Logic 2: Modal Logic

6Deontic Logic

6.1  Permission and obligation
6.2  Standard deontic logic
6.3  Norms and circumstances
6.4  Further challenges
6.5  Neighbourhood semantics

6.1Permission and obligation

Deontic logic studies formal properties of obligation, permission, prohibition, and related normative concepts. The box in deontic logic is usually written ‘\(\mathsf {O}\)’ (for ‘obligation’ or ‘ought’), the diamond ‘\(\mathsf {P}\)’ (for ‘permission’). If we read \(q\) as stating that you cook dinner, we might use \(\mathsf {O} q\) to express that you are obligated to cook dinner.

We assume that obligation and permission are duals. You are not obligated to cook dinner iff you are permitted to not cook dinner; you are not permitted to cook dinner iff you are obligated to not cook dinner.

There are many kinds of norms: legal norms, moral norms, prudential norms, social norms, and so on. There may also be overarching norms that combine some or all of the others. Deontic logic is applicable to norms of all kinds. We do not have to settle whether \(\mathsf {O}\) expresses legal obligation or moral obligation or some other kind of obligation. It is important, however, that we don’t equivocate. If the law requires \(q\) and morality \(\neg q\), we should not formalize this as \(\mathsf {O} q \land \mathsf {O}\neg q\). It would be better to use a multi-modal language with different operators for legal and moral obligation.

Obligations and permissions often vary from agent to agent. If it is your turn to cook dinner then you are obligated to cook dinner, but I am not. To capture this agent-relativity, we could add agent subscripts to the operators, as we did in epistemic logic. We could then express our different obligations as \(\mathsf {O}_1 \!q \land \neg \mathsf {O}_2 \!q\). But what does the sentence letter \(q\) stand for? When I say that you are obligated to cook dinner, the object of the obligation appears to be a type of act: cooking dinner. In the language of modal propositional logic, \(\mathsf {O}\) and \(\mathsf {P}\) are sentence operators. Unless we want to say that verb phrases in English (like ‘cook dinner’) should be translated into sentences of \(\mathfrak {L}_M\) – which is possible, but non-standard – we have to transform the acts that appear to be the objects of obligation and permission into propositions.

Consider sentence (1), which is arguably equivalent to (2).

In (2), the operator ‘you ought to see to it that’ attaches to a sentence, ‘you cook dinner’. So we can translate (1) via (2) as \(\mathsf {O}_1 \!q\), where \(q\) translates ‘you cook dinner’, and \(\mathsf {O}_1\) corresponds to ‘you ought to see to it that’.

The subject (you) is mentioned twice in (2). A common assumption in deontic logic is that we can drop the agent subscripts from deontic operators, since the embedded proposition will tell us upon whom the obligation or permission falls. Informally, the idea is that (2) is equivalent to (3), with an impersonal ‘ought’.

The impersonal ‘ought’ also figures in statements like (4).

When I say (4), I don’t mean that nobody is obligated to die of hunger. Nor do I mean that everybody is obligated to not die of hunger. Rather, I mean that a certain state of affairs – that nobody dies of hunger – ought to be the case. Without further assumptions, this does not impose any obligations on anyone.

There are reasons to question the equivalence between agent-relative ‘ought’ statements like (2) and impersonal ‘ought’ statements like (3). Suppose Amy has promised to play with Betty. Then Amy is obligated to play with Betty. But Betty is not thereby obligated to play with Amy. Betty may even have promised not to play with Amy. It is hard to express these facts in terms of impersonal oughts. If we say that it ought to be the case that Amy plays with Betty, we’re missing the fact that the obligation falls on Amy, not on Betty (who might be under a contrary obligation). So perhaps it would be better to keep the agent subscripts after all.

It can also be useful to make the ‘see to it that’ component in statements like (2) explicit. That Amy ought to play with Betty could then be translated as \(\mathsf {O}_a \mathsf {stit} p\), where \(\mathsf {stit}\) formalizes ‘sees to it that’. This allows us to distinguish between the following three possibilities.

\(\mathsf {O}_a \mathsf {stit} \neg p\) Amy ought to see to it that she doesn’t play with Betty.
\(\mathsf {O}_a \neg \mathsf {stit} p\) Amy ought to not see to it that she plays with Betty.
\(\neg \mathsf {O}_a \mathsf {stit} p\) It is not the case that Amy ought to see to it that she plays with

The \(\mathsf {stit}\) operator has proved useful to represent different concepts of rights and duties. In what follows, we will nonetheless stick to the simplest (and oldest) approach, without a \(\mathsf {stit}\) operator and without agent subscripts. This approach is sufficient for many applications, but its limitations should be kept in mind.

Exercise 6.1

Translate the following sentences into the standard language of deontic logic (without \(\mathsf {stit}\) or agent subscripts).
You must not go into the garden.
You may not go into the garden.
Jones ought to help his neighbours.
If Jones is going to help his neighbours, then he ought to tell them he’s coming.
If Jones isn’t going to help his neighbours, then he ought to not tell them he’s coming.

6.2Standard deontic logic

Think of a possible world as a history of events. For any such history, and any system of norms, we can ask whether the history conforms to the norms. Let’s call a world ideal (relative to some norms) if everything that happens at the world conforms to the norms. In an ideal world, everyone does what they ought to do.

