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In X bar theory, the first auxiliary is the head of a declarative clause:

(1) She will have finished it by tomorrow.

Here, subject she is a specifier, and verb phrase have finished it by tomorrow is the complement of the head will.

The first auxiliary remains the head of a corresponding interrogative clause through head movement:

(2) Will she have finished it by tomorrow?

Here, is the remainder she have finished it by tomorrow the complement of the head will?

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+50

The simple answer to the question is as follows: Yes, the complement of an auxiliary verb in a traditional X-bar-theoretic approach does view the entire string following the inverted auxiliary as the complement of the auxiliary. That this is so can be seen by examining an example. The next tree is taken from Haegeman (1991: 109). Haegeman’s book was written at a point in time when the influence of X-bar Theory was perhaps at its greatest.

enter image description here

This tree shows what is known as “head-to-head movement”. The auxiliary will moves out of the I head position and into the C head position. The entire IP is then the complement of the C head.

Taking Haegeman’s example as guidance, the X-bar-theoretic analysis of the sentence in the question would be as follows:

enter image description here

The same sort of movement occurs, although some sort of category adjustment might be necessary to accommodate the presence of the additional auxiliary, i.e. of have.

One thing to be aware of in all this is that X-bar Theory as understood here is something that has influenced the development of syntactic theory a lot, but many syntacticians today do not assume X-bar structures. Most modern approaches in the Chomskyan tradition assume some sort of reduced version of the traditional X-bar schema.

Consider further in this regard that what appears to be a straightforward and simple question about X-bar Theory, i.e. the current question, has gone unanswered in this forum for a few days. The fact that a bounty has been necessary to receive a response suggests to me that the sway of X-bar Theory in the field of syntax is by now quite minor.

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  • Thanks for the awesome answer. I understand that many syntacticians today, including even Chomsky himself, do not assume X-bar theory. But does that mean that many syntacticians today would not view the entire string following the inverted auxiliary as the complement of the auxiliary in the OP's example? In fact, I can't think of any such modern syntacticians, let alone many. I mean even if the question were not about X-bar theory but about any modern analysis, wouldn't the answer have been the same: "Yes"?
    – JK2
    Sep 4 '21 at 7:49
  • Certainly not! In my preferred framework for syntax, a version of dependency grammar (DG), the structure is flat after the auxiliary. The subject nominal and non-finite verb are sibling dependents. I suggest starting another question that asks this precise question, e.g. "Do all frameworks of syntax view the string following an inverted auxiliary verb in English as the complement of the auxiliary?" I can then provide illustration and discussion about an alternative way to view such structures. Sep 4 '21 at 10:05
  • Thanks. I just did: linguistics.stackexchange.com/q/42857/3380
    – JK2
    Sep 4 '21 at 10:27
  • I upvoted your answer and agree with everything except "The fact that a bounty has been necessary to receive a response suggests to me" - I think this question is so trivial and the answer can be easily found in any intro textbook, like Haegeman.
    – Alex B.
    Sep 4 '21 at 15:32
  • @Alex B. If the question were indeed so trivial, why didn't someone answer it before I did? A sentence or two in a comment would have taken care of the matter. And if one were willing to put those two sentences in a brief answer, one could have harvested some easy points. Sep 5 '21 at 4:10

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