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It's hot in summer, we can go swimming. It’s cold in winter, we can’t go swimming.

What is the function of can? Is it ability or possibility?

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It depends a bit on what the speaker has in mind and the circumstances. Generally, if you know how to swim, it's technically possible for you to swim regardless of what season it is. Whether you want to is a different question. You might also have a health condition that makes it dangerous to swim in certain cases, for example in the cold. Without more context, it can be hard to tell what the speaker wanted to convey.

"It's hot in summer, we can go swimming" - to me, this sentence is about the fact that swimming can cool you off when it's hot out. If you're the person suggesting that, you're presumably able to swim and have an idea where to go swimming.

"It's cold in winter, we can't go swimming" - this really depends on what the speaker means. I'd guess the underlying implication is "we would get sick if we go swimming when it's cold out", so even if it's possible (the beach is right there) and you're able to (you know how to swim), you'd choose not to.

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