How do the ideal worlds relate to permission and obligation? For a start, everything that happens at an ideal world is plausibly permitted, for we know that it conforms to the norms. The converse is plausible as well: whenever something is permitted then it happens at some ideal world. For suppose something doesn’t happen at any ideal world. Then the event entails the violation of some norm: it is incompatible with the satisfaction of all norms. And then it can’t be permitted.

We have a simple possible-worlds analysis of permission:

\(A\) is permitted (relative to some norms) iff \(A\) is the case at some world that is ideal (relative to these norms).

Given the duality of permission and obligation, we also have a possible-worlds analysis of obligation:

\(A\) is obligatory (relative to some norms) iff \(A\) is the case at all worlds that are ideal (relative to these norms).

These analyses resemble the simple possible-worlds analysis from chapter 2, where we assumed that \(A\) is possible iff it is the case at some world, and necessary iff it is the case at all worlds. The difference is that we now quantify only over ideal worlds.

We can capture this restriction with the help of Kripke models. In Kripke semantics, \(\Diamond A\) is true at a world \(w\) iff \(A\) is true at some world that is accessible from \(w\). Let’s assume that a world is accessible, from any world, iff it is ideal. Then Kripke semantics implies that \(\Diamond A\) is true at \(w\) iff \(A\) is true at some ideal world. That’s what we want.

The accessibility relation I’ve just defined is a little unusual: whether it holds between \(w\) and \(v\) does not at all depend on \(w\). We have \[ wRv \text { iff $v$ is ideal.} \] But that’s OK. The definition of a Kripke model allows for such degenerate relations.

Let’s investigate the formal properties of our degenerate accessibility relation. Is it, say, reflexive? Transitive? Symmetric? Euclidean?

Transitivity says that if \(wRv\) and \(vRu\) then \(wRu\). Now \(wRv\) means that \(v\) is ideal. And \(vRu\) means that \(u\) is ideal. \(wRu\) also means that \(u\) is ideal. So transitivity requires that if \(v\) is ideal and \(u\) is ideal then \(u\) is ideal. That’s obviously true. So our accessibility relation is transitive. The same reasoning shows that it is euclidean. Our possible-worlds analysis therefore validates the (intuitively somewhat elusive) schemas (4) and (5).

\(\mathsf {O} A \to \mathsf {O}\mathsf {O} A\)
\(\mathsf {P} A \to \mathsf {O}\mathsf {P} A\)

What about reflexivity? The hypothesis that every world has access to itself would mean that every world is ideal. When we reason about permission and obligation, we normally don’t take for granted that everyone does what they ought to do. We allow for the logical possibility that norms can be violated. So we don’t assume that every world is ideal. Equivalently, we don’t regard the (T)-schema \begin {equation} \tag {T}\mathsf {O} A \to A \end {equation} as valid.

Exercise 6.2

Is the accessibility relation (as defined above) symmetric?

We might, however, impose the weaker condition of seriality. This would validate the (D)-schema \begin {equation} \tag {D}\mathsf {O} A \to \mathsf {P} A. \end {equation} Intuitively, (D) says that the norms are consistent: if you’re obligated to do \(A\), then you are not obligated to do not-\(A\). (Remember that \(\mathsf {P} A\) is equivalent to \(\neg \mathsf {O}\neg A\).) Semantically, (D) corresponds to the assumption that there is at least one world at which all the norms are satisfied. If there were no such world, all sentences of the form \(\mathsf {O} A\) would come out true, and all sentences of the form \(\mathsf {P} A\) false. Everything would be obligatory, but nothing allowed. It is hard to make sense of such a scenario. If we use Kripke semantics for deontic logic, we should therefore rule out inconsistent norms and accept (D) as valid.

Here it may be important to distinguish prima facie obligations from actual, or all-things-considered obligations. If you’ve promised to cook dinner, you are under a prima facie obligation to cook dinner. But the obligation can be overridden by intervening circumstances or contrary obligations. If your child has an accident and needs urgent medical care, the right thing to do may well be to not cook dinner and instead bring your child to the hospital. In a sense, you are under conflicting obligations: you ought to cook dinner, and you ought to look after your child (and not cook dinner). There is no world at which you meet both of these obligations. This is not a counterexample to (D), if we understand \(\mathsf {O}\) as all-things-considered obligation. You are prima facie obligated to cook dinner, but all things considered, you should not cook dinner.

Another weakening of reflexivity is “shift reflexivity”. \(R\) is shift reflexive if \(wRv\) implies \(vRv\): every world that can be seen can see itself. Shift reflexivity corresponds to the following schema (U) (for “utopia”) \begin {equation} \tag {U}\mathsf {O}(\mathsf {O} A \to A) \end {equation} In words: it ought to be the case that whatever ought to be the case is the case. Shift reflexivity is entailed by euclidity, so our logic validates (U).

Exercise 6.3

Explain why euclidity entails shift reflexivity.

We could look at further properties of the accessibility relation, but we wouldn’t find any plausible candidates that are not entailed by seriality, transitivity, and euclidity. The complete logic of obligation and permission, assuming the above possible-worlds analysis, is plausibly KD45.

