In this paper we give an axiom system of a non-linear 4-valued logic which we call a de Morgan logic (ML), whose Lindenbaum algebra is the de Morgan algebra with implication (MI-algebra), and show that
(1) For every MI-algebra L, there is a quotient MI-algebra L^# such that it is embeddable to the simplest 4-valued MI-algebra M = {0, a, b, 1};
(2) The Lindenbaum algebra of ML is the MI-algebra;
(3) The completeness theorem of ML is established;
(4) ML is decidable.