Remark on Nicod's Reduction of Principia MathematicaAuthor(s): B. A. BernsteinSource: The Journal of Symbolic Logic, Vol. 2, No. 4 (Dec., 1937), pp. 165-166Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268282 .

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Tuz JogsNAL or Smouc Lowc Vohume 2, Number 4, December 1937

REMARK ON NICOD'S REDUCTION OF PRINCIPIA MATHEMATICA

B. A. BERNSTEIN

The object of this note is to close a slight gap, hitherto seemingly over- looked, in Nicod's derivation from his postulates' of the primitives of Principia mathematica. The gap consists in this: Nicod's propositions corresponding to * 1- 1- * 1.- 62 of the Principia, though they bear the same names and have the same symbol-forms as the Principia propositions, are not precisely * 1 -1- * 1* 6. Indeed, if we denote Nicod's "p 2 q" and "p . q" by "p < q" and "pq" respec- tively, and if we write p' for p1 p and p" for (p')', then in terms of the "stroke," we have

D. p <q. =.pjq D2. pq.- = . (pq)'.

But p 2 q and p . q of the Principia are

1.01. q P q =p'I I' 3*01. p . q. = * (P"1 q")'.

Accordingly, the Nicod-Principia propositions are not *1 1- * 1 6. They are * 1-1- * 1 6 in which < takes the place of z, and Nicod has not shown us explicitly how to pass from < to D.

It is true that Nicod also proves the following:

T1. 1-. (P1 q') I (p"I q') 1 - (f'f q') 1 (P Iq')', i.e.,

:p <q. <.p q, H :p q. <.p <q. This yields: If H . p < q then H . p n q, and conversely. This would seem to allow passing from < to n throughout a proposition. But, it is clear, T1 permits this change only in the case of the principal <. The change in the case of a minor < still has to be justified.

However, the passing from < to 2 throughout Nicod's propositions can be easily effected with the help of the auxiliary propositions (a)-(f) following. These propositions are all "rules."3

Received July 24, 1937. Presented to the American Mathematical Society April 3, 1937. lJ. G. P. Nicod, Proceedings of the Cambridge Philosophical Society, vol. 19 (1917), pp.

3241. 2 The proposition corresponding to * I I is not given explicitly by Nicod. ' There is no essential difference between a "rule" and a "postulate" in the Principia proposi-

tions * I1 1- * 1 71. The theory of deduction of the Principia seems to contain no "rule" except * I - 1. But * 3 * 03 is a "rule," and the authors might have obtained other "rules" ad libitum. For other "rules" derived from the primitives of the Principia see Proceedings of the Cambridge Philosophical Society, vol. 28 (1932), pp. 427432.

165

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166 B. A. BERNSTEIN

(a) If i . P| q' then H . I"1 q', and conversely. (b) If 1- . p, and H . pi q', then I- . q. (c) If F-. p| q', and F- . qj r', then F- . p r'. (d) If I- (p| q') I r' then F- . (P" q')|r'. (e) If I *p (q r)' then I-. (P |pr)'. (f) If . p (q r)' then F .p(q"Ir)'.

Propositions (a)-(f) follow from Nicod's proposition T, above, together with his Postulate II and his propositions T2, T3 following.

II. If F- .p (rj q), and H . p, then H.q. T2. . (p Iq') I [(q r') I (p Ir')']'. Ts. F-. [pI (qj r)'] 1 [qj (pi r)']'.

PROOFS OF (a)-(e). (a) by T1, II; (b) by II; (c) by T2, II; (d) by T1, (c); (e) by T3, I.

PROOF OF (f). If F- . PI (qI r)', then F- . q I (P| r)', by (e); hence F . q" I (p| r)', by (a); hence F- . P| (q"I r)', by (e).

If now we denote the Nicod correspondents of * 1- 1- * 1 6 by 1 1-1 6 re- spectively, the proofs of * 1 1- * 1 6 follow.

PROOF OF *1 1. Let F- . p and F- . pDq. Then H.p and F p" I q', by 1 01; hence F- . p and F- . p q', by (a); hence F- . q, by (b).

PROOFS OF *1 2-1* I5. 1 2 by 1 2, D1, (a), 10 1; *1 3 by 1 3, D1, (a), 1.01; * I*4 by 1 4, D1, (a), 1 01; * 1 5 by 1 5, D1, (a), 1 01.

PROOF OF *1 6. F-. (qIr')j [(pvq)j(pvr)']',by1 6, D1;hence F. (q"Ir')j [(p v q) i (p v r)']', by (d); hence F- . (q"| r') I [(P v q)"I (p v r)']', by (f); hence F-. (q"I o')"1 [(p v q)" (p v r)']', by (a); hence F :. q r . : p v q . n. p v r, by 1*01.

The revised Principia adopts Nicod's postulates as basis for the theory of deduction. The above discussion applies in essence to this revised theory. The discussion also applies to Jorgensen's exposition' of the Principia's revised theory.

THE UNIVERSITY OF CAIFORNIA

BERKELEY, CALIFORNIA

' J. Jorgensen, A treatise of formal logic, Copenhagen and London 1931, Vol. II, Chap. VII.

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