On axiom systems of propositional calculi, III
Yoshinari Arai · Proceedings of the Japan Academy Series A Mathematical Sciences · 1965
In this note, we shall concern with Lukasiewicz (L)axioms (see Y. Imai and K. Iski 1).As mentioned in our previous papers, we only use the rules of substitution and detachment.The fundamental axioms are the following three theses" 1CpCqp, 2 CCpCqrCCpqCpr, 3CCNpNqCqp.We shall first give a proof of (L)(L).From the (L)-system, we have the following theses:1 p/CCNqNpCpq, q/Np *C3 p/q,, q/p4, 4 CNpCCNqNpCpq.2 p/Np, q/CNqNp, r/Cpq *C4C1 p/Np, q/Nq5, 5 CNpCpq.2 p/Nq, v/p *65 p/q, q/p6, 6 CCNqqCNqp.1 p/CCNqqCNqp *C6--7, 7 CqCCNqqCNqp.2 p/q, q/CNqq, /CNqp *C7--C1 p/q, q/Nq--8, 8CqCNqp. 8 q/p, p/q *C9, 9 CpCNpq.i p/CCpCqrCCpqCpr, q/Cqr *C210, 10 CCqrCCpCqrCCpqCpr.2 p/Cqr, q/CpCqr, r/CCpqCpr *C10