Attitude verbs’ local context

Abstract

Schlenker (Semant Pragmat 2(3):1–78, 2009; Philos Stud 151(1):115–142, 2010a; Mind 119(474):377–391, 2010b) provides an algorithm for deriving the presupposition projection properties of an expression from that expression’s classical semantics. In this paper, we consider the predictions of Schlenker’s algorithm as applied to attitude verbs. More specifically, we compare Schlenker’s theory with a prominent view which maintains that attitudes exhibit belief projection, so that presupposition triggers in their scope imply that the attitude holder believes the presupposition (Karttunen in Theor Linguist 34(1):181, 1974; Heim in J Semant 9(3):183–221, 1992; Sudo in The art and craft of semantics: a festschrift for Irene Heim, MIT Press, 2014). We show that Schlenker’s theory does not predict belief projection, and discuss several consequences of this result.

1 Introduction

According to a prominent view of presuppositions, attitude verbs exhibit belief projection.¹ That is: presupposition triggers in the attitude’s scope imply that the subject believes the presupposition. For example, consider:

Bill stopped smoking presupposes that Bill used to smoke. Consequently, proponents of belief projection maintain that each report presupposes that Ann believes that Bill used to smoke. Theorists cite several kinds of evidence for belief projection. First, the belief entailment persists in embedded environments, which is diagnostic of presuppositions (Chierchia and McConnell-Ginet, 2000):

Second, the presupposition triggered by the attitude can be filtered by the subject’s beliefs. For instance, (3) presupposes nothing.

Third, denying the relevant beliefs results in infelicity:

The data in (2)–(4) are all explained by belief projection.

This paper is about the relationship between belief projection and the influential theory of presupposition projection from Schlenker (2009, 2010a, b). Schlenker’s account is a version of the satisfaction theory of presuppositions, which says that every presupposition must be entailed by the local context of its trigger, or must be “locally satisfied”.² Schlenker’s proposal is distinguished by a novel algorithm for computing the local context of each expression in a sentence from its truth conditional meaning. Roughly speaking, on this approach the local context of an expression in a sentence is the strongest piece of information that can be conjoined to that expression without affecting whether the sentence is true or false at any world in the context.

We show that Schlenker’s theory does not predict belief projection for verbs of desire such as ‘want’, ‘hope’, and ‘wish’, or fictives such as ‘imagine’ and ‘dream’. Instead, his theory predicts that when a presupposition trigger is embedded under an attitude, the sentence presupposes that the agent bears that same attitude to the presupposition. Thus, on Schlenker’s account the local context for the attitude is not the agent’s beliefs, but rather is the agent’s desires, imaginings, or dreams, as the case may be.

To be clear, our aims here are relatively modest. Our primary goal is to make explicit the predictions of Schlenker’s algorithm, and contrast them with belief projection. Although we will briefly consider the relative advantages of each theory, we will not be arguing for one account over another. Still, we hope that our discussion will help to clarify and focus the debate, and highlight some of the difficulties and outstanding problems in this fascinating area of semantic research.

Before we begin, it is worth remarking that our discussion touches on two recent strands of the literature on presuppositions. First, some of our arguments involving attitudes build on analogous results about conditionals from Mandelkern and Romoli (2017) and Mackay (2019). These authors maintain that Schlenker’s local context algorithm makes incorrect predictions for a range of conditional constructions. However, there are also significant differences between our conclusions and theirs. For instance, Mackay (2019) argues that his result can be avoided by modifying the semantics for conditionals. By contrast, we show there is no natural treatment of all verbs of desire and fictives that generates belief projection when combined with Schlenker’s algorithm.

Second, our arguments impact a widely discussed issue concerning whether it is legitimate to simply stipulate local contexts on a case by case basis, as a matter of the lexical semantics of various expressions, or whether the local context of an expression must follow from a more general theory.³ Schlenker (2009, 2010a, b) is perhaps the most sophisticated attempt to develop a general, systematic account of local contexts. However, Schlenker (2020) and Anvari and Blumberg (2021) have argued that Schlenker’s (2009) algorithm makes incorrect predictions for nominal modifiers and quantificational determiners. Consequently, Anvari and Blumberg (2021) suggest that the local contexts of these expressions might have to be stipulated. Similarly, we argue that if belief projection is ultimately correct, then the local context of attitude verbs will also need to be stipulated.⁴

The paper is structured as follows. Section 2 summarizes Schlenker’s theory of presupposition projection. Section 3 shows that the theory does not predict belief projection given a quantificational semantics for verbs of desire and fictives. Section 4 considers incorporating presuppositions involving the subject’s beliefs into these semantics. Section 5 extends the results beyond quantificational approaches. Section 6 briefly explores the prospects of abandoning belief projection. Section 7 concludes by considering how our results could impact the debate around the so-called “explanatory problem” for dynamic semantics.

2 Presupposition and local context

This section summarizes Schlenker’s (2009, 2010b) theory of presupposition projection. His account is a version of the satisfaction theory, which says that every presupposition must be entailed by the local context of its trigger, or must be “locally satisfied”.⁵ The local context of an expression $\textsf {E}$ aggregates information contributed by the common ground, together with the meaning of particular expressions in $\textsf {E}$’s syntactic environment. To illustrate, consider:

The second conjunct of (5) presupposes that Bill used to smoke. But (5) presupposes nothing. To explain this, many have claimed that the local context of $\textsf {q}$ in $\textsf {p}\,\textsf { and }\, \textsf {q}$ is the combination of the common ground and $\textsf {p}$. This guarantees that the presupposition of stopped in (5) is satisfied in its local context, regardless of what information is in the common ground.

