The following may be given as an example of elimination from universal affirmatives.
Let it be required to eliminate X (together with x) from the propositions All P is XQ or xR, Whatever is X or R is p or XQR. Combining these propositions, we have Everything is XQR or p ; therefore, by elimination, Everything is QR or p that is, All P is QR. It will be observed that P (together with p) cannot be eliminated from the above propositions.
491. Elimination from Universal Negatives.—Any universal negative proposition (or, by combination, any set of universal negative propositions) containing the term X and its contradictory x may by conversion be reduced to the form Nothing is PX or Qx or R, where P, Q, R are themselves simple or complex terms not involving either X or x. Here we might, in accordance with the rule given in section [448], simply omit the alternants PX, Qx, leaving us with the proposition Nothing is R. This, however, is but part of the information obtainable by the elimination of X. We have also No X is P, and No Q is x, that is, All Q is X ; whence by a syllogism in Celarent we may infer No Q is P. The full result of the elimination is, therefore, given by the proposition Nothing is PQ or R.[513]
[513] 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.
The following is an example: Let it be required to eliminate X from the propositions No P is Xq or xr, No X or R is xP or Pq or Pr. 511 Combining these propositions we have Nothing is XPq or XPr or xP or PqR ; therefore, by elimination in accordance with the above rule, Nothing is Pq or Pr, that is, No P is q or r.
492. Elimination from Particular Affirmatives.—Any particular affirmative proposition involving the term X may by conversion be reduced to the form Something is either PX or Qx or R, where P, Q, R are independent of X and x. We may here immediately apply the rule given in section [448] that a determinant may at any time be omitted from an undistributed term; and the result of eliminating X is accordingly Something is either P or Q or R.[514]
[514] Thus the rule for elimination from particular affirmatives is practically identical with the rule for elimination from universal affirmatives.
493. Elimination from Particular Negatives.—Any particular negative proposition involving the term may by contraposition be reduced to the form Something is not either PX or Qx or R. By the development of the first alternant with reference to Q and that of the second alternant with reference to P, this proposition becomes Something is not either PQX or PqX or PQx or pQx or R. But PQX or PQx is reducible to PQ and the alternants PqX, pQx may by the rule given in section [448] be omitted. Hence we get the proposition Something is not either PQ or R, from which X has been eliminated.[515]