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Inquisitive Semantics$

Ivano Ciardelli, Jeroen Groenendijk, and Floris Roelofsen

Print publication date: 2018

Print ISBN-13: 9780198814788

Published to Oxford Scholarship Online: December 2018

DOI: 10.1093/oso/9780198814788.001.0001

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Conditionals

Conditionals

Chapter:
(p.115) 7 Conditionals
Source:
Inquisitive Semantics
Author(s):

Ivano Ciardelli

Jeroen Groenendijk

Floris Roelofsen

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198814788.003.0007

Abstract and Keywords

Chapter 7 argues that inquisitive semantics is not only relevant for questions, but also for statements. The argument is based on recent experimental work on counterfactual conditionals, which shows that even if two premises A and B have exactly the same truth-conditions, the counterfactuals “If A then C” and “If B then C” may have different truth conditions. This means that it is impossible to give a compositional account of counterfactuals based on a purely truth-conditional notion of meaning. On the other hand, the relevant contrast finds a natural explanation once conditionals are analysed in inquisitive semantics. Further benefits of the account are discussed as well: it solves a well-known problem that classical analyses of conditionals have with disjunctive antecedents, and it naturally extends to unconditionals and conditional questions.

Keywords:   inquisitiveness beyond questions, counterfactual conditionals, disjunctive antecedents, unconditionals, conditional questions

In the previous chapter, we have seen that the inquisitive notion of meaning allows us to obtain a uniform semantic analysis of lexical and intonational elements that occur both in declarative and in interrogative sentences. However, we assumed that the logical form of an (ordinary, falling) declarative sentence is always headed by a projection operator, !, which makes the sentence non-inquisitive. This may suggest that, as long as we are only concerned with such sentences (and, therefore, not with treating operators like disjunction in a way that works uniformly across declaratives and interrogatives), the standard truth-conditional notion of meaning serves us well enough, and keeping track of inquisitive content is an unnecessary complication.

In this chapter, we will see that this is not the case: even for sentences which are not inquisitive, and whose meaning is therefore completely determined by their truth conditions, these truth conditions may depend crucially on the inquisitive content of some constituent within the sentence. Thus, to derive the right truth conditions for the whole sentence, the inquisitive content of the sentence’s constituents must be taken into account.

We will demonstrate this based on recent experimental work by Ciardelli, Zhang, and Champollion (2017c) on counterfactual conditionals. This work shows that even if two clauses φ‎ and φ‎′ have exactly the same truth conditions, the counterfactuals φ‎ > ψ‎ and φ‎′ > ψ‎ may have different truth conditions. In particular, the counterfactuals (1a) and (1b) have different truth-conditions, even though their antecedents are truth-conditionally equivalent.

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Conditionals

This means that it is impossible to give a compositional account of counterfactuals based on a purely truth-conditional notion of meaning.

(p.116) Ciardelli et al. (2017c) show that the relevant contrast finds a natural explanation once conditionals are analysed in inquisitive semantics. Moreover, Ciardelli (2016b) argues that an inquisitive analysis of conditionals has other merits as well: on the one hand, it solves a well-known problem that classical analyses of conditionals have with disjunctive antecedents; on the other hand, it does not only allow us to interpret run-of-the-mill conditional statements, but also two other classes of conditional constructions, namely, unconditionals such as (2a, b), and conditional questions such as (3a, b).

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Conditionals

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Conditionals

In this chapter, we will present the experimental results and theoretical arguments of Ciardelli (2016b) and Ciardelli et al. (2017c) in condensed form. Section 7.1 describes the experiment and explains why the obtained results are problematic for the standard view that equates meaning with truth-conditions. Section 7.2 introduces a recipe for lifting theories of conditionals from truth-conditional semantics to inquisitive semantics, and shows how the experimental results receive a natural explanation once we combine this inquisitive lifting with suitable assumptions about the process of making counterfactual assumptions. Finally, Section 7.3 discusses various further advantages of an inquisitive treatment of conditionals.

7.1 Evidence for truth-conditional effects

7.1.1 The experiment

Imagine a long hallway with a light in the middle and with two switches, one at each end. One switch is called switch A and the other one is called switch B. As the wiring diagram in Figure 7.1 shows, the light is on whenever both switches are in the same position (both up or both down); otherwise, the light is off. Right now, switch A and switch B are both up, and the light is on. But things could be different…

Conditionals

Figure 7.1 A multiway switch.

Which of the following counterfactual sentences are true in this scenario?

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Conditionals

(p.117)

Ciardelli et al. (2017c) conducted an experiment to test the intuitions of native speakers of English about this question. Participants were recruited online using Amazon’s Mechanical Turk platform; they were first shown the short text above and the diagram in Figure 7.1; they were then presented with one of the sentences in (4) and a filler sentence (one at a time, in random order), and they were asked to judge these sentences as either true, false, or indeterminate. Data from participants who failed to judge the filler correctly, or who otherwise failed to qualify for the task, were rejected. The remaining results are summarized in Table 7.1. For our purposes, the most important result is the contrast between sentences (4c) and (4d): (4c) was judged true by about 70% of participants, while only 22% of participants judged (4d) true.

Table 7.1 Results of Ciardelli et al.’s (2017c) main experiment

Sentence

Number

True

(%)

False

(%)

Indet.

(%)

(4a)

256

169

66.02%

6

2.34%

81

31.64%

(4b)

235

153

65.11%

7

2.98%

75

31.91%

(4c)

362

251

69.33%

14

3.87%

97

26.80%

(4d)

372

82

22.04%

136

36.56%

154

41.40%

(4e)

200

43

21.50%

63

31.50%

94

47.00%

7.1.2 A problem for the truth-conditional view on meaning

Assuming for the moment that the judgments found in the experiment are due to an actual difference in truth value between (4c) and (4d) in the given context, this is problematic for the standard view that equates (p.118) meaning with truth-conditions, regardless of the particular account of conditionals one assumes. To see why, consider the clauses (5a) and (5b), corresponding to the two antecedents of (4c) and (4d).

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Conditionals

Assuming that our switches can only take two positions, up and down, these clauses have the same truth conditions. If switch A or switch B is down, then clearly switch A and switch B are not both up. And conversely, if switch A and switch B are not both up, then either of them must be down. Under the view that the meaning of these clauses can be identified with their truth conditions, this means that (5a) and (5b) have the same meaning.

According to the principle of compositionality, the meaning of a sentence depends only on the meaning of its constituents and the way these constituents are combined. This implies that if, in a sentence φ‎, a constituent c is replaced by another constituent c′ with the same meaning, the resulting sentence φ‎[c′/c] must have the same meaning as φ‎.

Now, the counterfactual (4d) can be obtained from (4c) by replacing the sentential constituent corresponding to (5a) with (5b), which has the same meaning. Therefore, (4c) and (4d) must have the same meaning, and thus the same truth conditions. It follows that, in every particular context, these counterfactuals must have the same truth value. But this is not the case: in the context described in the experiment, (4c) is true, but (4d) is not.

