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Most Korean learners will learn the 아/어야 하다/되다/지/겠다 family of constructions fairly early on. Usually these end a phrase and simply mean that something must or should be done:

저는 밥을 먹어야 해요 - I have to eat
대학교에 가고 싶다면 열심히 공부해야 돼요 - If you want to go to University, you must study hard
무엇을 사 드릴까 결정해야갰어요 - We’ll have to decide what we’re going to get for him

Where I get confused is with 'conjunctive' uses of 야, which make me wonder what '야' means in general.

This instance seems easy enough to understand:

먹어야 삽니다 - to live, you must eat.

We can still see that what's on the left side of 야 (먹다) is something that should be done, so this is similar to the non-conjunctive examples above.

However, in cases like:

바보가 아니고야 왜 그 말도 모르니? - how do you not know that word unless you’re an idiot?

뛰어 봐야 이미 늦은걸요 - you can run around all you like; it’s too late now.

It's hard to see how '야' has the meaning of 'must'. Is there any way to rearrange those English translations so that I can see that '야' has a meaning of 'must'? Or does it actually mean something different here?

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I would try to see only if in the first type of use and even if in the second, as follows.

먹어야 삽니다
Only if you ate, could you live.

해야 한다.
It would be OK only if you did.

Only if of course can be related to a must. IF only if A then B, THEN whoever wants B must A.

As for the second type,

뛰어 봐야 늦었다
Even if we ran, we'd be late.

The scheme does not give you exactly what you wanted, i.e. a reduction of the usages to must, but one to if.

One of the examples does not appear to accommodate either only if or even if that well, but just a plain if seems to work:

바보가 아니고야 왜 그 말도 모르니?
If you are not a fool, why don't you know what that means?

Compare the sense of if in something like, 'The advice was good if unsolicited.' There it means something like although. What I am trying to suggest is that these critical connecting words can have subtly different senses in different contexts.

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The second definition of -어야:

  1. (뒤에 오는 부정어와 함께 쓰여) 아무리 가정하여도 영향이 없음을 나타내는 연결 어미.

    • 네가 아무리 울어야 소용이 없다. It doesn't matter how much you cry.

In the construction "A어야 B", if the following phrase B is a negative one, like 없다, 모르다, etc, it indicates that no matter how much you suppose A, B is always going to happen.

-고야 is probably -고 나서야 for short.

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