32 Mr JSicod, A Reduction in the nimiber 



A Reduction in the number of the Primitive Propositions of 

 Logic. By J. G. P. NicOD, Trinity College. (Communicated by 

 Mr G. H. Hardy.) 



[^Received and read 80 October 1916.] 



Of the four elementary truth-functions needed in logic, only 

 two are taken as indefinables in Principia Mathematica. These 

 two have now been defined by Mr Shefferf in terms of a single 

 new function p | q, " p stroke q." I propose to make use of Mr 

 Sheffer's discovery in order to reduce the number of the primitive 

 propositions needed for the logical calculus. 



There are two slightly different forms of the new indefinable, 

 for we may treat 2:)\q as meaning the same thing as either 

 ~jj . ~g, or <^p}/ ^qt- The definition of <^p is the same in 

 both cases, namely p \ p, while that of pv q simply changes from 

 p/q \p/q with the AND-form into p/p \ qjq with the 07^-form. 



However, the best course is for us to define all the four truth- 

 functions directly in terms of the new one. In so doing, we find 

 that, while the definition of ~j9 remains the same, and those of 

 pv q, p . q simply permute, as we pass from the ^iV^D-form to the 

 Oi^-form, the definition of pO q is simpler in the latter form. It 

 is p I qjq, as against j;/j) j q \p/p \ q. 



The OJ?-form is therefore to be preferred §. 



Definitions. 



f^p . = . p\p Df. pvq.^.plpiq/q Df. 



pO q . = . p\ qjq Df. p . q . — . p/q I p/q Df 



Remaeks on these Definitions, 



One ought not to aim at retaining before one's mind the 

 complex translation into the usual system, "-^pv^q" as the 

 "real meaning" of the stroke. For the stroke, in the stroke- 

 system, is simpler than either ~ or v, and fi-om it both of them 

 arise. We may not be able to think otherwise than in terms of 

 the four usual functions ; it will then be more in accordance with 

 the nature of the new system to think of the j , not as some fixed 

 compound of -^ and v, but as a bare structure, out of which, in 

 various ways, ~ and v will grow. 



+ Sheffer, Trans. Amer. Math. Soc. Vol. xiv. pp. 481—488. 

 X Sheffer, loc. cit., footnote f, p. 488. 



% p\q thus corresponds to what is termed the Disjunctive relation in Mr W. E. 

 Johnson's writincrs. 



|i 



