From Intuitionism to Many-Valued Logics Through Kripke Models

Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Godel (Kurt Godel collected works (Volume I) Publications 1929–1936, Oxford University Press, pp 222–225, 1932), and it is proved by Jaśkowski (Actes du Congres International de Philosophie Scientifique, VI. Philosophie des Mathematiques, Actualites Scientifiques et Industrielles 393:58–61, 1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Kripke (J Symbol Logic 24(1):1–14, 1959). Godel’s proof gave rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as Godel or the Godel-Dummett Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definability of propositional connectives in this logic.