Hi Appendix. 



Xlll 



transcend the region of the Probable — the term probable being here employed, 

 not in the more restricted sense of the popular usage, but in the wider 

 acceptation whereby it includes moral certainty. That all matter throughout 

 the universe gravitates ; that when the Sun shall rise to-morrow, his disc will 

 be circular ; that of the liquid in an unknown river, the main ingredients are 

 oxygen and hydrogen ; that no human being now alive will survive till the 

 year 2073 : all these are propositions as to wliich it may be presumed that no 

 person would entertain any doubt; but yet we may distinguish degrees of 

 diflerence in our assurance of them. They are results of accumulated evidence ; 

 and " probable evidence," as Bishop Butler justly observes, is that which 

 *' admits of degrees, and of all variety of them from the highest moral 

 certainty to the very lowest presumption." 



It is, then, an essential characteristic of inductive inference, that it does 

 not lead to absolute knowledge, though it can attain to what is practically 

 equivalent — moral cei'tainty. This essential characterestic, however, has been, 

 strange to say, ignored in the discussion of some of the fundamental questions 

 pertaining to the philosophy of Induction ; and yet, by due attention to it, we 

 may obtain, I am persuaded, the solution of that peculiar and various amount 

 of unsatisfactory paradox which still disfigures the basis of this department of 

 sciences. 



Inductive inference, as we have seen, conducts to general truths. "We 

 now, therefore, proceed to ask, are all general truths inductive? In other 

 words, are there any universal propositions wliich we know to be true, and 

 which are such that our knowledge, whether of those propositions themselves, 

 or of those from which they are deduced, is not dependent upon the fact of a 

 verifying test having been applied in a sufficiency of individual instances 1 



Let us take the arithmetical proposition, the square of 3 plus the square 

 of 4 is the square of 5. This proposition is unquestionably true. And it is a 

 general or univei'sal proposition; for the assertion virtually is, that in any 

 instance whatever of there being an aggregate of things the number of which 

 is the square of 3, and also an aggregate of things the number of which is the 

 square of 4, then in the sum of the two aggregates the number of those things 

 is the square of 5. Now, what is the foundation of our certainty that this 

 proposition is true 1 It may well be that none of us has ever formally tested 

 the truth of the proposition by actual exi)eriment ; and yet none of us doubts 

 it. It is deduced from other propositions ; and we are now to see whether those 

 others are inductive. The expression, the square of 3, means 3 taken 3 times ; 

 and so also as to 4 and 5. But what do we mean by these words, three, four, 

 five, etc.*? As the word two simply means 1 + 1, so the word three means 

 2 + 1 ; and so on until we have defined ten. And, further, the word eleven 

 means 10 + 1 ; the word twelve means 10 + 2 ; and so on until we have defined 



