Deontic Logic

Ernst Mally made the first attempts at a formal system for deontic logic in 1926; but his system turned out to trivialize the modality ("φ is obligatory" collapsed into just φ). Formal work on deontic logic got a new start with the work of G.H. von Wright starting with an article in Mind in 1951.

What has come to be known as "Standard Deontic Logic" is just the normal modal logic system D, with □φ interpreted as "it is obligatory/required that φ", and is usually written as Oφ. When φ describes you performing some action A, it's common to pronounce Oφ as "You ought to do A." However, this construal raises several difficulties, which we'll discuss below. (Nonetheless, I and you and other authors will no doubt talk this way regardless.) ◇φ is interpreted as "it is permitted that φ" (or "You may do A"), and is usually written as Pφ. □¬φ is interpretted as "it is forbidden that φ"; and ¬□φ (or ◇¬φ) is interpreted as "it is omissable that φ."


For reference, let me list some rules that a deontic logic may or may not include:

RE. From ⊢ φ ⊂⊃ ψ, infer ⊢ Oφ ⊂⊃ Oψ.
RM. From ⊢ φ ⊃ ψ, infer ⊢ Oφ ⊃ Oψ. (This rule is sometimes called "R*". It is equivalent to RE together with axiom M, described below.)
Nec[essitation]. From ⊢ φ, infer ⊢ Oφ.

Rule RE is among the least contested deontic principles; the other two rules (especially RM) are more controversial.

The modal system SDL or D can be axiomatized as the classical propositional logic plus the rule Nec (implying that "not everything is permitted"), together with the axiom:

K. O(φ ⊃ ψ) ⊃ (Oφ ⊃ Oψ)

and the axiom:

D. Oφ ⊃ Pφ

or equivalently, the axiom:

P. P⊤ ("something is permitted", also expressible as ¬O⊥)

D and P correspond to requiring that the accessibility relation in the semantics is extendable/serial.

Some authors accept P, interpreted as the claim that no single contradictory state of affairs is obligatory; but reject D, interpreted as the claim that if φ is obligatory, it cannot be that there is also some competing obligation that ¬φ. In SDL, these claims are equivalent; but perhaps they shouldn't be.

In SDL (as in any normal modal logic), the other rules RE and RM can be derived.

A different way of axiomatizing SDL is to include not the rule Nec, but rather the weaker rule RE, together with these four axioms:

M. O(φ ∧ ψ) ⊃ (Oφ ∧ Oψ)
C. O(φ ∧ ψ) ⊂ (Oφ ∧ Oψ) (written this way to demonstrate that it's the converse of M)
N. O⊤
P. P⊤ (or ¬O⊥)

Theorists who are reluctant to accept D because of the intelligibility of deontic conflicts will also be reluctant to accept C.


Some authors work with stronger systems than SDL. One idea is to add the converse of the 4 axiom, namely:

C4. Oφ ⊂ OOφ

This corresponds to requiring that the accessibility relation in the semantics (representing "deontic acceptability") is dense.

Some authors add the stronger axiom:

U. O(Oφ ⊃ φ) (this is the necessitation of axiom T)

or equivalently, O(φ ⊃ Pφ). In other words, it ought to be that only permissible things are true. (In a normal modal logic, this will entail C4.) This axiom corresponds to requiring that the accessibility relation is reflexive for worlds that are accessible to some other world. It has the effect of making it the case that no world where an obligation is violated can be an "acceptable alternative" to any world.

The logic SDL together with axiom U is called SDL+.

Another idea is to add the 4 axiom itself:

4. Oφ ⊃ OOφ

In other words, what is obligatory ought to be obligatory. I believe that this is independent of the two other strengthenings just described; though if we add both 4 and C4, that entails U.

As we'll see, in the face of some puzzles, there is some intuitive pressure to be working with a weaker modal logic than D/SDL, not a stronger one.

Issues of interpretation

Although a formula like "O(You wash the car)" is often pronounced as "You ought to wash the car," linguists have argued that there is a contrast in natural language between deontic modals like "ought" and "should", on the one hand, and modals like "must", "have to", and "are obligated/required to," on the other hand. The former are called "weak necessity modals" and the latter "strong necessity modals." Examples like this bring out the difference:

You ought to wash the car, but you don't have to.

If the O of deontic logic captures any natural English expression, it's most likely to be the stronger modals.

The labels "strong" and "weak" are also sometimes used to characterize two different kinds of permission: strong permission meaning the activity has explicitly been sanctioned, and weak permission meaning it merely hasn't been forbidden. (von Wright distinguishes these in his 1964 book.) If there is a genuine linguistic or normative difference here, it's an open issue what relation, if any, it has to the contrast between "weak" and "strong" deontic necessities. These labels are also sometimes used to characterize different views about the behavior of epistemic "must": one dominant view in linguistics saying that "Must φ" makes a "weaker" claim than simply φ, and the other view saying that it makes at least as strong a claim. von Fintel and Gillies defend the latter view. I mention these views about permission and epistemic "must" only to set them aside.

