Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/247

reasoning. This remark is equally applicable to the case of Syllogism, which we proceed next to consider.

5. The nature of syllogism is best seen in the particular instance. Suppose that we have the propositions,

All $\scriptstyle {X}$'s are $\scriptstyle {Y}$'s,
All $\scriptstyle {Y}$'s are $\scriptstyle {Z}$'s.

From these we may deduce the conclusion, All $\scriptstyle {X}$'s are $\scriptstyle {Z}$'s. This is a syllogistic inference. The terms $\scriptstyle {X}$ and $\scriptstyle {Z}$ are called the extremes, and $\scriptstyle {Y}$ is called the middle term. The function of the syllogism generally may now be defined. Given two propositions of the kind whose species are tabulated in (1), and involving one middle or common term $\scriptstyle {Y}$, which is connected in one of the propositions with an extreme $\scriptstyle {X}$, in the other with an extreme $\scriptstyle {Z}$; required the relation connecting the extremes $\scriptstyle {X}$ and $\scriptstyle {Z}$. The term $\scriptstyle {Y}$ may appear in its affirmative form, as, All $\scriptstyle {Y}$'s, Some $\scriptstyle {Y}$s; or in its negative form, as, All not-$\scriptstyle {Y}$'s, Some not-$\scriptstyle {Y}$'s; in either proposition, without regard to the particular form which it assumes in the other.

Nothing is easier than in particular instances to resolve the Syllogism by the method of this treatise. Its resolution is, indeed, a particular application of the process for the reduction of systems of propositions. Taking the examples above given, we have, ${\begin{array}{l}\scriptstyle {x=vy{\text{,}}}\\\scriptstyle {y=vz{\text{;}}}\end{array}}$^{[errata 1]} whence by substitution, $\scriptstyle {x=vv'z}$, which is interpreted into All $\scriptstyle {X}$'s are $\scriptstyle {Z}$'s. Or, proceeding rigorously in accordance with the method developed in (VIII. 7), we deduce $\scriptstyle {x(1-y)=0,\qquad y(1-z)=0}$. Adding these equations, and eliminating $\scriptstyle {y}$, we have $\scriptstyle {x(1-z)=0}$;

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1. Correction: ${\begin{array}{l}\scriptstyle {x=vy{\text{,}}}\\\scriptstyle {y=vz{\text{;}}}\end{array}}$ should be amended to ${\begin{array}{l}\scriptstyle {x=vy{\text{,}}}\\\scriptstyle {y=v'z{\text{;}}}\end{array}}$: detail