Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/146

 

CHAPTER IX.

ON CERTAIN METHODS OF ABBREVIATION.

1. THOUGH the three fundamental methods of development, elimination, and reduction, established and illustrated in the previous chapters, are sufficient for all the practical ends of Logic, yet there are certain cases in which they admit, and especially the method of elimination, of being simplified in an important degree; and to these I wish to direct attention in the present chapter. I shall first demonstrate some propositions in which the principles of the above methods of abbreviation are contained, and I shall afterwards apply them to particular examples.

Let us designate as class terms any terms which satisfy the fundamental law $\scriptstyle {V(1-V)=0}$. Such terms will individually be constituents; but, when occurring together, will not, as do the terms of a development, necessarily involve the same symbols in each. Thus $\scriptstyle {ax+bxy+cyz}$ may be described as an expression consisting of three class terms, $\scriptstyle {x}$, $\scriptstyle {xy}$, and $\scriptstyle {yz}$, multiplied by the coefficients $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$ respectively. The principle applied in the two following Propositions, and which, in some instances, greatly abbreviates the process of elimination, is that of the rejection of superfluous class terms; those being regarded as superfluous which do not add to the constituents of the final result.

Proposition I.

2. From any equation, $\scriptstyle {V=0}$, in which $\scriptstyle {V}$ consists of a series of class terms having positive coefficients, we are permitted to reject any term which contains another term as a factor, and to change every positive coefficient to unity.

For the significance of this series of positive terms depends only upon the number and nature of the constituents of its final expansion, i.e. of its expansion with reference to all the symbols