11 Deontic Logic

The language of deontic logic $\mathcal{L}^{\textsf{O}}$ supplements the language of propositional logic with a propositional operator, $\textsf{O}\varphi$, read: “it ought to be the case that $\varphi$”. Its dual, $\neg \textsf{O} \neg \varphi$ is symbolized: $\textsf{P}$ and read: “it is permissible that $\varphi$.” There are different interpretations of $\textsf{O}$ in line with different types of obligation, which may include prudential, legal, and moral obligation.

We may still use possible worlds models to interpret the language, but we will use a different heuristic. Given a frame $(W, R)$, we conceive of each member of $W$ as a possible world. We now gloss the accessibility relation $R$ in terms of the system of norms in force at each world. Of course the relevant system of norms will depend on the type of obligation we use to interpret the modal operator $\textsf{O}$. Given a world $w \in W$, we say that $u$ is accessible from $w$, $Rwu$, if, and only if, $w$ is admissible relative to the system norms in force at $w$. That is, whatever ought to be case according to the system of norms in force at $w$ is indeed the case at $u$. In that respect, $u$ involves no more, and perhaps less, of a departure from what is required by system of norms in force at $w$ than $u$ does.

Notice that the fact that $Rwu$ is not to say that whatever ought to be the case according to the system of norms at $u$ is the case at $u$. For the norms that are in force at $u$ may be more restrictive than those in place at $w$.

Obligation and Accessibility

There is a link between obligation, permissibility, and accessibility understood in terms of admissibility.

$\varphi$ ought to be the case at a world if, and only if, $\varphi$ is the case at all admissible worlds relative to the norms in force at the world of evaluation.
$\varphi$ is permissible at a world if, and only if, $\varphi$ is the case at some admissible world relative to the norms in force at the world of evaluation.

The use of possible worlds models to interpret the language of deontic logic is not inconsequential. For the model theory vindicates a rule of necessitation, and it validates all substitution instances of axiom $\textsf{K}$:

\[\tag{$\textsf{N}$} \varphi / \textsf{O} \varphi \]

\[\tag{$\textsf{K}$} \textsf{O}(p \to q) \to (\textsf{O}p \to \textsf{O}q) \]

But one may reasonably question either constraint. For it is difficult 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 be tempted to read $\textsf{O} \varphi$ in terms of entailment from what is required by a system of norms. Since anything entails a logical truth, it is to be expected that they follow from whatever is required by whatever norms may be in force.

11.1 The Logic of Obligation

We know that $\textsf{K}$ is sound and complete with respect to the class of all frames, but the question now is whether we should supplemented it with further axioms for obligation. Standard Deontic Logic supplements the axioms of $\textsf{K}$ with all substitution instances of the axiom:

\[ \tag{$\textsf{D}$} \textsf{O}p \to \neg \textsf{O} \neg p \]

Since $\textsf{P}p$ abbreviates: $\neg \textsf{O} \neg p$, standard deontic logic amounts to the normal modal logic $\textsf{KD}$ in $\mathcal{L}^\textsf{O}$. Axiom $\textsf{D}$ encodes the presupposition that something is obligatory only if it is permissible and modally defines the seriality condition on accessibility: no matter what world $w$ may be, some world $u$ must be accessible from $w$.

More controversial are the deontic counterparts of axioms $\textsf{4}$ and $\textsf{5}$, respectively. Consider the deontic counterpart of axiom $\textsf{4}$ first:

\[\tag{$\textsf{4}$} \textsf{O}p \to \textsf{O}\textsf{O}p \]

Whatever is obligatory ought to remain obligatory. We know that the condition modally defines the requirement that the accessibility relation be transitive. If $Rwu$ and $Rut$, then $Rwt$. That is, if whatever ought to be the case according to $w$ is the case at $u$ and whatever ought to be the case at $u$ is the case at $t$, then whatever ought to be the case according to $w$ is the case at $t$. One way to put it is that the system of norms in force at $w$ remain in force at $u$, which means that the system of norms can only be more restrictive from world to (admissible) world.

Let us look at $\textsf{5}$ now: \[\tag{5} \textsf{P}p \to \textsf{O}\textsf{P}p \]

Whatever is permissible ought to remain permissible. So, if $p$ is permissible at a world $u$, then $p$ 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 becomes more restrictive 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 $\textsf{4}$ and $\textsf{5}$ 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. At all events, $\textsf{KD45}$ is a deontic logic that is sound and complete with respect to the class of serial, transitive, and euclidean models.

There are intermediate systems between $\textsf{KD}$ and $\textsf{KD45}$. One plausible extension of $\textsf{KD}$ supplements its axioms with axiom $\textsf{U}$ below: \[\tag{$\textsf{U}$} \textsf{O}(\textsf{O}p \to p) \]

Notice that a world verifies $\textsf{O}p \to p$ only if it is an ideal world in which whatever ought to be the case relative to the norms in force at that world is in fact the case. Although it is not very plausible to think that our world is an ideal world, you may nonetheless think that it ought to be the case that the world is an ideal world. This constraint sometimes known as Utopia requires 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, $\textsf{KDU}$, is sound and complete with respect to the class of serial and weakly reflexive frames. A relation $R$ is weakly reflexive on $W$ if, and only if, for all $w\in W$ if $Rwu$, then $Ruu$.

