INTRODUCTION. 15 



plication, or the integral calculus rests upon the obser- 

 vation and remembrance of the results of the differential 

 calculus, so induction requires a prior knowledge of 

 deduction. An inverse process is the undoing of the 

 direct process. A person who enters a maze must either 

 trust to chance to lead him out again, or he must carefully 

 notice the road by which he entered. The facts furnished 

 to us by experience are a maze of particular results ; we 

 might by chance observe in them the fulfilment of a law, 

 but this is scarcely possible, unless we thoroughly learn 

 the effects which would attach to any particular law. 



Accordingly, the importance of deductive reasoning is 

 doubly supreme. Even when we gain the results of in- 

 duction they would be of little or no use without we 

 could deductively apply them. But before we can gain 

 them at all we must understand deduction, since it is the 

 inversion of deduction which constitutes induction. Our 

 first task then, in this work, must be to trace out fully the 

 nature of identity in all its forms of occurrence. Having 

 given any series of propositions we must be prepared to 

 develop the whole meaning embodied in them, and the 

 whole of the consequences which flow from them. 



Symbolic Expression of Logical Inference. 



In developing the results of the Principle of Inference 

 we require to use an appropriate language of signs. It 

 would indeed be quite possible to explain the processes of 

 reasoning merely by the use of words found in the ordinary 

 grammar and dictionary. Special examples of reasoning, 

 too, may seem to be more readily apprehended than general 

 and symbolic forms. But it has been abundantly proved 

 in the mathematical sciences that the attainment of truth 

 depends greatly upon the invention of a clear, brief, and 

 appropriate system of symbols. Not only is such a 



