The Universal Model for the Negation-free Fragment of IPC

We identify the universal n-model of the negation-free fragment of the intuitionistic propositional calculus IPC. We denote it by U(n) and show that it is isomorphic to a generated submodel of the universal n-model of IPC, which is denoted by U(n). We show that this close resemblance makes U(n) mirror many properties of U(n). Finally, using U(n), we give an alternative proof of Jankov’s Theorem that the intermediate logic KC, the logic of the weak excluded middle, is the greatest intermediate logic extending IPC that proves exactly the same negation-free formulas as IPC.