Universal grammar is a theory in linguistics that suggests that there are properties that all possible natural human languages have. Usually credited to Noam Chomsky, the theory suggests that some rules of grammar are hard-wired into the brain, and manifest without being taught.
If humans growing up under normal conditions (not conditions of extreme deprivation) always develop a language with property X (for example, distinguishing nouns from verbs, or distinguishing function words from lexical words) then property X is a property of universal grammar in this most general sense (here not capitalized).
There are theoretical senses of the term Universal Grammar as well (here capitalized). The most general of these would be that Universal Grammar is whatever properties of a normally developing human brain cause it to learn languages that conform to universal grammar (the non-capitalized, pretheoretical sense). Using the above examples, Universal Grammar would be the property that the brain has that causes it to posit a difference between nouns and verbs whenever presented with linguistic data.
As Chomsky puts it, "Evidently, development of language in the individual must involve three factors: (1) genetic endowment, which sets limits on the attainable languages, thereby making language acquisition possible; (2) external data, converted to the experience that selects one or another language within a narrow range; (3) principles not specific to FL."  [FL is the faculty of language, whatever properties of the brain cause it to learn language.] So (1) is Universal Grammar in the first theoretical sense, (2) is the linguistic data which the child is exposed to.
Sometimes aspects of Universal Grammar in this sense seem to be describable in terms of general facts about cognition. For example, if a predisposition to categorize events and objects as different classes of things is part of human cognition, and as a direct result nouns and verbs show up in all languages, then it could be said that this aspect of Universal Grammar is not specific to language, but is part of cognition more generally. To distinguish properties of languages that can be traced to other facts about cognition from properties of languages that cannot, the abbreviation UG* can be used. UG is the term often used by Chomsky for those aspects of the human brain which cause language to be the way it is (i.e. are Universal Grammar in the sense used here) but here for discussion it is used for those aspects which are furthermore specific to language (thus UG, as Chomsky uses it, is just an abbreviation for Universal Grammar, but UG* as used here is a subset of Universal Grammar).
In the same article, Chomsky casts the theme of a larger research program in terms of the following question: "How little can be attributed to UG while still accounting for the variety of I-languages attained, relying on third factor principles?" (I-languages meaning internal languages, the brain states that correspond to knowing how to speak and understand a particular language, and third factor principles meaning (3) in the previous quote).
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