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Contributions to a Mathematical Analysis of the English Verb-phrase 1

  • Joachim Lambek (a1)

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Ideally, the grammar of a language should tell us whkh sequences of words form sentences and which sequences don’t. It is doubtful whether any existing grammar achieves this end for one of the natural languages, such as English, Russian or Chinese. However, logicians have constructed certain artificial languages (e. g. the prepositional calculus) together with a set of rules which distinguish between sentences (or well-formed formulas, as logicians prefer to call them) and non-sentences. While it is premature to attempt a complete description of English, partial grammars have been constructed.

In this paper we shall consider a fragment of English containing the names John and Jane, the verbs must, work, call, have, be, the adverb today, the conjunctions but and while and a few other words of the same types. We also admit inflected forms such as works, worked, working, etc. We shall attempt to-decide which sequences of these word-forms are sentences and which are not. Our grammar, if successful, must tell us that “John must have been calling Jane today” is a sentence, and that “John has must call Jane” is not a sentence. However, we may as well admit that many sentences will escape our net, owing to the fact that certain constructions, e. g. the gerund, will not be considered here.

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1.

The present article arose from a short series of lectures given at the Summer Institute of Linguistics at the University of Montreal in 1956. Many of the ideas in it became crystallized during a correspondence between the author and N. Chomsky.

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2. Chomsky, N., Syntactic Structures. The Hague, 1957.

3. Ajdukiewicz, K., “Die syntaktische Konnexität”, Studia Philosophica, vol. I (1935): 127.

4. Bar-Hillel, Y., “A quasiarithmetical notation for syntactic description”, Language, vol. 29 (1953): 4758.

5. Lambek, J., “The mathematics of sentence structure”, American, Mathematical Monthly, vol. 65 (1958): 154170.

6. The non-mathematical reader is advised against inventing similar rules. Many plausible rules are in fact false. For example, the following are not valid: (x/y)/z → x/(y/z), (x/y)\z → x/(y\z), xy → yx, z → (z/y)y.

7. It is only fair to warn the reader that some experts in this field do not believe that the program envisaged here can be extended successfully beyond a small fragment of English. Thus Chomsky believes that only a small number of basic sentences in a language should be analyzed in this way and that other sentences may be obtained from them by certain transformations.

8. Furthermore, the passive auxiliary be may often be replaced by get, and the progressive auxiliary be seems to represent a large class of verbs, including at first sight start, begin, keep, continue, stop and finish.

9. This is not quite correct if we consider having in “having worked, John rested” as a participle.

10. “As regards being of type p/(q/n), while I have never heard John has been being called, John must be being called, these expressions become acceptable sentences if the first occurrence of be in each is replaced by stop or the second occurrence of be is replaced by pet”.

1. The present article arose from a short series of lectures given at the Summer Institute of Linguistics at the University of Montreal in 1956. Many of the ideas in it became crystallized during a correspondence between the author and N. Chomsky.

Contributions to a Mathematical Analysis of the English Verb-phrase 1

  • Joachim Lambek (a1)

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