Can someone help me understand why 是 is in this sentence?
I can't figure out what role it plays, and I think I usually see such a sentence without it. Does it form the passive or something? Thanks!
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是 does not mean "is" here. The reason is that the "is" required here is already contained in "解决了". The skeleton of this sentence is "事情解决了", that is, "解决了" is itself a verb and can be translated to "is resolved" or "has been resolved". No one would ever say "事情是解决了" to mean "the problem has been resolved" (for this sentence to make sense it must mean something complete different, I'll.mentioned it later).
In the sentence
BaiduBaike(百度百科) has an explanation and two examples of this usage (see 基本释义-2 in the linked page): 到底 means 经过种种变化或曲折最后出现某种结果. Examples are as follows.
新方法到底试验成功了 = The new method succeeded eventually through the tests.
我想了很久，到底明白了 = I thought about it for a very long time and understood eventually.
Clearly, you can replace 到底 here by 到底是 or 到底还是.
Notice that, the reason I say 是 does not mean "is" here is that, if it did mean "is", then after stripping the original sentence of unnecessary parts, you should still get a correct sentence without altering the basic meaning of the original sentence. But 事情是解决了 is neither natural nor correct, because 解决了 itself already means "is resolved" (yes this happens in Chinese), adding another "is" would be redundant.
This reasoning also suggests if you see 到底是+noun or 到底是+adjective (instead of 到底是+verb, which is the case above), then 是 may actually mean "is" in these cases, because the sentence needs a verb. But you still need to be careful about adjectives or nouns acting as verbs. Some examples:
这个包到底是你的还是他的？ = Is this your bag or his?
Here 是 means "is".
这幅画画的到底 是 什么？ = What exactly is this painting about?
Here 是 also means is.
You may wonder, what does 到底 mean in these cases? When 到底 is used in a question, it means you really want to know the answer, similar to on earth, the heck, exactly in English that people add to questions.
你今天到底是有什么毛病？ = What the heck is wrong with you today?
Here 到底是 goes together and 是 does not mean "is" because 有 is already the verb. A word for word translation is
Exactly what problems do you have today?
Finally let me remark on the sentence 事情是解决了. I said before that it is not a correct sentence to express "the matter has been resolved" because 是would be a redundant verb. But it can be a correct sentence if you interpret 是 as "although", for example,
事情是解决了，但是可能会留下不良影响 = Although the matter has been resolved, there may be some bad repercussions left behind.
I feel that it is a placeholder to fill up the sentence so that 到底 and 解決 won't stick together.
It is also a hybrid between "到底解決了" (which sounds slightly weird) and "到底還是解決了".
Another function would be to emphasize the fact that it is resolved, just like adding an unnecessary "do" before some sentences to emphasize the positive stance, such as "I do want you to resolve it now" rather than "I want you to resolve it now" (sorry, not a very good English example).
According to Chinese dictionary owned by Ministry of Education in Taiwan. 是 can be used when you affirm something or determine something, so 是 show that some status after 是 is used to describe things before 是 in this example. 解决了(solved) used to describe 这件事情(this matter) before 是.
Without the 是, the sentence may be mistaken for a question.
With the 是, it is unmistakably a statement of fact with some tonal emphasis when enunciating the 是 to underline the fact that the matter finally, (到底), is (是), resolved, (解决了)
Perhaps it may sound better to say 是, 这件事情到底解决了 which of course is answering a question, and since the sentence in question is a statement of fact, then, I suppose putting the 是 inside makes sense.