There is some awkwardness interpreting all the logician's uses of Oφ in terms of either the English expression "ought" or the English expression "obligatory." Notice the formula N, which is a theorem of SDL:

N. O⊤

What does it mean to say that it is obligatory that I be self-identical, or that I not be both mortal and immortal? These claims are hard to understand. Even harder to understand are claims of the form Oφ that make no reference to me, or any other agents at all. So perhaps Oφ should instead be understood to mean something like "It logically follows from all obligations being met that φ". This is the motivation for Kanger's and Anderson's version of deontic logic, Kd. Their systems work with an alethic modality □ (at least as strong as K) and a primitive constant d meaning something like "all obligations are met" or "no obligations have been violated". The only new axiom beyond those needed for □ is that ◇d, that is, all obligations might be met. We then define Oφ as □(d ⊃ φ). The resulting system is essentially equivalent to SDL. If the modality □ is assumed also to respect axiom T, then the resulting system is at least as strong as SDL+. (I don't know whether it's stronger.)

Another interpretive move is to deny that O should be an operator on arbitrary formula, but instead have it be an operator on a restricted set of predicates, expressing types of actions that agents can perform. This was in fact von Wright's initial strategy; but starting in 1964 he went along with the rest of the logic community (and Mally) in making O be an operator on sentential formulas.

At the surface level, in English, "ought" seems to combine with infinitival clauses like "to wash the car", to yield a predicate. But most linguists would analyze the infinitival clause as containing an unpronounced subject term (called PRO), and thus being syntactically a kind of sentence. Thus on their view "ought" combines with sentences to yield a predicate. The intepretation of the subject term PRO of the complement sentence is determined by what the argument is of the complex predicate "ought to wash the car."

Schroeder gives some arguments for thinking that "ought" in English doesn't in fact take sentential clauses as its argument. However, the evidence is more complex than he presents. We may look into this. Fogal discusses the issue briefly in his thesis.

Even if the dominant linguistic view is right, there is still the issue that "ought" in English doesn't just combine with a sentence, but (in "ought to wash the car") also requires a subject term. Perhaps we should instead consider claims of the form "it ought to be that you wash the car" or "it is obligatory that you wash the car." In English it's not clear that the first form always expresses something specifically deontic as opposed to evaluative in a broader sense ("it ought to be that you are better paid").

The idea that your obligation to do something is equivalent to there being a (non-agential, perhaps not even agent-relative) obligation that you do it is associated with Chisholm and Meinong.

In the SEP article on deontic logic, Paul McNamara presents other reasons for doubting that our natural language deontic vocabulary is correctly intepreted in terms of O in any standard deontic logic:


We'll discuss these in class. But for a quick overview, let's begin with some puzzles that challenge the rule:

RM. From ⊢ φ ⊃ ψ, infer ⊢ Oφ ⊃ Oψ.

which is a derived rule in any deontic logic (like SDL, SDL+, or Kd) that is a strengthening of K. It may be that some non-standard interpretation of Oφ could explain away some of the counter-intuitiveness of these results.

A second group of puzzles has to do with how to represent ideas like "A commits/requires you to B". Assume we're working within SDL. If we try to represent that as "A → OB", and interpret the → as ⊃, we'll get the results that false As commit you to everything, and that everything commits you obligatory Bs. If on the other hand we try to represent it as "O(A → B)", and interpret the → as ⊃, we'll again get the result that everything commits you obligatory Bs, and also the result that forbidden As commit you to everything.

Deontic logicians don't agree about much, but they do largely agree that these notions can't be represented in SDL using only the monadic operator O and the material conditional. Some of these theorists prefer to introduce a new dyadic operator O(• | •), where the old notion of Oφ can be expressed as O(φ | ⊤). Others prefer to stick with monadic O but to introduce a special, non-truth-functional conditional. If you're tempted to interpret it as the strict conditional , note that φ Oψ entails φ ∧ anything ⥽ Oψ. Bonevac instead interprets → as a non-monotonic conditional, with which we can say:

N is a bird → N can fly.
N is a bird that lives in Antarctica → N cannot fly.

Broome's strategy is to deny that "A commits/requires you to B" is fully analyzed by "O(A ⊃ B)" (his favored choice), but instead to say that it only entails this.

A related, third group of puzzles has to do with counter-to-duty obligations; that is, obligations having to do with what you should do if you violate some other obligation. Some things meeting that description might be weaker, "the least you can do" obligations as described before. Other obligations come into effect only in the cases where you've violated some other obligation; in fact, they might describe things you're obligated not to do if you've fulfilled the other obligations.

A classic example is Chisholm's Puzzle (1963). In plain English: You ought to visit your grandmother, though in fact you won't do so. If you do visit, you ought to tell her in advance that you're coming. If you don't visit, you ought to refrain from telling her in advance that you're coming. (This is the counter-to-duty conditional, because it applies to you in the case that you violate your duty to visit.) I've moved the discussion of this Puzzle to a separate page.

A puzzle similar to Chisholm's (and also the Good Samaritan and Robber Paradoxes, from above) is Forrester's "Gentle Murderer": O(not kill), but if you do kill you ought to kill gently, and in fact you will kill. Does it follow that O(kill gently)? Then in SDL it would follow that O(kill), which is not only counter-intuitive but according to SDL inconsistent with our starting assumptions.