36 Mr Nicod, A Reduction in the numher 



4. Proof of " Association," p \ q/r .D.q [p/s. 



5. Theorems equivalent to the definitions of p . q, p q in 

 Principia. 



Proof of Identity, t\t\t. 



As this first proof from a single formal premiss stands in a 

 unique position, I shall, without in any way obscuring the precise 

 play of the symbols, expound it after a more heuristic order than 

 is usually followed. 



We start with the Prop. P | tt | Q, and the Rule enabling us to 

 pass from the truth of P to that of Q ; and we have to prove tt. 

 This can only be reached through some proposition of the form 

 -4 1 5 1 TT, where A is a truth of logic f. The proof will thus consist 

 in passing from P | tt | Q to J. 1 5 | tt by some permutative process. 



A simple two-terms permutative law s 1 5' | ^ | 5, we do not yet 

 possess. Our Prop, yields only a roundabout three-terms per- 

 mutation, slglpjs, subject to the condition of ^jglr being a 

 truth of logic f. This, however, is enough for our purpose. 



In the Prop., write t. for p, q, r : 



(a) 7r|7r!Qi, 



Qi being s|^|^|s. Write now tt for p, q; Q^ for r: then by (a) 

 and the Rule, 



(b) S ! TT I TT I s. 



From (b), in the same manner. 



(c) u I tt/s I s/tt j u. 



This enables us to pass, by the Rule, from P | tt | Q to 



(d) Q|7r|P. 



In order to complete the proof of tt, we need only find some 

 expression which : (a) can be a value for P, i.e. is a case of p\q\ r, 

 and (/3) is implied in some truth of logic, say T. For, by T'lP | P, 

 the Prop., and the Rule, as above, 



(e) s\P[T\~s. 



In (e), write Q | tt for s: first by (d) and the Rule, then by T 

 and the Rule, we obtain T\Q\7r, and so 



(/) 



t This use of the Rule by anticipation, with still undetermined P's and Q's, is 

 in truth contrary to the nature of a non-formal rule, which must never be used to 

 build up the structure of an argument. It must always be possible to dispense 

 with all such ' anticipated ' assertions in the final form of a proof. This will be 

 seen to be very easy in the present case. 



