# (p.208) Appendix

# (p.208) Appendix

As I have emphasized throughout this book, my main concern is not with compositional semantics, but with the development of concepts for describing and theorizing about the dynamic structure of discourse. The aim is to do so in a way that makes minimal commitments about the details of semantic mechanisms, interesting as those questions are. But issues do not divide neatly, and one needs to make at least some tentative and speculative assumptions about semantics to get clear about pragmatic structures, and to engage with other work. I have made use of David Lewisâs abstract Context/Index framework at various places, and this appendix is mainly a series of digressions about this framework, and about some related issues.

# 1 The nuts and bolts of the CI framework

Here is a concise summary of David Lewisâs Context/Index (CI) semantic framework, with a few examples of the kind of semantic rule it licenses. A semantics of this kind for a language assigns to each sentence a function that takes a pair of abstract objects (each with its own complexity) to a truth-value. The first argument of the function is a *context* (what I have called a K-context), which is a centered possible world: a pair consisting of a possible world, and a center, where a center is itself a pair consisting of a person (the *agent* of the context) and a time. The K-context is intended to represent the context of utterance for an utterance in the world by the agent at the time. Since a K-context includes a whole possible world, it determines all the facts on which the content or the truth of an utterance might depend. The second argument of the function is an *index*, which will be an n-tuple consisting of a sequence of *coordinates*, each of which is a feature of a kind that is
(p.209)
determined by a K-context. In a simple example for illustration, an index will be a triple consisting of a time, a location, and a possible world. The coordinates of the index are independent of the K-context that is associated with it; that is, in a given context-index pair, ă*c*,*i*ă, the time, location, and possible world coordinates of *i* may be different from the time, location of the agent, and world of *c*. But for any K-context, there will be an initial index, the index *of* that context, for which the time coordinate is the time of the context, the world coordinate is the world of the context, and the place coordinate is the location of the agent of the context. Notation: â*c*â and â*i*â will refer, respectively, to an arbitrary context and index, *c* = ă*a _{c}, t_{c}, w_{c}*ă and

*i*= ă

*p*ă. The index of the context,

_{i}, t_{i}, w_{i}*c*, will be ă

*p*ă, where

_{c}, t_{c}, w_{c}*p*is the location of

_{c}*a*at

_{c}*t*in

_{c}*w*. I will use square brackets for the semantic value of a sentence. â[Ï]ă

_{c}*c*,

*i*ă = 1 (0)â says that sentence Ï is true (false), relative to the context/index pair, ă

*c*,

*i*ă.

The form of a semantic rule, for a temporal, locative or modal operator, â*o*â, will be something like this:

for some or all i*, where i* is defined in some specific way in terms of c and i.

For example: [somewhere Ï] ăc,iă = 1 iff[Ï] ăc,i*ă = 1

for some *i** that differs from *i* at most in the location term.

[Once Ï]ă*c*,*i*ă = 1 iff [Ï]ă*c*,*i**ă = 1 for some *i** that is like *i*, except that *t _{i}*

_{*}<

*t*.

_{i}[Now Ï]ă*c*,*i*ă = 1 iff [Ï]ă*c*,*i**ă = 1 where *i** is like *i*, except that *t _{i}*

_{*}=

*t*.

_{c}[Actually Ï]ă*c*,*i*ă = 1 iff [Ï]ă*c*,*i**ă = 1 where *i** is like *i*, except that *w _{i}*

_{*}=

*w*.

_{c}[âŠÏ]ă*c*,*i*ă = 1 iff [Ï]ă*c*,*i**ă = 1 where *i** differs from I at most at the world component, which must meet the condition that *w _{i}*R

*w*

_{i}_{*}.

(R is a binary accessibility relation, determined by the context.)^{1}

The rule for the first-person singular pronoun will, of course, be this:[I]_{ăc,iă} = *a _{c}*

The truth-value of a simple sentence, such as one that means the same as âit is rainingâ will depend entirely on the index. So if it is raining on
(p.210)
Tuesday in Pittsburgh in world *w*, then if the *index* time and place are Tuesday and Pittsburgh, and the *index* world is *w*, then âit is rainingâ will take the value 1 (true) even if the *context* time and place are Monday and Cambridge, where it is dry in the context world.

The semantics gives us a truth-value, relative to an arbitrary context-index pair, but how do we get from this to the truth-value of an actual token utterance? Specifically, how do we get the particular *c* and *i* that are relevant to determining the truth-value of the utterance? We need to connect the abstract object, the semantic value, to the concrete situation. The relevant *c* is the one determined by the circumstances of the utteranceâthe actual speaker and time of utterance, and the actual world. (Or if we are interpreting a counterfactual utterance event, an utterance that takes place at a certain time and place in another possible world, then that time, place and counterfactual world will be the relevant contextual parameters.) To get the relevant index for the interpretation of the token utterance, we note that any context determines a corresponding index, the index *of* the context. So the final step, to get an actual truth-value, will be to use the index of the context. The upshot is that an utterance of a sentence S by agent *a* at time *t* in world *w* will be true simpliciter if and only if S is true relative to the CI pair, ăă*a*,*t*,*w*ă, ă*p*,*t*,*w*ăă, where *p* is the location of *a* at *t* in *w*. For example, if a particular token of âit is rainingâ is uttered by RS on July 23, 2012 in the actual world, @, and if RS is in Cambridge at that time, then the utterance will be true just in case the sentence takes the value 1 relative to the context-index pair, ă*c*,*i*ă, where *c* = ăRS, July 23, 2012, @ă, and *I* = ăCambridge, July 23, 2012, @ă.

# 2 Monsters

The compositional rules of Lewisâs CI semantics all define the value, relative to ă*c*,*i*ă in terms of the value at ă*c*,*i**ă, where *i** may differ from *i*, but the *c* always remains the same. Why, one may ask, is the indexical semantics constrained in this way? This is the question, why the semantics rules out what David Kaplan called âmonstersâ. The answer I gave in Chapter 1 is that this restriction is no substantive constraint, since the general abstract theory puts no constraints on what can go in the index. The idea is that the index should include all of the âshiftableâ elements. If one wanted to use a Lewisian CI semantics to account for a language that shifted something that looked a lot like the abstract object that we use to
(p.211)
represent context, or something determined by it, then one can put an object of this kind into the index. For example, Philippe Schlenker, in his brief on behalf of monsters, reports that in Amharic, the first-person singular pronoun can shift, in the scope of a propositional attitude verb, to refer to the agent of the attitude, rather than to the speaker in the context of utterance. So the pronoun that means âIâ in a simple sentence that translates âI am happyâ, may refer to John, and be translated âheâ in a sentence that means that John_{1} believes that he_{1} is happy. This pronoun would not be a monster, in a CI semantics, since a semantics of this kind for this language will put an âagentâ place into the index. The agent coordinate of the index *of* the context will be the speaker, but that coordinate will shift in the scope of the attitude verb.^{2}

