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The Interpretation of Implication

Published online by Cambridge University Press:  14 March 2022

Charles E. Gauss
Charles E. Gauss*
Affiliation:
The Johns Hopkins University
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Extract

When the student of logic is first initiated into the problem of the relation of implication as expressed by the hypothetical proposition, he enters into a portion of logic which seems to him an impenetrable jungle. Elementary statements of the meaning of implication, as he is acquainted with them, are so meagre that they have failed to give any indication of the problems and paradoxes involved. Adequate discussions, on the other hand, are so abstruse that the initiate is lost in their incomprehensibility, for these are issues discussed by the technical logicians largely among themselves. When one considers the important bearings implicit in the interpretation of implication, such a state of affairs is greatly to be deplored. Through a simple but adequate presentation of the meaning of a hypothetical proposition one may be led not only to an understanding of a fundamental paradox of logic but to the whole question of modal logic and of the philosophy of logic. It is my purpose in what follows to present as simply as possible the various interpretations of the meaning of implication and to show their relations to modal logic. Having done this I believe it will be possible to state the case in favor of material implication in a new light.

Type
Research Article
Information
Philosophy of Science , Volume 10 , Issue 2 , April 1943 , pp. 95 - 103
Copyright
Copyright © The Philosophy of Science Association 1943

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References

1 Charles Sanders Peirce, Collected Papers, 3.374.

2 See below.

3 For eight paradoxes of material implication see C. I. Lewis, Survey of Symbolic Logic, pp. 325–326.

4 Cf. Susan K. Langer, Introduction to Symbolic Logic, pp. 276–278.

5 In this distinction between material and strict implication I am following the usual interpretation. Cf. C. West Churchman, Elements of Logic, pp. 21–22; Langer, op. cit., pp. 276–278; Susan Stebbing, A Modern Introduction to Logic, pp. 221–226. Concerning refinements and emendations on the notion of strict implication in its relation to entailment I have not thought necessary to speak here; cf. Lewis, op. cit., pp. 324ff; Paul Weiss, “Entailment and the Future of Logic,” Proceedings of the VII International Congress of Philosophy, pp. 143–150, and “Relativity in Logic,” The Monist, 1928, pp. 536–548.

6 I have met those who would so limit a hypothetical, yet would deny that the proposition, “If all men are bald, some men are bald,” is a genuine hypothetical. They seemed to object to it on the grounds that, the falsity of the antecedent and the truth of the consequent in most cases resulting in a situation in which the implication is neither true nor false, but only neutral, one may seriously doubt whether there is any implication in such cases. The fact that the material was in the logical system of class-inclusiveness did not to them seem to carry much weight. To them an implication depending on class-inclusiveness seemed as fruitless as the statement that “If pigs are pigs, pigs are pigs.”

7 It should, of course, be noted that it is not necessary to choose between these two interpretations only. As was observed above (note 6) the case in which the antecedent is false and the consequent true in a hypothetical proposition may be considered, as frequently seems more correct, to issue in an implication which is neither true, nor false, but only neutral. For instance, it is neither true nor false that if (in the sense of because, entailing a consequent) rocks are liquid, grass is green. The one has nothing to do with the other and the implication is no more false than it is true. Such an interpretation, however, would involve a three-valued logic, which means the denial of the principle of contradiction. Certainly, this offers an interesting way out of the dilemma, but since it would mean a whole review of the question from a newer approach than that which has been taken here, I omit any discussion of it. Besides it would lead us to the same issues which shall be discussed subsequently. Moreover, the quarrel over the proper interpretation for implication arose within the system of a two-valued logic. It is not so important to quarrel over the extension given to the meaning of truth as to determine just what is meant by truth as applied by someone within his system. It may have been observed that I have frequently used “meaning” and “truth” as synonymous as applied to implication. I have asked what is the meaning of a hypothetical, what does it include and exclude, and have answered that it excludes its contradictory, i.e. its contradictory falsifies it. I believe that for logical relations “truth” and “meaning” are synonymous. When I have pointed out the state of affairs that a relation expresses I have said when it is a true relation. Certainly it is false when it falsifies, i.e. does not express, the state of affairs it purports to express. The meaning of “true,” therefore, for the relation of implication is not the same as the meaning of “true” as applied to the truth or falsity of P or Q in the rule given by the material implicationist. This may be clearer after a perusal of the subsequent pages of the article.

8 C. S. Peirce, op. cit., 3.441-444.

9 Cf. Sextus Empiricus, Against the Logicians (Loeb Classical Library) II, 112–124.

10 Cf. E. Zeller, Socrates and the Socratic Schools, p. 261ff; also Aristotle, Metaphysics 1046b, 29ff; and W. D. Ross, Aristotle's Metaphysics, II, 243–244.

11 C. S. Peirce, op. cit., 3.374. Cf. also 3.442 and 3.527.

12 Louis Rougier, “The Relativity of Logic,” Philosophy and Phenomenological Research, December, 1941, p. 151.

13 Cf. Metaphysics, 1019b, 22ff; also 1047b, 3–30. Especially see Ross's commentaries on these passages (particularly the first), op. cit., I, 321, II, 243–245.

14 C. S. Peirce, op. cit., 3.527.