From Intuitionism to Many-Valued Logics through Kripke Models

Abstract: Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Kurt G\"odel (1932), and it is proved by Stanis{\l}aw Ja\'skowski (1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Saul Kripke (1959). G\"odel's proof gave rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as G\"odel or the G\"odel-Dummet Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definablility of propositional connectives in this logic.