X-bar theory is a component of linguistic theory which attempts to identify syntactic features common to all those human languages that fit in a presupposed (1965) framework. It claims that among their phrasal categories, all those languages share certain structural similarities, including one known as the "X-bar", which does not appear in traditional phrase structure rules for English or other natural languages. X-bar theory was first proposed by Noam Chomsky (1970) and further developed by Ray Jackendoff (1977). An X-bar theoretic understanding of sentence structure is possible in a constituency-based grammar only; it is not possible in a dependency-based grammar.
The letter X is used to signify an arbitrary lexical category (part of speech); when analyzing a specific utterance, specific categories are assigned. Thus, the X may become an N for noun, a V for verb, an A for adjective, or a P for preposition.
The term X-bar is derived from the notation representing this structure. Certain structures are represented by X (an X with a bar over it). Because this is difficult to typeset, this is often written as X′, using the prime symbol. In English, however, this is still read as "X bar". The notation XP stands for X Phrase, and is equivalent to X-bar-bar (X with a double overbar), written X″, usually read aloud as X double bar.
Read more about this on Wikipedia page on X-bar theory.
Watch this video which explains about the linguistic nuances in a detailed yet easy to understand manner.
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