The premisses may be written as follows: (1) All AB is Cd or cD ; (2) All BC is AD or ad ; (3) All ab is cd ; (4) All cd is ab.

Then, from (1), All A is b or Cd or cD ;
and from (2), AllA is b or c or D ;
therefore (by combining these),All A is b or cD ;
(3) gives no information regarding A (see thepreceding [section]);
but by (4), All A is C or D ;
therefore, All A is bC or bD or cD ;
and, since bD is by developmenteither bCD or bcD this becomes
All A is bC or cD.

This solves the problem as set. Proceeding also to determine a, we find that (1) gives no information with regard to this term; but by (2), All a is b or c or d ; and by (3), All a is B or cd ; therefore, All a is Bc or Bd or cd. Again by (4), All a is b or C or D. Therefore, All a is BCd or BcD or bcd ; and by contraposition, Whatever is Bcd or bC or bD or CD is A.[510]

[510] Taking into account the result arrived at above with regard to A, it will be seen that this may be resolved into Whatever is bC or bD is A and Nothing is BCD or Bcd. These two propositions taken together with the solution for A are equivalent to the original premisses.

489. The Problem of Elimination.—By elimination in logic is meant the omission of certain elements from a proposition or set of 509 propositions with the object of expressing more directly and concisely the connexion between the elements which remain. An example of the process is afforded by the ordinary categorical syllogism, where the so-called middle term is eliminated. Thus, given the premisses All M is P, All S is M, we may infer All S is MP ; but if we desire to know the relation between S and P independently of M we are content with the less precise but sufficient statement All S is P ; in other words, we eliminate M.

Elimination has been considered by some writers to be absolutely essential to logical reasoning. It is not, however, necessarily involved either in the process of contraposition or in the process discussed in the preceding section; and if formal inferences are recognised at all, the name of inference certainly cannot be denied to these processes. We must, therefore, refuse to regard elimination as of the essence of reasoning, although it may usually be involved therein.[511]

[511] Compare sections [207], 208.

490. Elimination from Universal Affirmations.—Any universal affirmative proposition (or, by combination, any set of universal affirmative propositions) involving the term X and its contradictory x may by contraposition be reduced to the form Everything is PX or Qx or R, where P, Q, R are themselves simple or complex terms not involving X or x ; and since by the rule given in section [448] a determinant may at any time be omitted from an undistributed term, we may eliminate X (and x) from this proposition by simply omitting them, and reducing the proposition to the form Everything is P or Q or R.[512]

[512] We might also proceed as follows: Solve for X and for x, as in section [488], so that we have All X is A, All x is B, where A and B are simple or complex terms not involving either X or x. Then, since Everything is X or x, we shall have Everything is A or B, and this will be a proposition containing neither X nor x.

We must, however, here admit the possibility of P, Q, R being of the forms A or a, Aa. These are equivalent respectively to the entire universe of discourse and to nothing. Thus, if P is of the form A or a, and Q is of the form Aa, our proposition will before elimination more naturally be written Everything is X or R ; if Q is of the form A or a, and R of the form Aa, it will more naturally be written Everything is PX or x. It follows that if either P or Q is of the form A or a (that is, if either P or Q is equivalent to the entire universe of discourse), the proposition resulting from elimination 510 will not afford any real information, since it is always true à priori that Everything is A or a or &c. Thus we are unable to eliminate X from such a proposition as All A is X or BC.