PP201406: 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 wellknown theorem that KC is conservative over the positive fragment of IPC is shown to generalize to many logics with positive axioms. In firstorder 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:  PP201406 
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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