Some Properties of a Lattice Generated by Implicational Logics

Abstract

Consider the following implicational formulae: A1 = (p ⊃ p), A2 = (((p ⊃ p) ⊃ p) ⊃ p) = ((A1 ⊃ p) ⊃ p), …, Ai+1 = ((Ai ⊃ p) ⊃ p), …, (i = 1, 2, 3, …). Using these formulae (axioms), we may construct the following logics: L1 =< A2i >, L2 =< (A2i ⊃ A2i+1) >, L3 =< A2i−1 >, L4 =< (A2i−1 ⊃ A2i) >, i = 1, 2, 3, … viz. the logic L1 is generated by the axioms A2i, i = 1, 2, 3, ...; the process is analogous for logics L2, L3, L4.