A modal does one of two jobs. It says how obliged somebody is, or it says how sure you are, and he must be tired and he must leave now share a word and nothing else at all.
English gives you no signal for which job is running. There is one usable clue and it is worth carrying: what follows the modal. An action after it tends to be obligation, and a state after it tends to be a deduction, which is why you must go is an instruction and you must be joking is a conclusion drawn from the evidence in the room.
Eight modal contrasts that change the meaning
- must, have to and should. Obligation from the speaker, obligation from outside, and advice. I must stop smoking is a decision; I have to wear a tie is somebody else's rule.
- he must be tired. The same word does deduction as well as obligation, and the deduction sense is the commoner one in conversation.
- you must not and you do not have to are opposites, not variants. The first forbids, the second releases, and confusing them reverses the instruction.
- could, may and might, expressing possibility with decreasing confidence, and may also asks permission.
- can and be able to. Can has no infinitive and no future of its own, so I will be able to fills the gap.
- he cannot have seen me. Deduction about the past, and the negative of must have, which learners rarely produce.
- would you mind, followed by an ing form, the politest ordinary request in English, and answered confusingly: no means yes.
- I should have called. Modal plus have plus participle is how English expresses a past that did not happen, and it is where regret and criticism live.
The negative flips to a different word
The clearest proof that these are two systems is what happens when you make one negative. Obligation negates as you would expect: you must go becomes you mustn't go for a prohibition or you don't have to go for a released obligation, and those two are opposites of each other rather than versions of the same thing.
Deduction does not negate that way. The opposite of he must be at home is he can't be at home. Mustn't is unavailable here, and a sentence like he mustn't be at home will be read as a prohibition on his being there, which is not what the speaker meant and is close to comic. Somebody who has learned only the obligation system produces exactly that sentence, because within that system it is the regular form.
They have no other forms, so the past belongs to another word
A modal takes no -s, no -ing, no infinitive and no past ending. Musted does not exist and never did. What that means practically is that each modal borrows its past from somewhere else, and which word it borrows depends on which of the two jobs it was doing.
An obligation in the past is had to: I had to leave early. A deduction about the past is must have: I must have left it at home. One word in the present, two unrelated words in the past, split along the same line as everything else on this page. A learner who has met only the obligation half will produce yesterday I must go, which is the regular past of a verb that has no regular past.
Where it stops: the deduction half never arrives
Coursebooks introduce ability, obligation and permission early because they are easy to set exercises for, and they leave the deduction meanings until an upper level that most learners never reach in a classroom. The result is a stall that shows up in listening rather than in speaking. You will be understood without ever producing a deduction modal in your life. You will not understand that'll be the postman, he can't have finished already or she should be there by now, and all three are ordinary English spoken at ordinary speed.
It stalls there rather than anywhere else because nothing in a conversation exposes it. A missed deduction sounds like a sentence you half heard, so it gets filed under speed or accent, and the gap that produced it is never located.
Where the two senses are written down separately
Oxford Learner's Dictionaries splits them for you. Look up must and the obligation sense and the deduction sense are separate numbered entries with their own examples, and the same is true of can't, should and will. It is free with no account, and it replaced the bilingual dictionary entry, which gives one gloss for must, almost always the obligation one, and stops there.
A dictionary can only take you to the edge of this. It lists the senses a word has; it cannot tell you which one a speaker just used, because that is settled by the rest of the sentence rather than by the word. The reading test above is faster than looking anything up, and it is right most of the time.
What it looks like when it goes wrong
Three failures are recognisable. I don't must, from treating a modal as an ordinary verb. Yesterday I must go, from a past that was never there. And an answer to you must be tired that treats it as an order, which is the deduction gap arriving in a real conversation and is the only one of the three that will confuse the other person.
The forms these words stand beside are covered in which form of the verb, and what each one says about time, and the same small set carries the result clause in if, would, and the four ways English talks about what might happen, both under the rules that decide what a sentence is allowed to mean.
Since this is a gap that shows itself in what you fail to hear rather than in what you say wrong, the useful next move is not more grammar. It is finding out how much of what you are missing is this and how much is speed: real speed, real accents, no subtitles.