A family of neighborhood contingency logics

TL;DR

This paper develops axiomatizations for various neighborhood contingency logics, providing sound and complete systems for monotone and regular classes, and clarifies the relationship between Kuhn's and Humberstone's functions.

Contribution

It introduces a canonical neighborhood function that achieves axiomatizations for multiple classes of neighborhood contingency logics, resolving open questions.

Findings

Axiomatizations for monotone and regular contingency logics are established.

Kuhn's neighborhood function is shown to be equivalent to Humberstone's function.

The work answers previously open questions in the field.

Abstract

This article proposes the axiomatizations of contingency logics of various natural classes of neighborhood frames. In particular, by defining a suitable canonical neighborhood function, we give sound and complete axiomatizations of monotone contingency logic and regular contingency logic, thereby answering two open questions raised by Bakhtiari, van Ditmarsch, and Hansen. The canonical function is inspired by a function proposed by Kuhn in~1995. We show that Kuhn's function is actually equal to a related function originally given by Humberstone.