34 Mr Nicod, A Reduction in the number 



Let us, however, take Mr Russell's eight propositions in the 

 form given in Principia. It is my object to reduce them to three 

 — two non-formal and one formal — by means of the stroke-defi- 

 nitions given above. 



It can be shown, as a first stage, that two formal propositions 

 are enough, namely : 



(1) p\l)/p. 



(2) p\q/q\s/q\^. 



The first proposition is the form of " Identity " (p D p) in the 

 stroke-system. It would, at first sight, appear more natural to 

 adopt the order q/s \ p/s in the left-hand side of (2), since 



p\qlq-'^-qls\p/s 



is the syllogistic principle of the stroke-system, giving " Syllogism," 

 pD q .D : q D s . D .pDs when s | s is written for s. 



It will however be found that the inverted order, s/q 1 p/s, is 

 much more advantageous than the normal syllogistic order, 

 q/s \p/s. For, owing to this " twist," Identity and (2) yield 

 " Permutation," s/p \ p/s, which now enables us to eliminate the 

 twist in (2), and revert to the normal order. From the three 

 propositions thus obtained, the rest follow. 



This, by the way, illustrates the following fundamental fact. 

 Which form of a given principle is the most general, and contains 

 the maximum assertion, is a function of the symbolic system used. 

 Thus, for instance, in Mr Russell's system, 



p .D . qwp (a) 

 is more general than p .0 . qD p (b) 



since (h) is (a) with <^q for q. In the stroke-system, on the 

 contrary, p \ q/q \ p/p, meaning the same thing as (a), is less general 



than p\ q \p/p, whose meaning is that of (b), since it is obtained 

 from it by writing q\q for q. 



A further step has to be made in order to be left with only one 

 formal primitive proposition. It consists in adapting to better 

 advantage the form of the primitive propositions to the properties 

 of the stroke-symbolism where implication is concerned. We had 

 above 



p'^q . = .p\ q/q Df 



If we look for the meaning of the general form p \ r/q, we find this 

 to be oo 29 V ~ (~ r V ~ 5'), i.e. p .D . r .q. We thus come to the 

 fundamental property that, in the new system, p"^ q is a case of 

 p .D . s . q, whereas in Principia the contrary relation of course 

 holds, 



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