Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/96

 

CHAPTER VI.

OF THE GENERAL INTERPRETATION OF LOGICAL EQUATIONS, AND THE RESULTING ANALYSIS OF PROPOSITIONS. ALSO, OF THE CONDITION OF INTERPRETABILITY OF LOGICAL FUNCTIONS.

1. IT has been observed that the complete expansion of any function by the general rule demonstrated in the last chapter, involves two distinct sets of elements, viz., the constituents of the expansion, and their coefficients. I propose in the present chapter to inquire, first, into the interpretation of constituents, and afterwards into the mode in which that interpretation is modified by the coefficients with which they are connected.

The terms "logical equation," "logical function," &c., will be employed generally to denote any equation or function involving the symbols $\scriptstyle {x}$, $\scriptstyle {y}$, &c., which may present itself either in the expression of a system of premises, or in the train of symbolical results which intervenes between the premises and the conclusion. If that function or equation is in a form not immediately interpretable in Logic, the symbols $\scriptstyle {x}$, $\scriptstyle {y}$, &c., must be regarded as quantitative symbols of the species described in previous chapters (II. 15), (V. 6), as satisfying the law, $\scriptstyle {x(1-x)=0}$.

By the problem, then, of the interpretation of any such logical function or equation, is meant the reduction of it to a form in which, when logical values are assigned to the symbols $\scriptstyle {x}$, $\scriptstyle {y}$, &c., it shall become interpretable, together with the resulting interpretation. These conventional definitions are in accordance with the general principles for the conducting of the method of this treatise, laid down in the previous chapter.