The inferences countenanced by the traditional rules of modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism, and the complex types of dilemma may be regarded as single applications of one rule of inference, “the rule of complementary elimination”. In the present paper, we shall discuss this rule informally and illustrate it in application to expressions written in the language of Principia Mathematica. Our illustrations will contain no connectives except for those for conjunction, disjunction, and negation; we use parentheses in place of dots; and we allow disjunction and conjunction to have any number of operands more than two.
In applying complementary elimination to a set of premises, we take the following three steps, (i) We form, merely by disjoining the premises, an expression which we shall call a premise disjunction, (ii) If we have n premises, we eliminate n minus one (or fewer) pairs of the following sort from our premise disjunction: each pair is such that one of its members is the negation of the other and both members are specific occurrences of disjuncts of our premise disjunction. We shall call such pairs complementary pairs, (iii) The formula obtained by means of our second step is one that may be made well-formed merely by eliminating parentheses or connectives other than negation; we make such elimination, and any formula we thus obtain is a consequence of our premises.