We might, however, reconsider our analysis. We’ve assumed that there is a fixed set of norms that divides the worlds into “ideal” ones, where all the norms are respected, and “non-ideal” ones, where some norms are violated. We might call this an absolutist conception of norms. A relativist conception, by contrast, would allow that the norms may vary from world to world.

Suppose, for example, that we want to reason about what is required by the traffic laws. The traffic laws evidently vary from world to world. Consider a world at which cyclists are required to wear top hats. Norman is cycling in this world, without a top hat. Is he violating the traffic laws? He is violating the laws of his world, but not the laws of our world. On an absolutist approach, we interpret \(\mathsf {O}\) and \(\mathsf {P}\) as always referring to the laws of our world, no matter what world is under consideration: \(\mathsf {O} p\) is true at \(w\) iff \(p\) is required by the laws of our world. On a relativist approach, we instead assume that \(\mathsf {O} p\) is true at \(w\) iff \(p\) is required by the laws at \(w\).

On the relativist approach, a world \(v\) is accessible from a world \(v\) iff everything that happens at \(v\) conforms to the (relevant) norms at \(w\).

Transitivity and euclidity now become implausible. Let \(w\) be a world in which the only relevant norm is that one must drive on the left. Let \(v\) be a world in which everyone drives on the left, but the law allows driving on either side. Let \(u\) be a world in which some people drive on the right. \(v\) is accessible from \(w\) and \(u\) from \(v\), but \(u\) is not accessible from \(w\). We don’t have transitivity.

Exercise 6.4

Show that the deontic accessibility relation is neither euclidean nor shift-reflexive, on the relativist approach.

As before, we probably don’t want to assume reflexivity, but we might want to assume seriality, which now means that there is no world at which the norms make inconsistent demands.

The relativist conception seems to be more common in deontic logic. So-called standard deontic logic assumes only that the accessibility relation is serial, making the system D the complete logic of obligation and permission.

The absolutist logic KD45 and the relativist logic D can be shown to disagree only about sentences in which a deontic operator occurs in the scope of another deontic operator. Any sentence that does not contain an \(\mathsf {O}\) or \(\mathsf {P}\) operator embedded under another \(\mathsf {O}\) or \(\mathsf {P}\) operator is D-valid iff it is KD45-valid.

Exercise 6.5

Use the tree method to check which of the following sentences are D-valid and which are KD45-valid.
\(\mathsf {P} (p \lor q) \to (\mathsf {P} p \land \mathsf {P} q)\)
\(\mathsf {O}\mathsf {P} p \to \mathsf {P} p\)
\(\neg \mathsf {P}(p \lor q) \to (\mathsf {P} \neg p \lor \mathsf {P}\neg q)\)
\(\mathsf {O}\mathsf {P} p \lor \mathsf {P}\mathsf {O} p\)

Exercise 6.6

Consider a world in which there are no sentient beings, and nothing else that could introduce norms or laws. Since there are no norms at this world, one might hold that nothing is obligatory relative to the world’s norms, and nothing is permitted. Explain why this casts doubt on the validity of (Dual1) and (Dual2) in the logic of relativist obligation and permission.

Exercise 6.7

A system of norms is intolerant if it requires of itself that it is in force and does not allow any other norms. That is, if the norms at \(w\) are intolerant, then only worlds with the same norms conform to these norms. Show that the relativist logic of intolerant norms validates (4) and (5).

6.3Norms and circumstances

We have assumed that something ought to be the case iff it is the case at all worlds where no (relevant) norms are violated. On closer inspection, many ordinary statements about oughts and obligations do not fit this analysis.

Suppose you are walking past a drowning baby. You ought to rescue the baby. But are you rescuing the baby at every world at which no norms are violated? Clearly not. There are worlds at which the baby never fell into the pond, and others at which you are overseas and have no means to rescue the baby. These worlds need not involve any violations of norms.

The example shows that even on an absolutist approach, obligations and permissions can vary from world to world. In worlds where you are passing by a drowning baby, you are obliged to save it. In other worlds, you are not. The relevant (moral) norms may well be the same in either case. What varies are your circumstances.

In general, what is required or permitted usually depends not just on the norms, but also on the circumstances – for example, on what you are able to do, and on what consequences the available options would have.

We can account for this dependence on the circumstances by changing our interpretation of the accessibility relation. Previously, we assumed that a world \(v\) is accessible from \(w\) iff all the norms (or all the norms at \(w\)) are respected at \(v\). Let’s add another condition: relevant circumstances at \(w\) must also obtain at \(v\). For example, if \(w\) is a world at which you come across a drowning baby then any accessible world must also be a world at which you come across a drowning baby. In all ideal worlds among these, you rescue the baby.

Here is the redefined accessibility relation, in terms of which we might try to analyse \(\mathsf {O}\) and \(\mathsf {P}\):

A world \(v\) is deontically accessible from a world \(w\) iff (a) \(v\) is circumstantially accessible from \(w\), and (b) no norms (at \(w\)) are violated at \(v\).

The parenthetical ’(at \(w\))’ must be included on a relativist approach, but not on an absolutist approach.

