# Government and Binding Theory/X-bar Theory

The X-bar theory is one of the core modules of GB theory, and as such, it will be the first we learn. Before we embark on our X-bar journey, though, it is necessary to look at why we need X-bar in the first place.

## Subcategorisation

In our introductory book on linguistics, we have devised a mini-grammar using rewrite rules:
* A mini-grammar of English
* NP → {PN, Pr, Det (Adj) N}
* VP → V (NP) (Adv)
* S → NP (Aux) VP
* PN → {Chomsky, Jackendoff, Pinker}
* Det → {this, that, the, a, my, some}
* Adj → {happy, green, lucky, colourless}
* N → {computer, book, homework idea}
* V → {defended, attacked, do, eat, slept, poisoned}
* Adv → {furiously, happily, noisily}
* Aux → {will, may, might do}
However, our mini-grammar can be very problematic. Refer to this sentence:
(1) *Chomsky slept a computer.
This sentence is syntactically unsound because slept is an intransitive verb. Whether a verb is transitive or intransitive depends, of course, on the verb. Thus we may conclude that a subcategorisation frame in the lexicon tells us whether a verb is transitive.
* Subcategorisation frames in verbs
* jump [__ Ø]
* eat [__ (NP)]
* wait [__ (PP$_{for}$)]
jump is intransitive, so nothing goes after it. eat can be both transitive or intranstive, so we added an optional (NP). wait can take an indirect object, so we added a prepositional phrase.
This phenomenon is not restricted to VPs by any means. We can similarly construct subcategorisation frames for APs:
* Subcategorisation frames in adjectives
* doubtful [__ (PP$_{of}$]
* satisfied [__ (PP$_{with}$)]
* obvious [__ (PP$_{tp}$)]
Now we encounter a dilemma. We need both the subcategorisation and phrase structure rules to generate sentences, but is it not redundant to have a rewrite rule that says VP → V (NP) (Adv) and individual subcategorisation rules? To solve this problem, linguists have proposed the X-bar theory.

## Projection

Instead of having both phrase structure and subcategorisation rules, linguists have suggested that the structure of a phrase is derived from the lexicon by a process called projection. When we take a lexical constituent and plug it into our structure, it will take the subcategorisation information with it to project a structure.
To understand this concept, we first need to know that there are only two phrase structure rules in X-bar theory. (Yes, all our old ones will be discarded. In a moment, we'll get a boot to kick them away.)
* Phrase structure rules in X-bar theory
* X′ → X YP
* X″ → spec X′
X is known as the head of the phrase, and is always a lexical category. Every phrase is named after the head. X itself is known as zero projection and can be written as X$^0$.
YP is the complement of the phrase. It is always a phrase. It is selected by the head with lexical information, i.e. it is subcategorised by the head.. X′ is known as intermediate projection.
X″ is just another way of writing XP, and spec is the specifier of the phrase. X″ is the maximal projection.
Let's return to the cheesy sentence we discussed in the introductory book. This time, we include the auxiliary in our verb phrase as the specifier of the VP:
Let's do the same for an NP. Since we're obsessed with homework, our NP will be the completion of homework:
Just to prove that it always works, let's draw bang on time (which, presumably, is how you will hand in your homework):
Cool with that? Good. Lastly, let's do the same with an AP, really obsessed with homework:
This gives rise to our first principle, in the words of Chomsky:
* Projection Principle
* Representations at each level of syntax are projected from the lexicon in that they observe the subcategorisation properties of lexical items.

## Binary branching revisited

Recall that in our introductory book, we met the Binary Branching Condition. Now we can make sense of this condition!
* Binary Branching Condition
* Each node must have at most two branches.
Terminal nodes have no branches at all. X″ without a specifier and X′ without a complement have one branch. X″ with a specifier and X′ with a complement have two branches. We cannot get any more.
Note that every phrase must have a head.

## Phrase structure rules get the boot

Seeing the great generalisation powers of X-bar theory, we can get rid of phrase structure rules now. Just for fun, we will use a boot from the 1600s:
You may be thinking that our booting is quite premature because our new rewrite rule would result in nonsensical phrases:
(2a) *the computer Chomsky (2b) *happy the idea
Our response to that is, well, deal with it. There is another module of GB theory related to this phenomenon, and we'll come back to it soon enough. For now, though, we need to deal with a rather more pressing problem. What the heck do we do with the residue of our old phrase structure days... the sentence? The rewrite rule is still here, and it is not in X-bar format:
(3) S → NP VP
What do we do with it?