PP-2014-06: de Jongh, Dick and Zhao, Zhiguang (2014) Positive Formulas in Intuitionistic and Minimal Logic. [Report]
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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.
| Item Type: | Report |
|---|---|
| Report Nr: | PP-2014-06 |
| Series Name: | Prepublication (PP) Series |
| Year: | 2014 |
| Uncontrolled Keywords: | intuitionistic logic; minimal logic; Jankov's logic; intermediate logics; positive formulas; interpolation; conservativity |
| Subjects: | Logic |
| Date Deposited: | 12 Oct 2016 14:37 |
| Last Modified: | 12 Oct 2016 14:37 |
| URI: | https://eprints.illc.uva.nl/id/eprint/494 |
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