Consider:
He might go to Beijing last month.
He might have gone to Beijing last month.
He could have gone to Beijing last month.
Any difference in meaning?
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He might go to Beijing last month.
He might have gone to Beijing last month.
He could have gone to Beijing last month.
Any difference in meaning?
He might go to Beijing.
Expresses a possible future action. Therefore the use of last month is impossible. You must use a time expression that goes with the future, for example:
He might go to Beijing next month.
He might have gone to Beijing last month.
He could have gone to Beijing last month.
Both express a possible situation in the past.
→ He might have gone to Beijing last month.
According to context could mean:
There's a possibility he went to Beijing last month, I'm not sure perhaps he did something else.
I know he went some place last month, I don't exactly remember where, it could be Beijing. (sentence stress on "to Beijing" when spoken)
I know he went to Beijing sometime or other, it could be last month but I'm not sure. (sentence stress on "last month" when spoken)
→ He could have gone to Beijing last month.
There was a possibility for him to go to Beijing last month but he didn't go.
He might go to Beijing last month.
This employs might as the past form of may. The expression last month (or week or year or Thursday) can only have present reference, so this sentence would normally occur only in reported speech (“indirect discourse”) where you are paraphrasing a statement ‘he’ made last month about his intentions then:
He said "I may go to Beijing" last month. =
He said last month he might go to Beijing.
He might have gone to Beijing last month.
He could have gone to Beijing last month.
Both of these employ the “pseudo-perfect” construction used to express the past tense of modal verbs when the ordinary past form signifies unreality rather than past tense. This construction is employed both in reported speech and in hypothetical statements about events which did not in fact occur.
He says that he might have gone to Beijing last month, but he wasn’t sure; he is going to check his appointment book.
He could have gone to Beijing last month, but a colleague got sick and he had to stay at his desk to cover the extra work.
The difference between can/could and may/might in present-day English† is the difference between ability and possibility.
I can go to Beijing any time I want to, and I may go next week. Or I may not; it depends on whether my wife can get time off from her job.
†As recently as fifty years ago may/might was also contrasted with can/could in formal usage as denoting permission rather than ability. But this distinction has been steadily declining for two hundred years, and today can/could is used freely to express permission.
Not correct grammar because it mixes future (might) and past (last month)
and 3. "might" versus "could"... "Could" implies "capability" - the ability to do something. Might, on the other hand, implies simply whether something can happen or not, regardless of whether a person makes it happen, or it happens to them outside their control, or whatever
So #2 means perhaps he went to Beijing last month, but we don't know. Maybe he went. Maybe he didn't.
And #3 means he was capable of making a trip to Beijing last month. It says and implies nothing about whether he did or not.