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(1) カードで払うことができますか。

(2) カードで払ってもいいですか。

In google these two sentences were both translated into: Can I pay by card?

Whether there exists any subtle differences? Or are they completely equivalent?

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They are not equivalent. The difference between the two is that of ability vs. permission. In English, both of these concepts can be expressed using "can", which is the reason why Google Translate offers the same result here. So these two sentences actually mean:

(1) Am I able to pay by card?

(2) Am I allowed to pay by card? / May I pay by card?

I suppose that in this context, the difference in practice is not very big, but in other cases it definitely can be. For example:

(1) 食べることができますか。 Am I / Are you / Is he etc. (physically) able to eat?

(2) 食べてもいいですか。May I eat? / May we eat?

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    However in some popular Japanese textbooks such as 'みんなの日本語', there are really a lot of sentences using "ことができますか" to express permission rather than ability. For example, A: カードで払うことができますか。(May I pay by card?) B: すみませんが、現金でお願いします。(Sorry, please pay by cash). Obviously in this sentence it expresses permission rather than ability. – billzt Dec 26 '19 at 11:34
  • How do you conclude that it obviously expresses permission here? I fail to see it. – Kaskade Dec 26 '19 at 11:39
  • The answer is "すみませんが、現金でお願いします". It means paying by card is not permit. – billzt Dec 26 '19 at 11:42
  • It just means that the cashier is asking the customer to pay in cash. The answer alone doesn't contain the reason for the cashier's request. But like I said, in this context, there's not really a difference, anyway... – Kaskade Dec 26 '19 at 11:49
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カードで払うことができますか。 am i able to pay with a credit card (at this store)?

カードで払ってもいいですか。 may i pay with a credit card (at this store)?

I agree with Kaskade's comment. Ability vs Permission. In the case of the credit card example, you are able to use 払うことができますか because the store may not be physically able to process your credit card (lack of a credit card machine). It's not a matter of permission. It's a matter of ability.

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