32 THE PRINCIPLES OF SCIENCE. [CHAP. 



immediate exclusion. The criterion of false reasoning, as we 

 shall find, is that it involves self-contradiction, the affirm- 

 ing and denying of the same statement. We might repre- 

 sent tne object of all reasoning as the separation of the 

 consistent and possible from the inconsistent and impossi- 

 ble ; and we cannot make any statement except a truism 

 without implying, that certain combinations of terms are 

 contradictory and excluded from thought. To assert 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 

 the 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 ABZ> = 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. 



Certain Special Conditions of Logical Symbols. 



In order that we may argue and infer truly we must 

 treat our logical symbols according to the fundamental 

 laws of Identity and Difference. But in thus using our 

 symbols we shall frequently meet with combinations of 

 which the meaning will not at first sight be apparent. If 

 in one case we learn that an object is " yellow and round," 

 and in another case that it is " round and yellow," there 

 arises the question whether these two descriptions are 

 identical in meaning or not. Again, if we proved that an 

 object was " round round," the meaning of such an expres- 

 sion would be open to doubt. Accordingly we must take 

 notice, before proceeding further, of certain special laws 

 which govern the combination of logical terms. 



In the first place the combination of a logical term with 

 itself is without effect, just as the repetition of a statement 

 does not alter the meaning of the statement ; " a round 

 round object" is simply " a round object." What is 

 yellow yellow is merely yellow; metallic metals cannot 

 differ from metals, nor circular circles from circles. In our 



