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On Nicod's reduction in the number of primitives of logic†

Published online by Cambridge University Press:  24 October 2008

B. A. Bernstein
B. A. Bernstein
Affiliation:
University of California
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Extract

Nicod, using Sheffer's operation “│” as a primitive (undefined) idea, obtained a drastic reduction in the number of primitive propositions (postulates) used by Whitehead and Russell in their theory of deduction for “elementary” propositions. In this reduction Nicod accepts Sheffer's replacement of the primitive ideas “∼” and “v” and the primitive propositions *1·7 and *1·71 of the Principia by a single primitive idea and a single primitive proposition, and, by adding two ingeniously devised postulates of his own, derives from the three postulates the Principia's six primitive propositions *1·1, *1·2, *1·3, *1·4, *1·5, *1·6. But the author does not consider the consistency or the independence of his three postulates; does not discuss the Principia's remaining primitives—the ideas “φx” and “⊢. φx” and the propositions *1·11 and *1·72; and fails to prove that his postulates are derivable from those of the Principia.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 4 , October 1932 , pp. 427 - 432
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

‡ Nicod, J. G. P., Proc. Camb. Phil. Soc., 19 (1917), 32–41.Google Scholar

§ Sheffer, H. M., Trans. Amer. Math. Soc., 14 (1913), 481–488.CrossRefGoogle Scholar

∥ Principia Mathematica, Vol. i, pp. 91–97.Google Scholar

The Principia, revised edition, Vol. i, p. xiii, says: “Nicod showed that one primitive proposition could replace the five primitive propositions *1·2·3·4·5·6”. One is a mistake. Nicod used his two postulates II and III in the derivation of the five propositions.

** Nicod does not derive *1·1 explicitly from his postulates, but the derivation is readily obtainable from his postulate II.

†† That Nicod's postulates, when properly interpreted, are facts of logic, can of course be easily verified. But that these facts are derivable from the Principia's theory of deduction requires proof. For there exist facts of logic that are not derivable from the theory of deduction. (See my review of the Principia, , Bull. Amer. Math. Soc., 32 (1926), 711–713Google Scholar, and my paper “On proposition *4·78 of Principia Mathematica”, ibid., 38 (1932), 388–391.

‡‡ The Principia, revised edition, Vol. i, p. xiii, instructs the reader to drop “⊢. øx” and *1·11 from the list of primitives, but does not justify the instruction. Note that Nicod's paper appeared long before the publication of the revised edition of the Principia.

* For a justification of this transformation of Nicod's theory, see my discussion of the theory of deduction in my “Whitehead and Russell's theory of deduction as a mathematical science”, Bull. Amer. Math. Soc., 37 (1931), 480–488.CrossRefGoogle Scholar

† Concretely, “p = u” means “p is a true proposition”, or simply “p is true”.

† In these systems, 0, 1, p′, p + q, pq denote respectively the Boolean zero, whole, negative of p, sum of p and q, product of p and q. In system S 1, the symbol p/q denotes the unique element x such that qx=p. In system S 2, postulate III′ is satisfied vacuously, since the element u of II′ does not exist. In system S 3, the element u of II′ is 1; III′ fails when p=0.

‡ In “Whitehead and Russell's theory of deduction as a mathematical science”, loc. cit.

§ See, for example, the proof of *3·03 in the Principia.

† See, for example, the note following the proof of *2·15 in the Principia. See also Principia, Vol. 1, revised edition, regarding Nicod's “rule of inference”.

‡ In these propositions, I write pq in place of the Principia's “p.q”. In accordance with my discussion of *1·11 above, I use propositions *1·1 and *3·03 in the forms:

*1·1. If ⊢. p and ⊢. p ⊃ q then ⊢. q,

*3·03. If ⊢. p and ⊢. q then ⊢. pq.

* In my review of the Principia, loc. cit., and in my paper on proposition *4·78, loc. cit.

† The inadequacy of Nicod's theory is seen directly from the fact that a class K consisting of a single element e, with e|e = e, satisfies postulates I′–III′, but contradicts the proposition p′ ≠ p, a proposition in the classic logic of propositions. This fact, of course, proves again the insufficiency of the theory of deduction for the logic of propositions.

‡ See my “Sets of postulates for the logic of propositions”, Trans. Amer. Math. Soc., 28 (1926), 472–478Google Scholar. Postulates A–C constitute one of a number of sets of postulates for the Boolean logic of propositions.

§ In a forthcoming paper. Compare the next footnote.

∥ See my “Sets of postulates for the logic of propositions”, loc. cit.

That the theory of deduction can be derived from postulates A–C, or from postulate P, can, of course, also be proved by deriving Nicod's postulates I′–III′ from A–C, or from P. This proof is essentially carried out in verifying that the system S 0, of § 4, satisfies I′–III′.