(58 THE PRINCIPLES OF SCIENCE. [CHAP. 



the sign +; but this sign should not be employed unless there 

 exists exact analogy between mathematical addition and 

 logical alternation. We shall find that the analogy is im- 

 perfect, and that there is such profound difference between 

 logical and mathematical terms as should prevent our 

 uniting them by the same symbol. Accordingly I have 

 chosen a sign -|- , which seems aptly to suggest whatever 

 degree of analogy may exist without implying more. 

 The exact meaning of the symbol we will now proceed to 

 investigate. 



Nature of the Alternative Relation. 



Before treating disjunctive propositions it is indispens- 

 able to decide whether the alternatives must be considered 

 exclusive or unexclusive. By exclusive alternatives we 

 mean those which cannot contain the same things. If we 

 say " Arches are circular or pointed," it is certainly to be 

 understood that the same arch cannot be described as both 

 circular and pointed. Many examples, on the other hand, 

 3an readily be suggested in which two or more alternatives 

 may hold true of the same object. Thus 



Luminous bodies are self-luminous or luminous by 



reflection. 



It is undoubtedly possible, by the laws of optics, that the 

 same surface may at one and the same moment give oft' 

 light of its own and reflect light from other bodies. We 

 speak familiarly of deaf or dumb persons, knowing that the 

 majority of those who are deaf from birth are also dumb. 



There can be no doubt that in a great many cases, 

 perhaps the greater number of cases, alternatives are 

 exclusive as a matter of fact. Any one number is 

 incompatible with any other ; one point of time or place 

 is exclusive of all others. Eoger Bacon died either in 

 1284 or 1292 ; it is certain that he could not die in both 

 years. Henry Fielding was born either in Dublin or 

 Somersetshire ; he could not be born in both places. 

 There is so much more precision and clearness in the use 

 of exclusive alternatives that we ought doubtless to select 

 them when possible. Old works on logic accordingly 

 wntained a rule directing that the Afembra dividenlia, the 



