Abstract
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 \(\mathsf {IPC}\).
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- 1.
- 2.
In fact, \(\Rightarrow \) is just the Heyting implication of the Heyting algebra of all upsets of W.
- 3.
Descriptive general frames are essentially the same as Esakia spaces (see e.g., [4, Sect. 2.3]). This topological perspective explains why compact general frames are called “compact” (the corresponding topology is compact). This also explains why \(\mathcal {Q}\) in Definition 3(3) is defined this way.
- 4.
Notice that some authors, e.g., [6] call such formulas negation-free.
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We would like to thank the referees of this paper for careful reading and many useful comments.
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Bezhanishvili, N., de Jongh, D., Tzimoulis, A., Zhao, Z. (2017). Universal Models for the Positive Fragment of Intuitionistic Logic. In: Hansen, H., Murray, S., Sadrzadeh, M., Zeevat, H. (eds) Logic, Language, and Computation. TbiLLC 2015. Lecture Notes in Computer Science(), vol 10148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54332-0_13
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