We might want to say more about the circumstantial accessibility relation in clause (a). Recall that a world \(v\) is circumstantially accessible from \(w\) if relevant circumstances that obtain in \(w\) also obtain in \(v\). Often, the “relevant circumstances” that we seem to hold fixed when we reason about norms comprise everything that is settled, in the sense of section 1.5 – everything that can no longer be changed. If the baby has fallen into the pond at \(w\), then there is nothing anyone can do to undo the falling; the falling is a “relevant circumstance” that takes place at every world accessible from \(w\). Arguably, however, there are cases in which we treat worlds as accessible that aren’t open. ‘Jones ought to be here’, for example, can be true even if it’s settled that Jones is somewhere else. Perhaps the circumstantial accessibility relation that figures in clause (a) varies with conversational context.

With the new definition of deontic accessibility, \(\mathsf {O} A\) says that among the circumstantially accessible worlds, all ideal worlds are \(A\)-worlds. We could make this more explicit. Let ‘\(\mathsf {N}\)’ be a propositional constant whose intended meaning is that all norms are satisfied. We can then use \(\Box (\mathsf {N} \to A)\) to express that \(A\) is required, where the box expresses the relevant kind of circumstantial necessity. This approach to formalizing obligation statements goes back to Leibniz.

Exercise 6.8

How could we define \(\mathsf {P}\) in terms of \(\Box \) and \(\mathsf {N}\), so that \(\mathsf {P}\) is the dual of \(\mathsf {O}\)?

Exercise 6.9

Show that the Leibnizian approach renders the (U)-schema valid, assuming that the circumstantial accessibility is reflexive. You have to first translate the schema into the Leibnizian language.

Whichever language we use to express it, our revised concept of obligation has a serious problem. It assumes that the circumstantially accessible worlds include ideal worlds, at which no norms are violated. For suppose there are no such worlds. Then no world is deontically accessible! We would have to say that everything is required and nothing permitted (because all instances of \(\mathsf {O} A\) are true and all instances of \(\mathsf {P} A\) false at worlds that can’t access anything).

Now remember that we don’t assume that all worlds are ideal. If a world is not ideal, then it is hard to see why the worlds that are circumstantially accessible from it should always include ideal worlds. Couldn’t the “relevant circumstances” that are held fixed include some norm violations?

The problem is brought ought by Arthur Prior’s Samaritan Paradox. Suppose someone (Smith) has been injured in a robbery, and Jones has the opportunity to help. We want to say that Jones ought to help the victim: he helps the victim at all deontically accessible worlds. But then the robbery must have taken place at all these worlds. (In a world without a robbery, there is no victim to help.) Here, the circumstantially accessible worlds all contain a violation of norms. In a truly ideal world, nobody would have been robbed and nobody would be in need of help.

We need to adjust our revised definition of deontic accessibility. How could we do that?

In the Samaritan Paradox, the robbery is settled; it has happened at all worlds that are compatible with the “relevant circumstances”. None of these worlds are ideal. Crucially, however, worlds at which Jones doesn’t help the victim are even worse, in terms of norm violations, than worlds at which he helps the victim. Both kinds of worlds are non-ideal, because the victim got robbed. But our norms don’t just divide the possible worlds into ideal and non-ideal; they allow for finer distinctions among non-ideal worlds. Jones ought to help the victim because that’s what he does in the best worlds among those he can bring about.

These considerations suggest that we should redefine deontic accessibility as follows, to properly account for the dependence of obligations and permissions on circumstances.

A world \(v\) is deontically accessible from a world \(w\) iff \(v\) is among the best worlds (by the norms at \(w\)) among those that are circumstantially accessible from \(w\).

As before, the parenthetical ‘(by the norms at \(w\))’ would be needed in a relativist account and not in an absolutist account.

It can be useful to factor out the circumstantial and deontic components that enter into the new definition. I don’t mean to separate them in the formal language, as in Leibniz’s proposal. Rather, I mean to separate them in the definition of a model.

Let’s define a new type of model. Instead of a deontic accessibility relation, we have two ingredients besides the worlds \(W\) and the interpretation function \(V\). One is a circumstantial accessibility relation. The other is a world-relative “order” that tells us which worlds are better than others, relative to the norms at any given world (which may be the norms at every world, on an absolutist approach).

Let ‘\(u \prec _w v\)’ mean that world \(u\) is better than world \(v\) relative to the norms at \(w\). The symbol ‘\(\prec \)’ hints at the idea that \(u\) contains fewer violations of norms than \(v\). We assume that for any world \(w\), the relation \(\prec _w\) is transitive. We also assume that it is asymmetric, meaning that if \(u \prec _w v\) then it is not the case that \(v \prec _w u\). Asymmetric and transitive relations are known as strict partial orders.

Definition 6.1

A deontic ordering model consists of

  • a non-empty set \(W\) (the worlds),
  • a binary relation \(R\) on \(W\) (the circumstantial accessibility relation),
  • for each world \(w\in W\), a strict partial order \(\prec _w\) on \(W\) (the world-relative ranking of worlds as better or worse), and
  • a function \(V\) that assigns to each sentence letter of \(\mathfrak {L}_M\) a subset of \(W\).

