184 Proceedings of the Roi/al Irish Academij. 



goes along, without inquiring whether they are consistent or superfluous. 

 This does not prevent the argument from being logical, though a finished 

 logical system aims at the artistic — though not always useful — ideal of 

 stating the minima of premisses required to prove a given set of con- 

 clusions. 



6. In all trains of reasoning, formulaj have to be remembered ; and thus 

 all human reasoning is subject to error. Good mathematicians often mistake 

 vivid memory for direct intmtion of fact, 



II. 



Logic, of course, cannot be defined without a circle, but its salient features 

 may be pointed out. It consists, as I understand it, in drawing conclusions 

 from the combination or synthesis of any number of premisses without 

 explicit reference to the question whether the premisses are exemplifiable. 

 The terms used in the premisses must, for the most part, refer to classes, i.e., 

 have a universal significance — otherwise the science would be useless. Any 

 representation may he used in logic if it can he universalized. The Aristotelian 

 theory recognizes this partly ; and modern logicians have wasted energy in 

 jeering at the traditional syllogism. The main differences between the new 

 and the old logic are, first, that the concept of Eelation is now introduced ; 

 secondly, that not only terms, but arbitrarily chosen types of inference, if not 

 self-contradictory, may be assumed as irreducible ideas ; thirdly, that a term 

 may be defined as a constituent of a proposition whose other elements are 

 known. This is an extension of the notion of predicational definition. 

 Fourthly, it is recognized that the indefinable or irreducible terms used 

 must be explicitly stated. All these developments leave us still within the 

 sphere of logic, because they treat everything from the universal point of 

 view. 



III. 



Certain misunderstandings exist as to the logical view of mathematics ; 

 and these were started, I think, by Leibniz, who, in some of his writings, 

 appears to claim that the Law of Contradiction is the only principle assumed 

 in mathematics. Practically, however, his real doctrine on the subject is 

 inconsistent with this view. It is obvious that the Law of Contradiction 

 cannot give premisses ; and it cannot be used without first assuming certain 

 fundamental ideas, as class, term, proposition, inference, and so forth. This 

 is true even of Aristotle's logic. Even a single syllogism involves more 

 than the Law of Contradiction : it is a synthesis of propositions. 



