of the Primitive Propositions of Logic 35 



This leads us to substitute p \ r/q for 'p \ q/q in the " left-hand 

 sides " of both the non-formal rule of implication and the syllo- 

 gistic proposition (2) above. The reform may be further extended 

 to the proposition (2) as a whole, which might be given the form 

 P ! S/Q instead of P \ Q/Q, with the proviso, if the proposition is to 

 remain true, that *S' must be implied in P. Now, for S, write the 

 pioposition (1) above, p\p/p ; for (as we at this early stage know 

 " unofficially ") a true proposition will be implied by everything. 



We then have the three primitive propositions of the stroke- 

 system : 



( I. If p is an elementary proposition, and q is an 

 Non- elementary proposition, then p\q is an elementary pro- 

 formal 1 position f. 



\ II. If J) [ r/q is true, and p is true, then q is true. 



This is the non-formal rule of implication, *1'1, with the modifi- 

 cation just explained. 



Formal III. p j q/r \t\t/t.\. s/q [p/s. 



I shall call II " the Rule," and III " the Prop." 



Remarks on these Primitive Propositions. 



Observe p r/q in II, while p | q/r in III. This alternance will 

 prove essential for the working of the calculus. 



In III, I shall use ir for 1 1 t/t, P for p j q/r, Q for s/q \p/s, and 

 shall speak of III as P \ tt/Q. 



P I ir/Q, by the Rule, yields the same result as the syllogistic 

 proposition (2) above, when the left-hand side P is a truth of 

 logic. This restriction of the syllogistic form to its categorical 

 use with an asserted premiss is a peculiar character of the first 

 proofs to follow, and is of some philosophical interest. 



One feels inclined to think that III merely asserts together 

 (1) and (2) above. This view, whatever may be the amount of 

 truth it contains, takes AND too much as a matter of course, 

 and tends to lose sight of (a) the fact that III, as a structui;^^s 

 simpler than (2) alone : for III is (2) with t \ t/t instead of s/q \p/s ; 

 and (y8) the very real step from p .q to q, together with the philo- 

 sophical difference between two assertions and only one. 



The main steps in the formal deduction are : 



1. Proof of " Identity," t \ t/t. 



2. Passage from P \ ir/Q to the u sual implicative form P [ Q/Q. 



3. Elimination of the twist s/q \p/s in Q, and return to the 

 normal order q/s \p/s. 



t This is the proposition shown by Sheffer to imply the analogous propositions 

 *1*7 and *1-71 in Principia. 



3—2 



