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 F_n and a desired conclusion G, we prove that F₁ &and F₂ &and ... &and F_n &and ¬ G is inconsistent. This is done by adding new clauses H_i, 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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