To make predictions about projection, Schlenker develops an algorithm for computing local contexts. The local context of an expression in a sentence is, roughly, the strongest information that can be conjoined to that expression without affecting whether the sentence is true or false at any world in the context. More precisely, Schlenker’s algorithm defines the local context of expression ${\textsf {E}}$ in a syntactic environment ${\textsf {a\_b}}$ and global context C. The local context is the strongest proposition (or ‘restriction’) that can be added to anything in ${\textsf {E}}$’s syntactic position without affecting whether the resulting sentence is true or false in C. The algorithm is incremental, allowing replacement of ${\textsf {b}}$ with any other “good final” ${\textsf {b'}}$ where ${\textsf {aEb'}}$ is grammatical.⁶ Say that sentences ${\textsf {s}}$ and ${\textsf {s'}}$ are equivalent in C (${\textsf {s}}\leftrightarrow _{\textit{C}}\, {\textsf {s}}'$) iff ${\textsf {s}}$ and ${\textsf {s'}}$ have the same truth value at every world in C. Then:

⁷

To illustrate, return to (5). Schlenker’s algorithm predicts that the local context of the second conjunct in (5) is the combination of the global context C and the worlds where the first conjunct (used) is true (call this set used). To see why, consider $\llbracket {\textsf {L}}\rrbracket = C \cap \, {\textbf {used}}$. First, for any sentence ${\textsf {q}}$, (7-a) and (7-b) are equivalent in C:

Second, $\llbracket {\textsf {L}}\rrbracket$ is the strongest restriction that creates contextual equivalence. To see why, consider a stronger restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world $w \in C$ in which Bill used to smoke. Let $\llbracket {\textsf {q}}\rrbracket = W$. Then contextual equivalence fails, because (8-a) is false and (8-b) is true at w:

So, according to Schlenker’s algorithm, $C \cap {\textbf {used}}$ is the local context of the second conjunct in (5).

In the next section we show that Schlenker’s theory does not predict belief projection in verbs of desire and fictives.

3 Attitude verbs’ local context

Since Hintikka (1962), many have analyzed attitude verbs as quantifiers over possible worlds. We argue that Schlenker’s theory does not predict belief projection when combined with a first pass version of these quantification theories.

On a popular semantics for want, this verb universally quantifies over the most desirable worlds consistent with what the agent believes.⁸ More precisely, let Bw be the worlds consistent with what the agent believes at w. Let BEST be a preference function that maps a set of worlds A to the subset of most desirable worlds in A.⁹

For example, Ann wants to smoke is true just in case all of Ann’s most preferred belief worlds are worlds where she smokes.

Now consider the result of applying Schlenker’s algorithm to (9). The local context for ${\textsf {p}}$ in ${\textsf {S wants p}}$ is the strongest information $\llbracket {\textsf {L}}\rrbracket$ that can be conjoined to any claim ${\textsf {q}}$ while guaranteeing that ${\textsf {S wants L}}\, \wedge \, {\textsf {q}}$ is contextually equivalent to ${\textsf {S wants q}}$. Applying Schlenker’s algorithm, this local context is the union, for any world w in the global context C, of the most preferred belief worlds at w, BEST(Bw).

To see why, let $\llbracket {\textsf {L}}\rrbracket = \bigcup \left\{ BEST(Bw) \mid w \in C\right\}$. First, for any sentence ${\textsf {q}}$, (11-a) and (11-b) are equivalent in C:

Second, $\llbracket {\textsf {L}}\rrbracket$ is the strongest restriction that creates contextual equivalence. To see why, consider a stronger restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world $v \in BEST(Bw)$ for some $w \in C$. Then contextual equivalence fails, because (12-a) is false and (12-b) is true at w, where $\llbracket {\textsf {q}}\rrbracket = BEST(Bw)$.

So, Schlenker’s algorithm predicts that $\bigcup \left\{ BEST(Bw) \mid w \in C\right\}$ is the local context of the complement of want.

This local context fails to predict belief projection. (1-b) would imply (13-a) rather than (13-b), because the presupposition of stopped would need to be satisfied by Ann’s most preferred belief worlds, rather than by all of Ann’s belief worlds.

In order to derive belief projection, the required local context would need to be $\bigcup \left\{ Bw \mid w \in C\right\}$, which is the agent’s belief worlds at any world in the global context.

Now turn from verbs of desire to fictives such as imagine. On a popular approach, fictives have a quantificational semantics involving quantification over a set of worlds that is possibly disjoint from the agent’s beliefs.¹⁰ Where Iw represents the worlds consistent with what the relevant agent imagines (again suppressing agent relativity):

For example, Ann imagines that Bill used to smoke is true just in case all of the worlds consistent with what Ann is imagining are worlds where Bill used to smoke.

Now consider the result of applying Schlenker’s algorithm to (9). By parity of reasoning from the case of want, we have:

The key fact here is that the set of worlds consistent with what the agent imagines is the strongest information that can be added to the complement of an imagination report without affecting truth values.

Again, this local context fails to predict belief projection. (1-b) would imply (13-a) rather than (13-b), because the presupposition of stopped would need to be satisfied by Ann’s imagination worlds, rather than by Ann’s belief worlds.¹¹

The results generalize beyond these examples. Quantificational attitude verbs are a type of Kratzerian modal. On the Kratzerian analysis, modals quantify over a domain of possibilities. This domain is fixed by two parameters: (i) the modal base f, which supplies a set of possibilities $\cap f(w)$, and (ii) the ordering source g, which determines the highest-ranked members of that set, $BEST(g(w), \cap f(w))$.¹² Where $\textsf {M}$ is a modal, and $Q_{\textsf {M}}$ is a generalized quantifier associated with $\textsf {M}$:

For instance, ${\textsf {must p}}$ is true just in case $\textsf {p}$ is true at all of the top-ranked worlds, and ${\textsf {may p}}$ is true just in case $\textsf {p}$ is true at some of the top-ranked worlds. Different modal flavors (epistemic, deliberative, deontic, bouletic, circumstantial) correspond to different choices of modal base and ordering source (see Kratzer, 1977, 1981, 1991, 2012).