This shows that, in combination with the principle of compositionality, the assumption that the meaning of a sentence can be identified with its truth conditions leads to wrong empirical predictions.

7.1.3 Ruling out alternative explanations

Ciardelli et al. (2017c) strengthen the argument made in the previous section by ruling out a number of alternative explanations for the data in Table 7.1.

First, one might worry that (4c) and (4d) are judged differently in spite of actually having the same truth value, due to the interference of other factors. The main concern that motivates this worry is that participants may have judged (4d) incorrectly for one of two reasons: they may have misread the phrase ‘not both up’ as ‘both not up’, that is, as ‘both down’; or they may have been confused by the higher (p.119) processing cost of the antecedent, which involves a negation scoping over a conjunction.

Neither of these hypotheses stands up to further scrutiny. According to the first hypothesis, many participants misread ‘not both up’ as ‘both down’. If so, we would expect many participants to judge the sentence (4e) as true, since the context explicitly specifies that the light is off when both switches are down. This is not what we observe: instead, (4e) is only judged true by about 20% of participants, just like (4d).

According to the second hypothesis, many participants fail to judge (4d) true as a consequence of some context-independent feature of this sentence, such as processing cost. This hypothesis predicts that many participants would also not judge this sentence true if the circuit had been wired differently. In a post-hoc experiment, participants were asked to judge the sentences in (4) in a modified scenario, where the light is on only when both switches are up. In this scenario, an overwhelming majority of participants (about 85%) judged (4d) to be true. The contrast between the results in the two scenarios shows that the reason why (4d) was not judged true in the main experiment does not have to do with intrinsic features of the sentence, but rather with the fact that if both switches were down, the light would not be off. For this is the only difference between the original scenario and the modified one.

Another way to resist the conclusion drawn in the previous section is to accept that the difference in truth values between (4c) and (4d) is genuine, but to deny that the antecedents of these sentences have the same truth conditions. There are two natural ways to do this: one may point out that down is not logically equivalent to not up, or hypothesize that the disjunction in the antecedent of (4c) is interpreted exclusively, i.e., as requiring that only one (and not both) of the disjuncts is true, for instance as a result of some exhaustification operation of the kind discussed by Chierchia et al. (2012).

Again, further control experiments render these explanations implausible. According to the first explanation, the contrast should vanish if the word down was replaced by the expression not up throughout the sentences in (4). A post-hoc experiment revealed that this prediction is not borne out: replacing down by not up does not modify the pattern exhibited by the results in Table 7.1.

According to the second explanation, the disjunction in the antecedent of (4c) is interpreted exclusively, possibly as a result of an exhaustification operator. If so, we would naturally expect the main disjunction in (5a) to be interpreted exclusively as well, and thus to be (p.120) judged as false or indeterminate in a scenario in which both switches are down. In a pre-test, participants were presented with a picture which displays the circuit with both switches down, and they were asked to judge the sentences (5a) and (5b) as true, false, or indeterminate. Both sentences were judged true by over 80% of participants. This shows that an exclusive reading of disjunction in (5a) is at best marginal, which makes it unlikely that it is responsible for the observed contrast.1

7.2 Conditionals in inquisitive semantics

In this section, we show that the findings discussed in the previous section have a natural explanation once we move from a truth-conditional semantic setting to inquisitive semantics. We start in Section 7.2.1 by showing how inquisitive semantics assigns the same truth-conditions but different meanings to the antecedents of (4c) and (4d), thus allowing for a compositional account that assigns different truth conditions to these counterfactuals. In Section 7.2.2, we introduce the inquisitive lifting operation developed in Ciardelli (2016b), and explain how a difference in inquisitive content between two antecedents can result in different truth conditions for the corresponding conditionals. Finally, in Section 7.2.3 we present the background theory of counterfactuals developed by Ciardelli et al. (2017c), and show that the inquisitive lifting of this theory yields the right predictions for the sentences in (4).

7.2.1 Breaking de Morgan’s law in inquisitive semantics

To see how inquisitive semantics allows us to explain the data in Table 7.1, let us first formalize our sentences in the system InqB equipped with an additional counterfactual connective >.2 We will assume a predicate Ux for ‘x is up’, an atomic sentence O for ‘the light is off’, and two (p.121) constants a, b which refer to the two switches. We will then analyse the sentences in (4) as follows:3

Conditionals

Let us consider the antecedents of these counterfactuals. We will assume that our model contains four possible worlds, corresponding to the four possible configurations of the switches. The propositions expressed by the different antecedents in this model are depicted in Figure 7.2.

Conditionals

Figure 7.2 The propositions expressed by the different antecedents. In world ↑↑, both switches are up, in world ↑↓ A is up and B is down, and so on.

Crucially, in inquisitive semantics, the antecedent of (4c), ¬Ua ∨¬Ub, and the antecedent of (4d), ¬(UaUb), are not semantically equivalent: while the two are assigned the same truth-conditions, the former is inquisitive, while the latter is not. ¬Ua ∨¬Ub generates two alternatives, namely, the set of worlds where A is down, and the set of worlds where B is down. By contrast, ¬(UaUb) generates a single alternative, namely, the set of worlds where the switches are not both up. This means that the problem we pointed out for truth-conditional semantics no longer arises in inquisitive semantics: the antecedents of (4c) and (4d) have different meanings, and can therefore make different semantic contributions.

7.2.2 Lifting conditionals to inquisitive semantics

We now want to explain how the difference in inquisitive content between ¬Ua ∨¬Ub and ¬(UaUb) can lead to a difference in truth-conditions for the counterfactuals in which these two clauses are (p.122) embedded as antecedents. For this, we adopt an idea of Alonso-Ovalle (2006, 2009) (see also van Rooij, 2006). We assume that an antecedent need not always specify a single counterfactual assumption; rather, when we have multiple alternatives for an antecedent, each of them is treated by the semantics as a distinct counterfactual assumption. In order for the counterfactual to be true, the consequent must follow on each of these assumptions.

