8Conditionals

8.1Material conditionals

‘What if?’ questions play an important role in our thinking. What will happen to the climate if we don’t reduce greenhouse gases? Will the curry taste better if we add lime? Could World War II have been avoided if the Treaty of Versailles had been less punitive?

A sentence stating that something is (or would be) the case if something else is (or would be) the case is called a conditional. Philosophers have debated the meaning and logic of conditionals for over 2000 years, with no agreement in sight.

One attractively simple view is that a conditional ‘if \(A\) then \(B\)’ is true iff the antecedent \(A\) is false or the consequent \(B\) is true. This would make conditionals truth-functional: ‘if \(A\) then \(B\)’ would be equivalent to \(\neg A \lor B\). Sentences with these truth-conditions are called material conditionals.

The conditionals \(A \to B\) of classical logic are material. The attractively simple view that English conditionals are material conditionals would mean that we can faithfully translate English conditionals into \(\mathfrak {L}_{M}\)-sentences of the form \(A \to B\). Is this correct? There are some arguments for a positive answer.

To begin, modus ponens looks valid for English conditionals: from \(A\) and ‘if \(A\) then \(B\)’ we can infer \(B\). It seems incoherent to accept \(A\) and ‘if \(A\) then \(B\)’ while denying \(B\). If this is right, ‘if \(A\) then \(B\)’ entails \(\neg A \lor B\). For suppose there is a case where ‘if \(A\) then \(B\)’ is true but \(\neg A \lor B\) false. The only way \(\neg A \lor B\) can be false is that \(A\) is true and \(B\) false. So any case where ‘if \(A\) then \(B\)’ is true but \(\neg A \lor B\) false would be a case where \(A\) and ‘if \(A\) then \(B\)’ are true while \(B\) is false: we would have a counterexample to modus ponens.

So ‘if \(A\) then \(B\)’ plausibly entails \(\neg A \lor B\). To show that English conditionals are material conditionals, it remains to show that the converse holds as well: that \(\neg A \lor B\) entails ‘if \(A\) then \(B\)’.

One argument to this effect draws on the observation that ‘\(A\) or \(B\)’ seems to entail ‘if not-\(A\) then \(B\)’. (This is sometimes called the or-to-if inference.) For example, suppose I tell you that Nadia is either in Rome or in Paris. Trusting me, you can infer that if she’s not in Rome then she’s in Paris. Suppose, then, that ‘\(A\) or \(B\)’ entails ‘if not-\(A\) then \(B\)’. Then ‘not-\(A\) or \(B\)’ entails ‘if not-not-\(A\) then \(B\)’: every instance of this schema is an instance of the or-to-if schema. Cancelling the double negation, and assuming that ‘not-\(A\) or \(B\)’ means \(\neg A \lor B\), we have reached the desired conclusion: \(\neg A \lor B\) entails ‘if \(A\) then \(B\)’.

Another argument starts with the judgement that ‘if \(A\) then if \(B\) then \(C\)’ is equivalent to ‘if \(A\) and \(B\) then \(C\)’. This is known as the Import-Export principle. Now take the sentence

(*)

If not-\(A\) or \(B\) then if \(A\) then \(B\).

By Import-Export, (*) is equivalent to ‘if [not-\(A\) or \(B\)] and \(A\) then \(B\)’. But ‘[not-\(A\) or \(B\)] and \(A\)’ is logically equivalent to \(B\). So (*) is equivalent to ‘if \(B\) then \(B\)’. So (*) is a logical truth. But if (*) is a logical truth and modus ponens is valid for English conditionals, then ‘not-\(A\) or \(B\)’ entails ‘if \(A\) then \(B\)’.

In sum, there are good arguments to believe that English conditionals are material conditionals. Confusingly, there are also good arguments for the opposite conclusion. Consider the following facts about logical consequence (in classical propositional logic).

(M1)

\(B \models A \to B\)

(M2)

\(\neg A \models A \to B\)

(M3)

\(\neg (A \to B) \models A\)

(M4)

\(A \to B \models \neg B\to \neg A\)

(M5)

\(A \to B \models (A\land C) \to B\)

If English conditionals were material conditionals, the following inferences, corresponding to (M1)–(M5), would be valid.

