Deontic logic studies the logical behaviour of normative concepts and normative reasoning. Normative concepts include the notions of obligation (“ought”), permission (“may”), and prohibition (“must not”), and related concepts. The contemporary study of deontic logic was founded in 1951 by G.H. von Wright after the failure of an earlier attempt by Ernst Mally.

The simplest systems of deontic logic comprise ordinary first-order logic plus the pair of interdefinable deontic operators “it is obligatory that,” expressed by O, and “it is permissible that,” expressed by P. Sometimes these operators are relativized to an agent, who is then expressed by a subscript to the operator, as in O_b or P_d. These operators obey many (but not all) of the same laws as operators for necessity and possibility, respectively. Indeed, these partial analogies are what originally inspired the development of deontic logic.

A semantics can be formulated for such a simple deontic logic along the same lines as possible-worlds semantics for modal or epistemic logic. The crucial idea of such semantics is the interpretation of the accessibility relation. The worlds accessible from a given world W₁ are the ones in which all the obligations that obtain in W₁ are fulfilled. On the basis of this interpretation, it is seen that in deontic logic the accessibility relation cannot be reflexive, for not all obligations are in fact fulfilled. Hence, the law O_p ⊃ p is not valid. At the same time, the more complex law O(O_p ⊃ p) is valid. It says that all obligations ought to be fulfilled. In general, one must distinguish the logical validity of a proposition p from its deontic validity, which consists simply of the logical validity of the proposition O_p. In ordinary informal thinking, these two notions are easily confused with each other. In fact, this confusion marred the first attempts to formulate an explicit deontic logic. Mally assumed as a purportedly valid axiom ((O_p & (p ⊃ O_q)) ⊃ O_q). Its consequent, O_q, can nevertheless be false, even though the antecedent, (O_p & (p ⊃ O_q)), is true if the obligation that p is not in fact fulfilled.

In general, the difficulties in basic deontic logic are due not to its structure, which is rather simple, but to the problems of formulating by its means the different deontic ideas that are naturally expressed in ordinary language. These difficulties take the form of different apparent paradoxes. They include what is known as Ross’s paradox, which consists of pointing out that an ordinary language proposition such as “Peter ought to mail a letter or burn it” cannot be of the logical form O_p (m ∨ b), for then it would be logically entailed by O_p m, which sounds absurd. A similar problem arises in formalizing disjunctive permissions, and other problems arise in trying to express conditional norms in the notation of basic deontic logic.

Suggestions have repeatedly been made to reduce deontic logic to the ordinary modal logic of necessity and possibility. These suggestions include defining the following

(1) p is obligatory for a if and only if it is necessary that p for a’s being a good person. (2) p is obligatory if and only if it is prescribed by morality. (3) p is obligatory if and only if failing to make it true implies a threat of a sanction.

These may be taken to have the following logical forms:

(1) N(G(a) ⊃ p) (2) N(m ⊃ p) (3) N(∼p ⊃ s)

where N is the necessity operator, G(a) means that a is a good person, m is a codification of the principles of morality, and s is the threat of a sanction.

The majority of actual norms do not concern how things ought to be but rather concern what someone ought to do or not to do. Furthermore, the important deontic concept of a right is relative to the person whose rights one is speaking of; it concerns what that person has a right to do or to enjoy. In order to systematize such norms and to discuss their logic, one therefore needs a logic of agency to supplement the basic deontic logic. One possible approach would be to treat agency by means of dynamic logic. However, logical analyses of agency have also been proposed by philosophers working in the tradition of deontic logic.

It is generally agreed that a single notion of agency is not enough. For example, von Wright distinguished the three notions of bringing about a state of affairs, omitting to do so, and sustaining an already obtaining state of affairs. Others have started from a single notion of “seeing to it that.” Still others have distinguished a’s doing p in the sense that p is necessary for something that a does and in the sense that it is sufficient for what a does.

It is also possible—and indeed useful—to make still finer distinctions—for example, by taking into account the means of doing something and the purpose of doing something. Then one can distinguish between sufficient doing (causing), expressed by C(x,m,r), where for x m suffices to make sure that r; instrumental action E(x,m,r,), where x sees to it that r by means of m; and purposive action, A(x,r,p), where x sees to it that r for the purpose that p.

There are interesting logical connections between these different notions and many logical laws holding for them. The main general difficulty in these studies is that the model-theoretic interpretation of the basic notions is far from clear. This also makes it difficult to determine which inferential relations hold between which deontic and action-theoretic propositions.