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I see that somebody must do something usually means that someone is obliged to do something. I also see that you must be kidding me means that it is highly likely that you are kidding me.

But can Alex must be working on the project mean an obligation (e.g. Alex must work on the project but with a sense of continuity) ?

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A frequent use of must is to express deduction, and I would say that that is what it is doing in Alex must be working on the project.

Obligation would be expressed by something like Alex really must work hard on the project (if he wants to get it completed in time).

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  • The intended senses are usually self-evident: I must be leaving (It is imperative that I leave) v You must be the manager (as Barrie says, expressing deduction). However, sometimes context is needed to disambiguate (We must be getting off [now] / We must be getting off [at the next stop]). Apr 5, 2014 at 13:25
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Must is a Modal Auxiliary Verb.
(The other modals in English are may, might, can, could, will, would, shall, and should.
What follows applies to all of them, not just must.)

All modals have two different types of meaning:,

  • one meaning, called the Epistemic sense of the modal,
    deals with possibility and probability
    e.g, (using may and must as examples):
    This may be the place (may = logically possible)
    This may not be the place (may not = possibly not)
    This must be the place (must = logically necessary)
    This must not be the place (must not = logically impossible)

and

  • another meaning, called the Deontic sense of the modal,
    deals with permission and obligation
    e.g,
    You may attend the prom. (may = permitted to)
    You may not attend the prom. (may not = forbidden to)
    You must attend the prom. (must = obliged to)
    You must not attend the prom. (must not = forbidden to)

As you can see, negatives have very complex patterns with modals,
but modals are plenty complex all by themselves.

Deciding which one is intended is a matter of interpretation and context, and certainly active predicates with human agent subjects, like He must do it, are more likely to be deontic, while stative predicates, like He must be late, are more likely to be epistemic.

However, this is not true in all cases; consider for example

  • He must do this every morning. (must = logically necessary)
  • He must be at work by 07:40. (must = obliged to)

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