Now we need to say under what conditions a sentence of the form \(\mathsf {O} A\) is true at a world in an ordering model. Informally, \(\mathsf {O} A\) will be true at \(w\) iff \(A\) is true at the best worlds among those that are circumstantially accessible. Let’s introduce one more piece of notation. For any set \(S\) and any partial order \(\prec \), let \(\mathrm {Min}^{\prec }(S)\) be the set of \(\prec \)-minimal members of \(S\): \[ \mathrm {Min}^{\prec }(S) =_\text {def} \{ v: v \in S \land \neg \exists u(u \in S \land u \prec v) \}. \] An expression of the form ‘\(\{ x: \ldots x \ldots \}\)’ denotes the set of all things \(x\) that satisfy the condition \(\ldots x \ldots \). So \(Min^{<}(S)\) is the set of all things \(v\) that are members of \(S\) and for which there are no members \(u\) of \(S\) for which \(u \prec v\).

Here, then, are the truth-conditions for \(\mathsf {O} A\) and \(\mathsf {P} A\) in deontic ordering models:

Definition 6.2: Ordering semantics

If \(M\) is a deontic ordering model and \(w\) a world in \(M\), then
\(M,w \models \mathsf {O} A\) iff \(M,v \models A\) for all \(v \in \mathrm {Min}^{\prec _w}(\{ u: wRu\})\)
\(M,w \models \mathsf {P} A\) iff \(M,v \models A\) for some \(v \in \mathrm {Min}^{\prec _w}(\{ u: wRu\})\)

This is just a formal way of saying that \(\mathsf {O} A\) is true at \(w\) iff \(A\) is true at the best worlds (by the norms at \(w\)) among the worlds that are circumstantially accessible at \(w\).

If we want the (D)-schema to be valid, we have to assume that there is always at least one best world among the circumstantially accessible worlds, so that \(\mathrm {Min}^{\prec _w}(\{ u: wRu\})\) is never empty. Let’s make this assumption.

The logic of obligation and permission now depends on formal properties of the circumstantial accessibility relation \(R\) and the deontic orderings \(\prec _w\). In section 1.5, I argued that the logic of historical necessity (of what is settled and open) is S5. This suggests that in normal contexts, \(R\) is an equivalence relation. If we adopt an absolutist approach, on which the orderings \(\prec _{w}\) are the same for every world \(w\), we then still get KD45. If we allow the orderings to vary from world to world, we still get D, unless we impose further restrictions on the orderings.

Exercise 6.10

Amy ought to promise to help Betty or to help Carla. She doesn’t make either promise. If she had promised to help Betty, she would be obligated to help Betty. If she had promised to help Carla, she would be obligated to help Carla. So it ought to be the case that Amy is either obligated to help Betty or obligated to help Carla. In fact, since Amy makes neither promise, she is neither obligated to help Betty nor to help Carla. Explain why this casts doubt on the assumption that deontic accessibility is euclidean.

Exercise 6.11

Suppose fatalism is true and the only world that is open (circumstantially accessible) relative to any world \(w\) is \(w\) itself. Can you describe the resulting deontic logic (on either an absolutist or a relativist approach)?

Ordering models prove useful when we want to formalize statements with modal operators and if-clauses, like (1)–(3).

How would you translate these into our language \(\mathfrak {L}_M\)? You seem to face a choice between (W) and (N).

In (W), the operator \(\mathsf {O}\) is said to have wide scope because it applies to the entire conditional \(p \to q\). In (N), the operator has narrow scope because it only applies to the consequent \(q\).

On reflection, neither translation is satisfactory. Starting with (N), note that \(p \to \mathsf {O} q\) and \(\neg \mathsf {O} q\) together entail \(\neg p\). But from (1), together with the assumption that you are not required to smoke (\(\neg \mathsf {O} q\)), we surely can’t infer that you do not in fact smoke.

(W) is not much better. For one, in our Kripke-style semantics, \(\mathsf {O}(p \to q)\) is entailed by \(\mathsf {O}(\neg p)\). But it is easy to imagine a scenario in which you must not smoke, or you must submit your tax return before the deadline, but in which (1) and (2) are false.

Both (N) and (W) would also license a problematic form of “strengthening the antecedent”. For example, they both suggest that (3) entails (4).

Exercise 6.12

Give tree proofs with the K-rules to show that \(p \to \mathsf {O} r\) entails \((p \land q) \to \mathsf {O} r\), and that \(\mathsf {O} (p \to r)\) entails \(\mathsf {O}((p \land q) \to r)\).

Let’s think about what is expressed by statements like (1)–(4). Intuitively, when we ask what must be done if \(p\) is the case, we are limiting our attention to situations in which \(p\) is the case, and consider which of these situations best conform to the relevant norms. It is irrelevant whether \(p\) is in fact the case or whether it ought to be the case. (1) says – roughly – that among worlds where you smoke, the best worlds are worlds where you smoke outside. Worlds where you smoke inside are worse than worlds where you smoke outside. Similarly for (2). A world at which you miss the deadline for tax returns and pay the fine contains only one violation of the tax rules. Worlds at which you miss the deadline and don’t pay the fine contain two. The best worlds among those at which you miss the deadline are worlds at which you pay the fine. Likewise for (3). Among worlds at which you have promised to call your parents, the best are worlds at which you keep the promise and call them.