(E1)

There won’t be a nuclear war. Therefore: If Russia attacks the US with nuclear weapons, there won’t be a nuclear war.

(E2)

There won’t be a nuclear war. Therefore: If there will be a nuclear war, nobody will die.

(E3)

It is not the case that if it will rain tomorrow then the Moon will fall onto the Earth. Therefore: It will rain tomorrow.

(E4)

If our opponents are cheating, we will never find out. Therefore: If we will find out that our opponents are cheating, then they aren’t cheating.

(E5)

If you add sugar to your coffee, it will taste good. Therefore: If you add sugar and vinegar to your coffee, it will taste good.

These inferences don’t seem valid. If we wanted to defend the material analysis of English conditionals, we would have to explain why the inferences sound bad even though they are actually valid. Exercise 8.7 will give you a hint of what such an explanation might involve.

In the meantime, let’s look at an alternative idea of how conditionals in natural language might be understood.

8.2Strict conditionals

The truth of a material conditional requires no connection between the antecedent and the consequent. By contrast, English conditionals usually convey that there is some such connection. Consider (1).

(1)

If we leave after 5, we will miss the train.

Intuitively, an utterance of (1) claims that missing the train is a necessary consequence of leaving after 5 – that it is impossible to leave after 5 and still catch the train, given certain facts about the distance to the station, the time it takes to get there, etc.

This suggests that (1) should be formalized not as \(p \to q\) but as \(\Box (p \to q)\) or, equivalently, \(\neg \Diamond (p \land \neg q)\).

Sentences that are equivalent to \(\Box (A \to B)\) are called strict conditionals. The label goes back to C.I. Lewis (1918), who also introduced the abbreviation \(A \strictif B\) for \(\Box (A \to B)\).

Lewis was not interested in ‘if …then …’ sentences. He introduced \(A \strictif B\) to formalize ‘\(A\) implies \(B\)’ or ‘\(A\) entails \(B\)’. His intended use of \(\strictif \) roughly matches our use of the double-barred turnstile ‘\(\models \)’. But there are important differences. The turnstile is an operator in our meta-language; Lewis’s \(\strictif \) is an object-language operator that, like \(\land \) or \(\to \), can be placed between any two sentences in a formal language to generate another sentence in the language. \(p \strictif (q \strictif p)\) is well-formed, whereas \(p \models (q\models p)\) is gibberish. Moreover, while \(p \models q\) is simply false – because there are models in which \(p\) is true and \(q\) false – Lewis’s \(p \strictif q\) is true on some interpretation of the sentence letters and false on others. If \(p\) means that it is raining heavily and \(q\) that it is raining, then \(p \strictif q\) is true because the hypothesis that it is raining heavily implies that it is raining.

Let’s set aside Lewis’s project of formalizing the concept of implication. Our goal is to find an object-language construction that functions like ‘if …then …’ in English. To see whether ‘\(\ldots \strictif \ldots \)’ can do the job, let’s have a closer look at the logic of strict conditionals.

Since \(A \strictif B\) is equivalent to \(\Box (A \to B)\), standard Kripke semantics for the box also provides a semantics for strict conditionals. In Kripke semantics, \(\Box (A \to B)\) is true at a world \(w\) iff \(A \to B\) is true at all worlds \(v\) accessible from \(w\). We also know that \(A \to B\) is true at \(v\) iff \(A\) is false at \(v\) or \(B\) is true at \(v\). We therefore have the following truth-conditions for strict conditionals.

As always, the logic of strict conditionals depends on what constraints we put on the accessibility relation. Without any constraints, \(\strictif \) does not validate modus ponens, in the sense that \(A \strictif B\) and \(A\) together do not entail \(B\). We can see this by translating \(A \strictif B\) back into \(\Box (A \to B)\) and setting up a tree. Recall that to test whether some premises entail a conclusion, we start the tree with the premises and the negated conclusion.

With the K-rules, where we don’t make any assumptions about the accessibility relation, node 1 can’t be expanded, so there is nothing more we can do.

