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I'm trying to write a syntax tree structure of a sentence with the word 'should'. It's a long sentence, so to simply, let's take this sentence instead:

I should play the piano.

I'm not sure whether 'should' is the tense marker in the sentence. To demonstrate (surface structure):

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My argument is that in the sentence I played the piano, it's clear that the T has merged with the V.... Thanks!

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Ask your teacher what they want you to call it; should is only a tense marker if your teacher says it is. For everybody else, should is a modal auxiliary verb, which, like all modals, is tenseless (though it is always the first auxiliary and thus should have tense if there were any) and takes an infinitive complement with Equi. –  jlawler Feb 19 '13 at 22:27
    
Thank you very much. –  pauliwago Feb 19 '13 at 23:04

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According to what I have learnt, I wouldn't say that should is a tense, but I would say that it carries a past tense inflection. And about your idea of T merging with V, I would suggest considering the following. Basically, there is a hypothesis saying that TPs are headed by a T constituent, either overt or null. Merging occurs, but that wouldn't fully explain what happens to T. In your fist example, the head of T is the modal auxiliary verb, but in the finite clause, the T would have a null phonetic spell out of a tense affix which also carries person and number features. The idea is that heads cannot be empty, and in this case the tense affix undergoes a morphological operation called affix hopping. Thus, in the sentence I played the piano, the T head would be something like Tense-3SgPr and it would be lowered onto the head V play. The derived structure would be [play+Affix-3SgPr], which in turn would be spelled out as played. See Radford (2006) for a more detailed explanation.

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Surely "I should play piano" carries a future tense inflection? "I should have played the piano" would be past tense, no? –  naught101 May 24 at 9:09

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