Likewise with attitude verbs. Fictives like imagine have an empty ordering source, so that every world in the modal base is among the best worlds in the modal base. Verbs of desire have an ordering source that reflects the agent’s preferences, and differ regarding the choice of modal base. In the case of want, the modal base is the agent’s beliefs. These choices of ordering sources and modal base recover the earlier meanings for want and imagine as special cases.

The general result is that Schlenker’s theory predicts that the local context for the prejacent of a Kratzerian modal is not the modal base, but rather is the set of best worlds in the modal base:¹³

The next section considers whether belief projection can be captured by enriching the quantificational theory with doxastic presuppositions. Then Sect. 5 considers whether belief projection can be captured by theories of attitudes that depart altogether from the quantificational framework.

4 Doxastic presuppositions

Section 3 treated want as a universal quantifier over the agent’s most preferred belief worlds. But some theorists think that this is too simplistic. They argue that want also contributes a further presupposition, requiring that its complement is consistent with the agent’s beliefs.¹⁴ In the setting of the quantificational semantics above, this produces the following meaning:

Combined with Schlenker’s algorithm, this revised entry accounts for belief projection.¹⁵

The key idea is that since the complement must be consistent with the agent’s beliefs, contextual equivalence is only guaranteed when all of the agent’s belief worlds are included in the local context.

However, we don’t think that appealing to doxastic presuppositions provides a compelling strategy to derive belief projection. First, several theorists have expressed skepticism that wants even has a compatibility requirement. For example, in recent work Grano and Phillips-Brown (2021) point to “counterfactual want ascriptions” such as:

¹⁶ Such examples are felicitous, but would not be if want reports carried a belief constraint.

Second, the purported compatibility requirement doesn’t project like a presupposition.¹⁷ After all, none of (23-a)–(23-c) suggest that Ann’s belief state leaves open whether Bill passed his test:

Finally, appealing to doxastic presuppositions does not explain belief projection in the full range of attitude verbs. Several theorists have argued that wish is counterfactual, presupposing that its complement is incompatible with the agent’s beliefs.¹⁸ When defined, ${\textsf {S wishes p}}$ is true only if ${\textsf {p}}$ is true throughout the most preferred worlds in some domain Dw.

The precise identity of Dw, plausibly some superset of the agent’s belief worlds, is not relevant for our purposes. However, it is important that the most preferred worlds in Dw are not belief worlds, i.e. $BEST(Dw) \cap Bw = \emptyset$. Consequently, the counterfactual presupposition derives a local context that is a strict superset of the subject’s belief set.¹⁹

This local context predicts that Ann wishes Bill would stop smoking implies not only that Ann believes Bill used to smoke, but also that Ann wishes Bill used to smoke. Note that although this explains some of the data motivating belief projection, it doesn’t capture all of it. For instance, consider the belief filtering effect in (26):

Intuitively, (26) presupposes nothing. However, given the above local context for the complement of ‘wish’, it can be shown that (26) is instead predicted to carry a non-trivial presupposition (roughly equivalent to If Ann thinks that Bill used to smoke, then she wishes he used to smoke).²⁰

Moreover, belief projection occurs not only in verbs of desire, but also in fictives like imagines, dreams, and supposes. But fictives do not constrain the subject’s beliefs. People can imagine things that are compatible with, entailed by, or contradicted by their beliefs. We see little hope of deriving belief projection for these attitudes by appealing to such constraints.²¹

5 Beyond quantification

Another attempt to derive belief projection replaces the quantificational analysis with a different kind of semantics for attitudes.

When it comes to desire verbs, there are two alternatives to the quantificational treatment. One approach builds on Stalnaker (1984), and substitutes quantification with comparative desirability. The basic idea is that S desires p when S prefers the closest p-worlds to the closest $\lnot p$-worlds. Perhaps the simplest comparative analysis goes as follows, where Sim(w, p) is the set of closest p-worlds to w and > is a preference relation over propositions.

However, this account predicts that the local context for the complement of want should be the set of all worlds, W. This essentially means that want reports should carry no presuppositions at all, which is clearly incorrect.²²

The key fact here is that each complement requires consideration of a different set of closest worlds, and so no particular candidate local context can be held fixed to guarantee contextual equivalence.

However, more sophisticated comparative analyses have been proposed. For instance, rather than considering the closest worlds outright, Heim (1992) instead proposes that we examine the closest belief worlds where the complement and its negation hold.

This semantics delivers belief projection.²³

In this way, the status of belief projection turns on whether the comparative semantics itself is sensitive in the right way to the agent’s beliefs.²⁴

The second alternative approach to verbs of desire models these expressions in terms of expected value. Suppose the subject’s credence function at w, Pw, assign a positive probability to all and only worlds in Bw. Let the subject’s utility function at w, Uw, assign a value to all worlds. Then on this analysis the agent wants whatever has greater expected value than its negation (here > denotes the natural ordering over real numbers):²⁵

This semantics delivers belief projection:²⁶

This result is less interesting than the corresponding result for the comparative desirability semantics in (29). This is because the expected value concept in (31) is defined in terms of conditionalizing the subject’s credence function. But on standard approaches to conditionalization, $Pw(w'|\llbracket \textsf {p}\rrbracket )$ will be undefined if the subject believes $\lnot p$, and $Pw(w'|\llbracket \lnot {\textsf {p}}\rrbracket )$ will be undefined if the subject believes p. So, the expected value semantics independently induces the kind of compatibility presupposition we discussed in Sect. 4. Consequently, it is not surprising that this entry derives belief projection.

