KiDD. — Induction a7id Necessary Truth. xliii 



twenty. These definitions being assumed, it is demonstrable by the mere 

 substitution of equivalent names, and so performing the enumeration in the 

 abstract, that the square of 3 is 9, that the square of 4 is 16, that 9 + 16 is 25, 

 and that the square of 5 is the same. But those fundamental assumptions, viz., 

 that two means 1 + 1, that three means 2 + 1-, etc., are not inductive propositions j 

 they are not merely results ascertained from repeated experiment. Our know- 

 ledge of the proposition thdXJive means 4 + 1 is not a moral certainty accruing 

 from accumulated evidence. These propositions are definitions, and nothing 

 more. "We find, therefore, that the proposition, the square of 3 + the square of 

 4 is the square of 5, is a universal proposition, the truth of which we know j 

 and that our knowledge of its truth is not dependent upon Induction. And a 

 similar analysis might be applied to all propositions whatever belonging to the 

 pure science of Number. 



But it may be said : Definitions are simply explanations of words ; and 

 how, then, can deductions from mere definitions demonstrate an objective 

 matter of fact % We must answer that they cannot do so j such a result is 

 impossible. That there are in real existence aggregates of things severally 

 numbering 9, and also aggregates of things severally numbering 16, as to such 

 propositions we have no absolute knowledge further than may be afibrded to 

 us by direct perception of individual things. But we have an absolute 

 knowledge of this, that if there exist any where an aggregate of things whose 

 number is 9, and also an aggregate of things whose number is 16, in that 

 case the sum of those things numbers 25. Such a proposition comprises a 

 condition as to the supposition of real existence j but we know absolutely the 

 truth of the conditional proposition, which truth consists in this, that the 

 predicate is coextensive with the subject. It may be, or may not be, a 

 matter of- fact that there is at the present moment in St. Giles's, London, a 

 mendicant having two pockets, in one of which are four, and in the other 

 three,- penny pieces. But it is a proposition of absolute certainty, that if 

 there be such a personage so circumstanced he has in his pockets coins to the 

 number of seven. All general propositions asserting unconditionally any 

 matter of fact, or any real existence, are, or ought to be, derived from 

 inductive data : but there are also general propositions expressly or virtually 

 hypothetic ; and many of these, in various departments of knowledge, are 

 necessary truths, that is, propositions whose truth is known absolutely, and 

 not merely probable or certain according to the amount of experience. 



We are now in a position enabling us to obtain a more distinct view of 

 what we mean by a necessary truth. A necessary truth is, of course, a species 

 of proposition or assertion : and all propositions, of whatsoever kind they may 

 be, and however much they may vary in other respects, agree in this, that 

 every proposition consists of a subject and a predicate; that is, a subject- 



