Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/109

CHAP. VI.] The latter function presupposes, as a condition of its interpretation, that the class represented by $$\scriptstyle{y}$$ is wholly contained in the class represented by $\scriptstyle{x}$; the former function does not imply any such requirement. Now if $$\scriptstyle{V}$$ independently interpretable, and if $$\scriptstyle{w}$$ represent the collection of individuals which it contains, the equation $$\scriptstyle{w=V}$$ will hold true without entailing as a consequence the vanishing of any of the constituents in the development of $\scriptstyle{V}$; since such vanishing of constituents would imply relations among the classes of things denoted by the symbols in $\scriptstyle{V}$. Hence the development of $$\scriptstyle{V}$$ will be of the form the coefficients $\scriptstyle{a_1}$, $\scriptstyle{a_2}$, &amp;c. all satisfying the condition Hence by the reasoning of Prop. 4, Chap. v. the function $$\scriptstyle{V}$$ will be subject to the law  This result, though evident  à priori from the fact that $$\scriptstyle{V}$$ is supposed to represent a class or collection of things, is thus seen to follow also from the properties of the constituents of which it is composed. The condition $$\scriptstyle{V(1-V)=0}$$ may be termed "the condition of interpretability of logical functions." The general form of solutions, or logical conclusions developed in the last Proposition, may be designated as a "Relation between terms." I use, as before, the word "terms" to denote the parts of a proposition, whether simple or complex, which are connected by the copula "is" or "are." The classes of things represented by the individual symbols may be called the elements of the proposition. —Resuming the definition of "clean beasts," (VI. 6), required a description of "unclean beasts." Here, as before, $$\scriptstyle{x}$$ standing for "clean beasts," $$\scriptstyle{y}$$ for "beasts dividing the hoof," $$\scriptstyle{z}$$ for "beasts chewing the cud," we have whence  and developing the second member,