Navigating Permission and Exceptions: A First-Order Perspective

Abstract

First-order deontic logic has received relatively little attention from both the deontic logic and first-order modal logic communities, particularly regarding the interaction between permission and quantifiers. In this work, we examine this interaction and introduce a logic that formally captures the notion of permission-as-exceptions and handles contrary-to-duty paradoxes, shedding new light on the advantages of adding quantifiers to a deontic logic. However, developing such a logic introduces challenges, particularly in managing the interplay between quantifiers and modal operators. Notably, avoiding problematic formulas—like the (converse) Barcan formulas—requires restricting seemingly intuitive rules in a first-order deontic proof system. To address this, we propose a varying domain neighbourhood semantics that captures our notion of permission but invalidates undesirable formulas while preserving key logical properties, including the monotonicity of permission and existential generalization for quantification. Additionally, we present a sound Hilbert-style proof system and show how our semantics prevents the derivation of problematic formulas.