14 Deontic Logic

The language of deontic logic $L^{O}$ supplements the language of propositional modla logic with a propositional operator, $O$ , read: “it is obligatory that”. Its dual, $\neg O \neg$ is symbolized: $P$ and read: “it is permissible that.” There are multiple interpretations available for the propositional operator $O$ depending on the specific type of obligation it is supposed to govern, e.g., legal obligation, moral obligation, etc.

A relational model $M$ for $L^{O}$ is a structure $(W, R, V)$ , where:

$W$ is a non-empty set of information states or states of the world,
$R$ is a binary accessibility relation on $W$ , and
$V$ is a function, which assigns to each propositional variable $p$ a set of states $V (p)$ .

We gloss the relation of deontic accessibility in terms of standards of obligation in force at each world. Given a world $w \in W$ , we will say that $u$ is deontically accessible from $w$ , $R w u$ , if, and only if, $w$ is admissible relative to the norms or standards of obligation in force at $w$ . In other words, $u$ is a world at which everything that ought to be case according to $w$ is the case. That is not to say that everything that ought to be the case according to $u$ is the case at $u$ . For the norms that are in force at $u$ may be more restrictive than those in place at $w$ . Nor does it follow that everything that ought to be the case according to the norms in force at $w$ is true at $w$ . Of course the relevant norms will depend on the type of obligation we use to interpret the modal operator $O$ .

Definition 14.1 (Truth at a World) $\begin{array}{lll} M, w ⊩ p & iff & w \in V (p) \\ M, w ⊩ \neg φ & iff & M, w ⊮ φ \\ M, w ⊩ (φ \to ψ) & iff & M, w ⊮ φ or M, w ⊩ ψ \\ M, w ⊩ O φ & iff & for every u \in W such that R w u, M, u ⊩ φ \end{array}$ We define $P$ as the dual of $O$ , that is, $\neg O \neg$ , which means: $\begin{array}{lll} M, w ⊩ P φ & iff & for some u \in W such that R w u, M, u ⊩ φ \end{array}$

We are now in a position to define truth in a model of deontic logic.

Definition 14.2 (Truth in a Model) A formula $φ$ is true in a model $M$ of the form $(W, R, V)$ , written $M ⊩ φ$ , if, and only if, for every $w \in W$ , $M, w ⊩ φ$ .

Obligation

These decisions are not inconsequential. For it follows that all logical truths are obligatory and that whatever follows from something obligatory is indeed obligatory. That is the framework vindicates a rule of necessitation and all substitution instances of axiom K: $\begin{matrix} (N) & φ / O φ \end{matrix}$

$\begin{matrix} (K) & O (p \to q) \to (O p \to O q) \end{matrix}$

Neither requirement seems plausible when it comes to obligation. For it is hard to find a sense in which a logical truth should be obligatory or, if you like, the content of a requirement or obligation. You may appeal to a modicum of idealization as we did in the case of knowledge and belief. That is, you may be tempted to read $O φ$ in terms of entailment from what is required by a system of norms or standards of obligation. Since everything entails a logical truth, it is to be expected that they follow from whatever is required by whatever norms may be in force.

The minimal deontic logic supplements $K$ with a counterpart of axiom $D$ for obligation: $\begin{matrix} (D) & O p \to P p \end{matrix}$

This axiom encodes the assumption that whatever is obligatory is permissible, and it corresponds to the seriality of deontic accessibility: no matter what the world is like, there is another world at which the actual standards of obligation are met.

More controversial are each of the deontic counterparts of axioms 4 and 5, respectively. Consider the deontic counterpart of axiom $4$ first: $\begin{matrix} (4) & O p \to O O p \end{matrix}$ This tells us that whatever is obligatory ought to be obligatory. So, if $φ$ is in line with the norms in place $u$ and $v$ is admissible relative to the norms in place at $u$ , then $O φ$ is in line with the norms in place at $v$ . One way to enforce this constraint is by making sure that the system of norms never decreases from world to (admissible) world: if $u$ sees $v$ , then $v$ should incorporate all of the norms that are in force at $u$ .

Let us look at $5$ now: $\begin{matrix} (5) & P p \to O P p \end{matrix}$ This now says that whatever is permissible ought to remain permissible. So, if $φ$ is permissible at a world $u$ , then $φ$ holds at some world $v$ that is admissible relative to the norms in force at $u$ . One way to implement this constraint is to make sure that the system of norms in force at a world never increase from world to (admissible) world. So, no admissible world relative to the norms at $u$ should incorporate further norms. One way to summarize the last two observations is that the combination of 4 and 5 requires the system of norms to remain constant across worlds, which is, on the face of it, not all that plausible when it comes to familiar types of obligation.

One more plausible extension of $K D$ supplements its axioms with axiom $U$ below: $\begin{matrix} (U) & O (O p \to p) \end{matrix}$ This principle, which is sometimes known as Utopia, tells us that it ought to be the case than whatever is obligatory is the case. So, this requires admissible worlds to be ones where the standard of obligation are in fact fulfilled.

The deontic logic that results, $K D U$ , is sound and complete with respect to the class of serial and weakly reflexive frames.

Classic Problems

The framework of deontic logic invites the formulation of a variety of puzzles. Some of these puzzles arise from the validity of the following derived rule of inference:

$\begin{matrix} (RK) & φ \to ψ / O φ \to O ψ \end{matrix}$

The Good Samaritan Paradox

The first difficulty arises by reflection on the following fact: $⊢_{K D} O (p \land q) \to O q$ The derivation is simple: $\begin{array}{lllll} 1 & p \land q \to q & PL \\ 2 & O (p \land q) \to O q & RK \end{array}$ The problem is that this observation appears to license the following inference:

Jones ought to help Smith, who has been robbed.
Smith ought to be robbed.

