Resolution Step for Propositional Calculus

A clause is a disjunction of literals (atoms or negations of atoms).

Select two clauses C1 and C2 that have exactly one atom that is positive in one clause and negated in the other.

Form a new clause, consisting of all literals of both clauses except for the two complementary literals, and add it to the set of clauses.

Theorem: The new clause produced by resolution is a logical consequence of the two parent clauses.

Proof: Let the parent clauses be C1 = L &or C1' and C2 = ¬ L &or C2'; the resolvent is H = C1' &or C2'. Suppose that C1 and C2 are true in an interpretation I.
Case 1: L = true in I. Then since C2 = ¬ L &or C2', C2' must be true in I and H is true in I.
Case 2: L = false in I. Then since C1 = L &or C1', C1' must be true in I and H is true in I.

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