If we assume that the accessibility relation is reflexive, the tree closes:

It is not hard to show that modus ponens for \(\strictif \) is valid on all and only the reflexive frames. Reflexivity is precisely what we need to render modus ponens valid. Since modus ponens looks plausible for English conditionals, we’ll probably want the relevant Kripke models to be reflexive.

We could now look at other conditions on the accessibility relation and decide whether they should be imposed, based on what they would imply for the logic of conditionals. But let’s take a shortcut.

I have suggested that sentence (1) might be understood as saying that it is impossible to leave after 5 and still make it to the train. Impossible in what sense? There are many possible worlds at which we leave after 5 and still make it to the train. There are, for example, worlds at which the train departs two hours later, worlds at which we live right next to the station, and so on. When I say that it is impossible to leave after 5 and still make it to the train, I arguably mean that it is impossible given what we know about the departure time, our location, etc.

Generalizing, a tempting proposal is that the accessibility relation relevant for conditionals like (1) is the epistemic accessibility relation that we studied in chapter 5, where a world \(v\) is accessible from \(w\) iff it is compatible with what is known at \(w\). On that hypothesis, the logic of conditionals is determined by the logic of epistemic necessity. We don’t need to figure out the relevant accessibility relation from scratch.

Since knowledge varies from agent to agent, the present idea implies that the truth-value of conditionals should be agent-relative. This seems to be confirmed by the following puzzle, due to Allan Gibbard.

Sly Pete and Mr. Stone are playing poker on a Mississippi riverboat. It is now up to Pete to call or fold. My henchman Zack sees Stone’s hand, which is quite good, and signals its content to Pete. My henchman Jack sees both hands, and sees that Pete’s hand is rather low, so that Stone’s is the winning hand. At this point the room is cleared. A few minutes later, Zack slips me a note which says ‘if Pete called, he won’, and Jack slips me a note which says ‘if Pete called, he lost’.

The puzzle is that Zack’s note and Jack’s note are intuitively contradictory, yet they both seem to be true.

We can resolve the puzzle if we understand the conditionals as strict conditionals with an agent-relative epistemic accessibility relation. Take Zack. Zack knows that Pete knows Stone’s hand. He also knows that Pete would not call unless he has the better hand. So among the worlds compatible with Zack’s knowledge, all worlds at which Pete calls are worlds at which Pete wins. If \(p\) translates ‘Pete called’ and \(q\) ‘Pete won’, then \(p \strictif q\) is true relative to Zack’s information state. Relative to Jack’s information state, however, the same sentence is false. Jack knows that Stone’s hand is better than Pete’s, but he doesn’t know that Pete knows Stone’s hand. Among the worlds compatible with Jack’s knowledge, all worlds at which Pete calls are therefore worlds at which Pete loses. Relative to Jack’s information state, \(p \strictif \neg q\) is true.

Another advantage of the “epistemically strict” interpretation is that it might explain why conditionals with antecedents that are known to be false often seem defective. For example, imagine a scenario in which Jones has gone to work. In that scenario, is (2) true or false?

(2)

If Jones has not gone to work, he is helping his neighbours.

The question is hard to answer – and not because we lack information about the scenario. Once we are told that Jones has gone to work, it is unclear how we are meant to assess whether Jones is helping his neighbours if he has not gone to work. On the epistemically strict interpretation, (2) says that Jones is helping his neighbours at all epistemically accessible worlds at which Jones hasn’t gone to work. Since we know that Jones has gone to work, there are no epistemically accessible worlds at which he hasn’t gone to work. And if there are no \(A\)-worlds, we naturally balk at the question whether all \(A\)-worlds are \(B\)-worlds. (In logic, we resolve to treat ‘all \(A\)s are \(B\)’ as true if there are no \(A\)s. Accordingly, (2) comes out true on the epistemically strict analysis. But we can still explain why it seems defective.)

So perhaps English conditionals should be understood not as material conditionals, but as “epistemically strict” conditionals – strict conditionals with an epistemic accessibility relation.

