- I cannot have decided which I will do.
Do you think that this sentence is correct? If it is wrong, please say why it is wrong. If have can be used after can, what will the form of verb? Can both present participle and past participle be used?
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All the modal verbs (can/could, may/might, must, shall/should, will/would) can take HAVE as a complement, and HAVE may act either as a lexical verb or as a perfect/pastmarking auxiliary.
LEXICAL = "possess": I cannot have everything I want.
LEXICAL = "cause": I must have John read this; he can explain it.
perfect AUXILIARY: I will have finished this by lunchtime.
pastmarking AUXILIARY: I would have told you if I had seen it.
The verbform which complements a modal verb is always an infinitive; no other verbform can follow it. And if the following verb is itself an auxiliary, it will likewise select the verbform which comes next.
Beyond these rules, the use of modal verbs is very complicated and idiosyncratic: each modal verb has a unique portfolio of uses and constraints. For instance, your example is not "ungrammatical", but it is very unlikely:
I cannot have decided which I will do.
If the sentence were in the third person, "He cannot have decided ...", it would be taken to be an epistemic use of can; that is, it would express an inference: "I regard it as impossible that he has decided ..." But in the first person it makes no sense to speak of inference: under ordinary circumstances you yourself know whether or not you have decided. Consequently, this must be taken in a dynamic sense, expressing your ability or opportunity, and it must have future reference: "I cannot have decided by next Tuesday; you must give me more time". But that is more naturally expressed, and with greater clarity, using a simple present construction: "I cannot decide by next Tuesday ...".
Note, moreover, that epistemic can, speaking of inferred possibility, is ordinarily employed only in the negative; the only situation in which we would say "He can have decided ..." to mean "I regard it as possible that he has decided ..." is echoically and emphatically to contradict someone's prior assertion that "He cannot ..."
A: "He cannot have decided yet which he will do."
B: "Oh yes he can have decided; in fact, he has decided."