I think this is the right way to think about monsters in the Lewis CI framework, but that is not the end of the story, since the framework in which Kaplan made his notorious remark about monsters was not exactly Lewisâs framework, and there are other interpretations of what a monster might be. In Kaplanâs semantics for demonstratives, the semantic value of a sentence is a *character*, which is a function from context to content. Kaplanâs prohibition of monsters was the claim that, while our compositional semantics should allow any operators or connectives that defines the *content* of the complex expression in terms of the *contents* of the parts, we do not allow operators that define the *character* of the complex expression in terms of the *character* of the parts. But in Lewisâs framework, the notion of propositional *content* plays no role at all, either in the compositional semantics, or in his pragmatic story of the conventions and rules governing speech. As a result, Kaplanâs notion of character also plays no role in the CI theory. I argued in Chapter 1 that it was a mistake to leave propositional content out of the pragmatic story, but it is a separate question whether a notion of content should play an essential role in the compositional semantics.

Despite the differences between Kaplanâs and Lewisâs frameworks, there is nevertheless a clear abstract analogy between Kaplanâs requirement that the language contain no monsters and the requirement of the CI framework that indices, but not contexts, can be shifted by the compositional rules. The Kaplanian dogma that there are no monsters
(p.212)
is essentially the assumption that the compositional process has the following structure: *context* determines the values of indexicals, which fix the *contents* of all the sentential constituents. Context has then finished its job: the compositional rules take over, giving the contents of complex expressions as a function of the contents of their parts. On the CI picture, neither content nor anything analogous to content plays an *essential* role, since Lewis argued that in a CI semantics there is only a notational difference between a two-stage process (function from context to function from index to truth-value) and a one-stage process (function from a context-index pair to a truth-value). But the two-stage representation facilitates comparison with the Kaplan picture, and if we describe the CI semantic framework that way, the abstract structure is exactly the same as it is on the Kaplan picture: the context determines, prior to the compositional process, the values of some parameters, and these will then determine functions from indices to truth-values (the abstract analogues of content), in context, for each of the sentential clauses. The job of context is then finished; the compositional process operates on the index-to-truth-value functions. The question is, why does the structure of a semantic theory have to take this form?

If a âcontextâ were really a *context*âthe concrete situation in which an utterance event takes placeâthen it would be hard to make sense of a compositional rule that shifted the context. But if we think of a âcontextâ as an abstract objectâa centered possible world that does not necessarily represent the situation in which an utterance being interpreted takes placeâthen it is not clear what the general motivation is for the assumption that the semantic process must have this structure. Is it then an empirical question (analogous to a question about the structure of universal grammar)? If so, then the evidence seems to suggest that the ban on monsters is empirically incorrect. But I have just argued that in the CI framework, with its abstract and unconstrained concept of an index, the imposition of this structure (the ban on monsters) puts no substantive constraint on the semantics. If the evidence shows that some parameter shifts, compositionally, then one can just put it (or a copy of it) into the index. This seems right, and unproblematic, for isolated parameters of a kind that are determined by context such as the agent parameter that is shifted in propositional attitude ascriptions in languages such as Amharic. But what if it is the whole âcontextâ that is shifted? Can we preserve the form of the CI framework by putting a
(p.213)
âcontextâ coordinate into the index, and if we can, is there any point in doing so? Letâs look at an example of a putative monster and see how the CI framework accommodates it. The example comes from a defense of monsters by Paolo Santorio.

The protagonists in Santorioâs story are familiar from the literature on indexical belief, with ancestry that goes back to Frege, but this time they are involved in a new and desperate situation. Rudolf Lingens and Gustav Lauben are two amnesiacs who each know that they are one of the two, but do not know which. They have been kidnapped and will be subjected to the following experiment:

First, they will be anesthetized. Then a coin will be tossed. If the outcome is tails, Lingens will be released in Main Library, Stanford, and Lauben will be killed. If the outcome is heads, Lauben will be released in Widener Library, Harvard, and Lingens will be killed. Lingens and Lauben are informed of the plan and the experiment is executed. Later, one of them wakes up in a library.

^{3}

The statements Santorio considers in this context are two indicative conditionals, and two corresponding epistemic possibility statements made by the person who wakes up.

(1) If the coin landed tails, I am in Main Library, Stanford.

(2) If the coin landed heads, I am in Widener Library, Harvard.

(3) I might be in Main Library, Stanford.

(4) I might be in Widener Library, Harvard.

The datum is that all of these statements are felicitous. If we take them to be statements that say something true or false, relative to a context, it seems clear that they are true in the scenario described, and known to be true by the speaker since he knows that the scenario is as described. The problem is that if we give them a compositional semantics in the CI style, we do not get the result that both conditionals, and both âmightâ statements are true unless we allow for a compositional âshiftâ in the context. If we do allow such a shift, we get the results we want easily and naturally, at least in this case. Intuitively, even in the world in which the coin landed heads, where the speaker is Lauben, his statement, âI might be in Main Library, Stanfordâ is true, since he does not know that he is not Lingens in the world where the coin landed tails. (More to the point,
(p.214)
it is an open possibility in the context that he is Lingens, and is in such a world.) If the possible worlds that are in the domain of the âmightâ are the *context* worlds, with the contextual parameters fixed as they are in that context world, then both statements come out true. That is, suppose the rule for the epistemic âmightâ were something like this:

where *g*(*c*) is the context set of the conversation taking place in K-context *c*. (We need to shift both the context and the world of the index in order to get the right result: that statement (2) is true when it is made by Lingens in the world where the coin landed tails, and where he is in fact in Main Library, Stanford.)

This rule gets the right result, but it turns the âmightâ into a monster. Could we avoid monstrosity and preserve the purity of the CI semantics by putting a copy of the âcontextâ into the index? Yes we can, but the required change will be more radical than it is in the case of the first-person pronoun in a language that works the way we are told that Amharic works. We will need to modify other rules so that values of indexicals and modifiers such as ânowâ and âactuallyâ are defined in terms of the âcontextâ coordinate of the index rather than the K-context itself. Is there any point in doing this, rather than simply removing the structural restriction that rules out compositional context shifts? From a technical point of view, it may make no difference exactly how one formulates the rule, but copying a context into the index and preserving the ban on Lewisian monsters has this conceptual point: the notion of a K-context is playing two different roles: (1) it models the *context of utterance*: the concrete situation in which the utterance being interpreted takes place. Contexts, in this sense, determine a body of information (the common ground). But (2) the elements of the common ground (in the given context of utterance) are themselves K-contexts, or something that, given an utterance, determines a unique K-context. A K-context, in its first role, cannot shift, compositionally, but there is no reason why K-contexts, qua abstract objects that are members of a set used to characterize an information state, cannot be subject to compositional shifts.