As you might expect, there are also problems with this proposal. Remember (E1)–(E5) here. If English conditionals are strict conditionals, then (E1)–(E3) are invalid. For example, while \(q\) entails \(p \to q\), it does not entail \(p \strictif q\). But the strict analogs of (M4) and (M5) still hold, no matter what we say about accessibility (see exercise 8.5): \begin {flalign*} \quad & A \strictif B \models \neg B\strictif \neg A; &\\ \quad & A \strictif B \models (A\land C) \strictif B. & \end {flalign*}

So we still predict that the inferences (E4) and (E5) are valid.

(E4)

If our opponents are cheating, we will never find out. Therefore: If we will find out that our opponents are cheating, then they aren’t cheating.

(E5)

If you add sugar to your coffee, it will taste good. Therefore: If you add sugar and vinegar to your coffee, it will taste good.

8.3Variably strict conditionals

So far, we’ve focussed on so-called indicative conditionals. Another important type of conditional are subjunctive conditionals. Compare the following two statements.

(2)

If Shakespeare didn’t write Hamlet, someone else did.

(3)

If Shakespeare hadn’t written Hamlet, someone else would have.

(2) seems true. Someone has written Hamlet; if it wasn’t Shakespeare, it must have been someone else. But (3) is almost certainly false. After all, it is very likely that Shakespeare did write Hamlet. And it is highly unlikely that if he hadn’t written Hamlet – if he got distracted by other projects, say – then someone else would have stepped in to write the exact same piece.

(2) is an indicative conditional. Intuitively, an indicative conditional states that something is in fact the case on the assumption that something else is the case. A subjunctive conditional like (3) states that something would be the case if something else were the case. Normally, we know that the “something else” is not in fact the case. We know, for example, that Shakespeare wrote Hamlet and therefore that the antecedent of (3) is false. For this reason, subjunctive conditionals are also called counterfactual conditionals or simply counterfactuals.

Since (2) is true and (3) false, subjunctive conditionals and indicative conditionals must have a different meaning. Indeed, it is clear that (3) is neither a material conditional nor an epistemically strict conditional: both of these readings would render (3) true.

To think about how we evaluate subjunctive conditionals, let’s look at an example that’s true. Suppose Jones has promised to hang up the laundry, and he does hang it up. The following is true:

(4)

If Jones hadn’t hung up the laundry, he would have broken a promise.

There is no logical connection between the antecedent and the consequent of (4). There are possible worlds where Jones doesn’t hang up the laundry without breaking any promise – because he never made the promise. When we evaluate what would have happened if Jones hadn’t hung up the laundry, we don’t consider such worlds. We hold fixed his promise.

This suggests that (4) expresses a strict conditional with an accessibility relation that excludes worlds at which the promise was never made. But what are the general rules for this accessibility relation?

Consider (5).

(5)

If Jones had gone on an expedition to the North Pole, he wouldn’t have promised to hang up the laundry.

This may also be true. When we evaluate (5), we don’t hold fixed Jones’s promise. Worlds where he doesn’t make the promise are now accessible.

So the accessibility relation for subjunctive conditionals appears to vary from conditional to conditional. As David Lewis put it, subjunctive conditionals seem to be not strict, but “variably strict”.

Let’s try to get a better grip on how this might work. (What follows is a slightly simplified version of an analysis developed by Robert Stalnaker and David Lewis in the 1960s. It is the most popular analysis of subjunctive conditionals.)

Intuitively, when we ask what would have been the case if a certain event had (or had not) occurred, we are considering worlds that are much like the actual world up to the time of the event. Then these worlds deviate in some minimal way to allow the event to occur (or not occur). Afterwards the worlds unfold in accordance with the general laws of the actual world.

For example, when we wonder what would have happened if Shakespeare hadn’t written Hamlet, we are attending to worlds that are like the actual world until 1599, at which point some mundane circumstances prevent Shakespeare from writing Hamlet. We are not interested in worlds at which Shakespeare was never born, or in which he was abducted by aliens. Plausibly, Shakespeare would have been a famous author even if he hadn’t written Hamlet, although he could hardly be famous in worlds in which he was never born.

Likewise for (4). Here we are considering worlds that are much like the actual world up to the point where Jones hangs up the laundry. Worlds where he never made the promise, perhaps because he has left for an arctic expedition, are ignored. Figuratively speaking, such worlds are “too remote”: they differ from the actual world in ways that are not required to make the antecedent true.

