Note 513 — Chapter V

Compare Mrs Ladd Franklin’s Essay on The Algebra of Logic (Studies in Logic by Members of the Johns Hopkins University). The same conclusion may be deduced by obversion from the result obtained in the preceding section. Nothing is PX or Qx or R becomes by obversion Everything is prX or qrx. Therefore, by the elimination of X, Everything is pr or qr ; and this proposition becomes by obversion Nothing is PQ or R.

Another method by which the same result may be obtained is as follows: By developing the first alternant with reference to Q and the second with reference to P, Nothing is PX or Qx or R becomes Nothing is PQX or PqX or PQx or pQx or R. But PQX or PQx is reducible to PQ, and on omitting PqX and pQx, we have Nothing is PQ or R.

It is interesting to observe that the above rule for elimination from negatives is equivalent to Boole’s famous rule for elimination. In order to eliminate X from the equation F(X) = 0, he gives the formula F(1) F(0) = 0. Now any equation containing X can be brought to the form AX + Bx + C = 0, where A, B, C are independent of X. Applying Boole’s rule we have (A + C)(B + C) = 0, that is, AB + C = 0; and this is precisely equivalent to the rule given in the text.