I have acquiesced in using the word âcontextâ to refer to centered possible worlds since the usage is so firmly entrenched, but in fact I think
(p.215)
this is extremely misleading terminology. K-contexts deserve the title, and can play their role in explaining how content is sensitive to context, only because they determine the common ground (since it is a fact about the world in which a speech act takes place that certain information is common ground among the relevant people at the relevant time). That is, a K-context contains or determines the contextual information to which context-sensitive expressions are sensitive, since it is a representation of a whole world, plus a center that determines the conversation in which certain information is common ground. But calling this abstract object (a centered possible world) a âcontextâ tends to blur the line between the distinctive information that is contextual information and other facts that may play a role in the compositional semantics. And it obscures the fact that centered possible worlds, in the general case, are not suitable to be contexts, since they need not be centered on an utterance, or on a conversation. If we were to stay with the intuitive idea motivating Kaplanâs character/content semantics, we would take character to be a function from the *CG*-context (rather than the K-context) to content. Where a CI semantics appeals to a feature of âcontextâ that is not available in the common ground (such as the world of the context, or in cases where the identity of the speaker or addressee is in question, or âat issueâ, the agent or addressee of the context), then we should say that what the context itselfâthe common groundâdetermines is not the specific feature (a world, or an agent), but instead a function whose argument is an element of the CG-context (a sequence-centered world), and whose value is the feature in question (a world, an agent, or an addressee). This is what happens, for example, with the first-person singular pronoun when one applies a CI semantics in the context of our sequence-centered worlds model of common ground. What context (CG-context) provides are the I-concepts, which are functions from elements of the common ground to agents. But if the semantic value (determined by context) of an indexical pronoun is a nonrigid individual concept, then we seem to have some deviation from the direct reference picture, and violations of the Kaplanian dogma that indexicals are always rigid designators. This deviation is what Santorioâs examples, which are robust and, I think, decisive, show.

The ban on monsters and the doctrine that indexicals such as âIâ and âthatâ are always rigid designators are tied together as a package in Kaplanâs theory, but they can be pulled apart. If our semantic theory (p.216) says that what context determines, for the interpretation of the pronoun âIâ is a possibly nonrigid individual concept (a function from possible worlds to individuals), then we have rejected the doctrine about indexicals, but we have not thereby lifted the ban on monstersâsemantic rules that shift the contextâeven in the Kaplan framework. Santorioâs examples are compatible with a compositional semantics for epistemic âmightâ in which context determines the content of the prejacent, and then the semantics for the âmightâ determines that this proposition is true in one of the contextually determined possible worlds. No context shifting need be involved.

# 3 Assertoric content

Lewisâs semantics deliver a truth-value for the utterance of a sentence as a function of the context of utterance: it is the value of the function that is the meaning of the sentence for the argument pair whose terms are the context of utterance and the index of that context. Lewis noted (somewhat dismissively) that, if you like, you can extract a proposition (a function from possible worlds to truth-values) from the semantics in either of the following two ways. First, for a given K-context, initialize all the coordinates of the index except the world coordinate, and abstract on the world coordinate. That gives you a function from K-contexts to propositions, and he labels the value of this function âthe *horizontal* propositionâ. A second way to extract a proposition from the semantic value of a sentence, which he calls âthe *diagonal* propositionâ, is a little more complicated. Here is what he says: âLet *c ^{w}* be that context, if there is one, that is located in world

*w*and indistinguishable from

*c*; then for all the speaker knows, he might inhabit

*w*, and

*c*might be the context of his utterance. (I ignore the case of two indistinguishable contexts in the same world.)â He then defines the diagonal proposition for a sentence Ï as the function that takes a world w to the truth-value of Ï, relative to the context/index pair ă

^{w}*c*ă, where

^{w}, i_{c}w*i*is the index of the context

_{c}w*c*. Now as I have noted, propositions do not play any role either in the compositional semantics, or in his pragmatic story about conventions of truth and trust, but if we want to identify a proposition that is the

^{w}*assertoric content*of an utterance, which of these should it be? Lewisâs diagonal proposition comes closest to what we want for the following reason: the (p.217) role of assertoric content, according to the general pragmatic theory that I am promoting, is to determine which are the possibilities that are to be eliminated from the context set when an assertion with that content is accepted. As Michael Dummett wrote, in introducing the distinction between assertoric content and ingredient sense, âto grasp the content of an assertion, one needs to know only what possibilities it rules out.â

^{4}In cases where the diagonal and horizontal propositions defined by Lewis differ (for possible worlds in the relevant context set), it is intuitively clear that the point of the assertion is to rule out those possible worlds in which the diagonal proposition is false. (I will give an example to illustrate this below.) Furthermore, this is not just a fact about usage that might have been different. As I have emphasized throughout the discussion of the role of context in determining content, the contextual features on which what is said depends must, in any appropriate speech act, be presumed to be available to the addressee. A K-context, centered on a speaker in a conversation, contains all of the information that is available, and that might be relevant, but it also determines information that is not available. To put the point in a slightly different way, a CG-context in which an utterance event is presupposed to be taking place is compatible with many K-contexts, and the information that distinguishes between the different K-contexts compatible with common ground is not presumed to be available. So we need a notion of assertoric content that meets the condition that the content of a speech act, relative to one K-context, is the same as the content relative to a different K-context that determines the same CG-context. The horizontal proposition, as defined by Lewis, does not, in general, meet this condition, but a version of the notion of a diagonal proposition does.

The definition of the diagonal proposition that I will give is slightly different from Lewisâs, relying on the notion of common ground, rather than a notion of indistinguishability, but the idea is essentially the same. Before stating the definition, I need to make some remarks about the relation between three things: (1) the uncentered possible worlds that *propositions* distinguish between, (2) the multiply-centered worlds (call them âs-worldsâ) that are the elements of the common ground, and (3) the singly-centered worlds that are K-contexts.

(p.218)
According to the account developed and defended in Chapter 5, the elements of the common ground are possible worlds centered on a time and a sequence of individual agentsâone individual agent for each of the participants in the conversation. It is an assumption of that account, which I defended in Chapter 5, that each uncentered world occurs only once in a common-ground set of s-worlds, and so there will be a one-one correspondence between the elements of the common ground and the context set, Câthe set of uncentered possible worlds that are compatible with the common ground. When an utterance event occurs, it takes place in each of the elements of the CG-context, and this determines a set of *singly*-centered worlds (K-contexts), centered on the speaker. So where C is the context set (the uncentered worlds compatible with the context) and Ï is the sentence uttered, it will be true that for each *w* â C there will be a unique K-context, *c _{w}* which is the world

*w*centered on the time of utterance, and the speaker. The

*diagonal proposition*for the utterance of Ï (defined relative to C) will be the proposition that takes w to [Ï]ă

*c*ă. That is, the diagonal proposition is the one that is true at

_{w}, i_{c}w*w*if and only if Ï is true, relative to the K-context determined by

*w*. This specifies only a

*partial*propositionâa function from possible worlds to truth-values that is defined only for the context set. We will consider below the ways in which the proposition can be extended.