To implement this idea in the inquisitive setting, Ciardelli (2016b) describes a general procedure for lifting accounts of conditionals to inquisitive semantics. This lifting procedure takes as its input a truth-conditional account of (indicative or counterfactual) conditionals, given in the form of a binary operation which maps any two classical propositions α‎ and γ‎ (expressed by the antecedent and the consequent of a conditional, respectively) to a third classical proposition αγ. All the standard theories of counterfactual conditionals, such as selection function semantics (Stalnaker, 1968), ordering semantics (Lewis, 1973), and premise semantics (Kratzer, 1981) yield such an operation .4

The output of the lifting procedure is an ‘inquisitivized’ version of this truth-conditional account, which interprets a conditional φ‎ > ψ‎ by means of the following support clause.5,6

Definition 7.1 (Inquisitive lifting)

sφ>ψiffαalt(φ)γalt(ψ)such thats(αγ)

(p.123) When φ‎ and ψ‎ are non-inquisitive, we have alt(φ‎) = {|φ‎|} and alt(ψ‎) = {|ψ‎|}, and the clause therefore boils down to:

sφ>ψα{|φ|}γ{|ψ|}such thats(αγ)s|φ||ψ|

Thus, the conditional φ‎ > ψ‎ is predicted to be a statement whose unique alternative is the classical proposition |φ||ψ| delivered by the given base account. Except for (4c), all of our counterfactuals have non-inquisitive antecedents and consequents, so they will be interpreted just as they are interpreted by the given base account. As for (4c), translated as ¬Ua ∨¬Ub > O, the clause yields the following:

s¬Ua¬Ub>Oα{|¬Ua|,|¬Ub|}γ{|O|}s.t.s(αγ)s|¬Ua||O|ands|¬Ub||O|s(|¬Ua||O|)(|¬Ub||O|)

As in the other cases, the conditional as a whole is a statement. However, the unique alternative for it, the set (|¬Ua||O|)(|¬Ua||O|), is not the same as the set |¬Ua¬Ub||O| that would be delivered by applying the base account directly, without lifting it to inquisitive semantics. Rather, the lifting procedure ensures that the base account is applied twice, once for each disjunct in the antecedent, and the results are then intersected. Thus, disjunctive antecedents are interpreted as providing multiple assumptions, and ¬Ua ∨¬Ub > O is predicted to be true just in case both ¬Ua > O and ¬Ub > O are true. This explains the strong similarity between the response pattern of (4c) and those of (4a) and (4b).

Now the majority judgments in Table 7.1 could be predicted if we could find a truth-conditional account of counterfactuals according to which (4a) and (4b) are true, but (4d) and (4e) are not. The inquisitive lift of this account would make the same predictions about these sentences, and it would also predict (4c) to be true —something that no purely truth-conditional account could do.

7.2.3 Background semantics for counterfactuals

Now that the problem of disentangling (4c) and (4d) is solved, one might expect that we can just take a standard account of counterfactuals, such as the ordering semantics of Lewis (1973), and lift it to inquisitive (p.124) semantics to obtain an account of our data. However, as Ciardelli et al. (2017c) discuss, this is not the case. The problem is that all standard accounts of counterfactuals validate the following entailment:

¬Ua>O,¬Ub>O¬(UaUb)>O

Thus, regardless of how the parameters needed to interpret counterfactuals are set in these theories, they can never predict that (4a) and (4b) are true but (4d) is not. Conceptually, the problem is that all the standard theories are based on the idea that, when making a counterfactual assumption, one is required to minimize the amount of change with respect to the actual world. This means that, when counterfactually assuming that A and B are not both up, one is required to retain the fact that at least one of them is up.7 This does not seem right: when asked to consider what would happen if the switches were not both up, we are naturally lead to consider the case that just one switch was down, as well as the case that both switches were down, which explains the observed judgments for (4d) and (4e).

To solve this problem, Ciardelli et al. (2017c) adopt a different perspective: they propose to replace the minimal change requirement by a qualitative distinction between aspects of the world that are in the foreground when making a counterfactual assumption, and aspects that are in the background. The latter are held fixed in the counterfactual scenario, while the former are allowed to change, and their change is not subject to any minimality requirement. We will refer to this account of counterfactuals as background semantics.

For a simple example, consider the sentences in (6):

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Conditionals

In both cases, when assuming that the antecedent is true, the length of the speaker’s hair is in the foreground, and we feel no pressure to imagine it to be as close as possible to the actual length; this explains why in normal circumstances we are not inclined to judge (6a) as true. On the other hand, in both cases the fact that people are able to pick up remarkable differences in hair length is in the background, and thus it (p.125) is held fixed when making the assumption; this explains why in normal circumstances we judge (6b) as true.

Now consider again (4a), (4b), and (4d). When we make the assumption that switch A is down, the position of switch B is naturally regarded as background, and therefore held fixed. This leads us to consider a counterfactual scenario in which A is down, but B is still up. Reasoning by the laws of the circuit, we therefore conclude that the light is off, and judge (4a) as true. Of course, the prediction is analogous for (4b). Now consider the assumption that the switches were not both up: in this case, the positions of both switches are at stake, and thus foregrounded. Therefore, we have no pressure to hold either of them fixed in the counterfactual scenario. This leads us to consider counterfactual scenarios where just one switch is down as well as scenarios where both switches are down: since these two kinds of scenarios do not agree on the state of the light, neither (4d) nor (4e) are judged true.

Let us now see how an account of the kind just sketched can be formalized, and verify that the predictions we just described are indeed derived. For conciseness, we present here a simplified version of the original background semantics proposed in Ciardelli et al. (2017c). This simplified version preserves the essence of the account of our sentences, although it is limited in scope.8

The account relies on a formal notion of causal structures inspired by the literature on causal reasoning (Pearl, 2009).9 For our purposes, such structures can be defined as follows.

Definition 7.2 (Causal structures)

A causal structure is a triple S = 〈V, E, L〉 where:

  • V is a set of atomic polar questions, the causal variables of the structure. The settings of a variable ?φ‎ are the sentences φ‎ and ¬φ‎.

  • V, E〉 is a directed acyclic graph, whose edges encode causal influence.

  • L is a set of statements, the causal laws of the structure; each causal law has the form φ1φnψ, where ψ‎ is a setting of a variable vV and φ1,,φn are settings of the parents of v in the graph 〈V, E〉.

(p.126) The electric circuit described in Section 7.1 can be modeled naturally as a causal structure as follows. The causal variables are ?Ua, ?Ub, and ?O, corresponding to the states of the switches and the light. The variables ?Ua and ?Ub have causal influence on ?O, but not on each other. Thus, the graph 〈V, E〉 looks as follows:

?Ua → ?O ← ?Ub

The causal laws are the following conditionals, encoding the behavior of the circuit:

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Conditionals

Within the context of a causal structure, we can associate a possible world with a set of facts—i.e., basic propositions that characterize the world. Moreover, we can equip the set of facts with some structure that reflects the causal relations between them.

Definition 7.3 (Facts)

A fact at a world w is a true setting of a causal variable. The set of facts at w is denoted Fw. The causal graph 〈V, E〉 naturally induces a corresponding graph on the set of facts. We say that a fact f is causally dependent on another fact f′ if f′ is an ancestor of f in this graph.

In our context, the facts are: that switch A is up; that switch B is up; and that the light is on. That is, Fw={Ua,Ub,¬O}. The fact that the light is on is dependent on the other facts, and no other dependencies hold.

We now want to specify, given a certain counterfactual assumption α‎, which of the facts in Fw are called into question by the assumption—and should therefore be considered as potentially different in the counterfactual scenario—and which facts can be regarded as background, and can therefore be held fixed. The basic idea is simple: an assumption α‎ calls into question those facts that are logically responsible for its falsity, as well as those facts that are causally dependent on them. The remaining facts can be regarded as background, although other factors in the context might lead to them being foregrounded as well, and thus varied in the counterfactual scenario.10

(p.127) The idea of a fact being responsible for the falsity of α‎ can be formalized as follows.

