Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/237

(1)

$\scriptstyle {xy+({\bar {t}}+tv+t{\bar {v}}{\bar {z}}+t{\bar {v}}zw)x{\bar {y}}+({\bar {v}}+tzwv){\bar {x}}y+tzw{\bar {x}}{\bar {y}}=0}$;

and this is the form of reduction sought.

2. Now according to the principle asserted in Prop, iii., Chap, x., the whole relation connecting any particular set of the symbols in the above equation may be deduced by developing that equation with reference to the particular symbols in question, and retaining in the result only those constituents whose coefficients are unity. Thus, if $\scriptstyle {x}$ and $\scriptstyle {y}$ are the symbols chosen, we are immediately conducted to the equation $\scriptstyle {xy=0}$, whence we have $\scriptstyle {y={\frac {0}{0}}(1-x)}$, with the interpretation, If gravitation is necessarily present, matter is not a necessary being.

Let us next seek the relation between $\scriptstyle {x}$ and $\scriptstyle {w}$. Developing (1) with respect to those symbols, we get ${\begin{array}{r}\scriptstyle {(y+{\bar {t}}{\bar {y}}+tv{\bar {y}}+t{\bar {v}}{\bar {z}}{\bar {y}}+t{\bar {v}}z{\bar {y}})xw+(y+{\bar {t}}{\bar {y}}+tv{\bar {y}}+t{\bar {v}}{\bar {z}}{\bar {y}})x{\bar {w}}\qquad }\\\scriptstyle {+(vy+tzvy+tz{\bar {y}}){\bar {x}}w+{\bar {v}}y{\bar {x}}{\bar {w}}=0{\text{.}}}\end{array}}$^{[errata 1]} The coefficient of $\scriptstyle {xw}$, and it alone, reduces to unity. For $\scriptstyle {t{\bar {v}}{\bar {z}}{\bar {y}}+t{\bar {v}}z{\bar {y}}=t{\bar {v}}{\bar {y}}}$, and $\scriptstyle {tv{\bar {y}}+t{\bar {v}}{\bar {y}}=t{\bar {y}}}$, and $\scriptstyle {{\bar {t}}{\bar {y}}+t{\bar {y}}={\bar {y}}}$, and lastly, $\scriptstyle {y+{\bar {y}}=1}$. This is always the mode in which such reductions take place. Hence we get $\scriptstyle {xw=0}$, $\scriptstyle {\therefore w={\frac {0}{0}}(1-x)}$, of which the interpretation is, If motion exists, matter is not a necessary being.

If, in like manner, we develop (1) with respect to $\scriptstyle {x}$ and $\scriptstyle {z}$, we get the equation $\scriptstyle {x{\bar {z}}=0}$, $\scriptstyle {\therefore x{\frac {0}{0}}z}$, with the interpretation, If matter is a necessary being, the world is merely material, and without a presiding intelligence.

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1. Correction: ${\begin{array}{r}\scriptstyle {(y+{\bar {t}}{\bar {y}}+tv{\bar {y}}+t{\bar {v}}{\bar {z}}{\bar {y}}+t{\bar {v}}z{\bar {y}})xw+(y+{\bar {t}}{\bar {y}}+tv{\bar {y}}+t{\bar {v}}{\bar {z}}{\bar {y}})x{\bar {w}}\qquad }\\\scriptstyle {+(vy+tzvy+tz{\bar {y}}){\bar {x}}w+{\bar {v}}y{\bar {x}}{\bar {w}}=0{\text{.}}}\end{array}}$ should be amended to ${\begin{array}{r}\scriptstyle {(y+{\bar {t}}{\bar {y}}+tv{\bar {y}}+t{\bar {v}}{\bar {z}}{\bar {y}}+t{\bar {v}}z{\bar {y}})xw+(y+{\bar {t}}{\bar {y}}+tv{\bar {y}}+t{\bar {v}}{\bar {z}}{\bar {y}})x{\bar {w}}\qquad }\\\scriptstyle {+({\bar {v}}y+tzvy+tz{\bar {y}}){\bar {x}}w+{\bar {v}}y{\bar {x}}{\bar {w}}=0{\text{.}}}\end{array}}$: detail