An Intermediate Logic Contained in Medvedev's Logic with Disjunction Property

2025/02/24 by Chen, Zhicheng · 1 citation
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2502.17242

Abstract

Let SU be the superintuitionistic logic defined by the axiom \boldsymbolsu = ((¬ p→ q)∧(¬ q→ p) → r \vee s) → ( p → r) \vee(q → s), or equivalently, by Andrew's axiom. It is easy to check that SU is contained in Medvedev's logic and contains both Kreisel-Putnam logic and Scott logic. We show that on S4 frames, \boldsymbolsu corresponds to a certain first-order property, called the ``strong union'' property. The strong completeness of SU, with respect to the class of S4 frames enjoying this property, is proved. Furthermore, we demonstrate that SU has the disjunction property. As a result, SU stands as the strongest logic currently known below Medvedev's logic that has both an axiomatization and the disjunction property.

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