Definition 7.4 (Contributing to the falsity of a classical proposition)

Let wW, αW a classical proposition, and fFw a fact in w. Then, if there exists some set FFw such that F is consistent with α‎, but F{f} is inconsistent with α‎, we say that f contributes to the falsity of α‎ in w.11

When making a counterfactual assumption α‎, we can no longer take for granted those facts that contribute to the falsity of α‎, nor anything that is causally dependent on such facts. We say that these facts are called into question by α‎.

Definition 7.5 (Calling a fact into question)

A classical proposition α‎ calls into question fFw if either (i) f is a fact that contributes to the falsity of α‎, or (ii) f is causally dependent on some such fact.

In our concrete setting, consider the classical proposition that switch A is down, |¬Ua|. It is easy to see that the only fact fFw that contributes to the falsity of this proposition is Ua. Thus, the counterfactual assumption |¬Ua| calls into question the fact that A is up, as well as the causally dependent fact that the light is on, but it does not call into question the fact that switch B is up. Similarly, the assumption that switch B is down calls into question the fact that B is up and the fact that the light is on, but not the fact that A is up.

Now consider the classical proposition that the switches are not both up, |¬(UaUb)|. It is easy to see that the fact that A is up and the Fact (p.128) that B is up both contribute to the falsity of this proposition. Thus, these facts are called into question, and so is the dependent fact that the light is on. Therefore, in this case the assumption calls into question all of the facts in our scenario.

The next step is to use these notions to determine which facts can be regarded as background for a given counterfactual assumption, and thus held fixed in making the assumption and assessing its consequences. We assume that only facts that are not called into question by the assumption can be backgrounded. Furthermore, we assume a requirement to avoid gratuitous changes, and thus to avoid foregrounding anything without a reason. In the absence of contextual cues providing a reason to foreground other facts, the background will consist of all and only the facts that are not called into question. We call this the maximal background for the given assumption.12

Definition 7.6 (Maximal background for an assumption)

The maximal background for a classical proposition α‎ at world w, denoted Bw(α), is the set of all facts which are not called into question by α‎.

Any fact that is part of the background for a given counterfactual assumption is held fixed in the counterfactual scenario. In other words, in making the assumption α‎, we imagine that α‎ is true in addition to all the background facts.

In our scenario, we have Bw(|¬Ua|)={Ub}. This explains why, when we suppose that switch A was down in our scenario, we envisage a situation where switch A is down but switch B is up. Similarly, Bw(|¬Ub|)={Ua}: when we suppose that switch B was down, we envisage a situation where switch B is down but switch A is still up. On the other hand, since the assumption |¬(UaUb)| calls all facts in Fw into question, we have Bw(|¬(UaUb)|)=, which means that when we suppose that the switches were not both up, no fact carries over from the actual state of affairs to the counterfactual scenario.

We can now define the information state that results from making a counterfactual assumption α‎ in a certain world w. This is the (p.129) information determined by the assumption itself, the background facts, and the underlying causal laws. Under a maximal background interpretation, this amounts to the following.

Definition 7.7 (Information state resulting from an assumption)

The information state that results from making an assumption α‎ at world w, denoted Sw(α‎), is the set of worlds in which the following are true: (i) the classical proposition α‎; (ii) all facts in Bw(α); and (iii) all laws in L.

The last step is to specify at which worlds the classical proposition αγ is true: this holds if the state that results from making the assumption, Sw(α‎), supports the conclusion γ‎, that is, if Sw(α)γ.

Definition 7.8 (Truth-conditional recipe for counterfactuals)

Given two classical propositions α‎ and γ‎, the counterfactual proposition αγ is true at a world w in case Sw(α)γ.

This completes the description of the truth-conditional map that we are going to use as the basis for our inquisitive account. Let us now check that the inquisitive account that results from lifting this map correctly predicts which of the counterfactuals in (4) are true in our scenario.

First consider the counterfactual assumption that switch A was down, |¬Ua|. We saw that Bw(|¬Ua|)={Ub}. Take any world vSw(|¬Ua|): at world v, (i) our assumption |¬Ua| is true, that is, switch A is down; (ii) the background facts are true, that is, switch B is up; (iii) all causal laws are true, in particular the law ¬UaUbO. Clearly, |O| must then be true in v. This shows that Sw(|¬Ua|)|O|, which means that |¬Ua||O| is true at w. Since |¬Ua||O| is the unique alternative that our inquisitive account assigns to ¬Ua > O, this counterfactual is correctly predicted to be true. Of course, the truth of the counterfactual ¬Ub > O is predicted in an analogous way.

Now consider the counterfactual ¬Ua ∨¬Ub > O. We saw in Section 7.2.2 that our inquisitive lifting account assigns a unique alternative to this sentence, namely, the intersection (|¬Ua||O|)(|¬Ub||O|). Since we have just seen that w belongs to both sets |¬Ua||O| and |¬Ub||O|, w also belongs to their intersection. Thus, ¬Ua ∨¬Ub > O is predicted to be true.

Finally, consider the counterfactuals ¬(UaUb) > O and ¬(UaUb) > ¬O. We saw that Bw(|¬(UaUb)|)=. Now, consider the state Sw(|¬(UaUb)|): this state consists of those worlds where the switches are not both up, and the causal laws hold; thus, this state contains (p.130) worlds where only one switch is down and the light is off, as well as worlds where both switches are down and the light is on. Therefore, Sw(|¬(UaUb)|)⫅̸|O| and Sw(|¬(UaUb)|)⫅̸|¬O|, which means that neither |¬(UaUb)||O| nor |¬(UaUb)||¬O| are true at w. Since these are, respectively, the unique alternative for ¬(UaUb) > O and the unique alternative for ¬(UaUb) > ¬O, we predict that neither of these counterfactuals is true in our scenario.

Summing up, combining the background semantics for counterfactuals described in this section with the inquisitive lifting procedure described in Section 7.2.2 we obtain an account that accurately predicts which of the counterfactuals in (4) are true in the given scenario. The crucial ingredients of this account are (i) the fine-grained notion of meaning given by inquisitive semantics, (ii) an account of conditionals which is sensitive to inquisitive content, and (iii) a procedure for making counterfactual assumptions which is not constrained by the requirement to minimize the difference with respect to the actual world.13

7.3 Further benefits

In the previous section, we have seen how any truth-conditional account of conditionals, whether indicative or counterfactual, can be lifted to inquisitive semantics. Moreover, we have applied this lifting procedure to a particular truth-conditional account of counterfactuals in order to explain the experimental findings in Table 7.1. In this section, based on Ciardelli (2016b), we will demonstrate some further general benefits of the lifting procedure. We will see that no matter what truth-conditional account of (indicative or counterfactual) conditionals we take as our starting point, the lifted inquisitive account will improve on it in three ways: first, it will give a more satisfactory account of conditionals with disjunctive antecedents, avoiding a shortcoming which affects all truth-conditional accounts; second, it will allow us to interpret not only standard if-then conditionals, but also so-called unconditionals; and (p.131) finally, it will allow us to interpret not only conditional statements, but also conditional questions. We will consider each of these topics in a separate sub-section.

