A Note on Nesting in Dyadic Deontic Logic

TL;DR

This paper investigates properties of Aqvist's dyadic deontic logic system G, demonstrating that nested modal formulas can be simplified and that the universal modality is definable within the system.

Contribution

It proves that nested modal operators in system G are equivalent to non-nested formulas and that the universal modality can be defined using the deontic modality.

Findings

Nested formulas are equivalent to non-nested ones in system G.

The universal modality is definable in terms of the deontic modality.

Properties of system G are clarified and simplified.

Abstract

The paper reports on some results concerning Aqvist's dyadic logic known as system G, which is one of the most influential logics for reasoning with dyadic obligations ("it ought to be the case that ... if it is the case that ..."). Although this logic has been known in the literature for a while, many of its properties still await in-depth consideration. In this short paper we show: that any formula in system G including nested modal operators is equivalent to some formula with no nesting; that the universal modality introduced by Aqvist in the first presentation of the system is definable in terms of the deontic modality.