Solving Open Questions and Other Challenge Problems Using Proof Sketches

In this article, we describe a set of procedures and strategies for searching for proofs in logical systems based on the inference rule condensed detachment. The procedures and strategies rely on the derivation of proof sketches – sequences of formulas that are used as hints to guide the search for sound proofs. In the simplest case, a proof sketch consists of a subproof – key lemmas to prove, for example – and the proof is completed by filling in the missing steps. In the more general case, a proof sketch consists of a sequence of formulas sufficient to find a proof, but it may include formulas that are not provable in the current theory. We find that even in this more general case, proof sketches can provide valuable guidance in finding sound proofs. Proof sketches have been used successfully for numerous problems coming from a variety of problem areas. We have, for example, used proof sketches to find several new two-axiom systems for Boolean algebra using the Sheffer stroke.