Here is a simple example to illustrate the contrast between the horizontal and diagonal propositions, and to show that what is asserted is the diagonal. Clara, on a game show, can choose to have the contents of one of two boxes, A or B. One contains a million dollars ($M), while the other contains only a thousand dollars ($T), but she has no idea which box is the one with the big prize. She chooses box A, but before she opens the box, the game show host observes to the audience, âClara could have gotten more money than she actually got.â It seems clear that this statement tells the audience that box A (the one Clara chose) is the one containing only $T. Now let us apply the CI semantics to the hostâs observation, which we will formalize this way: S is the sentence âClara got more money than she actually got,â so the sentence used to make the statement is formalized âŹ§S. (I use a solid diamond for the kind of causal modality that is expressed by the âcouldâ.) Since it is known that Clara chose box A, there are two relevant possible worlds compatible with the prior contextâthe CG-context as it was before the host made his statement. Call them Î± and ÎČ. In world Î± there is $M in box A, and $T in box B,
(p.219)
while in world ÎČ there is $T in box A and $M in box B. In both Î± and ÎČ, Clara chose box A. (It is common ground, in the prior context, that box A was chosen.) Since the sentence we are interpreting involves a (nonepistemic) modal âcouldâ, we will need to consider two other possible worlds that are outside of the context set: worlds Îł and ÎŽ are like Î± and ÎČ, respectively, in the amount of money in the boxes, but they are both worlds in which Clara selected the other box, B. Since Clara could have made either choice, but had no control over the amount of money in the boxes, Î± and Îł will be possible with respect to each other, as will ÎČ and ÎŽ, but neither Î± nor Îł will be possible with respect to ÎČ or ÎŽ. Let us assume that the world of the *context* (the K-context) is world ÎČ. So sentence S will be true (in that context) in index world *x* if and only if Clara gets more money, in world *x*, than she got in the context world, ÎČ. So S will be true (in the ÎČ context) in worlds Î± and ÎŽ, and false in ÎČ and Îł. Since Î± is possible relative to itself, and ÎŽ is possible relative to ÎČ, âŠS will be true, relative to both Î± and ÎČ. So the horizontal proposition expressed (in context world ÎČ) does not distinguish Î± from ÎČ; if this proposition were the assertoric content of what the host said, he would not have told the audience anything about where the money is. But if we evaluate the sentence relative to context world Î±, the result will be that it is false in both possible worlds. Since the sentence is false in world Î±, in context Î±, and true in world ÎČ, in context ÎČ, the diagonal proposition will be the one that excludes possible world Î±, and so it is intuitively clear that it is the diagonal that is the assertoric content of the hostâs statement.

âIt would be a convenience, nothing more,â Lewis wrote, âif we could take the propositional content of a sentence in context as its semantic value. But we cannot. The propositional contents do not obey the compositional principle, therefore they are not semantic values.â^{5} The point (to use Dummettâs terminology) is that we need to distinguish ingredient sense (the semantic value required for compositional purposes) from assertoric content (what a sentence is used to say). The main consideration Lewis appeals to in arguing for the need for this distinction is that to get the compositional semantics right, the index must contain coordinates other than a possible world, since the language contains operators that âshiftâ other features of the situation. One might think
(p.220)
that if we could somehow get rid of these other operators, and so get rid of the need for other coordinates of the index, we could preserve the simpler and more convenient semantics, where the semantic values (in context) of sentential clauses that are parts of more complex sentences were all the same as the assertoric contents that the corresponding sentences would have on their own. As noted in Chapter 1, Jeffrey King argued that we can and should eliminate temporal and locative sentential operators by using a semantics with variables in the logical forms of sentences that range over times and places, with tense and temporal and locative modifiers analyzed as object-language quantification over domains of times and places.^{6} He argues that we have independent motivation for adopting this kind of semantics, and that if we did so, we could avoid the distinction between assertoric content and ingredient sense. I am skeptical about the independent motivation, but I think that is a question best left to the syntacticians. (Despite its name, âlogical formâ is a syntactic notion.) I am also unclear how it would help to reduce all temporal and locative modification to explicit quantification over times and places, since in Lewisâs CI framework, quantifiers are themselves treated as index-shifting operators. (The CI semantics for quantification has a certain elegance, since it treats deictic pronouns and bound pronouns in a uniform way. Unbound, a pronounâs reference is fixed by the initial indexâthe index of the context. But when it is bound, the index value is shifted by the quantifier rule, and so depends on a domain supplied by the context, rather than specific values for the variables.) So even if we eliminate time and location coordinates, we will need, in the index, values for the variables that range over times and placesâvalues that are shifted by quantifiers. But my main point is that even in a CI semantics where the only coordinate of the index is a possible world, we still will need a distinction between assertoric content and ingredient sense. The point is made by the game show example, which does not involve temporal or locative modification, or quantifiers. It is essential to get the compositional semantics right that the argument of the causal âcouldâ be the âhorizontalâ proposition (or the function from index to truth-value, determined as a function of the context). But as the example made clear, the assertoric content of the whole sentence must be the diagonal proposition.

(p.221) The distinction between assertoric content and ingredient sense is obviously related to the issue about monsters, but the distinction does not by itself introduce monsters into the compositional process, or require that a âcontextâ coordinate be included in the index, since one can think of the extraction of assertoric content as a post-semantic operation. It is when assertoric contents appear in the scope of an operator, as seems to happen with epistemic modals and propositional attitudes that we get (in the CI framework) the problem about monsters.

