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I am familiar with indexing and co-indexing NP's but I've come across indexing that looks like this:

"Gregory(i) seems to enjoy (ti) Marvel Films." Does the (ti) indicate some kind of movement? Thanks!

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In transformation-based syntax theories, it is indeed assumed that there is a kind of movement involved.

to seem is a so-called raising-to-subject verb:

Semantically, Gregory (which is the subject of the main sentence) is also the subject of the verb phrase "to enjoy Marvel films".

It is therefore assumed that the underlying representation of the sentence is

It seems that [Gregory enjoys Marvel films]

and the surface form

Gregory_i seems to [t_i enjoy Marvel Films]

has been obtained by moving the NP Gregory out of the subject position of the embedded clause ("that...") to the subject position of the matrix clause. Hence, it leaves a trace t_i which we is conindexed with its surface position.

The verb to seem allows to raise the subject NP of the embedded sentence out to matrix sentence's subject position, where it replaces the dummy-subject it which is semantically empty and only served to occupy the subject position in the underlying structure (since in English, the subject of a declarative sentence may not be phonetically empty).

There are also raising-to-object verbs:

In

Gregory believes Henry_i to [t_i enjoy Marvel Films]

the NP Henry has been moved ("raised") from the subject position of the embedded clause to the object position of the matrix clause:

Gregory believes that [Henry enjoys Marvel Films]


And then there are so-called equi verbs, again coming in the two shapes:

In subject-oriented equi verbs, like to try,

Gregory_i tried to [t_i watch a Marvel film]

The subject of the matrix clause is simultaneously the subject of the embedded clause. Reconstructing the deep structure sounds a bit bumpy in that case:

Gregory tried that [Gregory watches a Marvel film]

The important thing is that while this reconstruction may not sound very natural, it's clear that semantically, Gregory is indeeed the subject to the watches as film verb phrase.

In contrast to raising verbs, Gregory is also semantically the subject of to try: to try requires an argument for the one who actively does the trying, while for to seem it is assumed that it is actually subject-less - there is not really anyone who does the seeming -, and the subject position is only filled syntactically (by a dummy-it or a raised subject).
This is different from equi verbs, which have in their lexical entry an argument position for a semantic subject (e.g. someone who tries).

And in object-oriented equi verbs, like to persuade,

Gregory persuaded Henry_i to [t_i watch a Marvel film]

the object of the matrix clause is underlyingly the subject of the embedded clause:

Gregory persuaded Henry that [Henry watches a Marvel film]


Not all syntactic theories assume that an actual movement operation took place; non-transformation based approaches will nevertheless often account in some way for the semantic coreference between the respective subjects or objects; for example, HPSG would use conindexed variables in the subcategorization information of the verb (for example, SUBCAT<1,VP[inf,SUBCAT<1>]:2> ... [SOA-ARG 2] in the case of subject-oriented raising verbs).

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    Actually, It seems that [Gregory enjoys Marvel films] isn't the underlying form. It's also a surface form, transformed by Extraposition instead of Raising. The underlying form of both would be the ungrammatical *That [Gregory enjoys Marvel films] seems. That's why seem is said to be an intransitive verb with a sentential subject that requires either Raising or Extraposition to get rid of the heavy subject. Traces are unnecessary; they're basically just pilpul. – jlawler May 6 '19 at 14:20

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