From Intuitionism to Many-Valued Logics through Kripke Models

TL;DR

This paper demonstrates that intuitionistic propositional logic can be characterized as an infinitely many-valued logic using Kripke models, offering alternative proofs and exploring its relation to G"odel logic.

Contribution

It provides new proofs of the many-valued nature of intuitionistic logic via Kripke models and investigates the inter-definability of propositional connectives within this framework.

Findings

Intuitionistic logic is an infinitely many-valued logic.

Alternative proofs using Kripke models are established.

Results on propositional connectives' inter-definability are presented.

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.