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LXIX. On Indirect Dermnstration. % Professor De Morgan*. 

 TJNDER the phrase indirect demonstration, mathematicians 

 J^- ,^^^ '^<=^^^st«™«l to inckide two things which are quite 

 distinct. J^rom this use of language springs confusion between 

 the different characters of different methods. Geometers have 

 seldom been very formal logicians; and their patent of exemp- 

 tion was signed by Euchd. 



Indirect demonstration, as commonly conceived, means demon- 

 stration ot the impossibility of all contradiction. But the fol- 

 iov^mg distinctions are required. Let the proposition to be 

 proved be Every A is B. To avoid using direct in two senses, 

 as opposed to converse, and as opposed to indirect, I shall take 

 thecovveinhves positive and contrapositive, direct and indirect. 



1. 1 he direct positive proposition is Every A is B. The direct 

 positive proof takes amj A, and shows that it is B. 



2^ The direct contrapositive proposition, identical with the last, 

 is Every not-B is not-A. The direct contrapositive proof takes 

 any not-B, and shows that it is not-A. 



3. The indirect positive proof attacks the positive contradic- 

 tion borne As are not-Bs, and taking an A assumed to be 

 not-ii, shows the assumption to have an absurdity for its neces- 

 sary consequence. 



4. The indirect contrapositive proof attacks the contrapositive 

 contradiction. Some not-Bs are As, and taking a not-B assumed 

 to be A, shows the assumption to have an absurdity for its neces- 

 sary consequence. 



The third and fourth have a slightness of distinction which I 

 maintain to exist also as to the first and second. Apijlviuo- the 

 notion t of form and matter to forms, the first and second differ 

 in form and also the third and fourth. But the first pair are 

 opposed to the second pair. The latter pair proceed from denial 

 of consequence to denial of hypothesis : the former pair proceed 

 from establishment of hypothesis to establishment of consequence 



VVhen the mathematician uses the second form, he usually 

 employs the third or fourth, subordinately, to connect it ^dth 

 the first. Is this necessary ? 



When we say a square is entirely contained within a circle do 

 we need an indirect process to establish that outside the circle is 

 outside the square 't Surely any attempt to establish this by 

 * Communicated by the Author. 



t The algebraist ought to be well aorustomcd to this an„lication In 

 anthmetic 1, 2, 3 &e are of the form, yar.ls, gallons, &c. are of tSatter 

 la common a gebra, 1 2, 3, &c. becon.e the matter, an.l a+b,abXc^e 

 d.8tmct.ons of fonn. In h.gher algebra a+b, ab, &c. bccomTmateHal,am1 

 tlt"'^'^' f '. •^'=i""f='« t« 'i''^. f«'!»- The distinction of form and nmtter 

 18 often concealed nnder the distmction of general and specific matter 

 2 F 2 