Although the semantics in (29) and (31) derive belief projection on Schlenker’s algorithm, it is worth pointing out that adopting alternative analyses won’t derive belief projection for desire verbs in general. For instance, since what a subject wishes can be incompatible with what they believe, on a comparative desirability account of wish it would be inappropriate to constrain the similarity calculation by intersecting with the subject’s beliefs, as in (29). Indeed, ignoring presuppositions, Heim’s (1992) semantics for wish is essentially the entry in (27). But as we have seen, that entry fails to derive belief projection.²⁷

Turning finally to fictives, we will be fairly brief since we aren’t aware of any serious alternatives to a quantificational analysis of these expressions. Attitudes such as imagining, dreaming and supposing do not involve any obvious comparative aspect, and it seems implausible that their analysis is tied to decision-theoretic concepts. So, entries modeled on (29) and (31) would be inappropriate for these attitudes. More generally, the prospects for deriving belief projection for fictives by appealing to alternative analyses appear fairly dim.²⁸$^{,}$²⁹

6 Rejecting belief projection

In this section, we briefly consider the prospects of rejecting belief projection, and survey possible strategies for explaining some of the data supporting it by other means.

First, we saw in Sect. 1 that beliefs filter the attitude presuppositions, which is exactly what belief projection would predict. In particular, the following sentences have no presuppositions:

However, as several theorists have observed, the same filtering occurs with want/want and imagine/imagine constructions:

These patterns are not predicted on belief projection. Instead, given belief projection an example such as (34-a) is predicted to carry a non-trivial presupposition (roughly equivalent to the conditional If Bill wants to own a cello, then he believes that he owns a cello). Defenders of belief projection have tried to account for these examples by appealing to modal subordination.³⁰ Roughly speaking, this is the phenomenon whereby possibilities raised to salience by earlier discourse shape the intensional environments relevant for the evaluation of later discourse. For example, (35) presupposes nothing, which is not predicted on standard accounts of the way presuppositions project from conditionals:

The idea is that the antecedent of the first conjunct in (35) raises to salience possibilities in which Bill comes to the party. And it is these possibilities against which the second conjunct is evaluated, which explains why the presupposition triggered by Mary comes too is satisfied. Similarly, the thought is that in conjunctions such as (34-a), the first conjunct raises to salience Bill’s most highly preferred belief worlds, and it is only those possibilities that are relevant for satisfying the presupposition triggered by the second conjunct.

For proponents of Schlenker’s algorithm, the dialectic with respect to (33) and (34) is reversed: Schlenker’s account incorrectly predicts that the examples in (33) should trigger non-trivial presuppositions (in the case of (33-b), for instance, the presupposition would be roughly equivalent to the conditional If Bill thinks that Ann used to smoke, then he is imagining that she used to smoke), but it correctly predicts that the examples in (34) should be presuppositionless.

In response, proponents of Schlenker’s theory could try to appeal to modal subordination in order to explain belief filtering. For instance, the idea is that the first conjunct in (33-a)/(33-b) raises to salience Bill’s belief worlds, and it is exactly those possibilities that are relevant for satisfying the presupposition triggered by ‘stop smoking’ in the second conjunct. Although modal subordination has not been applied in quite this way before, as far as we can see there is nothing conceptually incoherent about this response. We suspect that its ultimate tenability will rest on fairly fine-grained details in the final theory of modal subordination, and for those who endorse Schlenker’s theory we think that this could be an intriguing place for further research. Overall, then, when it comes to the belief-filtering data, proponents of Schlenker’s system are not at an obvious disadvantage compared to those who endorse belief projection.³¹

Second, we remarked in Sect. 1 that examples such as those in (36) give rise to a belief entailment; they all suggest that Ann believes Bill used to smoke.

A reviewer suggests that proponents of Schlenker’s algorithm could derive this effect as follows. First, assume the first-run quantificational entry for ‘want’ from Sect. 3 ($\llbracket {\textsf {S wants p}}\rrbracket ^{w} = 1$ iff $BEST(Bw) \subseteq \llbracket \textsf {p}\rrbracket$). Then Schlenker’s algorithm predicts that the sentences in (36) should all carry a presupposition that is too weak, namely that only Ann’s best belief worlds are ones where Bill used to smoke. However, it is known that presuppositions are strengthened in certain situations. For example, consider (37):

According to the standard satisfaction theory, (37) triggers a conditional presupposition to the effect that if Ann mowed the lawn, then she has a wife. But in fact (37) implies the unconditional claim that Ann has a wife. Somehow the conditional presupposition gets strengthened.³² The thought is that this strengthening mechanism is also at play in (36), converting the relatively weak presupposition predicted by Schlenker’s theory into the required belief entailment.

We think that this is an interesting response that deserves a more thorough discussion than we can undertake here. That said, we want to bring out two features of this package that could help to guide further research. For one thing, it is worth noting that Schlenker’s account in conjunction with strengthening doesn’t yield exactly the same predictions as belief projection. In some cases, the predictions are strictly stronger than those delivered by belief projection. For instance, consider fictives such as ‘imagine’:

Assuming the quantificational entry for ‘imagine’ from Sect. 3 ($\llbracket {\textsf {S imagines p}}\rrbracket ^{w} = 1$ iff $Iw \subseteq \llbracket \textsf {p}\rrbracket$), Schlenker’s algorithm predicts that the examples in (38) should carry the following presupposition: all of Ann’s imagination worlds are ones where Bill used to smoke. If we then apply the strengthening mechanism we have that, for example, (38-a) presupposes that all of Ann’s imagination worlds and all of Ann’s belief worlds are ones where Bill used to smoke. This presupposition is strictly stronger than that predicted by belief projection. One area for future work could try to use these divergent predictions to try to tease apart these two accounts empirically.³³$^{,}$³⁴

Finally, we observe that the strengthening mechanism appears to be obligatory. Consider the following examples:

These sentences sound incoherent; denying the belief entailment in an attitude report produces absurdity. This is just what belief projection predicts. By contrast, it appears that Schlenker’s theory can only make sense of the infelicity here if the operation of strengthening is obligatory. If this is correct, then the plausibility of Schlenker’s approach will partly turn on the tenability of mandatory presupposition strengthening.³⁵

In general, the data points considered here appear to be more easily explicable on a theory which validates belief projection compared to Schlenker’s approach.³⁶$^{,}$³⁷ However, there are still several avenues available to proponents of Schlenker’s algorithm in dealing with this data, and we have tried to outline those that seem most promising.

