Can has the following forms: can for the present tense and could for the past tense. Could is used in two ways: (a) in past-time contexts and (b) in present-time contexts to express unreality or as a milder and more polite form of can or as a form implying more uncertainty than can:
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He could speak English as a child.
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He could speak English if necessary.
Could I help you?
Could it be true?
The phrase to be able may be used to supply the missing forms of can:
When will you be able to pick her up?
I haven’t been able to sleep recently.
Meanings of can:
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1) mental or physical ability * He can do sums in her head.
2) possibility * He can easily carry the child.
2. 3) permission * You can go now, Marry.
asking for permission * Can I have another ice-cream?
4) prohibition * You can’t cross the street there.
5) request * Can you give me a lift?
6) disapproval, reproach * You could have sent me a word, couldn’t you?
3. 7) strong doubt * Can/ could they be waiting for us?
8) very strong doubt, improbability * She can’t have said it.
4. The modal verb CAN may be used to make a sentence more emotional. Such sentences have the form questions, but are not questions as the speaker doesn’t expect an answer. * Oh, how can you?
CAN/COULD is used in special questions for emotional colouring, for ex., to express puzzlement, impatience, etc.
NOTES
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In the meaning of ability and possibility CAN occurs in all kind of sentences, but followed only by the indefinite infinitive: *She can play the piano.
The form could sometimes is the past of can: *When we went into the house we could smell burning.
We use could to say that someone had the general ability to do sth: *My grandfather could speak 5 languages.
But if you mean that someone managed to do sth in a particular situation, you have to use was/were able to (not could): *The fire spread throughout the building very quickly, but everyone was able (=managed) to escape.
The negative couldn’t is possible in all situations: *My grandfather couldn’t swim.
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The form could + the indefinite infinitive may be used in present time contexts to express unreality with reference to the present or future: * Now she could do the work properly. (=могла бы)
In combination with the perfect infinitive could indicates that the action was not carried out in the past: *She could have done the work properly.
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CAN expresses permission in affirmative sentences. We use interrogative sentences to ask for permission or express a request and negative sentences to express prohibition.
In these meanings CAN is followed by the indefinite infinitive.
Could makes the sentence more polite.
Can’t, couldn’t make the sentence more emotional and insistent:
*Can/could you say it again? *Can/can’t/couldn’t I come too?
4. The meaning of uncertainty, doubt is found only in interrogative sentences (=general questions) ant the meaning of improbability only in negative sentences. CAN is followed by all forms of the infinitive.
If the reference is made to the present, the indefinite infinitive is found (with verbs that cannot have the continuous form) or the continuous form of the infinitive:
*Can he be really ill?
Сf: *Can he write poetry? (ability) *Can he be writing poetry? (doubt)
The perfect infinitive refers the situation to the past.
Could makes the sentence a little more emphatic, it implies more uncertainty.
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To express doubt about improbability of an action not taking place, we use the expression to fail to do, or the negative prefixes mis- and dis- :
*Can she have failed to see him? *She can’t have failed to see him. *Can she dislike me?