I was wondering what the difference was between a sentence with have and one with do. For example this one:
- They haven't started yet.
- They didn't start yet.
Is there a difference? When would one use the first or the latter?
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Wikipedia has a decent article on past tenses that explains a lot of this. To summarise:
"They didn't start yet" is the negative form of the simple past, "They started." In the positive form it indicates that the activity happened at some definite known point in the past, and is over and done with now. In the negative form this is a more nebulous idea, but in this example it would imply that "they" were supposed to have started at some time. In view of that, "They didn't start yet" sounds odd. "They didn't start" is a complete idea, and "yet" tries to extend it in time in a way that the simple past doesn't allow for.
"They haven't started yet" is the negative of the perfect tense, "They have started." As the article puts it, it is "used for describing a past action's effect on the present." It more or less requires that the idea gets extended in time, which "yet" does nicely.
“…haven't…yet” implies a statement about the present, i.e. what is currently happening. “Didn't” however implies a statement about the past, i.e. what has already happened.
In my view the “didn't” and “yet” in your second example therefore don't fit naturally together.
Note that there are constructions where “didn't” and “yet” would go together, for example “they didn't know yet.” The difference here is that the “yet” is implying some other past event (from the point of view of the writer) rather than something yet to happen.
I hope this is clear enough!
(One comment on your question: “first” and “latter” are not naturally complementary words; I would suggest “former” for the former.)
I don't think "didn't start yet" is really valid English in most contexts. I can't exactly say what it is about the construction that I don't like, but I decided to check written usage by others.
In Google Books I find just 14 hits for "doesn't start yet" and 91 for "didn't start yet", compared to tens of thousands for "haven't started yet", and for "hadn't started yet". Even if someone wants to argue that it's not actually "ungrammatical", doesn't/didn't is certainly avoided by most writers in this context; I suggest OP does the same.