82 OF INTERPRETATION. [CHAP. VI. 



are mutually distinct, through the possession of contrary qualities, 

 and they together make up the universe of discourse. 



3. These properties of constituents have their expression in 

 the theorems demonstrated in the conclusion of the last chapter, 

 and might thence have been deduced. From the fact that every 

 constituent satisfies the fundamental law of the individual sym- 

 bols, it might have been conjectured that each constituent would 

 represent a class. From the fact that the product of any two 

 constituents of an expansion vanishes, it might have been con- 

 cluded that the classes they represent are mutually exclusive. 

 Lastly, from the fact that the sum of the constituents of an ex- 

 pansion is unity, it might have been inferred, that the classes 

 which they represent, together make up the universe. 



4. Upon the laws of constituents and the mode of their in- 

 terpretation above determined, are founded the analysis and the 

 interpretation of logical equations. That all such equations ad- 

 mit of interpretation by the theorem of development has already 

 been stated. I propose here to investigate the forms of possible 

 solution which thus present themselves in the conclusion of a 

 train of reasoning, and to show how those forms arise. Although, 

 properly speaking, they are but manifestations of a single funda- 

 mental type or principle of expression, it will conduce to clearness 

 of apprehension if the minor varieties which they exhibit are 

 presented separately to the mind. 



The forms, which are three in number, are as follows : 



FORM I. 



5. The form we shall first consider arises when any logical 

 equation V= is developed, and the result, after resolution into 

 its component equations, is to be interpreted. The function is sup- 

 posed to involve the logical symbols #,?/,&c., in combinations Avhich 

 are not fractional. Fractional combinations indeed only arise in 

 the class of problems which will be considered when we come to 

 speak of the third of the forms of solution above referred to. 



PROPOSITION II. 



To itrierpret the logical equation V 0. 

 For simplicity let us suppose that V involves but two sym- 