Here we let $p$ be ‘Jones helps Smith’ and we let $q$ be ‘Smith is robbed’. But while the premise seems fine, the conclusion seems absurd.

One way out of course is to reject the formalization of the first premise by means of the conjunction $p \land q$ . Maybe we should think of the premise in terms of a conditional obligation: Jones ought to help Smith given that Smith has been robbed. It is one thing to say that the conjunction ought to be the case, which is false, and quite another to say that there is a conditional obligation for Jones to help Smith given that he has been robbed. Compare the distinction we just made with the distinction between a conditional probability and the probability of a conjunction. There is an important distinction to be drawn between ‘Probably, Smith will be robbed and Jones will help Smith’ and ’Probably, Jones will help Smith given that Smith has been robbed. The conjunction may remain improbable even if the conditional probability is very high. Conditional probability, we learn, is not the same as the probability of a conjunction.

Ross’ Paradox

The next issue involves another application of RK and begins with the following observation:

$⊢_{K} O p \to O (p \lor q)$ The derivation is simple: $\begin{array}{lllll} 1 & p \to p \lor q & PL \\ 2 & O p \to O (p \lor q) & RK \end{array}$

The problem is that this observation appears to license the following inference:

The letter ought to be mailed.
The letter ought to be mailed or burned.

Here we let $p$ be ‘the letter is mailed’ and we let $q$ be ‘the letter is burned’. But while the premise seems fine, the conclusion seems strange since one way to discharge the latter obligation is to burn the letter. That in turn is inconsistent with the first way to discharge the disjunctive obligation.

Chisholm’s Paradox

Four statements would appear to be perfectly consistent with each other:

You ought to help your neighbors.
You ought to tell your neighbors that you will help them if you plan to help them.
If you will not help your neighbors, you ought to not tell them that you will.
You will not help your neighbors.

Unfortunately, the scenario appears to be inconsistent when formalized against the background of the minimal deontic logic. For let $p$ stand for the proposition that you will help your neighbors and let $q$ stand for the proposition that you will tell them that you will help them. Then, we appear to have:

$O p$
$O (p \to q)$
$\neg p \to O \neg q$
$\neg p$

We now seem to be in a position to argue for a contradiction:

$\begin{array}{lllll} 1 & O (p \to q) \to (O p \to O q) & K \\ 2 & O (p \to q) & ii \\ 3 & O p \to O q & 1, PL \\ 4 & O q & i \\ 5 & O q \to \neg O \neg q & D \\ 6 & \neg O \neg q & 4, 5, PL \\ 7 & \neg p & iv \\ 8 & \neg p \to O \neg q & iii \\ 9 & O \neg q & 7, 8, PK \\ 10 & O \neg q \land \neg O \neg q & 6, 9, PL \end{array}$ This is unacceptable, unless, that is, we are prepared to concede that Chisholm’s scenario is incoherent, which means that we have to find a way to either live with the inconsistency of the first four claims or else find a way to resist the derivation of the contradiction.

One immediate suggestion is to question the formalization we proposed on the grounds perhaps that ii and iii ought to be treated on a par. That would result in two candidate alternatives to the original formulation:

$O p$
$O (p \to q)$
$O (\neg p \to O \neg q)$
$\neg p$

In this case, iii would become a consequence of i, which would make the original set of statements redundant.

$O p$
$p \to O q$
$\neg p \to O \neg q$
$\neg p$

In this case, ii would become a simple consequence of iv making the original set of statements redundant again.

Conditional Obligation

One common reaction to these problems is to invoke a distinction between obligation and conditional obligation akin to that between probability and conditional probability. The thought is that a conditional obligation is not the same as the obligation to make sure a conditional is true.

Let us tentatively enrich the language of deontic logic with a new binary operator into the language $O (p / q)$ , read: “it ought to be the case that $p$ given that $q$ ”. That would be the operator one would use to formalize the conditional obligation to tell your neighbors you will help them provided that you will. The question now becomes whether we are in a position to define conditional obligation in terms of unconditional obligation, and it seems clear that the salient alternatives will not do:

$O (p / q) := q \to O p$

This is much too weak to capture conditional obligation, since it makes them true whenever the antecedent is false. It would be trivially true, for example, that I ought to give you a million dollars given that I’m an engineer.
$O (p / q) := O (q \to p)$

The problem with this is that in the presence of $K D$ , we can derive: $\begin{array}{llll} ⊢_{K D} O p \to O (q \to p) & ⊢_{K D} O \neg q \to O (q \to p) \end{array}$ But in connection to the second observation, the fact that you ought not disturb your neighbors should not entail the conditional obligation to argue with them given that you disturb them.

There are further problems. Since $⊢ (q \to p) \to (q \land r \to p)$ , we learn: $\begin{array}{lllll} ⊢_{K D} O (q \to p) \to O (q \land r \to p) \end{array}$ So, if you ought to help your new neighbors move in given that they do, then you ought to help them move in given that they do and that they ask you to please not help them.

The alternative is to treat conditional obligation as a new binary operator governed by its own distinctive rules of inference: $\begin{array}{lllll} q \land O (p / q) \to O p & Factual Detachment \\ O q \land O (p / q) \to O p & Deontic Detachment \end{array}$ Given factual detachment, if your new neighbors will move in tomorrow, and you ought to help them provided that they do, then you ought to help them. On the other hand, the point of deontic detachment is to allow the move from the fact that it ought to be the case that they will move in tomorrow and the conditional obligation to the obligation to help them tomorrow.

Notice that once we do this, it becomes an option define the original obligation operator in terms of conditional obligation:

$O p := O (p / ⊤)$ The question, however, is how to specify truth conditions for such claims against the background of the possible worlds model theory.