Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/73

tion of a general law of expressions representing "classes or collections of things."

7. The results of these investigations may be embodied in the following rule of expression.

Rule.—Express simple names or qualities by the symbols $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$, &c., their contraries by $\scriptstyle {1-x}$, $\scriptstyle {1-y}$, $\scriptstyle {1-z}$, &c.; classes of things defined by common names or qualities, by connecting the corresponding symbols as in multiplication; collections of things, consisting of portions different from each other, by connecting the expressions of those portions by the sign $\scriptstyle {+}$. In particular, let the expression, "Either $\scriptstyle {x}$'s or $\scriptstyle {y}$'s," be expressed by $\scriptstyle {x(1-y)+y(1-x)}$, when the classes denoted by $\scriptstyle {x}$ and $\scriptstyle {y}$ are exclusive, by $\scriptstyle {x+y(1-x)}$ when they are not exclusive. Similarly let the expression, "Either $\scriptstyle {x}$'s, or $\scriptstyle {y}$'s, or $\scriptstyle {z}$'s," be expressed by $\scriptstyle {x(1-y)(1-z)+y(1-x)(1-z)+z(1-x)(1-y)}$, when the classes denoted by $\scriptstyle {x}$, $\scriptstyle {y}$, and $\scriptstyle {z}$, are designed to be mutually exclusive, by $\scriptstyle {x+y(1-x)+z(1-x)(1-y)}$, when they are not meant to be exclusive, and so on.

8. On this rule of expression is founded the converse rule of interpretation. Both these will be exemplified with, perhaps, sufficient fulness in the following instances. Omitting for brevity the universal subject "things," or "beings," let us assume $\scriptstyle {x=}$hard, $\scriptstyle {y=}$elastic, $\scriptstyle {z=}$metals; and we shall have the following results: "Non-elastic metals," will be expressed by $\scriptstyle {z(1-y)}$; "Elastic substances with non-elastic metals," by $\scriptstyle {y+z(1-y)}$; "Hard substances, except metals," by $\scriptstyle {x-y}$^{[errata 1]}; "Metallic substances, except those which are neither hard nor elastic," by $\scriptstyle {z-z(1-x)(1-y)}$, or by $\scriptstyle {z\{1-(1-x)(l-y)\}}$, vide (6), Chap. II. In the last example, what we had really to express was "Metals, except not hard, not elastic, metals." Conjunctions used between adjectives are usually superfluous, and, therefore, must not be expressed symbolically.

Thus, "Metals hard and elastic," is equivalent to "Hard elastic metals," and expressed by $\scriptstyle {xyz}$.

Take next the expression, "Hard substances, except those

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1. Correction: $\scriptstyle {x-y}$ should be amended to $\scriptstyle {x-z}$: detail