of the Primitive P7'opositions of Logic 41 



This uncertainty as to the status of S and is not without its 

 effect upoii the proofs. Consider, for instance, Th. 3. In the proof, 

 "1°: p true. By Axiom 3, pAp false" will be seen to require p Sp, 

 concerning the origin of which, and the relation it has to p D j^ 

 (Th. 4), which it indirectly serves to prove, Mr Van Horn says 

 nothing. 



(/3) In his extensive use of the Principle of Excluded Middle, 

 Mr Van Horn makes no explicit mention of the last steps, that 

 lead from pOq, ^^ pDq, to q. These steps would seem to require 

 several propositions: (1) those carrying us from ^^pyp to qvq 

 — " Summation," plus " Permutation," presumably — and (2) " Tau- 

 tology " qv q .D . q. As Mr Van Horn uses the principle of 

 Excluded Middle in this particular way in the first formal proof 

 given — that of Th. 3 — both the principle itself and the proposi- 

 tions required for its use ought, I think, to be deduced immediately 

 from Axiom 3 ; and I do not see how this is possible. 



