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Published online by Cambridge University Press: 14 March 2022
The controversy which has raged between those who accept Aristotelian Logic and those who reject it, seems to continue. It is, I believe, advisable to consider the arguments on both sides. Modern Logicians for the most part insist that Aristotelian Logic breaks down, i.e. it is not true for all meanings of the terms. On the other hand there are those who insist on the validity of Aristotelian Logic claiming that those values for which it fails are not permissible values. But there is one who stands almost entirely alone in his position—namely Professor H. B. Smith—who maintains that Aristotelian Logic breaks down because and only because its interpretation into Boolean terms has been accomplished by means of a definition that is not forced upon us and which, since it gives up desirable principles, is to be rejected.
1. “Opuscules et Fragments Inedits de Leibniz,” par L. Couturat, Paris, 1903, p. 375.
2. A similar interpretation is made by Whitehead and Russell in the forms A(ab) = (x)φ(X) ⊃ Ψ(y) I(ab) = ∃x. φ(x) ⊃3 Ψ(y) this actually stating that “there exists an X such that (φX) is true.” Cf. also Russell in Mind 14, 1905, p. 398, where Russell clearly states the existential import of particulars. ∃ means “there exists.”
3. Cf. Chapter XV on “Aristotelian Logic” on Boole's “Laws of Thought,” London, 1854, p. 226.
4. Formal Logic, A. DeMorgan, 1847, p. 111.
5. Ibid., p. 111.
6. Smith, Primer of Logic, (1917), Appendix II.
7. Formal Logic, J. N. Keynes, 1887, p. 137 ff.
8. Ibid., p. 155, and see below.
9. Venn attempted to symbolize I by XY = o which meant X and Y exists and refers to Leibniz A.B. ens. “Symbolic Logic,” London, 1894, p. 184. Venn also recognizes the breakdown of subalternation, but says that I follows from A only on the hypothesis that the subject of A exists. It is interesting to note that in 1879 Venn maintained a different position “When ... we say that ‘All A is B,‘ do we imply the existence of A and B? Certainly we do....
It is clear therefore that what we do is to take a license or make a convention for convenience sake. ...
In an affirmative proposition the subject and predicate distinctly imply the existence of these objects, but as we must appeal to experience to make sure of the existence of their contradictories, we have no right to contraposit such a proposition without due inquiry“ (Mind, IV, 1879, p. 41).
10. Cf. also the interpretation of A. Padoa in Rev. de Meta. et de Mor., V 20, 1912, p. 64 ff.
11. Despite McColl's insistence that the null-class is excluded from every class, and hence cannot be included in any class but itself. Cf. Symbolic Logic, London, 1906.
12. To show that even today the same argument in almost the same words persists, reference can be made to Lewis and Langford, “Symbolic Logic,” 1932, pp. 51–52 and 62, and in the new work of Korzybski, “Science and Sanity,” Lancaster, Pa., 1932, pp. 141–42.
13. How natural the existential interpretation is and how clearly it follows Leibniz can be seen by reference to Couturat “Des Propositions Particuleres” in Rev. de Met. et de Mor., 21, 1913, p. 256.
14. Is it not possible that the whole argument began as a result of a misunderstanding of the mediaeval distinction between “rational” and “natural” existence? The scorn with which moderns treat the mediaevalists has perhaps obscured the real origin of the problem. Certainly if one wishes to consider the problem from the point of view of Ontology, such a distinction must be made (and is made even today).
15. Cf. also the appendix I to Dr. Smith's “First Book in Logic” 2nd edition (1933) with the new chapters on Logic as the Art of Symbols. We do not here reproduce the consistency proof because it can easily be consulted in the original text.
16. A concrete example will show that there are expedient acts which are neither “just” nor “unjust.”
17. Cf. appendix to 2nd edition “First Book in Logic” and “Symbolic Logic.”
18. E. J. Nelson, “Square of Opposition,” Monist, April 1932, p. 271.
19. Formal Logic, p. 143 ff.
20. Ibid., p. 144.