7 Conclusion

We’ll close by revisiting the “explanatory problem” for dynamic semantics briefly discussed in Sect. 1. This debate is between those who stipulate projection properties as a matter of lexical semantics, and those who derive them from truth conditional meaning. Dynamic theories like Heim (1983, 1992) stipulate context change potentials for connectives and attitude verbs. However, some have objected to this framework on the grounds that it is insufficiently explanatory (Heim, 1990; Schlenker, 2009; Soames, 1982). This prompted Schlenker (2009) to offer an algorithm for deriving local contexts from truth conditional meanings.

Let us suppose that belief projection does in fact capture the behavior of attitude reports, and that Schlenker’s algorithm issues the wrong local contexts for attitude verbs. Then it is not obvious how to derive the local context for attitudes from their truth conditional meaning. As we have seen, this is perhaps most forceful in the case of fictives, such as imagine and dream. For instance, it is difficult to see how any theory could derive belief projection from the truth conditional meaning of imagine. Instead, from the perspective of the at-issue meaning of this verb, projection seems to be an arbitrary fact, and requires semantic stipulation.

If this is correct, then the “explanatory challenge” starts to look less urgent for dynamic semantics. Taken to its principled conclusion, this challenge requires that we provide a general, non-stipulative theory that can predict the appropriate local context for each expression, including those in the scope of attitude verbs. But if no such general account can be provided, because the local contexts of attitudes need to be stipulated in order to capture belief projection, then the explanatory challenge loses its bite. For if every account needs to stipulate the local contexts of certain expressions, then the differences between purportedly explanatory theories and stipulative accounts become less distinct.³⁸$^{,}$³⁹

Appendix

• (a) Where $\textsf {M}$ is a Kratzerian modal, the local context of $\textsf {p}$ in $\textsf {M}\textsf {p}$ in global context C is $\bigcup \left\{ \textsc {best}(g(w), \cap f(w)) \mid w \in C\right\}$. Proof. Let $\llbracket {\textsf {L}}\rrbracket = \bigcup \left\{ BEST(g(w), \cap f(w)) \mid w \in C\right\}$. First, for any sentence ${\textsf {q}}$, (41-a) and (41-b) are equivalent in C:

  

  To see this, suppose $w \models \textsf {M}({\textsf {L}} \,\wedge \,{\textsf {q}})$, with $w \in C$. Then $Q_{\textsf {M}}$ of the worlds in $BEST(g(w), \cap f(w))$ are $\llbracket {\textsf {L}} \,\wedge \,{\textsf {q}}\rrbracket$-worlds. Since $BEST(g(w), \cap f(w)) \subseteq \llbracket {\textsf {L}}\rrbracket$, it follows that $Q_{\textsf {M}}$ of the worlds in $BEST(g(w), \cap f(w))$ are $\llbracket {\textsf {q}}\rrbracket$-worlds. So, $w \models \textsf {M}{\textsf {q}}$ as well. Second, $\llbracket {\textsf {L}}\rrbracket$ is the strongest restriction that creates contextual equivalence. To see why, consider a stronger restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world $w' \in BEST(g(w), \cap f(w))$ for some $w \in C$. Then contextual equivalence fails. The precise form of the counterexample depends on the quantification force of the modal under discussion. If $Q_{\textsf {M}}$ quantifies universally, then let $\llbracket {\textsf {q}}\rrbracket = W$. Then (42-a) is false but (42-b) is true at w:

  

  For $\llbracket {\textsf {L'}}\rrbracket$ excludes $w'$, so $w' \not \in \llbracket {\textsf {L'}} \,\wedge {\textsf {q}}\rrbracket$. But of course $w' \in \llbracket {\textsf {q}}\rrbracket = W$. Analogous examples can be constructed for modals with different quantificational requirements. $\square$

• (b) Given the semantics for want in (20) below, the local context of ${\textsf {p}}$ in ${\textsf {S wants p}}$ in global context C is $\bigcup \left\{ Bw \mid w \in C\right\}$.

  

  Proof. We assume that at no world in C is the subject absolutely certain about which world they inhabit. We also assume that undefinedness is distinct from truth or falsity. Let $\llbracket {\textsf {L}}\rrbracket = \bigcup \left\{ Bw \mid w \in C\right\}$. First, it is clear that for any sentence ${\textsf {q}}$, (43-a) and (43-b) are equivalent in C:

  

  Second, $\llbracket {\textsf {L}}\rrbracket$ is the strongest restriction that creates contextual equivalence. For consider a stronger restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world $v \in Bw$ for some $w \in C$. Suppose, $\llbracket {\textsf {q}}\rrbracket = \left\{ v\right\}$. Then contextual equivalence fails, because $\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket = \emptyset$ so (44-a) will be undefined at w, but (44-b) will be true or false at w.$\square$

  

• (c) Given the semantics for wish in (24) below, the local context of ${\textsf {p}}$ in ${\textsf {S wishes p}}$ in global context C is $\bigcup \left\{ Bw \mid w \in C\right\} \cup \bigcup \left\{ BEST(Dw) \mid w \in C\right\}$.

