LAST EFFORTS OF LOGISTICIANS. 193 



tinguishable from the left ; it will be sufficient in fact to 

 give the number 2« - i to the right boot of the «''' 

 pair, and the number 2« to the left boot of the ;/* 

 pair. But it will not be so if the right boot is similar 

 to the left, because such an operation then becomes 

 impossible ; unless we admit Zermelo's axiom, since 

 in that case we can select at random from each pair 

 the boot we regard as the right. 



XL 



Conclusions. 



A demonstration really based upon the principles of 

 Analytical Logic will be composed of a succession of 

 propositions ; some, which will serve as premises, will 

 be identities or definitions ; others will be deduced 

 from the former step by step ; but although the con- 

 nexion between each proposition and the succeeding 

 proposition can be grasped immediately, it is not 

 obvious at a glance how it has been possible to pass 

 from the first to the last, which we may be tempted 

 to look upon as a new truth. But if we replace 

 successively the various expressions that are used by 

 their definitions, and if we pursue this operation to the 

 furthest possible limit, there will be nothing left at the 

 end but identities, so that all will be reduced to one 

 immense tautology. Logic therefore remains barren, 

 unless it is fertilized by intuition. 



This is what I wrote formerly. The logisticians 

 assert the contrary, and imagine that they have proved 

 it by effectively demonstrating new truths. But what 

 mechanism have they used ? 



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