Resolution[J. A. Robinson, A machine oriented logic based on the resolution principle, Journal of the ACM, 12, No. 1, 1965.] is a method of proof by contradiction.
Given a set of premises F1 &and F2 &and ... &and Fn and a desired conclusion G, we prove that F1 &and F2 &and ... &and Fn &and ¬ G is inconsistent. This is done by adding new clauses Hi, each of which is a logical consequence of the existing clauses, to the set. If we can add □ as a new clause, then the whole formula collapses to false and is thus inconsistent.
Theorem: If H is a logical consequence of F, F &and H = F. Proof: If F is false, then both sides are false. If F is true, then H must be true, so F &and H is true.
Thus, we are justified in adding any logical consequence of our set of formulas to the set without changing its truth value.
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