86 THE PKINCIPLES OF SCIENCE. [CHAP. 



c = cd 

 and if we substitute this expression for c in (2) we have 



AB = ABcd 



the full meaning of which is that " some nebulae do not 

 give continuous spectra and are not heated solids." 



We might similarly apply the contrapositive in many 

 other instances. Take the argument, " All fixed stars are 

 self-luminous ; but some of the heavenly bodies are not 

 self-luminous, and are therefore not fixed stars." Taking 

 our terms 



A = fixed stars 

 B = self-luminous 

 C = some 



D = heavenly bodies 

 we have the premises 



A = AB, (i) 



CD = &CD (2) 



Now from (i) we can draw the contrapositive 



b = ab 

 and substituting this expression for b in (2) we obtain 



CD = a&CD 



which expresses the conclusion of the argument that some 

 heavenly bodies are not fixed stars. 



Contrapositive of a Simple Identity. 



The reader should carefully note that when we apply 

 the process of Indirect Inference to a simple identity of 

 the form 



A = B 



we may obtain further results. If we wish to know what 

 is the term not-B, we have as before, by the Law of Duality, 



b = Ab }- ab 

 and substituting for A we obtain 



b = B6 -I- ab = ab. 



But we may now also draw a second contrapositive ; for 

 we have 



a = aB -|- ab, 

 and substituting for B its equivalent A we have 



a = aA. {' ab = ab. 



Hence from the single identity A = B we can draw 

 the two propositions 



