Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/145

nearly all disappear, and we have only left (5)

$\scriptstyle {xyz+x{\bar {y}}z=0}$^{[errata 1]};

whence

$\scriptstyle {z={\frac {0}{xy+{\bar {x}}{\bar {y}}}}}$

${\begin{array}{l}\scriptstyle {=0xy+{\frac {0}{0}}x{\bar {y}}+{\frac {0}{0}}{\bar {x}}y+0{\bar {x}}{\bar {y}}}\\\scriptstyle {={\frac {0}{0}}x{\bar {y}}+{\frac {0}{0}}{\bar {x}}y{\text{,}}}\end{array}}$

furnishing the interpretation. Wherever the property $\scriptstyle {C}$ is found, either the property $\scriptstyle {A}$ or the property $\scriptstyle {B}$ will be found with it, but not both of them together.

From the equation (5) we may readily deduce the result arrived at in the previous investigation by the method of arbitrary constant multipliers, as well as any other proposed forms of the relation between $\scriptstyle {x}$, $\scriptstyle {y}$, and $\scriptstyle {z}$; e.g. If the property $\scriptstyle {B}$ is absent, either $\scriptstyle {A}$ and $\scriptstyle {C}$ will be jointly present, or $\scriptstyle {C}$ will be absent. And conversely, If $\scriptstyle {A}$ and $\scriptstyle {C}$ are jointly present, $\scriptstyle {B}$ will be absent. The converse part of this conclusion is founded on the presence of a term $\scriptstyle {xz}$ with unity for its coefficient in the developed value of $\scriptstyle {\bar {y}}$.

---

1. Correction: $\scriptstyle {xyz+x{\bar {y}}z=0}$ should be amended to $\scriptstyle {xyz+{\bar {x}}{\bar {y}}z=0}$: detail