More proofs of an axiom of Łukasiewicz

This paper reports results and some new problems in one of the domains to which automatic first-order theorem provers have been most successfully applied: axiomatics of nonclassical propositional logics. It is well known that one of the standard axioms of the denumerable-valued pure implication logic of Łukasiewicz becomes derivable from the remainder in the presence of negation. Here it is shown that the same axiom is similarly derivable using conjunction and disjunction instead of negation. This closes a problem left open by Harris and Fitelson (Journal of Automated Reasoning 27, 2001). Related problems are discussed, and five such open problems are proposed as challenges to the automated reasoning community.