of the Primitive Propositions of Logic 33 



The above definitions give clear expression to the symmetiy 

 between OR and AND ; and this, notwithstanding the choice that 

 we had to make between an Oi?-forni, and an AND-iorva. This 

 is of some interest, because, in general, the very symmetry forces 

 upon us an arbitrary choice, which, in turn, quite obscures the 

 symmetry. 



I shall use q for q\q whenever convenient. Observe that 

 p I q, i.e. pD q, forms a natural symbol | for implication, 



allowing of permutation ~q\ p. We may notice in general that 

 the new system brings the four functions into relations far closer 

 than those in Mr Russell's system. For instance, in 



p/p\p/p-\.p/p 

 the two propositions pv p .D . p and r^pv p coincide. 



Every stroke-formula falls into two parts on the right and left 

 of a central stem. It will, therefore, add to clearness to use black 

 type instead of dots to indicate the central symbol. Further, 

 slanting strokes are covered by straight ones : thus p/q j p/q stands 

 for (p\q)\ (pj q). 



The definition of the two primitive notions of the Principia 

 in terms of a single new one tends to reduce the number of the 

 primitive propositions needed. But how far does this reduction 

 actually occur ? Does it extend beyond the obvious substitution 

 of " If p and q are elementary propositions, p\q is an elementary 

 prop." (Sheffer, p. 488) for *r7 and *1'71, stating the same for 

 ~ p and py q respectively ? The reduction goes, as we shall 

 presently find, very much farther. 



It has first to be said, in order that we may be as precise as 

 possible, that the tuhole amount gained in applying the stroke- 

 definitions cannot with complete certainty be attributed to them. 

 For Mr Russell's system, as it now stands, has not said its last 

 word in that matter. 



Incidentally, I found that *1'4, pv q .D . q y p, can be proved 

 by means of the other four, with the unimportant change of *1'3, 

 q . "^ . pv q into q . "^ . q v p. In "Association," *1*5, writers for r : 



p y {q y p) . 1^ . qv ij) V p). 



The left-hand side, by the help of q ."D . qvp and " Summation," 

 will be found to be implied in pv q. The right-hand side, like- 

 wise, hy p V p . D r p, and " Summation," will be found to imply 

 qvj). The result then follows by using "Syllogism" (obtained 



from " Summation " with the transformation — - f) twice. 



p 



P V p' 

 t By - or ^-^ I mean (following Mr Russell) the substitution of p for q or 



p, p' for q, q'. By {e.g.) P~ I mean the result of effecting the substitution in P. 

 VOL. XIX. PT. I, 3 