7.3.1 Simplification of disjunctive antecedents

Consider the sentences in (8). One seems justified in inferring (8b) from (8a), but certainly not in inferring (8c) from (8b).

(8)

Conditionals

The inference from (8a) to (8b) is an instance of a principle called simplification of disjunctive antecedents (SDA); the inference from (8b) to (8c) is an instance of a principle called strengthening of the antecedent (SA).

AB>CA>C(SDA)A>CAB>C(SA)

Intuitively, SDA is valid. Indeed, a conditional like (8a) seems to mean exactly the same as the conjunction in (9).

(9)

Conditionals

However, classical theories of counterfactuals, such as Stalnaker (1968); Lewis (1973) and Kratzer (1981), fail to validate this principle. This has been widely regarded as a problem for these theories (see, e.g., Fine, 1975; Nute, 1975; Ellis et al., 1977; Alonso-Ovalle, 2009; Fine, 2012) and it has also been clear since Fine (1975) that this problem is more than an accidental shortcoming. Indeed, based on a truth-conditional view on meaning and the classical treatment of connectives, a compositional account that validates SDA is bound to validate SA as well, and this is undesirable in view of the invalid inference from (8b) to (8c).14

In recent years, this problem has motivated approaches to counterfactuals which rely on a more fine-grained semantic representation of (p.132) antecedents than the truth-conditional one. Perhaps the most prominent account of this kind is due to Alonso-Ovalle (2006, 2009), which we already mentioned above as an inspiration for the inquisitive lifting procedure (for accounts in the same spirit, see also van Rooij, 2006; Fine, 2012; Willer, 2015). The fundamental idea of this account is that disjunctive sentences denote sets of classical propositions, rather than single propositions, and that each proposition in the set serves as a separate counterfactual assumption. Clearly, this approach validates SDA: evaluating a counterfactual with a disjunctive antecedent, AB > C, effectively amounts to evaluating the conjunction (A > C) ∧ (B > C). On the other hand, SA is invalid: for non-disjunctive antecedents, Alonso-Ovalle’s account coincides with the ordering semantics of Lewis (1973), which invalidates SA.

The inquisitive lifting recipe achieves essentially the same: when an antecedent is associated with multiple alternatives, the lifted account leads us to run the base account separately for each of these alternatives. This holds in particular for disjunctive antecedents, which typically present one alternative for each disjunct. Thus, no matter what account of conditionals we take as our starting point, the lifting of this account will interpret AB > C as equivalent with (A > C) ∧ (B > C), validating SDA.15 On the other hand, if the base account does not validate SA, neither will its lifting, since the two will coincide in the absence of inquisitiveness. Thus, inquisitive semantics provides a way to disentangle SDA from SA and to avoid one of the central problems faced by standard theories of conditionals.

(p.133) With respect to Alonso-Ovalle’s own account, the inquisitive treatment of conditionals can be seen as a generalization in three different ways. First, on the inquisitive approach, disjunction is not treated as a special, non-standard connective; instead, all connectives are taken to operate on inquisitive propositions, rather than on classical propositions. As we saw in Chapter 4, this allows us to retain a principled and well-behaved theory of propositional connectives, which preserves the attractive features of the classical theory.

Second, Alonso-Ovalle’s account is based on a specific account of conditionals, namely, the ordering semantics of Lewis (1973). By contrast, the inquisitive lifting recipe can be applied to any base account of conditionals, provided that it is compositional and operates in a truth-conditional setting. In the previous section, we have already made use of this degree of freedom. As we saw, independently of the issue of disjunctive antecedents, minimal change theories could not possibly predict the majority judgments in Table 7.1, as these run against the very logic of these theories. The modularity of the inquisitive lifting strategy allowed us to disentangle the problem of dealing with disjunctive antecedents from the problem of determining the right procedure for making counterfactual assumptions.

Finally, inquisitive lifting is not specifically designed to deal with disjunctive antecedents; rather, it provides a general treatment of the interaction between conditionals and inquisitiveness—an interaction which is manifested not just in conditionals with disjunctive antecedents, but in other classes of conditional sentences as well, as we will discuss in Section 7.3.2 and 7.3.3.

Before turning to the next topic, let us spend a few words on some examples that seem to show that SDA is not in fact generally valid. These examples have a special form, with the consequent coinciding with one of the disjuncts in the antecedent. The most famous such example is (10), due to McKay and Van Inwagen (1977):

(10)

Conditionals

This sentence seems true, even though it is certainly not the case that if Spain had fought with the Allies she would have fought with the Axis. On a standard theory like the one of Lewis (1973), the truth of this sentence would be explained by saying that some worlds where Spain fought with the Axis are more similar to the actual world than any world (p.134) where Spain fought with the Allies. However, this diagnosis leads us to expect that (11a) is true, since it effectively boils down to (11b).

(11)

Conditionals

As Nute (1980) notes, this is wrong: (11a) is naturally interpreted as implying that Germany would have been pleased if Spain fought with the Allies, in accordance with SDA. This problem, together with the fact that these counterexamples have a very special form, suggests that these cases involve some kind of anomaly.

In our inquisitive account, these counterexamples could be accounted for by stipulating that it is in principle possible to insert a projection operator ! in the antecedent. Thus, (10) would be translated as !(AxAl)> Ax, and analysed as a basic conditional with a non-inquisitive antecedent, which would block SDA in this case. However, the possibility to insert ! should be restricted, in order to account for the apparent lack of ambiguity of ordinary conditionals such as (8a) and (11a). One way of explaining why ! is inserted in (10) is based on the observation that a logical form such as AxAl > Ax is equivalent with the simpler form Al > Ax.16 Assuming a general ban against structural redundancy, of the kind proposed by Meyer (2014), this would make the logical form AxAl > Ax unavailable for a conditional such as (10), justifying the insertion of ! as a repair strategy. This explanation would account for why SDA only seems to fail in sentences where the consequent coincides with one of the disjuncts in the antecedent.

7.3.2 Unconditionals

In the previous section, we mentioned that our account derives the behavior of disjunctive antecedents as a particular case of a more general pattern of interaction between conditionals and inquisitiveness. Another class of sentences in which this interaction is manifested is that of unconditionals. These are sentences such as the following:

(12)

Conditionals

(p.135) Following Rawlins (2008), we will analyse unconditionals as conditional constructions where the ‘antecedent’ is an interrogative clause. According to the compositional account given in Chapter 6, we translate the polar interrogative whether they play Bach or not as Pb ∨¬Pb, which is equivalent to ?Pb, and the disjunctive interrogative whether they play Bach or Handel as PbPh. Moreover, we assume that the antecedent of (12c) corresponds to the interrogative what they play, which we translate as ∃xPx.17 This gives the following translation for our unconditionals.

