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On the Logic of Imperatives

Published online by Cambridge University Press:  14 March 2022

Albert Hofstadter  and
J. C. C. McKinsey
Albert Hofstadter
Affiliation:
New York University
J. C. C. McKinsey
Affiliation:
New York University
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Extract

It is the purpose of this paper to carry out a partial syntactical analysis of imperatives. Imperatives form a large body of linguistic expressions, appearing, e.g. in mathematical proofs (“Let f(x) be a continuous function!”), laws, moral injunctions, instruction, etc. For analytical purposes we distinguish between two forms of imperatives, the fiat and the directive. By a directive we mean an imperative which includes an indication of the agent who is to carry it out. For example, “Henry, don't forget to stop at the grocery!” is a directive. By a fiat we mean an imperative which includes no reference to an agent who is to carry it out. For example, “Let there be light!” is a fiat. This is a distinction made in terms of meaning. If, however, proper symbolic devices were introduced for the formalization of imperatives, it could be made in a syntactical manner. Thus we could distinguish between imperatives which possess a certain operator—the directive operator formed by putting a name within square brackets—as “[Henry] (Let it be the case that Henry does not forget to stop at the grocery)!”, and imperatives which do not possess this operator, e.g. “Let it be the case that Henry does not forget to stop at the grocery!” In the following we pay attention only to fiats. This, of course, involves a great limitation of subject matter and excludes topics of great interest.

Type
Research Article
Information
Philosophy of Science , Volume 6 , Issue 4 , October 1939 , pp. 446 - 457
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

1

Received November 7, 1938.

References

Notes

2 Since this paper was written, Jørgen Jørgenson has remarked (see “Imperatives and logic”, Erkenntnis, Band 7, Heft 4, pp. 288-296) that “there seems to be no reason for, indeed hardly any possibility of, constructing a specific ‘logic of imperatives‘“. We are of the opinion, that Jørgenson made this statement largely because he had not taken account of the fact, that more than a single verb in an imperative can be in the imperative mood. Thus, for example, one often hears conjunctive commands such as: “Close the door, and open the window!” We believe it is reasonable, and possible, to provide formal rules for the manipulation of such complex imperatives.

3 These restrictions (of the definitions and numerical expressions of Language IC) are made, not out of any sort of necessity, but just to simplify the subsequent discussion. It would be easy enough to give a material interpretation, for example, of the bounded K-operator standing before a sentence which contained occurrences of “!”, but we have felt that the generality to be gained in such a way would hardly be worth the technical complexities thus introduced.

4 We have not investigated the question, whether these primitive sentences could be weakened, or reduced in number.

5 If we had chosen to add imperatives to a different language than Language I, we should have different syntactical terms at hand. If, for example, the initial syntax contained P-terms, we could define the corresponding P-terms for commands.

6 Since this paper was written, Karl Menger has published a paper (“A logic of the doubtful. On optative and imperative logic,” Reports of a Mathematical Colloquium, Second Series, Issue I, pp. 53-64) in which he maintains that there ought not to be so close a correlation between imperatives and declarative sentences. He also seems to believe, that a three-valued logic is necessary, in order to give an adequate representation of imperatives. We do not feel that this is the case, since we believe that the classification of imperatives into analytic, contradictory, and synthetic satisfies much the same purpose.

Our treatment differs further from Menger's, in that Menger treats an imperative as a certain sort of declarative sentence, whereas we treat imperatives as distinct from declarative sentences.