In the field of natural language semantics, the term fragment usually refers to a limited portion of a language, for which explicit syntactic and semantic rules are given. For example, Montague (1973) opens with the statement "The aim of this paper is to present in a rigorous way the syntax and semantics of a certain fragment of a certain dialect of English."
This use of the term fragment seems obviously to be based on an earlier usage in logic, for a logical system obtained by limiting the language of some other logical system to just those formulae meeting certain syntactic conditions. Often, such fragments are identified in order to show that one system has at least the expressive or deductive power of some other system. For example, by showing that formulae from some modal logic can be systematically translated into corresponding formulae of first-order predicate logic, you can show that the modal logic is equivalent to a specific "fragment" of first-order predicate logic, and therefore has the same expressive power (which might be less than the expressive power of first-order predicate logic as a whole).
I have been unable to identify with certainty when or by whom this use of fragment was first introduced, but an early appearance is in Quine (1940), p. 317. (The relevant passage is in the first edition only, and is omitted from later editions.)
More interesting to me (and I suspect, to many readers) is the question of when and by whom fragment was first used for a formally defined portion of a natural language like English (or its grammar). Until recently, I would have guessed that this originated with Montague — probably Montague (1970), where it appears in the second paragraph. It turns out this is not the case.
I was quite surprised to find that that Noam Chomsky appears to be earliest author to use the term fragment in this way. In 1958, he presented a paper titled 'A Fragment of English Grammar' at the Third Texas Conference on Problems of Linguistic Analysis in English. The conference proceedings were published in 1962; unfortunately, I have not been able to obtain a copy.
Other early (pre-Montague) uses of fragment for natural language may be found in descriptions of a series of research projects conducted by the Data Processing Systems Division of the U.S. National Bureau of Standards. Here is an example from 1960.
- Chomsky (1962) 'A Fragment of English Grammar', in Third Texas Conference on Problems of Linguistic Analysis in English, ed. by Archibald Hill. University of Texas at Austin.
- Montague, Richard (1970) 'English as a Formal Language', Linguaggi nella Società e nella Tecnica, ed. by B. Visenni, et al. Edizioni di Comunità.
- Montague, Richard (1973) 'The Proper Treatment of Quantification in Ordinary English', Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics, ed. by J. Hintikka, et al. D. Reidel.
- National Science Foundation (1960) Current Research and Development in Scientific Documentation, no. 7. NSF-60-65.
- Quine, W.V.O. (1940) Mathematical Logic. W.W. Norton.