Deontic Logic

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by Hugo Zuleta

I. Introduction

Deontic logic is the branch of symbolic logic that studies the formal properties of normative concepts.
It contributes to the general theory of law with a formal analysis of such concepts as obligation, permission, prohibition, commitment, rule, authority, power, rights, and responsibility. It analyses the formal properties of normative systems, helping to clarify notions such as legal gaps and legal contradictions.
The first viable system of deontic logic was presented by G.H. von Wright in his classic essay ‘Deontic Logic’ (1951). There was a previous attempt to build a formal theory by Ernst Mally, in 1926, but it was unsuccessful. As a matter of historic curiosity, it can be added that it is possible to find suggestions of a logic treatment of normative concepts as far as Aristotle, the Stoics, a modern philosopher like Leibniz, and also in Bentham.
After von Wright’s seminal paper, many systems of deontic logic were developed, even by von Wright himself. Many of them were designed to avoid certain paradoxical results that were seen to arise in his original system.
Many problems remain open. From a philosophical point of view, the main one concerns the interpretation and validity of its basic principles.
There is also a great deal of controversy about the proper way to represent some basic deontic notions such as those of commitment and conditional obligation.
Many contemporary studies in the field are oriented to the formal representation of legal knowledge, the analysis of legal argumentation and the links between deontic logic and computer science, artificial intelligence, and organization theory. In this line of research, important efforts are focused on applications, such as the formal specification of systems for the management of bureaucratic processes in public or private administration, database integrity constraints, computer security protocols, electronic institutions, and norm-regulated multi-agent systems.
Most systems of deontic logic are built upon propositional logic, and lack the expressive resources of quantification. As a consequence, their applicability to real-life normative discourse in moral o legal contexts is rather imperfect. In order to represent legal knowledge, it seems that deontic logic languages must be enriched not only with quantification but also with notions for agency and temporal devices. In recent years many lines of research were headed in those directions.

II. The Classic and the Standard Systems

Von Wright’s approach (1951) is generally known as ‘the Classic System’. It is based upon the observation that there is an analogy between the deontic notions ‘obligation’ and ‘permission’ and the alethic modal notions ‘necessity’ and ‘possibility’. The deontic concepts are interdefinable: If ‘permission’ is taken as primitive, as in von Wright’s essay, ‘Op’ (it is obligatory to p) can be defined as ‘¬P¬p’ (it is not the case that not to do p is permitted), where p stands for the name of a generic act, like swimming or smoking. Conversely, if obligation is taken as primitive, ‘Pp’ can be defined as ‘¬O¬p’. The concepts of ‘necessity’ and ‘possibility’ are related in the same way. If we substitute ‘possible’ for ‘permitted’ and ‘necessary’ for ‘obligatory’ in the aforementioned definitions, we obtain parallel interdefinitions for the alethic modal concepts.[1]
Other similarities between alethic modal notions and deontic notions are represented in the following graphic.

The vertical arrows express subalternation (entailment from the upper to the lower sentence), the diagonals indicate contradictions, the upper horizontal line means contrariness (both cannot be true), and the bottom horizontal line expresses subcontrariness (both cannot be false). The deontic expressions (in blue) maintain subalternation, contradiction, contrariness, and subcontrariness in the same way as the alethic modal expressions (in red).
Moreover, the notions of ‘permission’ and ‘possibility’ behave in the same way with respect to conjunction and disjunction, and the same happens between ‘obligation’ and ‘necessity’. Permission can be distributed into the disjunction and obligation can be distributed into the conjunction. On the other hand, ‘it is permitted to p and q’ entails ‘it is permitted to p and it is permitted to q’, but the converse is not valid; and ‘it is obligatory to p or it is obligatory to q’ entails ‘it is obligatory to p or to q’, but, again, the converse is not valid.
The most significant difference between deontic and alethic modal concepts lies in the fact that whereas alethic modal logic accepts as valid the implication from ‘p’ to ‘it is possible that p’, and from ‘it is necessary that p’ to ‘p’, any sound deontic logic must reject their analogues: that p is the case does not imply that it is permitted, and that p is obligatory does not imply that it is the case. Obligations can be violated and impermissible things do hold. Von Wright’s system is included in many other systems of deontic logic. Among them, the best known and most discussed, and one of the first axiomatically specified, is the so called ‘Standard System.
There are two important differences between standard deontic logic (SDL) and the classic system: First, in SDL the deontic operators are understood as operating on propositions, not on names of generic acts. So, ‘Op’ is to be understood as stating the obligation that some state of affairs –the one described by p– be the case, not as expressing the obligation to perform a certain type of action. It must be read as ‘it is obligatory that p’, not as ‘it is obligatory to p’. SDL may be conceived as a logic about what ought to be the case, as opposed to a logic about what ought to be done. This feature seems to obscure the agent relativity of legal obligations, permissions, and prohibitions. Second, there is a rule of inference to the effect that every logically true proposition (e.g. every tautology), is obligatory, and, as a consequence, every contradiction is forbidden (this rule resembles the rule of necessitation of alethic modal logic). Both modifications strengthen the analogy with modal logic.