This suggests that a subjunctive conditional is true iff the consequent is true at the “closest” worlds at which the antecedent is true – where “closeness” is a matter of similarity in certain respects. The closest worlds (to the actual world) at which Shakespeare didn’t write Hamlet are worlds that match the actual world until 1599, then deviate a little so that Shakespeare didn’t write Hamlet, and afterwards still resemble the actual world with respect to the general laws of nature. We will not try to spell out in full generality what the relevant closeness measure should look like.

Let ‘\(v \prec _w u\)’ mean that \(v\) is closer to \(w\) than \(u\), in the sense that \(v\) differs less than \(u\) from \(w\) in whatever respects are relevant to the interpretation of subjunctive conditionals.

We make the following structural assumptions about the world-relative ordering \(\prec \).

1.

If \(v \prec _w u\) then \(u \nprec _w v\). (Asymmetry)

2.

If \(v \prec _w u\), then for all \(t\) either \(v \prec _w t\) or \(t \prec _w u\). (Quasi-connectedness)

3.

For any non-empty set of worlds \(X\) and world \(w\) there is a \(v\) in \(X\) such that there is no \(u\) in \(X\) with \(u \prec _w v\).

Asymmetric and quasi-connected relations are known as weak orders. Asymmetry is self-explanatory. Quasi-connectedness is more often called negative transitivity, because it is equivalent to the assumption that if \(t \nprec s\) and \(s\nprec r\) then \(t\nprec r\). It ensures that the “equidistance” relation that holds between \(v\) and \(u\) if neither \(v \prec _w u\) nor \(u \prec _w v\) is an equivalence relation. With these two assumptions, we can picture each world \(w\) as associated with nested spheres of worlds; \(v \prec _w u\) means that \(v\) is in a more narrow \(w\)-sphere than \(u\).

Assumption 3 is known as the Limit Assumption. It ensures that for any consistent proposition \(A\) and world \(w\), there is a set of closest \(A\)-worlds. Without the Limit Assumption, there could be an infinite chain of ever closer \(A\)-worlds, with no world being maximally close.

We are going to introduce a variably strict operator \(\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \) so that \(A\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) is true at a world \(w\) iff \(B\) is true at the closest worlds to \(w\) at which \(A\) is true. Models for a language with the \(\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \) operator must contain closeness orderings \(\prec \) on the set of worlds.

To formally state the semantics of \(\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \), we can re-use a concept from section 6.3. Let \(S\) be an arbitrary set of worlds, and let \(w\) be some world (that may or may not be in \(S\)). It will be useful to have an expression that picks out the most similar worlds to \(w\), among all the worlds in \(S\). This expression is \(\mathrm {Min}^{\prec _w}(S)\), which we have defined as follows in section 6.3: \[ \mathrm {Min}^{\prec _w}(S) =_\text {def} \{ v: v \in S \land \neg \exists u (u \in S \land u \prec _w v) \}. \]

Now \(\{ u : M,u\models A \}\) is the set of worlds (in model \(M\)) at which \(A\) is true. So \(\mathrm {Min}^{\prec _w}(\{ u : M,u\models A \})\) is the set of those \(A\)-worlds that are closest to \(w\). We want \(A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) to be true at \(w\) iff \(B\) is true at the closest \(A\)-worlds to \(w\).

You may notice that \(A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) works almost exactly like \(\mathsf {O}(B/A)\) from section 6.3. There, I said that for any world \(w\) in any deontic ordering model \(M\),

 \(M,w \models \mathsf {O} (B/A) \text { iff } M,v \models B\text { for all $v$ in $\mathrm {Min}^{\prec _w}(\{ u: wRu $ and $M,u\models A \})$}\).

The main difference is that conditional obligation is sensitive to an accessibility relation. If that relation is an equivalence relation, this makes no difference to the logic.

Of course, the order \(\prec \) in deontic ordering models is supposed to represent degree of conformity to norms, while the order \(\prec \) in similarity models represents a certain similarity ranking in the evaluation of subjunctive conditionals. A different type of ordering might be in play when we evaluate indicative conditionals, which some have argued should also be interpreted as variably strict. But again, these differences in interpretation don’t affect the logic.