The if-clause in sentences like (1)–(3) therefore seems to restrict the worlds over which the modal operator quantifies. Whereas ‘ought \(q\)’ alone says that \(q\) is true at the best of the open worlds, ‘if \(p\) then ought \(q\)’ says that \(q\) is true at the best of the open worlds at which \(p\) is true.

There is no way to express these truth-conditions with the resources of \(\mathfrak {L}_M\). But we can introduce a new, binary operator for conditional obligation. The operator is often written ‘\(\mathsf {O}(\cdot /\cdot )\)’, with a slash separating the two argument places. Intuitively, \(\mathsf {O}(B/A)\) means that \(B\) ought to be the case if \(A\) is the case.

The formal truth-conditions for \(\mathsf {O}(B/A)\) are much like those for \(\mathsf {O} B\), except that we add the assumption \(A\) to the circumstances that are held fixed:

Definition 6.3: Ordering semantics for conditional obligation

If \(M\) is a deontic ordering model and \(w\) a world in \(M\), then
\(M,w \models \mathsf {O} (B/A) \text { iff $M,v \models B$ for all $v \in \mathrm {Min}^{\prec _w}(\{ u: wRu \text { and } M,u\models A \})$}\).

Here, \(\{ u: wRu \text { and } M,u\models A \}\) is the set of worlds \(u\) that are circumstantially accessible from \(w\) and at which \(A\) is true. \(\mathrm {Min}^{\prec _{w}}(\{ u: wRu \text { and } M,u\models A \})\) is the set that comprises the best of these worlds. So \(\mathsf {O}(B/A)\) is true at \(w\) iff \(B\) is true at all of the best \(A\)-worlds that are accessible at \(w\).

Exercise 6.13

“Deontic detachment” is the inference from \(\mathsf {O} A \) and \(\mathsf {O}(B/A)\) to \(\mathsf {O} B\). “Factual detachment” is the inference from \(A\) and \(\mathsf {O}(B/A)\) to \(\mathsf {O} B\). Which of these are valid on the present semantics?

Exercise 6.14

In exercise 6.1, you were asked to translate the following statements.
Jones ought to help his neighbours.
If Jones is going to help his neighbours, then he ought to tell them he’s coming.
If Jones isn’t going to help his neighbours, then he ought to not tell them he’s coming.

Let’s add a fourth statement:

Jones is not going to help his neighbours.

Intuitively, none of these four statements is entailed by one of the others. Moreover, they don’t impose contradictory requirements on Jones. This shows that your translations in exercise 6.1 were incorrect. Explain. (This puzzle is due to Roderick Chisholm.)

Exercise 6.15

The dual of conditional obligation is conditional permission. Spell out truth-conditions for \(\mathsf {P}(B/A)\) that parallel the truth-conditions I have given for \(\mathsf {O}(B/A)\), so that \(\mathsf {P}(B/A)\) is equivalent to \(\neg \mathsf {O}(\neg B/A)\).

6.4Further challenges

Many apparent problems for standard deontic logic arise from the dependence of obligations on circumstances. We can avoid these problems by using deontic ordering models and formalizing conditional obligation statements with the binary \(\mathsf {O}(\cdot /\cdot )\) operator. There are, however, other problems and “paradoxes” for which this move doesn’t help. I will mention three.

First, we already saw that standard deontic logic does not allow for conflicting obligations. Suppose you have promised your family to be home for dinner and your friends to join them at the pub. You are under conflicting prima facie obligations. It is not clear that one of them overrides the other. Legal systems can also contain contradictory rules, without any higher-level rules for how to resolve such contradictions.

We can, of course, drop principle (D). But even in the minimal logic K, \(\mathsf {O} p\) and \(\mathsf {O} \neg p\) entail \(\mathsf {O} A\), for any sentence \(A\). Intuitively, however, the fact that you have given incompatible promises does not entail that you are obligated to, say, kill the Prime Minister.

Another family of problems arises from the fact that in any logic defined in terms of Kripke models, \(\mathsf {O}\) is closed under logical consequence, meaning that if \(\mathsf {O} A\) is true and \(A\) entails \(B\), then \(\mathsf {O} B\) is true. Since logical truths are logically entailed by everything, it follows that all logical truths come out as obligatory. (This is easy to see semantically. A logical truth is true at all worlds; so it is true at all deontically accessible worlds.) But ought it to be the case that it either rains or doesn’t rain?

In response, one might argue that the relevant statements sound wrong not because they are false, but because their utterance would violate a pragmatic norm of cooperative communication. A basic norm of pragmatics is that utterances should make a helpful contribution to the relevant conversation. In a normal conversational context, it would be pointless to say that something ought (or ought not) to be the case if it is logically guaranteed to be the case anyway. An utterance of ‘it ought to be that \(p\)’ is pragmatically appropriate only if \(p\) could be false. This might explain why it sounds wrong to say that it ought to either rain or not rain.

Note also that by duality, \(\neg \mathsf {O}(p \lor \neg p)\) entails \(\mathsf {P} \neg (p \lor \neg p)\). If we deny that it ought to either rain or not rain, and we accept the duality of obligation and permission, we have to say that it is permissible that it neither rains nor doesn’t rain. That sounds even worse.

