In deontic logic, an operator intended to correspond to the modalities ‘it is obligatory that $\phi$ hold’, ‘it is obligatory that an agent $\alpha$ ensures that $\phi$ is true’, etc. Obligation is frequently treated as dual to permissibility $\text{P}$, so that when $\neg$ is a negation operator, $\text{O}\phi$ is treated as equivalent to (or definable as) $\neg \text{P}\neg \phi$, e.g.:

• • It is obligatory that an individual gives to the poor.

is arguably equivalent to the sentence:

• • It is not permissible that an individual should not give to the poor.

Formal deontic logics deal with subtler theses concerning the behaviour of the obligation operator. For example, an important thesis concerning obligation is Kant’s law, i.e., the thesis that the claim that $\phi$ is obligatory entails that $\phi$ is possible, and multi-modal logics with both modalities are capable of examining such a claim. Furthermore, logics dealing with contrary-to-duty obligations formalize circumstances in which an agent may have inconsistent obligations or that an agent failed to meet an obligation triggers a secondary obligation and conditional obligation formalizes obligations that hold only when some ancillary condition is met.

Although the name of the modal logic $\text{D}$ (for ‘deontic’) suggests that the necessity operator $\square$ is intended to formally capture the notion of the obligation operator, doubt has been cast upon the suitability of semantics with possible worlds as models of obligation. For example, although for many modalities, the accessibility relation $R$ can be given an intuitive reading (e.g., moving forward in time, adding new information to a state of knowledge), the reading of $wR{w}^{\prime }$ as ‘$w^{\prime }$ is a permissible alternative to $w$’ is far more obscure.