38 THjj PRINCIPLES OF SCIENCE. 



Symbolic Expression of the Law of Contradiction. 



The synthesis of terms is subject to the all-important 

 Law of Thought, described in a previous section (p. 6) 

 and called the Law of Contradiction. It is self-evident 

 that no quality or circumstance can be both present and 

 absent at the same time and place. This fundamental 

 condition of all thought and all existence is expressed 

 symbolically by a rule that a term and its negative shall 

 never be allowed to come into combination. Such com- 

 bined terms as A a, B6, Cc, &c. are self-contradictory and 

 devoid of all meaning. If they represented anything, it 

 would be what cannot exist, and cannot even be imagined 

 in the mind. They can therefore only enter into our con- 

 sideration to suffer immediate exclusion. The criterion 

 of false reasoning, as we shall find, is that it involves 

 self-contradiction, the affirming and denying of the same 

 statement. Thus we might represent the object of all 

 reasoning as the separation of the consistent and possible 

 from the inconsistent and impossible ; and we cannot make 

 any inference without implying that certain combinations 

 of terms are contradictory and excluded from thought. 

 To conclude that 'all A's are B's' is equivalent to the 

 assertion that ' A's which are not B's cannot exist.' 



It will be convenient to have the means of indicating 

 this exclusion of the self-contradictory ; and we may use 

 the familiar sign for nothing, the cipher o. Thus the 

 second law of thought may be symbolised in the forms 



Aa = o AB6 = o ABCa = o. 

 We may variously describe the meaning of o in logic as 

 the non-existent, the impossible, the self -inconsistent, the 

 inconceivable. Close analogy exists between this meaning 

 and its mathematical signification. 



O 



