DEMONSTRATION, AND NECESSARY TRUTHS. 187 



CHAPTER VI. 



THE SAME SUBJECT CONTINUED. 



§ 1. In the examination which formed the subject of the last chapter, 

 into the nature of the evidence of those deductive sciences which are com- 

 monly represented to be systems of necessary truth, we have been led to 

 the following conclusions. The results of those sciences are indeed nec- 

 essary, in the sense of necessarily following from certain first principles, 

 commonly called axioms and definitions; that is, of being certainly true 

 if those axioms and definitions are so ; for the word necessity, even in this 

 acceptation of it, means no more than certainty. But their claim to the 

 character of necessity in any sense beyond this, as implying an evidence 

 independent of and superior to observation and experience, must depend 

 on the previous establishment of such a claim in favor of the definitions 

 and axioms themselves. With regard to axioms, we found that, consid- 

 ered as experimental truths, they rest on superabundant and obvious ev- 

 idence. We inquired, whether, since this is the case, it be imperative to 

 suppose any other evidence of those truths than experimental evidence, any 

 other origin for our belief of them than an experimental origin. We de- 

 cided, that the burden of proof lies with those who maintain the affirma- 

 tive, and we examined, at considerable length, such arguments as they have 

 produced. The examination having led to the rejection of those arguments, 

 we have thought ourselves warranted in concluding that axioms are but a 

 class, the most universal class, of inductions from experience; the simplest 

 and easiest cases of generalization from the facts furnished to us by our 

 senses or by our internal consciousness. 



While the axioms of demonstrative sciences thus appeared to be exper- 

 imental truths, the definitions, as they are incorrectly called, in those sci- 

 ences, were found by us to be generalizations from experience which are 

 not even, accurately speaking, truths ; being propositions in which, while 

 we assert of some kind of object, some property or properties which ob- 

 servation shows to belong to it, we at the same time deny that it possesses 

 any other properties, though in truth other properties do in every individ- 

 ual instance accompany, and in almost all instances modify, the property 

 thus exclusively predicated. The denial, therefore, is a mere fiction, or sup- 

 position, made for the purpose of excluding the consideration of those mod- 

 ifying circumstances, when their influence is of too trifling amount to be 

 worth considering, or adjourning it, when important to a more convenient 

 moment. _«^ 



From these considerations it would appear that Deductive or Demon- • 

 strative Sciences are all, without exception. Inductive Sciences ; that their 

 evidence is that of experience ; but that they are also, in virtue of the pe- 

 culiar character of one indispensable portion of the general formulae ac- 

 cording to which their inductions are made. Hypothetical Sciences. Their 

 conclusions are only true on certain suppositions, which are, or ought to / 

 be, approximations to the truth, but are seldom, if ever, exactly true; and J 

 to this hypothetical character is to be ascribed the pecuUar certainty, which 

 is supposed to be inherent in demonstration. 