(13)

Conditionals

Let us now consider the semantics that our inquisitive account of conditionals assigns to these sentences. Let us start with (12a). The antecedent is inquisitive, while the consequent is not: alt(?Pb) = {|Pb|, |¬Pb|}, alt(G) = {|G|}. Our support clause gives:

s?Pb>Gα{|Pb|,|¬Pb|}γ{|G|}such thats(αγ)s(|Pb||G|)(|¬Pb||G|)s(Pb>G)(¬Pb>G)

According to this analysis, (12a) is non-inquisitive, and it is true in case Alice will go if they play Bach, and she will go if they do not. This is precisely the analysis we expect for the unconditional (12a). Similarly, for (12b) and (12c) we obtain the following predictions:

sPbBh>Gα{|Pb|,|Ph|}γ{|G|}such thats(αγ)s(|Pb||G|)(|Ph||G|)s(Pb>G)(Ph>G)sxPx>Gα{|Pd||dD}γ{|G|}such thats(αγ)sdD(|Pd||G|)sx(Px>G)

It is derived that both (12b) and (12c) are non-inquisitive; further, (12b) is true in case Alice will go both if they play Bach and if they play (p.136) Handel, and (12c) is true if for every x in the domain, Alice will go if they play x. Again, these are indeed the natural truth conditions for these sentences.

Thus, our inquisitive account of conditionals extends naturally to a general analysis of unconditional sentences. The resulting analysis is in line with the one proposed by Rawlins (2008), and shares its core idea. However, a nice feature of the inquisitive approach is that it is modular: it does not commit us to a specific account of conditionals, but is compatible with a wide range of accounts. A second advantage is that nothing special had to be stipulated to analyse unconditionals: the desired analysis follows for free from the semantics of questions and the conditional operator, once unconditionals are analysed as conditionals. Thus, the approach is not merely descriptive, but also has some explanatory power. Another way to put this last point is this: we have given a uniform account of disjunctive and interrogative antecedents as introducing multiple assumptions, and provided an explanation for this commonality based on a feature shared by disjunctive and interrogative clauses, namely, inquisitiveness.

One may complain that, in giving this uniform explanation, we have gone too far: at this point, we have given exactly the same translation for the standard if-conditional in (14a) and the unconditional in (14b).

(14)

Conditionals

There is a sense in which this prediction is correct. Both (14a) and (14b) are not inquisitive, and they have the same truth conditions: both are true in case Alice will go if they play Bach, and also if they play Handel. Yet, intuitively there is also a difference between these sentences, which is not reflected in our translation.

The idea pursued in Ciardelli (2016b), proposed already by Zaefferer (1991), is that the difference between (14a) and (14b) is one of presupposition: the unconditional in (14b) presupposes that they will play either Bach or Handel, whereas the conditional in (14a) lacks this presupposition. This can be seen with the following pair of examples, the first of which is adapted from Zaefferer:

(15)

Conditionals

(p.137)

(16)

Conditionals

In the case of (15), the first sentence in the discourse indicates that the speaker cannot presuppose that the meeting is in Rome or in Paris. It is then odd for her to continue with an unconditional which carries this presupposition. By contrast, in the case of (16), it would be natural for a speaker to presuppose that the baby will be a boy or a girl. This makes her use of a standard conditional form odd as a result of the principle maximize presupposition, which requires speakers to prefer equivalent forms with stronger presuppositions whenever these presuppositions are satisfied (see Ciardelli, 2016b, for a more detailed discussion of these data).18

Importantly, in the analysis we described, this semantic difference between standard conditionals and unconditionals does not have to be stipulated, but can be derived from two standard generalizations about the presuppositions of interrogatives, and the way presuppositions project from conditional antecedents.

  1. 1. Interrogative clauses presuppose that one of their alternatives is true.

    (see, e.g., Belnap, 1966)19

  2. 2. Conditionals inherit the presuppositions of their antecedent.

    (see, e.g., Karttunen, 1973, 1974)

Since we view unconditionals as conditionals with an interrogative clause as their antecedent, it follows from (1) and (2) that unconditionals always presuppose that one of the alternatives for their antecedent is true. For a formalization of these ideas in a system that captures presuppositions, see Ciardelli (2016b).

7.3.3 Conditional questions

Another class of sentences which involve the interplay of conditionals and inquisitiveness is given by conditional questions, such as those in (17) and (18).

(17)

Conditionals

(p.138)

(18)

Conditionals

Standard theories of conditionals, being couched in a truth-conditional semantic framework, cannot be directly applied to analyse these sentences. By contrast, the inquisitive lifting of such theories can be applied directly to these questions, yielding natural results. Let us see how. Here, we will not take a stance on what the semantic difference is between indicative and counterfactual conditionals; we will just suppose that we are given two maps i and c which correspond to these two different classes of conditionals, and we will assume two operators >i and >c which are interpreted by lifting these maps to inquisitive semantics.20

As above, we translate the clause whether they play Bach as ?Pb, and the clause what they play as ∃xPx. This gives the following translations for our sentences:

Conditionals

Let us now see what predictions this yields for the conditional questions in (17) and (18). Since the lifting recipe works in the same way for indicative and counterfactual conditionals, we will suppress subscripts in the derivation. Let us start with the conditional polar questions in (17a) and (18a).

sG>?Pbα{|G|}γ{|Pb|,|¬Pb|}such thats(αγ)s|G||Pb|ors|G||¬Pb|sG>PborsG>¬Pb

Thus, (17a) and (18a) are predicted to be inquisitive. A state supports G >i?Pb iff it supports G >iPb, or it supports G >i¬Pb; this means that in order to resolve (17a), one must establish either that if Alice goes they will play Bach, or that if Alice goes they will not play Bach. These are precisely the resolution conditions that we expect for (17a). Similarly, (18a) is supported iff either of G >cPb and G >c¬Pb is supported, which again gives the natural resolution conditions for this question. Now let us consider the conditional wh-questions in (17b) and (18b). (p.139)

sG>xPxα{|G|}γ{|Pd||dD}such thats(αγ)s|G||Pd|for somedDsG>Pdfor somedD

Thus, (17b) and (18b) are predicted to be inquisitive. An information state supports G >ixPx iff it supports G >iPd for some dD; this means that in order to resolve (17b), one must establish for some specific d that if Alice goes, they will play d. Similarly, (18b) is supported iff G >cPd is supported for some dD. Again, these are precisely the resolution conditions that we would intuitively assign to these questions.

These examples illustrate how lifting an account of conditionals to inquisitive semantics immediately yields an extension of this account to conditional questions. This approach differs from previous accounts of conditional questions such as Velissaratou (2000) and Isaacs and Rawlins (2008), which focus on indicative conditional questions like those in (17) and cannot be used directly to analyse counterfactual conditional questions like those in (18). As we have seen, inquisitive lifting applies uniformly to indicative and counterfactual questions. Additionally, inquisitive lifting leaves us with a choice as to the underlying theory of conditionals that we use to interpret these questions.