The problem of closure under entailment has special bite when obligation statements are restricted by circumstances. Return to the Samaritan puzzle. Suppose the victim is bleeding, and Jones ought to stop the blood flow. It is logically impossible to stop a blood flow if no blood is flowing. In all the deontic logics we have so far considered, the claim that Jones ought to stop the victim’s blood flow therefore entails that the victim ought to be bleeding. But wouldn’t it be better if the victim weren’t bleeding?

Here, too, one might appeal to a pragmatic explanation. When we say that Jones ought to stop the blood flow, we take for granted that the victim is bleeding. We are interested in what should be done given the state in which Jones found the victim. Worlds where the victim isn’t injured are set aside; they are not circumstantially accessible. But circumstantial accessibility can shift with conversational context. The claim that the victim ought to be bleeding is pointless if we hold fixed the victim’s state of injury. So when we evaluate this claim, we naturally assume that the relevant circumstantial accessibility relation does not hold fixed the injuries. Intuitively, we are no longer considering what should be done given the state in which Jones found the victim, but whether that state itself should have obtained. Worlds in which the state doesn’t obtain become circumstantially accessible.

A third family of problems arises from disjunctive statements of permission and obligation. Consider (1).

Intuitively, (1) suggests that both mailing the letter and burning it are permitted. In standard deontic logic, however, \(\mathsf {O}(A \lor B)\) does not entail \(\mathsf {P} A \land \mathsf {P} B\). (This puzzle was first noticed by Alf Ross and is known as “Ross’s Paradox”.)

A similar puzzle arises for permissions. (This one is known as the “Paradox of Free Choice”.)

Intuitively, (2) implies that beer and wine are both permitted. But in standard deontic logic, \(\mathsf {P}(A \lor B)\) does not entail \(\mathsf {P} A \land \mathsf {P} B\).

We could add the missing principles.

\(\mathsf {O}(A \lor B) \to (\mathsf {P} A \land \mathsf {P} B)\)
\(\mathsf {P}(A \lor B) \to (\mathsf {P} A \land \mathsf {P} B)\)

But both of these have unacceptable consequences when added to the minimal modal logic K. With the help of (R), we could show that \(\mathsf {O} A\) entails \(\mathsf {P} B\): \(\mathsf {O} A\) entails \(\mathsf {O} (A \lor B)\), which by (R) entails \(\mathsf {P} B\). But clearly ‘you ought to mail the letter’ does not entail ‘you may burn the letter’. Similarly for (FC). In K, \(\mathsf {P} A\) entails \(\mathsf {P}(A \lor B)\); by (FC), \(\mathsf {P}(A \lor B)\) entails \(\mathsf {P} B\). But ‘you may have beer’ does not entail ‘you may have wine’.

Exercise 6.16

Analogous puzzles to those raised by Ross’s Paradox and the Paradox of Free Choice arise for epistemic ‘must’ and ‘might’. Can you give examples?

6.5Neighbourhood semantics

In reaction to apparent problems for standard deontic logic, some have argued that we should not interpret obligation and permission in terms of quantification over possible worlds. If we give up this core tenet of Kripke semantics, we can define “non-normal” logics weaker than K. (A normal modal logic is a modal logic that can be defined in terms of classes of Kripke frames.)

A popular alternative to Kripke semantic is neighbourhood semantics, also known as Scott-Montague semantics, after its inventors Dana Scott and Richard Montague.

Models in neighbourhood semantics still involve possible worlds. Validity is still defined as truth at all worlds in all (suitable) models. But the box and the diamond are no longer interpreted as quantifiers over accessible worlds. Instead, we simply assume that at every world, some propositions are “necessary” and others are not. \(\Box A\) is true at a world if \(A\) expresses one of the necessary propositions at that world.

Formally, the accessibility relation in Kripke models is replaced by a neighbourhood function \(N\) that associates each world in a model with the propositions that are necessary relative to \(w\). Propositions are identified with sets of possible worlds. Thus \(N(w)\) is a set of sets of worlds. Each set of world in \(N(w)\) is necessary at \(w\).

Definition 6.4

A neighbourhood model consists of

  • a non-empty set \(W\),
  • a function \(N\) that assigns to each member of \(W\) a set of subsets of \(W\), and
  • a function \(V\) that assigns to each sentence letter of \(\mathfrak {L}_M\) a subset of \(W\).

The interpretation of non-modal sentences at neighbourhood models works just as in Kripke semantics (definition 3.2). To state the semantics for modal sentences, let \([A]^M\) be the set of worlds in model \(M\) at which \(A\) is true. This is our proxy for the proposition expressed by \(A\). Then: \begin {align*} M,w \models \Box A &\text { \;iff\; $[A]^M$ is in $N(w)$}.\\ M,w \models \Diamond A &\text { \;iff\; $[\neg A]^M$ is not in $N(w)$.} \end {align*}

Intuitively, the clause for the box says that \(\Box A\) is true at \(w\) iff the proposition expressed by \(A\) is one of those that are necessary at \(w\). The clause for the diamond ensures that the box and the diamond are duals.