11.2 Classic Problems in Deontic Logic

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:

\[\tag{$\textsf{RK}$} \varphi \to \psi / \textsf{O} \varphi \to \textsf{O}\psi \]

The Good Samaritan Paradox

The first difficulty arises by reflection on the following fact: \[ \vdash_{\textsf{K}} \textsf{O}(p \wedge q) \to \textsf{O}q \] The derivation is simple: \[ \begin{array}{lllll} 1 & & p \wedge q \to q & & \textsf{PL} \\ 2 & & \textsf{O}(p \wedge q) \to \textsf{O}q & & \textsf{RK}, 1 \end{array} \] 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 the proposition that Jones helps Smith and we let $q$ be the proposition that Smith is robbed. You may then take the first premise to express the conjunction of those propositions, which appears to be true. The conclusion, however, is unacceptable.

One way out is to reject the formalization of the first premise in terms of $\textsf{O}(p \wedge q)$. 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 Jones is subject to a conditional obligation to help Smith given that he has been robbed.

Compare the distinction just now made with the distinction between a conditional probability and the probability of a conjunction. There is an important distinction to be drawn between the probability of the conjunction that Smith will be robbed and Jones will help Smith, which is low, we may assume, versus the conditional probability that Jones will help Smith given that he has been robbed, which is high given the nature of Jones. Conditional probability, we learn, is not the same as the probability of a conjunction. Likewise, one may say, conditional obligation is not the same as the obligation attached to a conjunction.

Ross’ Paradox

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

\[ \vdash_{\textsf{K}} \textsf{O}p \to \textsf{O}(p \vee q) \] The derivation is again quite simple: \[ \begin{array}{lllll} 1 & & p \to p \vee q & & \text{PL} \\ 2 & & \textsf{O}p \to \textsf{O}(p \vee q) & & \textsf{RK}, 1 \end{array} \]

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 proposition that the letter is mailed and we let $q$ be the proposition that 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 will help them.
If you will not help your neighbors, you ought to not tell them that you will.
You will not help your neighbors.

The problem is that the scenario turns out 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:

$\textsf{O}p$
$\textsf{O}(p \to q)$
$\neg p \to \textsf{O}\neg q$
$\neg p$

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

\[ \begin{array}{lllll} 1 & & \textsf{O}(p \to q) \to (\textsf{O}p \to \textsf{O}q)& & \textsf{K} \\ 2 & & \textsf{O}(p \to q) & & \text{ii} \\ 3 & & \textsf{O}p \to \textsf{O}q & & \textsf{PL}, 1 \\ 4 & & \textsf{O}q & & \text{i} \\ 5 & & \textsf{O} q \to \neg \textsf{O} \neg q & & \text{D} \\ 6 & & \neg \textsf{O} \neg q & & \textsf{PL}, 4, 5\\ 7 & & \neg p & & \text{iv} \\ 8 & & \neg p \to \textsf{O} \neg q & & \text{iii} \\ 9 & & \textsf{O} \neg q & &\textsf{PL}, 7, 8 \\ 10 & & \textsf{O} \neg q \wedge \neg \textsf{O} \neg q & & \textsf{PL}, 6, 9\\ \end{array} \]

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:

$\textsf{O}p$
$\textsf{O}(p \to q)$
$\textsf{O}(\neg p \to \neg q)$
$\neg p$

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

$\textsf{O}p$
$p \to \textsf{O}q$
$\neg p \to \textsf{O}\neg q$
$\neg p$

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

11.3 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 $O(p/q)$, to express the proposition that 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:

$\textsf{O}(p/q):= q \to \textsf{O}p$

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 overslept today.
$\textsf{O}(p/q):= \textsf{O}(q \to p)$

The problem with this is that in the presence of $\textsf{KD}$, we can derive: \[ \begin{array}{llll} \vdash_{\textsf{KD}} \textsf{O}p \to \textsf{O}(q \to p) & & \vdash_{\textsf{KD}} \textsf{O} \ \neg q \to \textsf{O}(q \to p)\\ \end{array} \]

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 $\vdash (q \to p) \to ((q \wedge r) \to p)$, we learn: \[ \begin{array}{lllll} \vdash_{\textsf{KD}} \textsf{O}(q \to p) \to \textsf{O}((q \wedge r) \to p) & & & & \\ \end{array} \]

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: \[ \begin{array}{lllll} q \wedge \textsf{O}(p/q) \to \textsf{O}p & & & & \text{Factual Detachment}\\ \textsf{O}q \wedge \textsf{O}(p/q) \to Op & & & & \text{Deontic Detachment}\\ \end{array} \]

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:

\[ \textsf{O}p := \textsf{O}(p/\top) \] The question, however, is how to specify truth conditions for such claims against the background of the possible worlds model theory.