III. The paradoxes of deontic logic

There is a large number of problems and limitations attributed to standard deontic logic. They are usually called ‘paradoxes’, but that word is used here in a loose sense. Some of them are not real paradoxes but results that could be found counter-intuitive. At any rate, they show that the formal language does not reflect faithfully the way in which some normative statements are generally understood in ordinary language.

Ross’s paradox[2]: In SDL, O(pvq) (it ought to be the case that p or q) can be derived from Op. So, if it ought to be the case that a letter is mailed, then it ought to be the case that the letter is mailed or burnt.
It seems rather odd to say that an obligation to mail a letter entails an obligation that can be fulfilled by burning it. However, this is a misunderstanding. The implication does not mean that the original obligation can be fulfilled by burning the letter. By propositional logic, whenever ‘p’ is true, it is also true any disjunction of which ‘p’ forms part. So, if it is obligatory to see to it that p is the case, it is obligatory to see to it that the disjunction of ‘p’ with any proposition is the case.
The air of paradox derives from the fact that, in ordinary language, a disjunctive obligation is generally understood as one in which the agent is free to choose any of the alternatives; but this is not the meaning of O(pvq) in SDL.

The paradox of derived obligation[3]: In the Classic System, as well as in SDL, the idea of conditional obligation (or commitment) is represented by ‘O(p⊃q)’, where ‘p⊃q’ is understood as a material conditional. It can be proved that, if some state of affairs, say p, is forbidden, then it is obligatory any conditional in which p is the antecedent. So, if it is forbidden that I steal a gun, then it ought to be that if I steal a gun I kill someone.
Now, if we substitute ‘¬pvq’ for ‘p⊃q’, which is logically equivalent, it is easy to see that this paradox is but a variation of Ross’s.

Contrary-to-Duty Paradox[4]: Consider the following: (1) It ought to be that John visits his mother. (2) It ought to be that if John visits his mother then he tells her he is coming. (3) If John doesn't visit his mother, then he ought not to tell her he is coming. (4) John doesn't visit his mother.
Proposition (3) expresses what Chisholm named ‘a contrary-to-duty imperative’, it says what a person ought to do if she has violated her duties.
It is reasonable to expect that (1)–(4) constitute a mutually consistent and logically independent set of sentences. Yet, it can be shown that, if we represent the logical form of (2) as O(p⊃q), and represent (3) as ¬p⊃O¬q, a contradiction can be derived in SDL, as can be easily shown: From (1) and (2) we obtain, by deontic detachment (or deontic Modus Ponens), ‘it ought to be the case that John tells his mother he is coming’, and from (3) and (4) we get, by factual detachment (or factual Modus Ponens), ‘John ought not to tell his mother he is coming’. So, if norms (1)-(3) hold, it is logically impossible that John doesn’t visit his mother, which is absurd.
The cause of the paradox seems to be that SDL allows both factual and deontic detachments. This led to attempts to block or modify one or both of those detachment principles, to introduce temporal restrictions to the deontic operators, and also to a reconsideration of the formalization of conditional obligations.
This last idea was explored, among others, by von Wright. In von Wright (1956) he presented a new system of deontic logic in which the deontic operators are intrinsically associated with conditionality. The atomic expressions have the form ‘O(p/q)’ and ‘P(p/q)’, which can be read ‘it is obligatory that p given q’ and ‘it is permitted that p given q’ respectively. The systems that use this kind of deontic operators are named ‘dyadic deontic logics'.
Nowadays it seems to be generally admitted that material implication does not express the notion of conditional obligation faithfully, and the dyadic approach tends to be the one most commonly followed.

