3i 8 THE SCIENCE OF LOGIC 



by assuming a particular to prove the universal which involves it ; 

 (4) by assuming successively, in parts, the proposition to be 

 proved ; (5) by assuming, without independent proof j a proposition 

 which is the reciprocal of the proposition to be proved. This 

 last is too simple to need notice. It is arguing, for instance, that 

 Cork is south of Dublin because Dublin is north of Cork. The 

 fourth mode is merely a variety of the first. Aristotle instances 

 the attempt to prove &quot; that the art of healing is knowledge of what 

 is wholesome and unwholesome &quot; by assuming it successively to be 

 a knowledge of each. The third mode is really the inductive 

 fallacy of supposing that enumerative induction establishes a uni 

 versal truth (207). We have, therefore, to consider the first two 

 modes, which are the really important ones. 



(i) The very proposition to be proved is rarely assumed as a 

 premiss, except under cover of some circumlocution. As simpler 

 examples we may take the following : &quot; The House of Lords is 

 out of date because an upper chamber in England is an anachron 

 ism &quot;. &quot; The bill before the house is well calculated to elevate 

 the character of education in the country, for the general standard 

 of instruction in all the schools will be raised by it.&quot; 



De Morgan, in his Budget of Paradoxes (p. 327), gives an interesting 

 example, from an attempt at squaring the circle, by a Mr. James Smith, in a 

 work entitled Nut to Crack. Smith attempted to prove that the ratio of circum 

 ference to diameter is 3$, by assuming that it is so, and showing that every other 

 ratio is, on this hypothesis^ absurd : &quot; I think you will not dare to dispute my 

 right to this hypothesis, when I can prove by means of it that every other value 

 of IT will lead to the grossest absurdities ; unless indeed you are disposed to 

 dispute the right of Euclid to adopt a false line hypothetical ly, for the purpose 

 of a reductio ad absurdum demonstration in pure geometry &quot;. He thus con 

 founds his own fallacious procedure with Euclid s process of indirect proof. 

 He argues that &quot; if 3! be the right ratio, then all other ratios will be wrong ; 

 but they will be wrong, on the hypothesis ; therefore the hypothesis is right &quot; ; 

 whereas he should have shown, independently of his hypothesis, that they will 

 be all wrong. He argues that &quot; If A is true, B will be false ; but B will be 

 false if A is true ; therefore A is true&quot; instead of arguing &quot; If A is true, B 

 will be false, but B is false, therefore (since B includes all suppositions other 

 than A} A is true.&quot; In the reductio ad absurdum Euclid argues that &quot; If A 

 is false B will be true ; but B is false ; therefore A is true.&quot; &quot; Euclid assumes 

 what he wants to disprove, and shows that his assumption leads to absurdity, 

 and so upsets itself. Mr. Smith assumes what he wants to prove, and shows 

 that his assumption makes other propositions lead to absurdity. This is 

 enough for all who can reason.&quot; (De Morgan, ibid.). 



The example just given suggests a method of procedure which is an exceed 

 ingly seductive form of the fallacy, a method which apparently entraps honest 

 reasoners themselves just as frequently as it is knowingly used by dishonest 