Suppose we add the \(\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \) operator to the language of standard propositional logic. The set of sentences in the resulting language that are true at all worlds in all similarity models is known as system V. There are tree rules and axiomatic calculi for this system, but they aren’t very user-friendly. We will only explore the system semantically.

To begin, we can check whether modus ponens is valid for \(\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \). That is, we check whether the truth of \(A\) and \(A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) at a world in a similarity model entails the truth of \(B\).

Assume that \(A\) and \(A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) are true at a world \(w\). By definition 8.3, the latter means that \(B\) is true at all the closest \(A\)-worlds to \(w\) (at all worlds in \(\mathrm {Min}^{\prec _w}(\{u: M,u\models A\})\)). The world \(w\) itself is an \(A\)-world. If we could show that \(w\) is among the closest \(A\)-worlds to itself then we could infer that \(B\) is true at \(w\).

Without further assumptions, however, we can’t show this. If we want to validate modus ponens, we must add a further constraint on our models: that every world is among the closest worlds to itself. More precisely, \[ \text {for all worlds $w$ and $v$, $v \nprec _{w} w$.} \] This assumption is known as Weak Centring. The logic we get if we impose this constraint is system VC.

None of the problematic inferences (E1)–(E5) are valid if the relevant conditionals are interpreted as variably strict. (E5), for example, would assume that \(p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r \) entails \((p \land q) \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r\). But it does not. We can give a countermodel with two worlds   \(w\) and \(v\); \(p\) is true at both worlds, \(q\) is true only at \(v\), and \(r\) only at \(w\); if \(w\) is closer to itself than \(v\), then \(p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r\) is true at \(w\) (because the closest \(p\)-worlds to \(w\) are all \(r\)-worlds), but \((p \land q) \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r\) is false at \(w\) (because the closest \((p\land q)\)-worlds to \(w\) aren’t all \(r\)-worlds).

The diagram on the right represents this model. The circles around \(w\) depict the similarity spheres. \(w\) is closer to \(w\) than \(v\) because it is in the innermost sphere around \(w\), while \(v\) is only in the second sphere. (If \(v\) were also in the innermost sphere then the two worlds would be equally close to \(w\). That’s allowed.) In general, we can represent the assumption that a world \(v\) is closer to a world \(w\) than a world \(u\) (\(v \prec _w u\)) by putting \(v\) in a closer sphere around \(w\) than \(u\). I have not drawn any spheres around \(v\) because it doesn’t matter what these look like.

The logic of variably strict conditionals is weaker than the logic of strict conditionals. Some have argued that it is too weak to explain our reasoning with conditionals. For example, the following statements are all false. (The corresponding statements for \(\strictif \) are true; see exercise 8.5.)

1.

\(p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow q, q \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r \models p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r\)

2.

\(((p \lor q) \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r) \models (p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r) \land (q \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r)\)

3.

\(p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow (q \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r) \models (p \land q) \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow r\)

If English conditionals are variably strict, this means (for example) that we can’t infer ‘if \(p\) then \(r\)’ from ‘if \(p\) then \(q\)’ and ‘if \(q\) then \(r\)’. But isn’t this a valid inference?

Well, perhaps not. Stalnaker gave the following counterexample, using cold-war era subjunctive conditionals.

If J. Edgar Hoover had been born a Russian, he would be a communist.
If Hoover were a communist, he would be a traitor.
Therefore, if Hoover had been born a Russian, he would be a traitor.

The semantics I have presented for \(\mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \) is a middle ground between that of Lewis and Stalnaker. Stalnaker assumes that \(\prec _w\) is not just quasi-connected, but connected: for any \(w,v,u\), either \(v \prec _w u\) or \(v=u\) or \(u \prec _w v\). (‘\(v=u\)’ means that \(v\) and \(u\) are the same world.) This rules out ties in similarity: no sphere contains more than one world.

