40 Mr Nicod, A Reduction in the nmnher 



Devi. : Taut. SylL; then, Perm., Taut., and SylL, or S'. 



Reciprocal theorem by Add, ^-^ instead of Taut. 



^ . g . D . ~ {"^p V '^q) and reciprocal theorem. 

 That is, p .q .D . p/q \ p/q. 



Dem. : Id,, Def, of ~, preceding theorem, and Syll. 

 Reciprocal theorem in the same manner. 



Appendix, 



After the substance of this paper had been written, I was 

 given the opportunity of seeing Mr Van Horn's very interesting 

 and original paper dealing with what is practically the same 

 subject, Mr Van Horn recognises clearly the superiority of what 

 has been called above the Oi^-form over the j4iVD-form chosen 

 in Sheffer's text. This deserves the more notice, as Mr Van 

 Horn, I understand, had not Sheffer's article at hand in the time 

 he was writing his own paper. His A, as will be seen from the 

 definitions he gives, is indistinguishable from |. I was much 

 attracted by the harmonious character of Mr Van Horn's third 

 Axiom. It seems to me therefore all the more desirable that 

 certain objections, which Mr Van Horn's proofs in their present 

 form naturally suggest to the reader, should be dealt M'ith, 



(a) It is not quite plain to me whether " of the same truth- 

 value " (say S for short), " of opposite truth-values " (say 0), are 

 used as indefinables, or as abbreviations. If the former, we have 

 no right to go, e.g., from p q, and '^p, to q, etc., without some 

 axiom to that effect, connecting and S with A, If, on the 

 other hand, S and are abbreviations — as it seems to me they 

 are — the two parts of Axiom 3 stand for not less than four 

 propositions : 



1, If jj and q, '^{pAq). 



2. If (^p and ~(/, pAq. 



3. If p and ^q, pAq. 



4, If ^p and q, pAq. 



We cannot assert the first two, or the last two, or all four, 

 propositions together, because we should then need p . q . D . p, 

 p . q . D . q, before we could make any use of such a synthetic 

 Axiom, 



