Universal Models for the Positive Fragment of Intuitionistic Logic

We describe the n-universal model $$\mathcal {U}^\star n$$ of the positive fragment of the intuitionistic propositional calculus $$\mathsf {IPC}$$. We show that $$\mathcal {U}^\star n$$ is isomorphic to a generated submodel of $$\mathcal {U}n$$ --- the n-universal model of $$\mathsf {IPC}$$. Using $$\mathcal {U}^\star n$$, we give an alternative proof of Jankov's theorem stating that the intermediate logic $$\mathsf {KC}$$, the logic of the weak law of excluded middle, is the greatest intermediate logic extending $$\mathsf {IPC}$$ that proves exactly the same positive formulas as $$\mathsf {IPC}$$.