It is too hard to understand modal verbs because different sites say different things, or maybe they say the same things but using different terms.
Here is what I think, but I am not sure I am right.
According to this site:
1-1 We use the negative "can’t or cannot" to show that something is impossible:
That can’t be true.
You cannot be serious.
1-2 We use "couldn’t/could not" to talk about the past:
We knew it could not be true.
He was obviously joking. He could not be serious.
According to this site,
2- "can't have + Ved" expresses impossibility in the past. "can't have + Ved" is the opposite of "must have + V-ed"
She can't have stayed at home. (it is impossible that she stayed at home)
She must have stayed at home. (it is highly possible (95-99%) that she stayed at home)
The question is that:
Are (2) & (1-2) the same?
Is "She can't have stayed at home."="She couldn't stay at home"?
This site says:
3- Couldn't have + past participle means that something wasn't possible in the past, even if you had wanted to do it. This structure is used hypothetically, to talk about things that didn't really happen in the past.
He couldn't have passed the exam, even if he had studied harder. It's a really, really difficult exam. (we just gave this hypothesis, 95% sure but not 100% sure)
Are (3) & (1-2) the same?
"He couldn't have passed the exam" vs "He couldn't pass the exam"?
I would think "He couldn't have passed the exam" is a hypothetical statement while "He couldn't pass the exam" is a fact in the past.
This site says
4- Couldn't have + past participle can be used in unreal condition.
He couldn't have passed the exam if he hadn't looked at your notes
This site says:
5-1 couldn't (for general ability)
My grandfather couldn't swim.
5-2 couldn't (for specific ability)
I couldn't open the window.
The question is:
is (5-2) (inability) the same as (1-2) (impossible)?
This is what I think but I am not sure I am right!!