Do I always have to use might when talk about the past and may does not belong there?
I may beat him now / tomorrow.
maymight beat him a year ago.
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The past-tense forms of the modal auxiliaries can, may, shall and will are always used in sentences with past reference. But they are used in different ways. That is because these verbs have dual uses which present a special problem.
Just like the past-tense forms of other verbs, they are used to represent present-tense forms 'backshifted' into the past:
You are going to fight. I think You were going to fight. I thought ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ you can beat him. ⇨ you could beat him. you may beat him. ⇨ you might beat him. ?you shall beat him. ⇨ you should beat him. (this use of 'shall' is dying) you will beat him. ⇨ you would beat him.
But these auxiliaries, unlike ‘lexical’ verbs also employ their past-tense forms with present or future reference. This is particularly visible in so-called 'first conditional' constructions: non-past conditionals in which the condition is regarded as possible:
If the fight is fair, I think If the fight is fair, I think ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ you can beat him. AND you could beat him. you may beat him. AND you might beat him. you shall beat him. AND you should beat him. you will beat him. ? you would beat him. (This one's a bit iffy; but it would certainly be acceptable in a context like "If the contests are fair, I think you would beat him sixty times out of a hundred.")
In many cases, the difference between 'present-reference/present-tense' and 'present-reference/past-tense' use is so slight that the simple past form may be used for either when the sentence is backshifted to past-reference—for instance, in a backshifted ‘first conditional’ (we might call it a ‘one-and-a-halfth conditional’):
Back in '06 I thought that if the fight was fair you could / might / should / would beat him.
In the case of higher-level conditionals, however—what are sometimes called counterfactual or irrealis conditionals, in which the condition is regarded as impossible—this does not suffice. These constructions must be strongly marked with a past-tense form.
When an irrealis sentence has present reference, the so-called ‘second conditional’, this marking is accomplished by casting the verb into a past-tense form undifferentiated for person, what used to be called the past subjunctive:
If the fight were fair, you could / might / should / would beat him.
But now all the verb sequences have been cast into past-tense forms. How, then, do you backshift this sentence into past reference?
The workaround which English has evolved is to mark both sequences with a HAVE + past participle construction. This marker is placed on the finite verb which heads each verb sequence if it is capable of taking it. But modal verbs are 'defective', they have no past participles with which this marker can realized; so the marker is moved to the next verb in the sequence:
PRESENT: If the fight [ were ] fair you [might beat ] him, but it’s not; past ↓ ↳ → marker ↓ (oops!) ↴ PAST: If the fight [had been] fair you [might have beaten] him; but it wasn’t.
The last sentence looks like a perfect construction, but it’s not. It's the form taken by a subjunctive past with past reference. You must be careful to distinguish this from such sentences as:
He may have won; I'm not sure and its past version
He might have won; I wasn't sure.
He may have been cheated; I'm not sure and its past version
He might have been cheated; I wasn't sure.
These are true perfect constructions, with current (past or present) reference, designating his current state of "championship" resulting from his past action winning or of "cheatedness" resulting from somebody's past action of cheating him. They are cast into infinitive form because they are headed by the modal MAY, indicating uncertainty. They are thus equivalent to:
It is possible that he has won ... and
It was possible that he had won.
It is possible that he has been cheated ... and
It was possible that he had been cheated.
Might + have + past participle
to express an action that didn't happen in the past. May is not used in this way.
If I had been as fit as I am today, I might have beaten him a year ago.
A: (in the car) Careful! There's a cat crossing the road!
B: Yes, don't worry. I saw it.
A: No, you didn't. You might have killed it!
B: But I didn't.
to express possibility in the past. Both may and might can be used this way.
A: Where's Derek, he should be here by now?
B: He's always been a late riser, he may / might have missed the train.
A: I had to take the bus to work today.
B: Why? Don't you have a car?
A: Yes, but I couldn't find the keys anywhere.
B: You may / might have left them in the gym locker. You did the last time.
It feels more natural, but I don't think it is a strict requirement.
Counter-example in The Independent:
He may have been coached, but was Sir David Nicholson ready for these Reservoir Dogs?