Many Valued Logic of Gödel and Łukasiewicz

Gödel and Łukasiewicz proposed the three-valued logic by adding the third logical situation, which includes uncertainty and ambiguity, to the classical two logical values, true or false. These logical systems were generalized to the many types of many valued logics, and especially, Gödel’s many valued logic was developed to Heyting algebra and Łukasiewicz’s one to lattice implication algebra. In this paper, we introduce the many valued logics of Gödel and Łukasiewicz, and Heyting’s algebra and lattice implication algebra that are generalizations of Gödel’s and Łukasiewicz’s logic, respectively. Also, we research the properties and relationship of Heyting algebras and lattice implication algebras, especially by defining another implication on a finite lattice implication algebra, we prove finite implication algebra is a special case of Heying algebras.