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I am not sure whether participle or gerund of modal verbs is possible in english. For example we have the sentence: Having introduced the speaker, the chairman opened the meeting. What would be the correct form when using the modal verb "must"? Is it "having had to introduce the speaker…"?

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You have it exactly right. The full modals (can/could,may/might,must,shall/should,will/would) are 'defective': they have no non-finite forms (participles and infinitives) and do not inflect for person and number. That's one reason why the 'periphrastic modals' like HAVE to, BE going to, BE able to have increased in use over the last couple of centuries.

  • +1 :) But, what does 'periphrastic' mean (in this context)? – Araucaria Apr 26 '15 at 12:24
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    @Araucaria: Presumably, the same as it would in any other context (circumlocutory, using roundabout phrasing). Thus, in English, the past tense is conveyed using inflection, but the future can only be indicated by periphrasis or idiomatically established context (contextual I am going to die someday, as opposed to periphrastic I will die). – FumbleFingers Apr 26 '15 at 13:24
  • @FumbleFingers Thanks, suddenly realised I've been reading that for ages but didn't really know exactly what it meant ... – Araucaria Apr 26 '15 at 19:55
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    @Araucaria Most of what schoolteachers call 'tenses' in English - progr - essive, passive, perfect - .are periphrastic. It's the old-fashioned term for what is now usually expressed by the synthetic/analytic contrast. And the periphrastic modals are those which express the same sense as the ordinary modals by periphrasis. – StoneyB Apr 26 '15 at 21:15

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