Can

followed by the non-perfect common aspect infinitive expresses:

Physical and mental ability or capacity

John can keep a secret if he wants to (can = to be capable of).

Possibility.

possibility due to circumstances

Anybody can make a mistake. Ошибаться может каждый

possibility due to the existing rules of laws:

In old days a man could be sentenced to death for a small crime. В старые времена можно было приговорить человека к смерти за небольшое преступление.

possibility of the idea (the so-called “theoretical” possibility):

The railways can be improved. (It is possible for the railways to be improved, as they are not yet perfect.)

Permission.

Can we go home, Miss? Можно идти домой, мисс?

Prohibition. It corresponds to the Russian нельзя, не надо.

You can’t cross the street here. Здесь нельзя переходить улицу.

Request.

Can I have some water?

set expressions

Cannot/can’t help doing smth. - He могу не делать что-то

When I saw him I couldn’t help laughing. - Когда я увидел его, я не мог не засмеяться.

Cannot/can’t but do smth. - не могу не ...

We cannot but hope he is right. - Нам остается только надеяться, что... (не можем не надеяться...)

One cannot but wonder - нельзя не задуматься

As can be - an intensifying expression

They are as pleased as can be. - Они очень (страшно) довольны.

Как произносить CAN и CAN'T
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followed by any form of the infinitive express

Strong doubt, improbability, incredulity

This meaning occurs only with the negative form of the modal verb + perfect infinitive, continuous infinitive, or be

He can’t have seen it (it’s impossible that he saw it).
- He может быть, чтобы он видел это.

Could is used instead of can to express greater doubt. Thus the difference between can and could is in the degree of expressiveness, could showing a greater degree of doubt or incredulity.

He Can’t \ Couldn’t be so old.
- He может быть, что он так стар.

Surprise, when can/could is used in questions. It corresponds to the Russian неужели

Can it be so late as all that? Неужели уже так поздно?
Therefore the Russian negative questions of the type - нeyжeли он не ... is translated into English in different ways:

a) by complex sentences:

Can if be that you haven’t seen him?
Неужели вы не видели его?

b) by different lexical means:

Can you have failed to see him?
Неужели вы не видели его?
Can you dislike the book?
Неужели вам не нравится эта книга?

. Reproach, implying that a person should have done something, or behaved in a certain way, but didn’t do it. This meaning is found only with the form could.

You could at least have met me at the station, couldn’t you?

Purpose. This meaning occurs only with the form could in clauses of purpose.

I wrote down the telephone number so that I could remember it.

The action was possible but not carried out.

She could have married anybody she chose but she stayed single.

Модальные глаголы в английском языке: can, can't, could, to be able to (видео с youtube)