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Published online by Cambridge University Press: 14 March 2022
The purpose of this note is to present a set of rules for the syllogism which not only is equivalent with the set ordinarily used, but also is the dual of the latter. It must be emphasized, however, that the discussion of both of these sets presupposes the hypothetical interpretation of universal propositions, and would not hold true of the existential interpretation of such propositions. A universal proposition is interpreted hypothetically, rather than existentially, when it is not assumed that the class denoted by its subject must have members. Thus the hypothetical interpretation of “All unicorns are mammals” would be just “If anything is a unicorn, then it is a mammal—but it is not necessarily true that unicorns exist.” One important consequence of the distinction between these two modes of interpretation is that rule 3, below, is true for the hypothetical interpretation, but not for the existential.
For advice on the organization of this paper, I am indebted to Professor C. West Churchman, to the referee, and to Professor William Craig of The Pennsylvania State University. Professor Haskell B. Curry of The Pennsylvania State University has also assisted me on some details.
2 Rule 2 is often given in the stronger form, “One premise is negative if and only if the conclusion is negative”; but when rule 3 is assumed as above, the converse of rule 2 can be proved. See the proof of rule 3′, below.
3 See, for example, J. N. Keynes, Studies and Exercises in Formal Logic, Fourth Edition, 1906, pp. 289–90.
4 There are, of course, other senses of “independence” in which the rules are not independent. See, for example, Keynes, op. cit., pp. 291–295.
5 For all the special rules of the figures, see Keynes, op. cit., pp. 309–313.