  

  Proof. Let $\llbracket {\textsf {L}}\rrbracket = \bigcup \left\{ Bw \mid w \in C\right\} \cup \bigcup \left\{ BEST(Dw) \mid w \in C\right\}$. First, for any sentence ${\textsf {q}}$, (45-a) and (45-b) are equivalent in C:

  

  ${\textsf {S wishes q}}$ is defined in C iff ${\textsf {S wishes (L}} \wedge {\textsf {q)}}$ is defined : since $\llbracket {\textsf {L}}\rrbracket$ contains every belief world in the context, $\llbracket {\textsf {L}} \wedge {\textsf {q}}\rrbracket$ is incompatible with the belief worlds iff $\llbracket {\textsf {q}}\rrbracket$ is. Now suppose ${\textsf {S wishes q}}$ is true at w in C. Then $BEST(Dw) \subseteq \llbracket \textsf {q}\rrbracket$. Since $BEST(Dw) \subseteq \llbracket {\textsf {L}}\rrbracket$, ${\textsf {S wishes (L}}\, \wedge {\textsf {q)}}$ is true at w in C as well. Conversely, if ${\textsf {S wishes (L}}\, \wedge {\textsf {q)}}$ is true at w in C, then $BEST(Dw) \subseteq \llbracket {\textsf {L}}\rrbracket \cap \llbracket \textsf {q}\rrbracket$ and so $BEST(Dw) \subseteq \llbracket \textsf {q}\rrbracket$. Thus, ${\textsf {S wishes q}}$ is true at w in C. Second, $\llbracket {\textsf {L}}\rrbracket$ is the strongest restriction that creates contextual equivalence. To see why, consider a stronger restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world v in either Bw or Best(Dw) for some $w \in C$. First, suppose $v \in Bw$. Then contextual equivalence fails, because (46-a) is false at w, while (46-b) is undefined, where $\llbracket {\textsf {q}}\rrbracket = \left\{ v\right\}$.

  

  Second, suppose $v \in Best(Dw)$. Then contextual equivalence fails, because (47-a) is false at w, while (47-b) is true, where $\llbracket {\textsf {q}}\rrbracket = W - Bw$.

  

  $\square$

• (d) Given the semantics for want in (27) below, the local context of ${\textsf {p}}$ in ${\textsf {S wants p}}$ in global context C is W.

  

  Proof. It will be helpful to make explicit our assumptions about the similarity function Sim. First, we assume that for any w, $Sim(w, \emptyset ) = \emptyset$.⁴⁰ Second, we assume Strong Centering: if $w \in p$, then $Sim(w, p) = w$. Finally, we assume that any world in a subject’s belief set is more similar to the other worlds in the belief set than any worlds outside of it. That is, we assume that $Sim(u, p) \subseteq Bw$, if $u \in Bw$ and $p \cap Bw \ne \emptyset$. Let us call this assumption Belief Similarity. We also follow Heim (1992) in defining >, an ordering over propositions, in terms of $\succ$, a strict partial ordering over worlds. More explicitly: for any propositions p, q: $p > q$ iff $\forall w \in p$, $\forall w' \in q: w \succ w'$. First, let $\llbracket {\textsf {L}}\rrbracket = W$. Then for any sentence ${\textsf {q}}$, (48-a) and (48-b) are equivalent in C:

  

  Second, $\llbracket {\textsf {L}}\rrbracket$ is the strongest restriction that creates contextual equivalence. To see why, consider a stronger restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world v. We reason by cases, depending on whether $v \in \bigcup _{w \in C}Bw$ or not. (A) First, suppose $v \in \bigcup _{w \in C}Bw$. Then there is some $w \in C$ such that $v \in Bw$. Now we reason by cases again, depending on where v lies in $\succ$ with respect to the other elements in Bw. (A1) Suppose that there is some $v' \in Bw$ such that $v' \ne v$ and $v \not \succ v'$. Then let $\llbracket {\textsf {q}}\rrbracket = \{v\}$. Then (49-a) is true and (49-b) is false at w:

  

