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Inference by complementary elimination
Published online by Cambridge University Press: 12 March 2014
Extract
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.
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- Research Article
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- Copyright © Association for Symbolic Logic 1957
References
1 We interpret “disjunct” in such a way that a disjunct of a disjunct of a disjunction may also be called a disjunct of that disjunction.
2 Instead of working with a disjunction formed by disjoining our premises, we could work with the negation of a conjunction formed by conjoining the negations of our premises.
3 (Added July 14, 1957: We are indebted to the referee for simplifying the above paragraph.) If some of our n premises are conjunctions, we may eliminate more than n—1 complementary pairs from our premise disjunction. Let us say that a formula is an A-premise of a set S of premises if it is either a member of S or a conjunct of a member of S; and let us say that a formula is a B-premise of S if it is an A-premise of S and is not a conjunction. We may make our complementary eliminations from a disjunction formed merely by disjoining the B-premises of S; if S has n B-premises, we may make up to n—1 complementary eliminations from a disjunction formed merely by disjoining the B-premises of S.
4 We may, of course, eliminate from any disjunction all but one occurrence of any disjunct which is reiterated in that disjunction; the result of such elimination will be a consequence of that disjunction. But if such eliminations are made from a premise disjunction, we may not be justified in applying complementary elimination to the result. Consider this example:
Had we eliminated the occurrence of p in the third premise, on the ground that it is a reiterated disjunct, we would have derived the invalid consequence r.
5 We interpret “conjunct” in such a way that a conjunct of a conjunct of a conjunction may also be called a conjunct of that conjunction.
6 Any application of the elimination introduced in (9) may be thought of as an application of complementary elimination to an expanded premise disjunction. In place of the premise disjunction used in (10), we could use an expanded premise disjunction in this way:
