Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/98

82 are mutually distinct, through the possession of contrary qualities, and they together make up the universe of discourse. 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 symbols, 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 concluded that the classes they represent are mutually exclusive. Lastly, from the fact that the sum of the constituents of an expansion is unity, it might have been inferred, that the classes which they represent, together make up the universe. Upon the laws of constituents and the mode of their interpretation above determined, are founded the analysis and the interpretation of logical equations. That all such equations admit 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 fundamental 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: The form we shall first consider arises when any logical equation $$\scriptstyle{V=0}$$ is developed, and the result, after resolution into its component equations, is to be interpreted. The function is supposed to involve the logical symbols $\scriptstyle{x}$, $\scriptstyle{y}$, &amp;c., in combinations which 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. To interpret the logical equation $\scriptstyle{V=0}$. For simplicity let us suppose that $$\scriptstyle{V}$$ involves but two