I noted that the definition I have given for the diagonal proposition (the assertoric content) determines only a partial proposition: as specified so far, the function from possible worlds to truth-values is defined only relative to the domain of possible worlds in the context set. If we stop there, we have a semantic/pragmatic account that is essentially like the dynamic semantics that takes semantic values to be context-change potentialsâfunctions taking prior contexts into posterior contexts. I have resisted the move to dynamic semantics, favoring instead a dynamic pragmatic story, and part of my reason for the resistance, as I have said, is that I think we need a notion of propositional content that is defined for possible worlds beyond those compatible with the current local context. We want to be able to detach information we have been told from the immediate context in which we were told it, and we want at least in some cases to be able to assess statement (as true or false) in a retrospective context that is different from the one in which the statement was made. So it is important for my general view that the assertoric content expressed in a speech act be defined, at least in some cases, for a wider range of possibilities. In many cases, the extension of the domain of the function that is the proposition will be unproblematic, since with many expressions the semantic value may not depend on context, or may depend only on features that are shared by a wide range of possible contexts. But the extension of the assertoric content of even paradigm indexical sentences will often be unproblematic. For example, the context (the common ground) may provide a natural extension of the relevant I-concept to possible worlds outside the context set. In contexts where the identity of the speaker is not âat issueâ, (which is the normal case) the relevant I-concept will be the ordinary rigid concept, picking out the actual speaker in all possible worlds. Even when the identity of the speaker *is* at issue, the relevant I-concept clearly picks out the actual speaker in the actual world (or in any possible world compatible with a
(p.222)
retrospective context in which the proposition is assessed), even if the worlds in the retrospective context are not compatible with the context in which the statement was made. In the derived contexts relevant to the interpretation of subjunctive conditionals, the extension of the relevant I-concept is clear, since the derived context is defied by an imaging function (a mapping of worlds to worlds), and the referent of âIâ in the image world will be the referent in the base world. To interpret propositional attitude ascriptions (such as, for example, âRalph believes that I am a spy,â said by Ortcutt), a proposition will be determined in the relevant derived context only if the basic contextâthe common groundâdetermines a uniquely most salient way that the subject identifies the speaker. A salient I-concept *may* be undetermined for some possible worlds (outside of the basic context set), and so the assertoric content of utterances of sentences with this pronoun in the scope of a attitude verb may remain partial, but appropriate speech will require that the I-concept be defined, not only for all the possible situations compatible with the basic context, but also for possible situations compatible with any of the subordinate contexts that are relevant to the interpretation of what is said.^{7}

# 4 Two-dimensional âsemanticsâ

In distinguishing the two ways of determining a proposition, and in using the âhorizontal/diagonalâ terminology for this distinction, Lewis was, as he said, âborrowing and modifyingâ some apparatus that I used in my âAssertionâ paper. But I put this apparatus to use, in that paper and others, in a somewhat different way. I thought of the two-dimensional structures (what I called âpropositional conceptsâââfunctions from possible worlds to propositions, which were themselves functions from possible worlds to truth-values) as a descriptive apparatus for representing the different ways in which what is said is a function of the facts. That a particular utterance expresses a proposition is a matter of fact, which means that the truth-value of an utterance will depend on the facts in two different ways: the (p.223) facts determine what is said, and the facts also determine whether what is said is true or false. (The directional terminology derives from the matrices used to represent the two dimensions: vertical for the dimension that represents the role of facts in determining what is said, and horizontal to represent the role of the facts in determining whether what is said is true or false.) This application of the apparatus is not a part of the semantics for any language in which propositions are expressedâit abstracts away from that, representing the facts on which the determination of propositional content depends simply by a function from possible worlds to propositions. So at this level of abstraction, the representation is neutral on the question, what sort of fact is it that determines content, and it says or assumes nothing about the compositional semantics that determines what propositions are expressed. The facts that determine content will presumably include facts about the semantics of the language in which the content is expressed, as well as facts about the context, however that notion is understood, in cases where the expressions of the language are context-sensitive.

The word âsemanticsâ is in quotation marks in the title to this section in order to highlight a way that this label for the two-dimensional apparatus, in the use to which I put it, may be misleading. The two-dimensional apparatus was taken from a formal semantics, or model theory, for a two-dimensional modal logic. This semantics gives compositional rules for the interpretation of sentences of an artificial language with two-dimensional modal operators, but in the use to which I put them, the structures defined in that model theory were not used to give a two-dimensional semantics for any natural language. Whether natural languages are appropriately given any kind of two-dimensional semantics is a separate and independent question. In both Kaplanâs semantics in *Demonstratives* and Lewisâs CI theory a two-dimensional framework *is* used for a different and complementary purpose: for giving a compositional semantics for a language. The two ways of using two-dimensional structures interact and may overlap, but it is important not to conflate them.^{8}

In the abstract two-dimensional model theory that I used in âAssertionâ,^{9} the two parameters (the two dimensions relative to which truth-
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values are defined) were the same: truth was defined relative to an ordered pair of possible worlds. A two-dimensional concept associated with a particular utterance token can be defined for any set of possible worlds in which that utterance event occurs, including worlds that are not compatible with the common ground, or with what a speaker is presupposing. The set of worlds relative to which a propositional concept is defined will include all the worlds compatible with the common ground, since it is assumed that it is common ground that the utterance event took place. But one can use a propositional concept to represent ignorance and error about what is said, either the ignorance or error of a participant in a conversation, or of an outside observer. The example used in Chapter 1 (and in my original âPragmaticsâ paper) to illustrate the contrast between the two roles that factual information plays in determining the truth-value of an utterance was this kind of case: Daniels thinks I said something false when I said âYou are a foolâ to OâLeary, because he mistakenly thought that I was talking to him. In this example, the speaker was *presupposing* that OâLeary was the addressee, so the worlds that are compatible with what Daniels believed were outside of the context set. Perhaps OâLeary misunderstood as well, in which case I might correct him this way: âHey! OâLeary! Iâm talking to *you.*â The assertoric content of this correction statement is that OâLeary is the addressee. Since I recognized that OâLeary had misunderstood me, I am no longer presupposing, in making this statement, that OâLeary is the addressee. This assertion was intended to exclude possible situations in which the âyouâ refers to Daniels, which means that the âyouâ refers to Daniels relative to the possible situations that are excluded. Of course there are no limits to what OâLeary can misunderstand, so he might miss the point, even of my correction, and remain ignorant of the assertoric content of what I said. One might use a two-dimensional matrix to represent OâLearyâs further misunderstanding, in which case the âhorizontalâ propositions in this representation will be the alternative assertoric contents that for all OâLeary knows, are the contents of what I said. These same propositionsâthe âhorizontalsâ of one representationâmight be determined as diagonal propositions of the kind defined for a two-dimensional, CI semantics.

