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I'm trying to understand the meanings of must with a lexical verb in present continuous. Usually "must" means the judgement of the speaker (epistemic modality).

Can it have a non-epistemic meaning (obligation or necessity) with a lexical verb in present continuous? For example,

It’s 10.30, he must be running now, why isn't he here?

Is this a correct sentence?

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No. Idiomatically, he must be VERBing is epistemic; for a deontic meaning you should use he is supposed to be VERBing. – StoneyB Jun 18 '13 at 18:26
He has to be running now, by the rules. He's not allowed to walk longer than 5 minutes or he's disqualified. – John Lawler Jun 18 '13 at 18:43
sen, what John said, but there is to add that epistemic form of descriptions, like here's the case, is misleading, as it does not correlate their ontological and semantic meaning. AIWK fallacies could happen. – user19148 Jun 18 '13 at 20:33

No, this sentence isn't correct in the obligatory modality. It is correct in the epistemic modality, and could mean:

  • It's 10:30, and he is late, so I suppose that he is running to get here.
  • It's 10:30, which is the time that he usually runs, so I suppose he is running now.

Both of these interpretations are a little unusual, but they are valid sentences. However, in the obligatory modality, the sentence must be intended to mean:

  • It's 10:30, which is the time that he is required to run.

In this case, there are two better ways of writing the sentence:

  • It's 10:30, he should be running now, why isn't he here?
  • It's 10:30, he's supposed to be running now, why isn't he here?

However, "must" can be used with the present continuous in the obligatory sense, when speaking directly to the runner about the future: "If you want to get there at 10:35, you must be running at 10:30."

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