  For $\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket = \emptyset$, and so for all $u \in Bw: Sim(u, \llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket ) = \emptyset > Sim(u, W/\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket )$. But $\{v\} = Sim(v', \llbracket {\textsf {q}}\rrbracket ) \not > Sim(v', W/\llbracket {\textsf {q}}\rrbracket ) = \{v'\}$, by Strong Centering and our assumptions about the relative ranking of v and $v'$. (A2) Now suppose that for all $v' \in Bw$ such that $v' \ne v$: $v \succ v'$. We reason by cases again, depending on whether $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$ or not. (A2i) Suppose that $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$. Then let $\llbracket {\textsf {q}}\rrbracket = Bw/\{v\}$. Given Belief Similarity, $Sim(v, \llbracket {\textsf {q}}\rrbracket ) \subseteq Bw$, and by Strong Centering we have $Sim(v, W/\llbracket {\textsf {q}}\rrbracket )$ = $\{v\}$. Given our assumption about the ranking of v, it follows that $Sim(v, \llbracket {\textsf {q}}\rrbracket ) \not > Sim(v, W/\llbracket {\textsf {q}}\rrbracket )$ and so (49-b) is false at w. But (49-a) is trivially true at w, since $\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket = \emptyset$ given our assumption that $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$. (A2ii) Now suppose that $\llbracket {\textsf {L'}}\rrbracket \cap Bw \ne \emptyset$. Let $\llbracket {\textsf {q}}\rrbracket = W$. Then (49-b) is trivially true at w. But by Belief Similarity, $Sim(v, \llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket ) \subseteq Bw$, and by Strong Centering we have $Sim(v, W/\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket )$ = $\{v\}$. Given our assumption about the ranking of v, it follows that $Sim(v, \llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket ) \not > Sim(v, W/\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket )$, and so (49-a) is false at w. This completes the case where $v \in \bigcup _{w \in C}Bw$. (B) Now suppose that $v \not \in \bigcup _{w \in C}Bw$. We reason by cases again, depending on whether v outranks all of the elements in $\bigcup _{w \in C}Bw$ or not. (B1) Suppose that for some $v' \in \bigcup _{w \in C}Bw$, $v \not \succ v'$. Then there is some $w \in C$ such that $v' \in Bw$. Let $\llbracket {\textsf {q}}\rrbracket = \{v\}$. Then $\{v\} = Sim(v', \llbracket {\textsf {q}}\rrbracket ) \not > Sim(v', W/\llbracket {\textsf {q}}\rrbracket ) = \{v'\}$, given Strong Centering and our assumption about the ranking of v. So, (49-b) is false at w. But (49-a) is trivially true at w. (B2) Now suppose that for all $v' \in \bigcup _{w \in C}Bw$, $v \succ v'$. We reason by cases again, depending on whether every $s \in W/\llbracket {\textsf {L'}}\rrbracket$ outranks every $v' \in \bigcup _{w \in C}Bw$ or not. (B2i) Suppose that there is some $s \in W/\llbracket {\textsf {L'}}\rrbracket$, and some $v' \in \bigcup _{w \in C}Bw$, such that $s \not \succ v'$. Then we reason by cases again, essentially recapitulating the cases that we have already seen. To make things readable, let us denote the subcases for (B2i) with strings beginning with capital roman numerals, e.g. I, II, etc. (I) Suppose that $s \in \bigcup _{w \in C}Bw$. Then there is some $w \in C$ such that $s \in Bw$. Now we reason by cases again, depending on where s lies in $\succ$ with respect to the other elements in Bw. (I1) Suppose that there is some $s' \in Bw$ such that $s' \ne s$ and $s \not \succ s'$. Then let $\llbracket {\textsf {q}}\rrbracket = \{s\}$. Then (49-a) is true and (49-b) is false at w [the justification is essentially the same as for the analogous claims made in (A1)]. (I2) Now suppose that for all $s' \in Bw$ such that $s' \ne s$: $s \succ s'$. We reason by cases again, depending on whether $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$ or not. (I2i) Suppose that $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$. Then let $\llbracket {\textsf {q}}\rrbracket = Bw/\{s\}$. For reasons similar to those given in (A2i), (49-b) is false at w, but (49-a) is trivially true at w. (I2ii) Now suppose that $\llbracket {\textsf {L'}}\rrbracket \cap Bw \ne \emptyset$. Let $\llbracket {\textsf {q}}\rrbracket = W$. Then (49-b) is trivially true at w. But (49-a) is false at w, for reasons similar to those given in (A2ii). (II) Now suppose that $s \not \in \bigcup _{w \in C}Bw$. Recall that we are assuming that there is some $v' \in \bigcup _{w \in C}Bw$, such that $s \not \succ v'$. Then there is some $w \in C$ such that $v' \in Bw$. Let $\llbracket {\textsf {q}}\rrbracket = \{s\}$. Then for reasons similar to those given in (B1), (49-b) is false at w, but (49-a) is trivially true at w. This completes the case where there is some $s \in W/\llbracket {\textsf {L'}}\rrbracket$, and some $v' \in \bigcup _{w \in C}Bw$, such that $s \not \succ v'$. (B2ii) Suppose that for all $s \in W/\llbracket {\textsf {L'}}\rrbracket$, and for all $v' \in \bigcup _{w \in C}Bw$: $s \succ v'$. It follows that $\bigcup _{w \in C}Bw \subseteq \llbracket {\textsf {L'}}\rrbracket$. Now let $\llbracket {\textsf {q}}\rrbracket = W$. Let $w \in C$ be arbitrary. Then (49-b) is trivially true at w. But given Belief Similarity, for any $u \in Bw$: $Sim(u, \llbracket {\textsf {L'}} \,\wedge {\textsf {q}}\rrbracket ) \subseteq Bw$, since $Bw \subseteq \llbracket {\textsf {L'}}\rrbracket$. Moreover, since $W/\llbracket {\textsf {L'}}\rrbracket$ is comprised of elements that dominate $\bigcup _{w \in C}Bw$ and thus Bw, $Sim(u, W/\llbracket {\textsf {L'}} \,\wedge \,{\textsf {q}}\rrbracket ) = Sim(u, W/\llbracket {\textsf {L'}}\rrbracket )$ is comprised of elements that dominate Bw. Hence, $Sim(u, \llbracket {\textsf {L'}} \,\wedge \,{\textsf {q}}\rrbracket ) \not > Sim(u, W/\llbracket {\textsf {L'}} \,\wedge \,{\textsf {q}}\rrbracket )$, and (49-a) is false at w. This completes the proof. $\square$

• (e) Given the semantics for want in (29) below, the local context of ${\textsf {p}}$ in ${\textsf {S wants p}}$ in global context C is $\bigcup \left\{ Bw \mid w \in C\right\}$.⁴¹

  

  Proof. First, let $\llbracket {\textsf {L}}\rrbracket = \bigcup \left\{ Bw \mid w \in C\right\}$. Then for any sentence ${\textsf {q}}$, the following are equivalent in C:

  