In calling some semantic theory (either for a formal language, or for a natural language âtwo-dimensionalâ I mean only that it defines truth as a function of two parameters. In Segerbergâs abstract two-dimensional
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modal semantics, and in my original application of it, the two parameters were drawn from the same domainâa set of possible worldsâso truth is defined relative to an ordered pair of things of the same kind. In this case, the concept of a diagonal is straightforward: if *f* is a propositional concept (a binary function taking an ordered pair of possible worlds to a truth-value), then the corresponding diagonal proposition is the singulary function Î»*xf*(*x*,*x*). If ââ â is the diagonalization operator (in a formal language such as the one defined by Segerberg), then the rule for this operator will be this: [â Ï]ă*x*,*y*ă = [Ï]ă*y*,*y*ă. But in the more general case, the two dimensions (the two arguments of the binary function) may be drawn from distinct domains, as is the case with a CI semantics, where the dimensions are K-contexts (centered possible worlds) and indices (sequences of coordinates of a kind that context determines). In these cases, there is no straightforward notion of a diagonal, and whether a notion of a diagonal can be defined at all depends on the relationship between the two different domains of the binary function. The definition of a diagonal proposition defined in section 3 above depended on the fact that one component both of a K-context and of an index is a possible world, and on the constraint that within the set of possible worlds compatible with the common ground, there is a function from index worlds (of the index of the context) to context worlds. In David Chalmersâs two-dimensional framework^{10} one parameter is a centered world, and the other is an uncentered world. The diagonal is not a proposition, but a centered-worlds propositionâa function from K-contexts to truth-values. If *f* is a two-dimensional function taking a centered world *c* and an uncentered world *w* to a truth-value, and if *w _{c}* is the world of

*c*, without its center, then the diagonal (centered-world) proposition is defined as follows: Î»

*cf*(

*c*,

*w*).

_{c}In Seth Yalcinâs semantics for epistemic modals, which I will discuss just below, he defines both a diagonal and a horizontal, but in this case, neither is a proposition. The diagonal is, as in Chalmersâs two-dimensional semantics, a centered-worlds proposition, or a function from centered worlds, or K-contexts, to truth-values. Yalcinâs horizontals are functions from indices to truth-values; the semantics determines a horizontal for an utterance as a function of its K-context.

# (p.226) 5 Yalcinâs semantics for epistemic modals

Seth Yalcin (in Yalcin 2007) proposed a semantics for epistemic modals in Lewisâs CI framework. The main innovation was to include as one coordinate of the index an information state parameter, represented by a set of possible worlds and a probability function on it. Assuming for simplicity that the only elements of the index are a possible world and an information state, the rule for the epistemic âmightâ is this:

The information state coordinate may shift when the âââ occurs within the scope of a supposition or propositional attitude operator. Specifically, a supposition will shift the information state to the set of possible worlds compatible with what is being supposed.

The abstract semantics leaves open the specific question, (a)âwhat determines the value of the information parameter, in application?âas well as a more general question about the role of an epistemic modal claim in communication. Yalcin puts the general question this way: âWhat informational content do utterances of epistemic modal sentences communicate?â (p.1006), and he contrasts two ways of answering it, which he calls âthe diagonal viewâ and âthe informational viewâ.

The diagonal view follows the general pattern of Lewisâs story about the role of assertive utterances in discourse: the context of utterance *c* provides an âinitialâ index, *i _{c}*, the index

*of*the context. An utterance of a sentence in a context c will be true just in case it is true relative to the context-index pair, ă

*c, i*ă. To complete the story, we need to specify the initial information state coordinateâthe information state of the index of the context. (This is question (a).) Yalcin leaves this open, in his characterization of the diagonal view, but a natural proposal that he considers is that it should be the context setâthe possibilities compatible with the common ground of the K-context.

_{c}^{11}To get the informational content of the epistemic modal statement, we follow the general account of assertoric content. Yalcin takes this to be a centered-worlds proposition: a function from K-contexts to truth values, defined as follows for (p.227) arbitrary sentence Ï: Î»

*c*[Ï]ă

*c, w*ă. So the diagonal view treats utterances of epistemic âmightâ sentences exactly as it treats ordinary assertions.

_{c}, s_{c}The contrasting informational view rejects the assumption that epistemic modal sentences have propositional content, and are used to say what the world is like. The relevant abstract object, in application, is not a function from centered worlds to truth-values, but a function (defined relative to a given context *c*) from *information states* to truth-values, defined (again, for arbitrary sentence Ï) as follows: Î»*s*[Ï]ă*c, w _{c}, s*ă. On the informational view, the role of the utterance is to put a constraint on the context set: it must be an information state

*s*that this function takes to the value 1 (true). On the informational view, there is no need to specify an initial value for the information state of the index of the context.

Yalcin remarks that if we add to the neutral diagonal view (which leaves it unspecified what the initial information state is), the proposal to identify the information state of the context as the context set of that context, then we âeffectively collapse it into the informational view.âŠ The two pragmatic views technically come to the same thing.â (p. 1012) Let me spell out the sense in which this is true:

For any K-context Î± and any sentence Ï, âÏ will be true in K-context Î± (according to the diagonal view) if and only if the context set of Î± meets the constraint, in context Î±, imposed on the informational view. Formally,

Yalcin expresses a preference for the informational view, but given this equivalence, why do we have to choose between the two views?

There remains a difference between the *neutral* diagonal view (that leaves open the question, what is the information state coordinate of the index of the K-context) and the informational view (that makes no use of an initialized information state), but the difference is that the diagonal view retains a flexibility that is not available to the informational view. That is, the informational view is equivalent to a special case of the diagonal view. If the special case is the only one with any application, then we can ignore the general case, and treat the two views as two ways of representing the same thing. But if there are cases where an epistemic âmightâ should be interpreted relative to an information state that is
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different from the context set, then there is an advantage to the formulation that allows the additional flexibility. The account of epistemic modals sketched in Chapters 6 and 7 suggest that in cases of disagreement, we need to allow for this divergence.

Let me elaborate on this last point. The equivalence of the special case of the diagonal view and informational view might be put this way: a modal sentence is *acceptable*, relative to a given K-context Î± if and only if it is true in that K-context (on the diagonal view) and if and only if the context set of Î± conforms to the constraint on information states (on the informational view). These two characterizations of acceptability are equivalent. But what happens if an epistemic modal claim is made in a (prior) context in which it is unacceptable? On either way of viewing the situation, there are two options: (1) the context will adjust so as to make the statement acceptable; or (2) the statement will be rejected. Case (2) has two subcases: (2a) the statement is then retracted; or (2b) the speaker sticks to her claim, while the interlocutor sticks to his rejection. (2b)âthe disagreement caseâis the case where (I argued in Chapter 7) the information state relevant to the interpretation of the epistemic modal comes apart from the common groundâthe kind of case for which we need the flexibility of the diagonal view.

