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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!!

  • Believe it or not, Stack Exchange is not a good place for learning in general, including learning the English language in general, or even for learning about modals in general. This is not a teaching service and it is not set up to be one. – AmE speaker Feb 6 '17 at 14:25
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Multiple questions here so I'll try to address all of them.

Are (2) & (1-2) the same?

Is "She can't have stayed at home."="She couldn't stay at home"?

The difference between these two statements is the first implies that it was impossible for her to have stayed at home, eg. "She was in New York on business, so she can't have stayed at home".

The second statement is a bit more nuanced, it depends on context. The way I read it is that, whilst it may not be logistically impossible for her to stay at home, the current situation would not warrant her staying at home, eg. "She was not on good terms with her family, so she couldn't stay at home".

Are (3) & (1-2) the same?

"He couldn't have passed the exam" vs "He couldn't pass the exam"?

Again, the first statement implies that he couldn't have possibly passed the exam. eg. "He didn't attend the exam, so he couldn't have passed the exam". or, like the example in your question "He couldn't possibly have passed the exam without cheating".

The second statement implies that he attempted the exam but didn't manage to pass.

is (5-2) (inability) the same as (1-2) (impossible)?

This is contextual.

"Couldn't pass the exam" "Couldn't open the window" "Couldn't swim"

Are all the same thing, implying that they tried to pass the exam / open the window / swim but couldn't.

However:

"She couldn't stay at home" "He couldn't tell her the truth"

These types of statements would usually imply some sort of emotional reasoning rather than lack of ability, you'd need more context to understand the reasoning.

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