Stalnaker’s logic (called C2) is stronger than Lewis’s VC. The following principle of “Conditional Excluded Middle” is C2-valid but not VC-valid: \begin {equation} \tag {CEM}(A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B) \lor (A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \neg B) \end {equation}

Whether conditionals in natural language satisfy Conditional Excluded Middle is a matter of ongoing debate. On the one hand, it is natural think that ‘it is not the case that if \(p\) then \(q\)’ entails ‘if \(p\) then not \(q\)’, which suggests that the principle is valid. On the other hand, suppose I have a number of coins in my pocket, none of which I have tossed. What would have happened if I had tossed one of the coins? Arguably, I might have gotten heads and I might have gotten tails. Either result is possible, but neither would have come about.

Lewis not only rejects connectedness, but also the Limit Assumption. He argued that there might be an infinite chain of ever closer \(A\)-worlds. Definition 8.3 implies that if there are no closest \(A\)-worlds then any sentence of the form \(A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) is true. That does not seem right. Lewis therefore gives a more complicated semantics:

\(M,w \models A \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow B\) iff either there is no \(v\) for which \(M,v\models A\) or there is some world \(v\) such that \(M,v\models A\) and for all \(u \prec _w v\), \(M,u \models A \to B\).

It turns out that it makes no difference to the logic whether we impose the Limit Assumption and use the old definition or don’t impose the Limit Assumption and use Lewis’s new definition. The same sentences are valid either way.

8.4Restrictors

Consider these two statements.

(1)

If it rains we always stay inside.

(2)

If it rains we sometimes stay inside.

On its most natural reading, (1) says that we stay inside at all times at which it rains. We can express this in \(\mathfrak {L}_{M}\), using the box as a universal quantifier over the relevant times. (So \(\Box A\) now means ‘always \(A\)’.) The translation would be \(\Box (r \to s)\).

One might expect that (2) should then be translated as \(\Diamond (r \to s)\), where the diamond is an existential quantifier over the relevant times (‘sometimes’). But \(\Diamond (r \to s)\) is equivalent to \(\Diamond (\neg r \lor s)\). This is true whenever \(\Diamond \neg r\) is true. (2), however, isn’t true simply because it doesn’t always rain. On its most salient reading, (2) says there are times at which it rains and we stay inside. Its correct translation is \(\Diamond (r \land s)\).

This is a little surprising, given that (2) seems to contain a conditional. Does the conditional here express a conjunction?

Things get worse if we look at (3).

(3)

If it rains we usually stay inside.

Let’s introduce an operator \(\mathsf {M}\) for ‘usually’, so that \(\mathsf {M} A\) is true at a time iff \(A\) is true at most times. Can you translate (3) with the help of \(\mathsf {M}\)?

You can’t. Neither \(\mathsf {M}(r \to s)\) nor \(\mathsf {M}(r \land s)\) capture the intended meaning of (3). \(\mathsf {M}(r \land s)\) entails that \(r\) is usually true. But (3) doesn’t entail that it usually rains. \(\mathsf {M}(r \to s)\) is true as long as \(r\) is usually false, even if we’re always outside when it is raining. You could try to bring in some of the new kinds of conditional that we’ve encountered in the previous sections. How about \(\mathsf {M}(r \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow s)\), or \(\mathsf {M}(r \strictif s)\), or \(r \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow \mathsf {M} s\), or \(r \strictif \mathsf {M} s\)? None of these are adequate.

The problem is that (3) doesn’t say, of any particular proposition, that it is true at most times. It doesn’t say that among all times, most are such-and-such. Rather, it says that among times at which it rains, most times are times at which we stay inside. The function of the ‘if’-clause in (3) is to restrict the domain of times over which the ‘usually’ operator quantifies.

Now return to (1) and (2). Suppose that here, too, the ‘if’-clause serves to restrict the domain of times, so that ‘always’ and ‘sometimes’ only quantify over times at which it rains. On that hypothesis, (1) says that among times at which it rains, all times are times at which we stay inside, and (2) says that among times at which it rains, some times are times at which we stay inside. This is indeed what (1) and (2) mean, on their most salient interpretation.

