Positive Formulas in Intuitionistic and Minimal Logic
Dick de Jongh, Zhiguang Zhao
Abstract:
In this article we investigate the positive, i.e.\ $\neg,\bot$-free formulas of intuitionistic propositional and predicate logic, IPC and IQC, and minimal logic, MPC and MQC. For each formula $\varphi$ of IQC we define the positive formula $\varphi^+$ that represents the positive content of $\varphi$. The formulas $\varphi$ and $\varphi^+$ exhibit the same behavior on top models, models with a largest world that makes all atomic sentences true. We characterize the positive formulas of IPC and IQC as the formulas that are immune to the operation of turning a model into a top model. With the +-operation we show, using the uniform interpolation theorem for IPC, that both the positive fragment of IPC and MPC respect a revised version of uniform interpolation. In propositional logic the well-known theorem that KC is conservative over the positive fragment of IPC is shown to generalize to many logics with positive axioms. In first-order logic, we show that IQC + DNS (double negation shift) + KC is conservative over the positive fragment of IQC and similar results as for IPC.