I suspect that one reason for Yalcinâs resistance to the diagonal view is that by treating epistemic modal sentences exactly like ordinary assertions, it fails to explain the expressive character of epistemic modals. I agree that one needs to clarify the sense in which epistemic modal sentences are (often at least) used not to make claims about what the world is like, but to guide the direction of the discourse, and to express epistemic priorities. The account I sketched tries to explain the expressive character of epistemic modals and indicative conditionals in terms of the prospective character of the speech acts. But the application of this explanation of speech act force to cases of disagreement led us to make distinctions between the possibilities that may go beyond what there is a fact of the matter about. In a case where Alice accepts that it might be that Ï, while Bert rejects this possibility, we need a possibility in which she is right and he is wrong. I described this refinement of the points of evaluation relative to which the contents of speech acts are defined as a kind of projection of epistemic priorities onto a world. Sometimes this kind of projection is a way of getting richer resources for describing the facts, but sometimes the projection is temporary, and disappears in a retrospective context. That is, sometimes when we ask later about such a (p.229) disagreement, âWho turned out to be right?â the question has no answer. Yalcinâs semantics for epistemic modals provides a way to model the more fine-grained possibilities as possible-world/information-state pairs. I will say a little more about this below in the section on expressivist semantics.

A second reason, I think, for Yalcinâs preference for the informational view is that he is concerned with questions about logical structure and semantic consequence, and the informational view goes naturally with a conception of informational consequence that is useful for explaining some of the subtleties and puzzles about reasoning with epistemic modals, and other constructions that involve the kind of index shifting that epistemic modals illustrate. But just as the equivalence between the two views (in the special case of the diagonal view) means that we donât have to choose between them (at least in contexts that donât involve disagreement), it also means that we donât have to choose between a more classical notion of consequence and the notion of informational consequence that Yalcin defines. We can use both of them, and explore in rigorous detail the relations between them.

# 6 Expressivist semantics

The semantics for Lewisâs language game of command and permission involves a normative componentâa function whose value is a âsphere of permissibilityâ determining what the slave is obliged and permitted to do. As Lewis defines the game, the normative component is wholly determined by (and so supervenient on) the nonnormative facts about a possible world. Specifically, what is required and permitted at a given time in a given world is wholly determined by what utterances the master has produced at or before that time in that world. But one might have a more flexible semantics. After all, in these enlightened times, we might question whether the slave is really obliged to do what the master commands. (Imagine the subversive kibitzer saying to the slave, âYou know, you donât really have to do what the master tells you to do.â Even the conservative, who thinks the kibitzerâs subversive claim is false could allow that it is not a contradiction.) We might keep the game the same (with it being a rule of the game that the slave must do what the master commands), but generalize the semantics so that this rule is not a *semantic* rule, but a substantive normative claim (!the slave does
(p.230)
whatever the master commands). This more flexible semantics would be essentially the same as the semantics that Allan Gibbard proposed in his original presentation of a semantics for normative discourse.^{12} A system of norms, in Gibbardâs sense, can be defined as a function taking a world-time pair to a sphere of permissibility. The points of evaluation for the interpretation of the sentences of the mixed factual/normative language will be pairs consisting of a possible world and a system of norms. The âpropositionsâ expressed by the sentences of this language will be functions from points of evaluation of this kind to truth-values.

It is a substantive meta-normative question what it means for an agent to *accept* a set of norms, just as it is a substantive question in the philosophy of mind what it is to *believe* a proposition. But both factual propositions and systems of norms are abstract objects definable independently of any propositional attitudes, or any substantive account of what it is to have a cognitive, conative, or evaluative attitude.^{13}

Yalcin describes his account of epistemic modals as âexpressivistâ, and it is natural to see his semantics for epistemic modals as a variation on the kind of semantics that Gibbard proposed for normative discourse. As with Gibbardâs semantics, in Yalcinâs the points of evaluation will be pairs of a possible world and a nonfactual component, but in this case the nonfactual component is an information state. Just as Gibbardâs general meta-normative theory needs an account of what it is to accept a system of norms, so Yalcinâs needs an account of what it is to accept a âpropositionâ defined in terms of this kind of point of evaluation. But the compositional semantics does not depend on the way this substantive question is answered.

(p.231) As should be clear, I am sympathetic with both kinds of expressivist semantic frameworks, though I would emphasize, in both cases, that the sharp separation of the factual and normative components of a point of evaluation is a severe idealization. One benefit of doing semantics in terms of points of evaluation that mix factual and normative elements is that one can explain how discourse and inquiry can proceed without answering in advance questions about what there is a fact of the matter about.

# 7 Common ground contexts, basic and derived

I have made a number of distinctions between different kinds of common-ground contexts, basic and derived. This section aims just to summarize in one place these distinctions, and some of the formal properties of the different kinds of context sets.

In the formal representation of common ground developed in Chapters 2, 4 and 5, the elements are possible worlds centered on a sequence of individualsâthe participants in the communicative situation. The worlds are those compatible with the shared information of the participantsâwith their presumed common knowledge, including common knowledge of who and where they are. The centering represents the presumed shared ways that they all locate themselves and each other in the relevant possible worlds.

So the elements of a common-ground context are multiply-centered worlds (which I labeled s-worlds), but any given (uncentered) possible world occurs only once in such a set. So there is a one-one correspondence between the s-worlds that define a CG-context and a corresponding set of uncentered worlds. âThe context setâ sometimes refers to this corresponding set of uncentered worlds.

The common ground is a state defined in terms of a propositional attitude (acceptance for the purposes of the conversation) of the parties, but it is not itself any one personâs propositional attitude. What a party to a conversation *presupposes* is defined as what that party accepts to be common ground: in a conversation involving just Alice and Bert, Alice *presupposes* that Ï if and only if Alice accepts that Alice and Bert commonly accept that Ï. The iterative structure implies that if what Alice and Bert presuppose is the same, then the set of possibilities compatible with what each presupposes will be the same as the set that is
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compatible with what the other presupposes. Their presuppositions diverge if and only if at least one is mistaken about what is common ground. Since successful communication depends on agreement about shared information, we say that a context is *defective* if the presuppositions diverge.

Presupposition and common ground are not factive notions: even in nondefective contexts, what is presupposed may be factually false, which means that a (centered) possible world in which certain things are common ground may not itself be a possible world that is compatible with the common ground. Where pretense is involved, it may be known, even common knowledge, that some presuppositions are false. Even in nondefective contexts without any pretense or error where all parties know and accept that the common ground is what it in fact is, it may not be *common ground* that the common ground is what it in fact is.

In the idealized model, we assume that the basic attitude in terms of which common ground and speaker presupposition are defined (acceptance for the purposes of the conversation) is a *transparent* attitude, conforming to the principles of positive and negative introspection. (If Alice accepts that Ï, she accepts that she accepts it, and if she does not accept it, she accepts that she does not accept it.)^{14} Positive introspection carries over to the iterated attitudes: If it is common ground that Ï, then it is common ground that it is common ground that Ï. Negative introspection does not, however, carry over: Ï might fail to be common ground, even though one of the parties mistakenly thinks that it is. In this case, it will not be common ground that it is not common ground that Ï. The negative introspection principle for common ground may fail even in some nondefective contexts,^{15} although it is true that if an agent *presupposes* that the context is nondefective, then the negative introspection principle will hold for that agentâs presuppositions. (And if it is
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common ground that the context is nondefective, then the negative introspection principle will hold for the common ground.)