  Here, the key observation is that $Sim(w', \llbracket {\textsf {q}}\rrbracket \cap Bw) = Sim(w', \llbracket {\textsf {L}} \,\wedge \,{\textsf {q}}\rrbracket \cap Bw)$, and $Sim(w', \llbracket \lnot {\textsf {q}}\rrbracket \cap Bw) = Sim(w', \llbracket \lnot {\textsf {(L}} \wedge {\textsf {q)}}\rrbracket \cap Bw)$. After all, for any $w \in C$, $Bw \subseteq \llbracket {\textsf {L}}\rrbracket$. It follows that $\llbracket {\textsf {q}}\rrbracket \cap Bw = \llbracket {\textsf {q}}\rrbracket \cap (\llbracket {\textsf {L}}\rrbracket \cap Bw) = (\llbracket {\textsf {L}}\rrbracket \cap \llbracket {\textsf {q}}\rrbracket ) \cap Bw = \llbracket {\textsf {L}} \wedge {\textsf {q}}\rrbracket \cap Bw$; and that $\llbracket \lnot {\textsf {q}}\rrbracket \cap Bw = W/\llbracket {\textsf {q}}\rrbracket \cap Bw = (W/\llbracket {\textsf {q}}\rrbracket \cap Bw) \cup \emptyset = (W/\llbracket {\textsf {q}}\rrbracket \cap Bw) \cup (W/\llbracket {\textsf {L}}\rrbracket \cap Bw) = (W/\llbracket {\textsf {q}}\rrbracket \cup W/\llbracket {\textsf {L}}\rrbracket ) \cap Bw = (W/(\llbracket {\textsf {q}}\rrbracket \cap \llbracket {\textsf {L}}\rrbracket )) \cap Bw = (W/(\llbracket {\textsf {L}} \,\wedge \,{\textsf {q}}\rrbracket ) )\cap Bw = (\llbracket \lnot ({\textsf {L}} \wedge {\textsf {q}})\rrbracket ) \cap Bw$. Second, consider another restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world $w' \in \bigcup \left\{ Bw \mid w \in C\right\}$. There must be some $w \in C$ such that $w' \in Bw$. Now we reason by cases, depending on whether $\llbracket {\textsf {L'}}\rrbracket$ contains worlds in Bw or not. First, suppose $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$. Then we reason by cases again, depending on where $w'$ lies in $\succ$ (we assume that $\succ$ is restricted to Bw). First, suppose that $w'$ outranks the other elements in Bw. Let $\llbracket {\textsf {q}}\rrbracket = W/\{w'\}$. Then the semantic values of (51-a) and (51-b) will diverge at w. The former will be trivially true, since $\llbracket {\textsf {L'}}\rrbracket \cap Bw = \emptyset$, and so for all $v \in Bw: Sim(v, \llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket \cap Bw) = \emptyset > Sim(v, W/\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket \cap Bw)$. But the latter will be false, since $Sim(w', \llbracket {\textsf {q}}\rrbracket \cap Bw) \not > Sim(w', W/\llbracket {\textsf {q}}\rrbracket \cap Bw) = \{w'\}$ (assuming Strong Centering, and given our assumption that $w'$ is the greatest element).

  

  Now suppose $w'$ is not greatest. Then there is some $w'' \in Bw$ such that $w' \not \succ w''$. Let $\llbracket {\textsf {q}}\rrbracket = \{w'\}$. Then the semantic values of (51-a) and (51-b) will again diverge at w: the former will be trivially true, but the latter will be false since $\{w'\} = Sim(w'', \llbracket {\textsf {q}}\rrbracket \cap Bw) \not > Sim(w'', W/\llbracket {\textsf {q}}\rrbracket \cap Bw) = \{w''\}$ (again given Strong Centering and our assumption that $w'$ is not greatest). Now suppose that $\llbracket {\textsf {L'}}\rrbracket \cap Bw \not = \emptyset$. Suppose that $w'$ is greatest. Let $\llbracket {\textsf {q}}\rrbracket = W$. Then the semantic values of (51-a) and (51-b) will diverge at w. The former will be false, since $w' \in W/\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket \cap Bw$, and so $\emptyset \ne Sim(w', \llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket \cap Bw) \not > Sim(w', W/\llbracket {\textsf {L'}} \wedge {\textsf {q}}\rrbracket \cap Bw) = \{w'\}$ (again given Strong Centering and assuming that $w'$ is greatest). But the latter will be trivially true. Now suppose $w'$ is not greatest. Then there is some $w'' \in Bw$ such that $w' \not > w''$. Let $\llbracket {\textsf {q}}\rrbracket = \{w'\}$. Then the semantic values of (51-a) and (51-b) will diverge at w. The former will be trivially true, since $\llbracket {\textsf {L'}}\rrbracket \cap \llbracket {\textsf {q}}\rrbracket = \emptyset$. But the latter will be false since $\{w'\} = Sim(w'', \llbracket {\textsf {q}}\rrbracket \cap Bw) \not > Sim(w'', W/\llbracket {\textsf {q}}\rrbracket \cap Bw) = \{w''\}$ (again given Strong Centering and our assumption that $w'$ is not greatest). This completes the proof. $\square$

• (f) Given the semantics for ${\textsf {want}}$ in (31), the local context of ${\textsf {p}}$ in ${\textsf {S wants p}}$ in global context C is $\bigcup \left\{ Bw \mid w \in C\right\}$.

  

  Proof. First, let $\llbracket {\textsf {L}}\rrbracket = \bigcup \left\{ Bw \mid w \in C\right\}$. Then since Pw assigns positive probability to all and only belief worlds, it is clear that for any sentence ${\textsf {q}}$, (52-a) and (52-b) are equivalent in C:

  

  Second, consider another restriction $\llbracket {\textsf {L'}}\rrbracket$ that excludes some world $w' \in \bigcup \left\{ Bw \mid w \in C\right\}$. There must be some $w \in C$ such that $w' \in Bw$. Let $\llbracket {\textsf {q}}\rrbracket = \{w'\}$. Then $\llbracket {\textsf {L'}} \ \wedge \ {\textsf {q}}\rrbracket = \emptyset$. Now, for any credence function Pw, and proposition p, $Pw(p | \emptyset )$ is undefined. It follows that (53-a) will be undefined at w, but (53-b) will be true or false at w.

  

  $\square$