1)EM0NSTRAT10N AND NECESSARY TRUTHS. 



171 



eluded if we find, and in proportion as 

 we find, the assumptions to be true — 

 may be performed once for all, and 

 the results held ready to be employed 

 as the occasions turn up for use. We 

 thus do all beforehand that can be so 

 done, and leave the least possible work 

 to be performed when cases arise and 

 press for a decision. This inquiry 

 into the inferences which can be drawn 

 from assumptions is what properly 

 constitutes Demonstrative Science. 



It is of course quite as practicable 

 to arrive at new conclusions from facts 

 assumed, as from facts observed ; 

 from fictitious, as from real, induc- 

 tions. Deduction, as we have seen, 

 consists of a series of inferences in 

 this form — a is a mark of 6, 6 of c, c 

 of c?, therefore a is a mark of d, which 

 last may be a truth inaccessible to 

 direct observation. In like manner 

 it is allowable to say, suppose that a 

 were a mark of h, h of c, and c oi d, a 

 would be a mark of rf, which last 

 conclusion was not thought of by 

 those who laid down the premises. 

 A system of propositions as compli- 

 cated as geometry might be deduced 

 from assumptions which are false ; as 

 was done by Ptolemy, Descartes, and 

 others, in their attempts to explain 

 synthetically the phenomena of th' 

 solar system on the supposition that 

 the apparent motions of the heavenly 

 bodies were the real motions, or were 

 produced in some way more or less 

 different from the true one. Some- 

 times the same thing is knowingly 

 done for the purpose of showing the 

 falsity of the assumption ; which is 

 called a reductio ad absurdum. In 

 such cases the reasoning is as fol- 

 lows : a is a mark of 6, and 6 of c ; 

 now if c were also a mark of d, a 

 would be a mark of d ; but d is 

 known to be a mark of the absence of 

 a ; consequently a would be a mark 

 of its own absence, which is a contra- 

 diction ; therefore c is not a mark of d. 



§ 5. It has even been held by some 

 writers that all ratiocination rests in 

 the last resort on a reductio ad ab- 



surdum, since the way to enforce 

 assent to it, in case of obscurity, would 

 be to show that if the conclusion be 

 denied we must deny some one at 

 least of the premises, which, as they 

 are all supposed true, wotild be a con- 

 tradiction. And in accordance with 

 this, many have thought that the 

 peculiar nature of the evidence of 

 ratiocination consisted in the impos- 

 sibility of admitting the premises and 

 rejecting the conclusion without a 

 contradiction in terms. This theory, 

 however, is admissible as an explana- 

 tion of the grounds on which ratiocina- 

 tion itself rests. If any one denies 

 the conclusion notwithstanding his 

 admission of the premises, he is not 

 involved in any direct and express 

 contradiction until he is compelled to 

 deny some premise ; and he can only 

 be forced to do this by a reductio ad 

 absurdum, that is, by another ratioci- 

 nation : now, if he denies the validity 

 of the reasoning process itself, he can 

 no more be forced to assent to the 

 second syllogism than to the first. In 

 truth, therefore, no one is ever forced 

 to a contradiction in terms : he can 

 only be forced to a contradiction (or 

 rather an infringement) of the funda- 

 mentalmaximof ratiocination, namely, 

 that whatever has a mark, has what it 

 is a mark of ; or, (in the case of uni- 

 versal propositions,) that whatever is 

 a mark of anything, is a mark of what- 

 ever else that thing is a mark of. 

 For in the case of every correct argu- 

 ment, as soon as thrown into the 

 syllogistic form, it is evident without 

 the aid of any other syllogism, that he 

 who, admitting the premises, fails to 

 draw the conclusion, does not conform 

 to the above axiom. 



We have now proceeded as far in 

 the theory of Deduction as we can 

 advance in the present stage of our 

 inquiry. Any further insight into the 

 subject requires that the foundation 

 shall have been laid of the philosophic 

 theory of Induction itself ; in which 

 theory that of Deduction, as a mode 

 of Induction, which we have now 

 shown it to be, will assume spontane- 