As it turns out, ‘among \(r\)-times, all times are \(s\)-times’ is equivalent to ‘all times are not-\(r\)-times or \(s\)-times’. That’s why we can formalize (1) as \(\Box (r \to s)\). ‘Among \(r\)-times, some times are \(s\)-times’, on the other hand, is equivalent to ‘some times are \(r\)-times and \(s\)-times’. That’s why we can formalize (2) as \(\Diamond (r \land s)\). It would be wrong to think that the conditional in (1) is material, the conditional in (2) is a conjunction, and the conditional in (3) is something else altogether. A much better explanation is that the ‘if’-clause in (1) does the exact same thing as in (2) and (3). In each case, it restricts the domain of times over which the relevant operators quantify.

We can arguably see the same effect in (4) and (5).

(4)

If the lights are on, Ada must be in her office.

(5)

If the lights are on, Ada might be in her office.

Letting the box express epistemic necessity, we can translate (4) as \(\Box (p \to q)\). But (5) can’t be translated as \(\Diamond (p \to q)\), which would be equivalent to \(\Diamond (\neg p \lor q)\). Nor can we translate (5) as \(p \to \Diamond q\), which is entailed by \(\Diamond q\). It is easy to think of scenarios in which (5) is false even though ‘Ada might be in her office’ is true. The correct translation of (5) is plausibly \(\Diamond (p \land q)\). The sentence is true iff there is an epistemically accessible world at which the lights are on and Ada is in her office.

As before, we can understand what is going on if we assume that the ‘if’-clause in (4) and (5) functions as a restrictor. The ‘if’-clause restricts the domain of worlds over which ‘must’ and ‘might’ quantify. (4) says that among epistemically possible worlds at which the lights are on, all worlds are worlds at which Ada is in her office. (5) says that among epistemically possible worlds at which the lights are on, some worlds are worlds at which Ada is in her office.

The hypothesis that ‘if’-clauses are restrictors also sheds light on the problem of conditional obligation.

(6)

Jones ought to help his neighbours.

(7)

If Jones doesn’t help his neighbours, he ought to not tell them that he’s coming.

In chapter 6, we analysed ‘ought’ as a quantifier over the best of the circumstantially accessible worlds. On this approach, (6) says that among the accessible worlds, all the best ones are worlds at which Jones helps his neighbours. Suppose the ‘if’-clause in (7) serves to restrict the domain of worlds, excluding worlds at which Jones helps his neighbours. We then predict (7) to state that among the accessible worlds at which Jones doesn’t help his neighbours, all the best worlds are worlds at which Jones doesn’t tell his neighbours that he’s coming. This can’t be expressed by combining the monadic \(\mathsf {O}\) quantifier with truth-functional connectives. Hence we had to introduce a primitive binary operator \(\mathsf {O}(\cdot /\cdot )\).

The upshot of all this is that we can make sense of a wide range of puzzling phenomena by assuming that ‘if’-clauses are restrictors. Their function is to restrict the domain of worlds or times over which modal operators quantify.

What, then, is the purpose of ‘if’-clauses in “bare” conditionals like (8) and (9), where there are no modal operators to restrict?

(8)

If Shakespeare didn’t write Hamlet, then someone else did.

(9)

If Shakespeare hadn’t written Hamlet, then someone else would have.

Here opinions vary. One possibility, prominently defended by the linguist Angelika Kratzer, is that even bare conditionals contain modal operators. Arguably, ‘would’ in (9) functions as a kind of box. If this box is a simple quantifier over circumstantially accessible worlds, and the ‘if’-clause in (9) restricts its domain, then (9) can be formalized as \(\Box (p \to q)\). If, on the other hand, ‘would’ in (9) works more like ‘ought’ – if it quantifies over the closest of the accessible worlds –, and the ‘if’-clause restricts the domain of accessible worlds, then the resulting truth-conditions are those of \(p \mathrel {\mathop \Box }\mathrel {\mkern -2.5mu}\rightarrow q\). Both the strict and the variably strict analysis of (9) are therefore compatible with the hypothesis that ‘if’-clauses are restrictors.

What about (8)? This sentence really doesn’t appear to contain a relevant modal. Kratzer suggests that it contains an unpronounced epistemic ‘must’: (8) says that if Shakespeare didn’t write Hamlet then someone else must have written Hamlet. Assuming that the ‘if’-clause restricts the domain of this operator, bare indicative conditionals would be equivalent to strict epistemic conditionals.