I defined various kinds of subordinate or derived contexts. While these are relevant to the interpretation of complex expressions of a language, and may be appealed to (as the basic context is) in the compositional rules of a language, the subordinate contexts are information states that are definable wholly in terms of features of the basic contextâof what is taken to be common ground. That is why they are appropriately called âderived contextsâ. First, there are derived contexts used in the interpretation of propositional attitude ascriptions. In a context in which what Charlie believes is at issue, it will be true in each of the possible worlds compatible with the basic context that Charlie is in a particular belief state, represented by a set of possible worlds, those compatible with what Charlie believes in that world. The derived context will be the union of all of those sets. That is, if the basic context set is C and R is the accessibility relation for Charlieâs belief states in a simple Hintikka-style âsemanticsâ^{16} for belief, then the derived context is defined as follows: {*y*: *x*R*y* for some *x* â C}.^{17} In doing the compositional semantics for an attitude ascription, one must determine a value (a proposition) denoted by the clause that is the object of belief. If context-sensitive expressions occur in the clause, one may appeal to information in either the basic context or the derived context to fix the content or reference of those expressions. Although this kind of derived context is an information state determined by the basic context that is available for the interpretation of clauses embedded in propositional attitude ascriptions, it is only part of the information about what is presupposed about the attitude state of the subject of the ascription. It might, for example, be presupposed that Ralph believes that a particular person is a spy without it being presupposed of any particular person that Ralph believes that he or she is a spy. That kind of information will not be reflected in a set of
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worlds that is the union of those that might, for all that is presupposed, be compatible with Ralphâs beliefs.

One special case of a propositional attitude context that got some attention, both in Chapter 4 and in Chapter 7, was a case where the attitude was *acceptance* (the same kind acceptance whose iterated extension is used to define the common ground) and the subject of the attitude was one of the parties to the conversation. In this case, the derived context, defined as above, will be a simple one, which is to say it will be a subset of the basic context set.

We also considered derived contexts determined by a supposition (either counterfactual or epistemic). In the general case, this kind of derived context is defined by an imaging function that takes a possible world from the basic context set to a possible world in which the content of the supposition clause is true. An imaging function of this kind must be determined by the basic context; that is, in order for suppositions to be interpretable, it must be presupposed in the basic context that a certain imaging function is uniquely salient, though it may be a âblurredâ imaging function (to use terminology David Lewis introduced), which means that it may be only partially defined. In the case of indicative suppositions, it is a constraint that from any base world within the basic context set, the image must also be from that set. The upshot is that the derived context set determined by an indicative supposition will be simple in the sense defined.

So there are two kinds of derived contexts whose context sets are subsets of the basic context set. I labeled these â*simple* derived contextsâ. Simple derived contexts contrast with others (labeled â*parallel* derived contextsâ) in that the full iterative structure, and the centering, is preserved. In the possible worlds in counterfactual context sets, and in the context sets relevant to propositional attitude ascriptions for agents that are not parties to the conversation, the utterance event whose content the semantics is aiming to determine is not necessarily taking place, and so these possible worlds are not components of a âcontext of utteranceâ. This means that the contextual features determined by a K-context (for example a speaker and time of utterance) are not determined. But in determining the proposition expressed in a subordinate clause, one may appeal to the information in either the basic context or the derived context. Suppose, for example, the consequent clause of a counterfactual contained the pronoun âIâ. There may be no speaker in the possible
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worlds in the parallel derived context, but we can and will take the indexical âIâ to refer to the speaker in the base world. That is, if world *y* is the image of world *x* (from the basic context), then the âIâ will denote, in *y*, the speaker in world *x*.

In Chapter 7, I introduced one further distinction between different basic common-ground contexts. Even a *nondefective* context may fail to be an *equilibrium* context. A context is in equilibrium when the parties agree, not only about what they accept in common, but also agree about what they *ought* to accept in commonâwhat they are in a position to accept. A context is in equilibrium when what is manifestly accepted by the different parties is the same. That is, a context is in equilibrium when a proposition is common ground if and only if it is common ground that each party accepts that proposition. This is the same as to say that the simple derived contexts, what it is common ground that X accepts for each X that is a party to the conversation will all be the same, and will coincide with the common ground.

## Notes:

(^{1})
I use a solid diamond, rather than an open one in order to contrast a metaphysical or causal modality with the epistemic âmightâ, represented âââ, which will come in for some discussion later.

(^{7})
The extension of the domain of possible worlds over which a proposition is defined beyond those in the context set (to possible worlds incompatible with what is being presupposed) is related to the notion Stephen Yablo develops of ânoncatastrophic presupposition failureâ (see Yablo 2006).

(^{11})
There is a problem for this suggestion, noted by Yalcin, when the world of the context is not itself compatible with the common ground (that is, when something false is being presupposed). We can set this problem aside by restricting the proposal to the case where the K-context is compatible with the common ground.

(^{13})
Mark Schroeder (in Schroeder 2008) takes it to be a commitment of an expressivist theory that the semantics must be done by associating sentences with attitudes, and then giving compositional rules for those attitude states. The problems for expressivism that he develops in the book all derive from this commitment. But I think this is not a commitment that an expressivist need or should accept. If one does the compositional semantics for abstract objects that are definable independently of any attitudes, and separates the semantics from substantive meta-normative and meta-semantic questions about what it is to accept or to communicate a proposition, in the extended sense of proposition defined in the semantics, then one avoids all of the problems that he explores in the book, problems that he argues cannot be given a satisfactory solution. I take the main lesson of Schroederâs book to be that one should not do expressivist semantics in this way.

(^{14})
This is of course an idealization, but even in the idealization, it does not imply that she can reflect on or articulate that she does or does not accept what she accepts. Higher order attitudes can be tacit and inarticulate, just as easily as ground level attitudes can be.

(^{15})
This last remark conflicts with a claim made in Stalnaker 2002: âIf a speaker believes the context is nondefective, then that speakerâs presuppositions will satisfy both negative and positive introspection. In terms of the possible worlds model, this is equivalent to the assumption that if world y is compatible with what the agent is presupposing in world *x*, then what the agent is presupposing in world y is the same as what she is presupposing in world *x*.â Thanks to Harvey Lederman, who constructed some models that are counterexamples to this claim. But I believe the weaker claim is sufficient for the points made in that paper.

(^{16})
âSemanticsâ is in scare quotes for the same reason as it was put in scare quotes in the heading of section 4 of this appendix: the Hintikka-style semantics is a semantics for a formal language used to represent relationships between propositional attitude states. It may also play a role in the compositional semantics for belief ascriptions in a natural language, but we are not appealing to semantics, in that sense, here.