TbiLLC 2013: Logic, Language, and Computation pp 175-189

Positive Formulas in Intuitionistic and Minimal Logic

• Dick de Jongh
• Zhiguang Zhao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8984)

Abstract

In this article we investigate the positive, i.e. $$\lnot ,\bot$$-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 on formulas 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.

Keywords

Intuitionistic logic Minimal logic Jankov’s logic Intermediate logics Positive formulas Interpolation Conservativity

Notes

Acknowledgement

We thank Albert Visser, Nick Bezhanishvili, Rosalie Iemhoff, Grisha Mints and Anne Troelstra for informative discussions on the subject. We thank the referees for their corrections and Linde Frölke for her preparatory work.

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