Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses ${\displaystyle {\mathsf {O}}A}$ to mean it is obligatory that ${\displaystyle A}$, or it ought to be (the case) that ${\displaystyle A}$, and ${\displaystyle {\mathsf {P}}A}$ to mean it is permitted (or permissible) that ${\displaystyle A}$, which is defined as ${\displaystyle {\mathsf {P}}A\equiv \neg {\mathsf {O}}\neg A}$.

In natural language, the statement "You may go to the zoo OR the park" should be understood as ${\displaystyle {\mathsf {P}}z\land {\mathsf {P}}p}$ instead of ${\displaystyle {\mathsf {P}}z\lor {\mathsf {P}}p}$, as both options are permitted by the statement. When there are multiple agents involved in the domain of discourse, the deontic modal operator can be specified to each agent to express their individual obligations and permissions. For example, by using a subscript ${\displaystyle {\mathsf {O}}_{i}}$ for agent ${\displaystyle a_{i}}$, ${\displaystyle {\mathsf {O}}_{i}A}$ means that "It is an obligation for agent ${\displaystyle a_{i}}$ (to bring it about/make it happen) that ${\displaystyle A}$". Note that ${\displaystyle A}$ could be stated as an action by another agent; One example is "It is an obligation for Adam that Bob doesn't crash the car", which would be represented as ${\displaystyle {\mathsf {P}}_{\mathit {Adam}}B}$, where ${\displaystyle B}$ denotes "Bob doesn't crash the car".

The term deontic is derived from the Ancient Greek: δέον, romanized: déon (gen.: δέοντος, déontos), meaning "that which is binding or proper."

In Georg Henrik von Wright's first system, obligatoriness and permissibility were treated as features of acts. Soon after this, it was found that a deontic logic of propositions could be given a simple and elegant Kripke-style semantics, and von Wright himself joined this movement. The deontic logic so specified came to be known as "standard deontic logic," often referred to as SDL, KD, or simply D. It can be axiomatized by adding the following axioms to a standard axiomatization of classical propositional logic:

In English, these axioms say, respectively:

${\displaystyle {\mathsf {F}}A}$, meaning it is forbidden that A, can be defined (equivalently) as ${\displaystyle {\mathsf {O}}\lnot A}$ or ${\displaystyle \lnot {\mathsf {P}}A}$.

There are two main extensions of SDL that are usually considered. The first results by adding an alethic modal operator ${\displaystyle \Box }$ in order to express the Kantian claim that "ought implies can":

where ${\displaystyle \Diamond \equiv \lnot \Box \lnot }$. It is generally assumed that ${\displaystyle \Box }$ is at least a KT operator, but most commonly it is taken to be an S5 operator. In practical situations, obligations are usually assigned in anticipation of future events, in which case alethic possibilities can be hard to judge. Therefore, obligation assignments may be performed under the assumption of different conditions on different branches of timelines in the future, and past obligation assignments may be updated due to unforeseen developments that happened along the timeline.