Before concluding this section, an important issue remains to be addressed. At this point, the reader might be worried that the conditional statement (19a) might end up being assigned the same meaning as the conditional question (19b).

(19)

Conditionals

This problem does not arise, however, since as discussed in Chapter 6, the LF of a declarative or interrogative clause always involves a complementizer which contributes a corresponding operator ! or 〈?〉. Thus, we translate (19a) to our formal language as G > !(PbPh), and we translate (19b) as G > 〈?〉(PbPh), which is equivalent to G > (PbPh). In all the other examples discussed in this chapter inserting the operators ! and 〈?〉 in the main clause would have a vacuous effect, which is why we could safely disregard these operators. However, it is crucial for us to assume that the interpretation of an ‘if’ clause does not involve any projection operator. This is justified by the observation that if-clauses are syntactically distinct from both declarative and interrogative clauses; the former are headed by the complementizer ‘if’, while the latter are (p.140) taken to be headed by declarative or interrogative complementizers, which in English main clauses are not lexicalized, but affect word order.21

Summing up, in this section we saw that lifting an account of conditionals to inquisitive semantics leads to an account which improves on the original one in various ways: first, it gives a more satisfactory treatment of disjunctive antecedents, which are interpreted as providing multiple assumptions; second, it extends the scope of the original account beyond standard conditional statements, allowing us to analyse two other classes of conditional constructions: unconditionals, and conditional questions.

7.4 Summary

Our main goal in this chapter was to show that inquisitive content is relevant even for phenomena that have no obvious link to questions, and that the inquisitive content of a constituent can sometimes play a crucial role in determining the truth conditions of a sentence. We have illustrated this point with conditionals, which provide an especially interesting and rich domain of application. In this domain, taking inquisitive content into account provides a natural explanation for some otherwise puzzling data (such as those in Table 7.1), solves some long-standing logical problems (the inter-derivability between SDA and SA), and allows for a substantial extension of the scope of standard theories (bringing unconditionals and conditional questions within reach).22

We think that conditionals are not an isolated case, but only one of many environments where inquisitive content plays a role. To give one example, it has been argued by Simons (2005), Aloni (2007), and Willer (2017), among others, that something like inquisitive content is responsible for the free-choice inferences triggered by disjunctions under (p.141) modals, exemplified in (20), which are not predicted under standard theories of modals.

(20)

Conditionals

Interestingly, the same contrast between disjunctions and negated conjunctions that we discussed above in the case of counterfactual antecedents is found in the domain of modals. For instance, (21a) does not have the free choice inference in (21b). To see this, consider a context where we are looking for someone to translate from Dutch to French, and where it is known that Alice speaks Dutch, but it is not known whether she also speaks French; in this context, (21a) is true, but (21b) is not.

(21)

Conditionals

If free choice inferences stem from the presence of multiple alternatives, then the contrast is expected from the inquisitive semantics perspective, since disjunctions are typically inquisitive, but negated conjunctions are not.

Clearly, more work, both empirical and theoretical, is needed to investigate exactly in which linguistic environments inquisitive content plays a role, and to provide formal accounts of the relevant phenomena.

7.5 Exercises

Exercise 7.1 Lifting material implication

Show that inquisitive implication is the lifting of material implication. That is, show that if is defined as material implication (i.e., for every two classical propositions p and q, pq amounts to p¯q), then the support conditions assigned to φ‎ > ψ‎ by the inquisitive lifting recipe coincide with the support conditions of φ‎ψ‎ in InqB.

Exercise 7.2 Background semantics

Consider sentence (22) in the following two scenarios (Tichý, 1976):

  • Context 1: Jones has the following habits as regards wearing his hat. Bad weather invariably induces him to wear his hat. Fine weather, on the other hand, affects him neither way: on fine days he puts his hat on or leaves it on the peg, completely at random. Suppose moreover that actually the weather is bad, so Jones is wearing his hat.

  • (p.142) Context 2: Jones always flips a coin before he opens the curtains to see what the weather is like. Heads means he is going to wear his hat in case the weather is fine, whereas tails means he is not going to wear his hat in that case. Like above, bad weather invariably makes him wear his hat. Today heads came up when he flipped the coin, and it is raining. So Jones is wearing his hat.

(22)

Conditionals

Intuitively, the sentence is true in context 2 but not in context 1. Show how this is derived in background semantics of counterfactuals, modeling the causal structure of each context.

Exercise 7.3 Quantification in the antecedent of a counterfactual

Consider an electrical circuit with four switches and one light. The light is on if and only if an even number of switches is up. Currently, all switches are up, so the light is on. Now consider the following sentences:

(23)

Conditionals

(24)

Conditionals

Intuitively, (23) is true in the given scenario, but (24) is not. Suppose that the sentences are translated as (∃xUx) > O and (¬∀x.Ux) > O, respectively.

  1. 1. Show that the given intuitions cannot be captured by any truth-conditional compositional account of counterfactuals.

  2. 2. Show that they are captured by the inquisitive account described above.

Exercise 7.4 Conditional questions with disjunctive antecedents

Consider the following indicative conditional question:

(25)

Conditionals

  1. 1. Translate the sentence into a suitable first-order logical language.

  2. 2. Assuming a truth-conditional map for indicative conditionals, derive the support conditions that the inquisitive lifting of assigns to the question in (25).

  3. 3. What does this predict about the circumstances under which the question is resolved?

Notes:

(1) To maintain that exhaustive strengthening is responsible for the observed effects, one would have to assume that exhaustification takes place in conditional antecedents more often than in matrix clauses. As far as we know, there is no evidence supporting this assumption.

(2) Clearly, the implication connective → provided by InqB is not suitable as a translation of counterfactuals. When applied to statements, this operator yields the same truth conditions as the standard material implication connective of classical logic. Since the antecedents of our counterfactual sentences are all false, this would immediately render all these sentences true.

(3) Of course, we do not mean here that the logical form of a sentence like (4a) has to contain a negation operator. We could introduce another predicate Dx for ‘x is down’, and another atomic sentence On for ‘the light is on’. However, we would then have to introduce some meaning postulates to enforce that Dx is true exactly when Ux is false, and On is true exactly when O is false. This would then lead to the same results that our analysis gives.

(4) In each of these theories, the definition of αγ makes use of some additional piece of structure: a selection function in Stalnaker (1968), a similarity ordering in Lewis (1973), an ordering source in Kratzer (1981). However, our lifting recipe only needs access to the resulting operation on propositions—not to this underlying structure.

