102 THE PRINCIPLES OF SCIENCE. [chap. 



A = ABc -I- AbO, (I) 



Be = DeBC, (2) 



DE = DEbc. (3) 



As five terms enter into these premises it is requisite to 

 treat their thirty-two combinations, and it will be found 

 that fourteen of them remain consistent with the premises, 

 namely 



ABcdE aBCBe ahCdE 



ABcde oBCdE abCde 



AbCdE aBCde ahcDF. 



AbCde aBcdE abcdE 



aBcde ahcde. 



If we examine the first four combinations, all of which 

 contain A, we find that they none of them contain D ; or 

 again, if we select those which contain D, we have only 

 two, thus — 



D = aBCDe ■{■ abcDE. 

 Hence it is clear that no A is D, and vice verad no D is A. 

 We might draw many other conclusions from the same 

 premises ; for instance — 



DE = ahcDE, 

 or D and E never meet but in the absence of A, B, and C. 



Fallacies analysed by the Indirect Method. 



It has been sufficiently shown, perhaps, that we can by 

 the Indirect Method of Inference extract the whole truth 

 from a series of propositions, and exhibit it anew in any 

 required form of conclusion. But it may also need to be 

 sliown by examples that so long as we follow correctly 

 the almost mechanical rules of the method, we cannot i'all 

 into any of the fallacies or paralogisms which are often 

 committed in ordinary discussion. Let us take the example 

 of a fallacii)us argument, previously treated by the Method 

 of Direct Inference (p. 62), 



Granite is not a sedimentaiy rock, (i) 



Basalt is not a sedimentary rock, (2) 



and let us ascertain whether any precise conclusion can be 

 drawn concerning the relation of granite and basalt. 

 Taking as before 



A = granite, 



B = sedimentary rock, 



C = basalt, 



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I 