In neighbourhood semantics, the modal operators are not closed under logical consequence. The neighbourhood function \(N\) can easily make \(p\) necessary at a world without making \(p\lor q\) necessary, even thought \(p\) entails \(p \lor q\). If we interpret \(\mathsf {O}\) and \(\mathsf {P}\) as the box and the diamond in neighbourhood semantics, we can therefore say that Jones ought to tend to the victim’s injuries even thought it is not the case that someone ought to be injured.

We can also allow for conflicting obligations. If the laws at \(w\) require both \(p\) and \(\neg p\), we simply have \([p]^M \in N(w)\) and \([\neg p]^M \in N(w)\). It no longer follows that any proposition whatsoever is obligatory.

We may further hope to escape the problems from section 6.3 that led us to introduce a primitive conditional obligation operator. I argued that the wide-scope translation \(\mathsf {O}(A \to B)\) of conditional obligation sentences is problematic because \(\mathsf {O}(A \to B)\) is entailed by \(\mathsf {O}(\neg A)\). In neighbourhood semantics, this entailment fails.

Bare neighbourhood semantics determines a very weak logic called E. It is axiomatized by (Dual), (CPL), and a rule (called “RN”) that allows inferring \(\Box A \leftrightarrow \Box B\) from \(A \leftrightarrow B\). We can get stronger logics, with more validities, by imposing conditions on the neighbourhood function \(N\).

For example, suppose we want to maintain that if something is logically guaranteed to be true then it can’t be forbidden. Equivalently, any logically necessary truth should be permitted. By the neighbourhood semantics for \(\mathsf {P}\), \(A\) is permitted at a world \(w\) in a model \(M\) iff \([\neg A]^M\) is not in \(N(w)\). If \(A\) is a logical truth, then \(A\) is true at all worlds; in that case, \(\neg A\) is true at no worlds, and \([\neg A]^M\) is the empty set. If we want logical truths to be permitted, we therefore have to stipulate that \(N(w)\) never contains the empty set.

In Kripke semantics, the assumption that logically necessary truths are permitted is equivalent to the assumption that (every instance of) the (D)-schema \(\mathsf {O} A \to \mathsf {P} A\) is valid. Both assumptions correspond to seriality of the accessibility relation. In neighbourhood semantics, we can distinguish between the two assumptions. While the permissibility of logical truths requires that \(N(w)\) doesn’t contain the empty set, the validity of \(\mathsf {O} A \to \mathsf {P} A\) requires that \(N(w)\) doesn’t contains contradictory propositions \([A]^M\) and \([\neg A]^M\).

If we assume that the neighbourhood function is closed under intersection, in the sense that whenever two sets \(X\) and \(Y\) are in \(N(w)\) then so is their intersection \(X\cap Y\), then \((\Box A \land \Box B) \to \Box (A \land B)\) becomes valid. If we also require the converse, that whenever \(X\cap Y \in N(w)\) then \(X \in N(w)\) and \(Y\in N(w)\), and in addition that \(W \in N(w)\), we get back the minimal normal logic K.

Exercise 6.17

Can you find a condition on the neighbourhood function that renders the (T)-schema valid?

For some purposes, even the minimal logic of neighbourhood semantics is too strong. Return to the intuitive “Free Choice” principle from the previous section: \begin {equation} \tag {FC}\mathsf {P}(A \lor B) \to (\mathsf {P} A \land \mathsf {P} B) \end {equation} We have seen that this principle is untenable in Kripke semantics. It is still untenable in neighbourhood semantics.

To see why, note first that whenever two sentences \(A\) and \(B\) are logically equivalent, then in neighbourhood semantics \(\mathsf {P} A\) and \(\mathsf {P} B\) are also equivalent. The reason is that the modal operators in neighbourhood semantics operate on the set of worlds at which the embedded sentence is true. If \(A\) and \(B\) are logically equivalent, then in any model \(M\), the set \([A]^M\) is the same set as \([B]^M\), and so \([A]^M\) is in \(N(w)\) iff \([B]^M\) is in \(N(w)\). Likewise, \([\neg A]^M\) is in \(N(w)\) iff \([\neg B]^M\) is in \(N(w)\).

Now any sentence \(A\) is logically equivalent to \((A \land B) \lor (A \land \neg B)\), for any \(B\). In the logic E, \(\mathsf {P} A\) therefore entails \(\mathsf {P} ((A \land B) \lor (A \land \neg B))\). By (FC), \(\mathsf {P} ((A \land B) \lor (A \land \neg B))\) entails \(\mathsf {P} (A \land B)\). We could still reason from ‘you may have a cookie’ to ‘you may have a cookie and burn down the house’.

Exercise 6.18

Rational beliefs come in degrees, which are often assumed to satisfy the formal rules of probability. Suppose we say that someone believes \(A\) iff their degree of belief in \(A\) is above a certain threshold – say, 0.9. Explain why one can’t give a Kripke semantics for this concept of belief. (Although one can give a neighbourhood semantics.) Hint: One rule of probability says that if \(p\) and \(q\) are independent propositions, then the probability of their conjunction \(p \land q\) is the product of their individual probabilities.

Next chapter: 7 Temporal Logic