(5) Note that the interpretation of φ‎ > ψ‎ is specified in terms of support conditions. Recall that the proposition expressed by a sentence in InqB is the set of all states that support it; see Section 4.2 for discussion. Also note that the support conditions of φ‎ > ψ‎ are formulated here in terms of alt(φ‎) and alt(ψ‎). This formulation assumes that alt(φ‎) and alt(ψ‎) completely determine the meaning of φ‎ and ψ‎, respectively, i.e., that [φ]={ssαfor someαalt(φ)} and similarly for ψ‎. This will indeed be the case for all the examples that we will consider in this chapter, but it does not always hold in InqB (see footnote 3 on page 21). For the general case, the support conditions of φ‎ > ψ‎ can be formulated as follows, without making reference to alternatives:

sφ>ψiffα[φ]α[φ]γ[ψ]:ααands(αγ)

Provided that the map is upward monotonic in its second argument, which is true of all truth-conditional theories of conditionals we are aware of, this clause boils down to the clause of Definition 7.1 whenever alt(φ‎) and alt(ψ‎) completely determine the meaning of φ‎ and ψ‎.

(6) The resulting inquisitive account is called the lifting of the original account because, while the latter operates on classical propositions, the former operates on propositions in the inquisitive semantics sense, which are objects of a higher semantic type.

(7) This may seem like an over-simplification since, e.g., in ordering semantics, one may well stipulate that toggling two switches is not to be counted as a bigger change than toggling just one. However, this stipulation would make it impossible to account for the truth of ¬Ua > O and ¬Ub > O. A similar argument applies to the other standard accounts.

(8) In particular, unlike the original version of the semantics, the simplified version is not equipped to deal with cases in which assumptions ‘intervene’ on causal laws, in the sense of Pearl (2009).

(9) For related theories of counterfactuals using causal structures, see Schulz (2011), Kaufmann (2013), and Santorio (2016).

(10) Ciardelli et al. (2017c) discuss in detail the fact that seeing the filler sentence can affect the way the target sentences are judged. Among the participants who saw the target sentence first, less than 20% judged (4a) and (4b) as indeterminate. By contrast, among the participants who saw the filler sentence before the target sentence, about 45% judged (4a) and (4b) as indeterminate. Background semantics offers a natural explanation for this finding. The filler sentence used in the experiment was (i), whose antecedent calls into question the positions of both switches.

((i))

Conditionals

It is natural to assume that after seeing the filler, some of the participants kept thinking of the position of switch B as foregrounded even when making the assumption that A was down. Thus, they ended up considering the possibility that both switches are down, leading to the high proportion of ‘indeterminate’ judgments for (4a) (and similarly for (4b)). For a more systematic discussion of order effects, we refer the reader to Ciardelli et al. (2017c).

(11) We say that a set of facts F is consistent with α‎ if there is some world wW where α‎ is true (i.e., wα‎) and all facts in F are true. Notice that if some fact ‘contributes to the falsity of α‎ in w’, then α‎ indeed has to be false in w. Otherwise no set of facts in Fw could be inconsistent with α‎.

(12) We should emphasize that the requirement to maximize the background is only a default. In interpreting a counterfactual, a hearer may foreground other facts besides those called into question by the antecedent, for instance because the possibility of those facts changing is salient in the context (for some evidence that this is indeed possible, see Section 5 of Ciardelli et al., 2017c). Here we focus on the case in which the background is maximized, since we assume that it is this default interpretation that accounts for the majority judgments about our counterfactuals.

(13) The semantics described here is only concerned with predicting when a sentence is true. For a complete account of the data in Table 7.1, one would have to complement it with a component that explains when non-true sentences are judged as indeterminate, as opposed to simply false. It is natural to suppose that ‘indeterminate’ judgments result from the failure of a homogeneity presupposition to the effect that making a counterfactual assumption should lead to a state which settles whether the consequent is true (von Fintel, 1997). However, the issue of how failures of semantic presuppositions are reflected in truth value intuitions is a notoriously tricky one (see von Fintel, 2004).

(14) In fact, the problem is not limited to counterfactuals, but concerns conditionals more generally. The intuitions about the indicative conditionals in (i) are exactly the same as those for the corresponding counterfactuals in (8).

((i))

Conditionals

The proof given by Fine (1975) for counterfactuals also shows that a compositional account of indicative conditionals based on truth-conditions is bound to make SDA and SA inter-derivable.

(15) Disjunctive antecedents where one of the disjuncts entails the other, either logically or contextually, form an exception to this claim. If A, B are atomic sentences with |A||B|, then ABB in inquisitive semantics, and as a consequence, (AB > C) ≡ (B > C). We take this to be a welcome result. For consider a conditional of this special form, such as (i):

((i))

Conditionals

This sentence is odd if uttered by someone who is aware of the fact that Californians are Americans. This is commonly explained in terms of a ban against logical forms that contain structural redundancy (Katzir and Singh, 2013; Meyer, 2014). In Alonso-Ovalle’s account, this explanation is no longer available: even when |A||B|, we have (AB > C) ≢ (B > C), so a sentence like (i) does not involve any structural redundancy. By contrast, on our account we have that (AB > C) ≡ (B > C), which allows us to preserve the standard explanation for the oddity of (i). This observation is not specific to conditionals, but it points to an underlying difference between inquisitive semantics and alternative semantics, the framework in which Alonso-Ovalle’s account is cast. For extensive discussion of this point, see Ciardelli and Roelofsen (2017a). We will also come back to this in Section 9.1, where we compare inquisitive and alternative semantics in detail.

(16) We are assuming here that the underlying account of conditionals makes any proposition αα tautological. This is a minimal desideratum for an account of conditionals, both indicative and counterfactual, and it holds in any theory that we are aware of.

(17) Just as in Chapter 6, we disregard at this point the presuppositional component of these sentences. We will return to this later on in this section.

(18) Ciardelli (2016b) also notes that the maximize presupposition principle explains the oddness of a conditional like If they play Bach or they don’t, Alice will go. Since one can always presuppose that either they will or they will not play Bach, the principle requires that, in any context, a speaker should choose the corresponding unconditional form, Whether they play Bach or not, Alice will go.

(19) For further discussion of this generalization, see Ciardelli (2016d, pp. 21–7).

(20) This assumption does not preclude the possibility of having a uniform semantics for both classes of conditionals: in this case, the maps i and c will be derived from the same underlying account, perhaps by setting some parameters differently in the two cases.

(21) In the case of unconditionals, we assume that the antecedent is an interrogative, and thus involves the operator 〈?〉. In all the cases that we discussed in this chapter, the presence of this operator would not affect the meaning we predict. Nevertheless, as we discussed in Section 7.3.2, we take the fact that the antecedent is interrogative to be responsible for the existential presupposition associated with unconditionals. In a presuppositional refinement of our analysis of interrogative complementizers (see Roelofsen, 2015a), this presupposition would be derived automatically.

(22) The analysis of indicative conditional statements and questions has also given rise to further refinements of the basic inquisitive notion of meaning presented here (Groenendijk and Roelofsen, 2010, 2015; Aher and Groenendijk, 2015). These refinements address empirical issues that are orthogonal to the ones considered here.