By Marcy Sheiner
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Hence if the theorem holds for sums and products of n terms, so that “ (fli + 0,2 + dz + ... + dii) = then “•[(c&l + 0,2 + Q>z + * * . + d n ) + fln+l] = (“ & i * ""#3 * • • • - a „ ) - f l n+i. Thus, since the theorem holds for two terms, and is such that if it hold for n terms, it will hold for n + 1 terms, it will, therefore, hold for any finite number of terms. , for any finite number of terms. The validity of this procedure depends upon such extension of the laws of the system, as given above, by use of the principle of mathematical induction.
9] [4*92] [4 •93] If A x + B - x 9^ 0 , then A + B 9^ 0 . [5*5] (A x + B -x) c (A + B). Hence the theorem follows from 6 *6 . Laws such as 6-3, 6-31, and 6*32, according to which an equation and an inequation, combined, give an inequation, are particularly important: they apply to cases in which a universal premise (an equation) and a particular premise (an inequation) together give a particular conclusion. Examples will be given in the next chapter. A great many laws of the algebra which have been proved for sums and products of two terms extend to sums and products of any number of terms.
Description of a Notation for the Logic of Relatives,” Memoirs of the American Academy of Arts and Sciences, IX (1870), 317-378. ------------------ , “ On the Algebra of Logic,” American Journal of Mathe matics, III (1880), 15-57. ------------------ , “ On the Logic of Relatives,” in Studies in Logic by Members of Johns Hopkins University (Boston, 1883). ------------------ , “ On the Algebra of Logic; a Contribution to the Philosophy of Notation,” American Journal of Mathematics, VII (1885), 180-202.