are ambiguous. 'Lift an arm' is ambiguous between actively lifting an arm and passively lifting an arm; 'opening the safe' is ambiguous between opening the safe deliberately and opening it by luck. But this doesn't seem right. If I opened the safe by luck, the claim that I did not open the safe is unambiguously false.
Explanation 3. The strengthened prejacent is an implicature. After all, it can be cancelled: `I can open the safe, but only by luck'; `I can play the piano, but only poorly'; `I can lift my arm, but only with the other arm'. But I can't think of a Gricean explanation for the supposed inference. In the three examples, a sentence is uttered that is literally false (on the present explanation). What kind of reasoning leads us from there to the conclusion that the speaker wanted to convey an alternative proposition that is true? Paradigm examples of implicatures strengthen the content of an assertion; here it weakens the content. Also, the supposed implicatures are equally present in questions (`can you raise your arm?'), which makes it hard to explain them in terms of norms of assertion.
Explanation 4: The prejacent is strengthened by a process of "free enrichment". Recanati, Bach, Carston and other have argued that when we process utterances, we often supplement the uttered sentence by further, unarticulated constituents that don't have to be pronounced because they can be taken for granted in the relevant context. Perhaps this happens in the prejacent of our modals. For example, `I can't raise my arm' is understood as `I can't raise my arm actively' – with the adverb `actively' supplemented in the contextual processing. This would explain the observed effects, but the whole idea of free enrichment is controversial, and there is (to my knowledge) no precise model that would predict when an enrichment occurs, and what kind of enrichments can occur.
Explanation 5: Modals have a hidden parameter of adverbial type that can either be left empty or supplied by conversational context. If the parameter is supplied, it restricts the interpretation of the prejacent. In the context of example 1, "active" actions are relevant, so an unarticulated `actively' modifier is passed to `can', which restricts not the accessible worlds but the interpretation of the prejacent. This makes the right predictions, but one would like to have some independent evidence for the postulated mechanism.