The Logics of Individual Medvedev Frames

TL;DR

This paper studies the logic of Medvedev frames based on non-empty subsets of a set of size n, providing axiomatizations and analyzing properties like compactness and completeness.

Contribution

It offers a uniform axiomatization of n-Medvedev logics and explores their properties, extending understanding beyond the classical Medvedev logic.

Findings

Characterization of n-Medvedev frames via maximal points

Axiomatization using Gabbay-style rules

Analysis of properties like compactness and structural completeness

Abstract

Let $n$-Medvedev's logic $\mathbf{ML}_n$ be the intuitionistic logic of Medvedev frames based on the non-empty subsets of a set of size $n$, which we call $n$-Medvedev frames. While these are tabular logics, after characterizing $n$-Medvedev frames using the property of having at least $n$ maximal points, we offer a uniform axiomatization of them through a Gabbay-style rule corresponding to this property. Further properties including compactness, disjunction property, and structural completeness of $\mathbf{ML}_n$ are explored and compared to those of Medvedev's logic $\mathbf{ML}$.