IV. The problem of interpretation

In standard logic, to say that a sentence derives from a set of sentences means asserting that, in every possible interpretation in which all the sentences appearing as premises are true, so it is the sentence appearing as conclusion. The meaning of the logical connectives is also characterized by the truth-value given to the propositional compounds taking into account the truth value of the component propositions. Now, if norms do not have any truth-values, it is not clear what could possibly mean to say that a norm logically follows from other norms or that two norms are contradictory; nor is it clearer what the conjunction of two norms means. However, it seems intuitively true that there are logical relations between norms, that some norms are incompatible, and that the conjunction of two norms makes sense.
The problem is usually framed as a dilemma –called ‘Jørgensen’s dilemma’–: either there are no logical relations between norms and logical connectives cannot be applied to norms or else the logical relations and the logical connectives can be characterized in a way that does not involve the notions of truth and falsity.
The dilemma is based on the philosophical assumption that norms cannot have truth-values. So, an obvious way out would be to deny this claim. As a matter of fact, most presentations of deontic logic treat norms as if they could bear truth-values. This is usually explained by a semantic theory of possible worlds.
In von Wright (1963) a distinction is made between norms, norm-formulations, and norm-propositions. Norm-formulations have a characteristic ambiguity: the same sentence may be used both prescriptively to rule other people’s behaviour and descriptively for stating that some norm exists. In the descriptive use norm-formulations do have truth-value; then their logical relations can be accounted for in the standard way. Accordingly, the author distinguishes between a prescriptive and a descriptive interpretation of the deontic formulas. He considers deontic logic as a logic of descriptively interpreted expressions. “But –he adds– the laws (principles, rules) which are peculiar to this logic, concern logical properties of the norms themselves, which are then reflected in logical properties of norm-propositions. Thus, in a sense, the ‘basis’ of Deontic Logic is a logical theory of prescriptively interpreted O- and P- expressions” [p.134].
Von Wright seems to have thought that deontic operators have the same logical properties under a descriptive as under a prescriptive interpretation. However, Alchourrón (1969) showed that the logic of normative propositions differs from that of norms in some important aspects. In that paper a logical system of normative propositions is presented for the first time. Yet, it is explicitly based on the logic of norms.
Alchourrón and Bulygin (1981) explore the consequences of characterizing norms as the result of the prescriptive use of language. In this view, named ‘the expressive conception’, norms only differ from assertions or questions on the pragmatic level; there is no difference on a semantic level. The possibility of a logic of norms is then precluded, but they develop a logic of the descriptive norm-contents, i.e. of normative propositions. The system obtained looks very much like the classic system. However, it is dubious that a system of normative propositions could replace a real deontic logic because, in order to justify or criticize a decision, it seems necessary to use norms. This is not the same as using descriptive sentences about the existence of norms in some normative system.
In von Wright (1983) the possibility of deontic logic is rejected. However, an analogous system is derived from a set of minimal criteria for rational legislation. Alchourrón and Martino (1990) grasp the second horn of Jørgensen’s dilemma. They contend that the notion of deductive consequence can be characterized by some formal properties, without any reference to truth values. Then, taking that notion as primitive, the deontic operators, as well as the logical connectives, can be defined by sintactic rules that indicate how to introduce and how to eliminate them within a deductive context.

V. Conluding remarks

Since deontic logic was launched by von Wright as an academic specialization the proliferation of competing formal systems is impressive. They can be classified by different criteria. They can be divided into monadic or dyadic deontic logics according to the deontic operator they use. According to their treatment of conditional norms, they can be characterized as deontic logics with material implication, with necessary implication, with defeasible conditionals, with counterfactual conditionals, etc. They can also be divided into different groups according to the kind of detachment they admit (factual, deontic, or both).
It is an area in which there is still a good deal of disagreement about fundamental matters.


Alchourrón, C. (1969), “Logic of Norms and Logic of Normative Propositions”, Logique et Analyse 12, 242-268.

Alchourrón C. and Bulygin, E. (1981), “The Expressive Conception of Norms”. See Hilpinen, R. (1981), pp. 95-124.

Alchourrón, C. and Martino, A. (1990), “Logic without Truth”, Ratio Juris 3: 46-67.

Åqvist, L. (2002), “Deontic Logic”. In Gabbay, D. and Guenthner, F. (eds.), Handbook of Philosophical Logic, Kluwer, Dordrecht, vol. 8.

Hilpinen, R. (1971), Deontic Logic: Introductory and Systematic Readings, D. Reidel, Dordrecht.

Hilpinen, R. (1981), New Studies in Deontic Logic, D. Reidel, Dordrecht.

Horty, J. (2001), Agency and Deontic Logic, Oxford U.P., Oxford-New York.

McNamara, P. (2006), “Deontic Logic”, The Stanford Encyclopedia of Philosophy, electronic publication.

Nute, D. (1997), Defeasible Deontic Logic, Kluwer, Dordrecht.

Goble, L. and Meyer, J. (eds.) (2006), Deontic Logic and Artificial Normative Systems, Springer, Berlin-Heidelberg.

Von Wright, G. (1951), “Deontic Logic”, Mind 60, 1-15.

Von Wrhigt, G. (1956). “A Note on Deontic Logic and Derived Obligation”, Mind 65: 507-509.

Von Wright, G. (1963), Norm and Action, Routledge & Kegan Paul, New York.

Von Wright, G. (1983), “Norms, Truth, and Logic”. In von Wright, G., Practical Reason, Cornell U.P., Ithaca, New York.


[1] The analogies of interdefinability were first noticed by Leibniz in Elementa Iuris Natralis, 1672.

[2] Ross, A. (1941), “Imperatives and Logic,” Theoria 7: 53-71.

[3] Prior, A. N. (1954), “The Paradoxes of Derived Obligation,” Mind 63: 64-65.

[4] Chisholm, R. (1963), “Contrary-to-Duty Imperatives and Deontic Logic,” Analysis 24: 33-36.

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