Why is two-dimensional semantics important? One can think of it as the most recent act in a drama involving three of the central concepts of philosophy: meaning, reason, and modality. First, Kant linked reason and modality, by suggesting that what is necessary is knowable a priori, and vice versa. Second, Frege linked reason and meaning, by proposing an aspect of meaning (sense) that is constitutively tied to cognitive significance. Third, Carnap linked meaning and modality, by proposing an aspect of meaning (intension) that is constitutively tied to possibility and necessity.

Carnap's proposal was intended as something of a vindication of Frege's. Frege's notion of sense is somewhat obscure, but Carnap's notion of intension is more clearly defined. And given the Kantian connection between reason and modality, it follows that intensions have many of the properties of Fregean senses. In effect, Carnap's link between meaning and modality, in conjunction with Kant's link between modality and reason, could be seen as building a Fregean link between meaning and reason. The result was a golden triangle of constitutive connections between meaning, reason, and modality.

Some years later, Kripke severed the Kantian link between apriority and necessity, thus severing the link between reason and modality. Carnap's link between meaning and modality was left intact, but it no longer grounded a Fregean link between meaning and reason. In this way the golden triangle was broken: meaning and modality were dissociated from reason.

Two-dimensional semantics promises to restore the golden triangle. While acknowledging the aspects of meaning and modality that derive from Kripke, it promises to explicate further aspects of meaning and modality that are more closely tied to the rational domain. In particular it promises to look at the space of possibilities in a different way, and to erect a notion of meaning on that basis. In this way, we might once again have a grip on an aspect of meaning that is constitutively tied to reason.

To date, this restoration has been incomplete. Many different ways of understanding two-dimensional semantics have been proposed, and many of them restore the triangle at best partially. It is controversial whether two-dimensional semantics can be understood in such a way that the triangle is fully restored. To see this is possible, we need to investigate the foundations of two-dimensional semantics, and explore the many different ways in which the framework can be understood. I think that when the framework is understood in the right way, it can reinstantiate the links between meaning, reason, and modality.

We can begin with some more detailed background. If we squint at history from just the right angle, focusing on one strand of thought and setting aside others, we obtain a simplified rational reconstruction that brings out the key points.

It is useful to start with Frege. Frege held that an expression in a language typically has a referent — or what I will here call an extension. The extension of a singular term is an individual: for example, the extension of the name 'Hesperus' is the planet Venus, and the extension of the description 'the teacher of Aristotle' is Plato. The extension of a general term is a class. And the extension of a sentence is its truth-value.

Frege noted that the extension of an expression does not in general determine its cognitive significance: the role it plays in reasoning and in knowledge. For example, 'Hesperus' (the name used for the evening star) and 'Phosphorus' (the name used for the morning star) have the same referent but have different cognitive significance, as witnessed by the fact that 'Hesperus is Hesperus' is cognitively trivial, while 'Hesperus is Phosphorus' is nontrivial. The same goes for many other pairs of expressions: perhaps 'renate' (creature with a kidney) and 'cordate' (creature with a heart), or 'water' and 'H₂O', or 'I' (as used by me) and 'David Chalmers'. In each pair, the members are co-extensive (they have the same extension), but they are cognitively and rationally distinct.

Frege held that meaning is tied constitutively to cognitive significance, so that if two expressions have different cognitive significance, they have different meaning. It follows that there must be more to meaning than extension. Frege postulated a second aspect to meaning: sense. When two expressions are cognitively distinct, they have different senses. For example, the nontriviality of 'Hesperus is Phosphorus' entails that although 'Hesperus' and 'Phosphorus' have the same extension, they have a different sense. We can put the general idea as follows:

Fregean Thesis: Two expressions 'A' and 'B' have the same sense iff 'A == B' is cognitively insignificant.

Here, 'A == B' is a claim that is true if and only if 'A' and 'B' have the same extension. Where 'A' and 'B' are singular terms, this will be the identity 'A=B'; where 'A' and 'B' are sentences, this will be the material biconditional 'A iff B'; and so on. As for cognitive significance, we can say at a first approximation that a claim is cognitively insignificant when it can be known trivially by a rational being. As such, we can see this characterization of sense as providing a first bridge between meaning and reason.

The idea that expressions have senses is attractive, but senses are nevertheless elusive. What exactly is a sense? What exactly is cognitive significance? How does one analyze meanings beyond extensions? In the middle part of the 20th century, a number of philosophers, notably Carnap, had an insight. We can use the notions of possibility and necessity to help understand meaning, and in particular to help understand sense.

There are many possible ways the world might be; and we can use language to describe these possibilities. An expression can be applied to the actual state of the world, yielding an actual extension, or it can be applied to alternative possible states of the world, yielding alternative possible extensions. Take expressions such as 'renate' and 'cordate'. In the world as it actually is, all renates are cordates, so these terms have the same extension. But it is not necessary that all renates are cordates: if the world had been different, some renates might have failed to be cordates. Applied to such an alternative possibility, the two terms have a different extension. We can say: 'renate' and 'cordate' are co-extensive, but they are not necessarily co-extensive. Carnap suggested that we say two expressions have the same intension if and only if they are necessarily co-extensive.[*] So 'renate' and 'cordate' have the same extension, but different intension. We can put the general claim as follows:

Carnapian Thesis: 'A' and 'B' have the same intension iff 'A == B' is necessary.

*[[See Carnap 1947. The idea is also present in Lewis 1944.]]

What exactly is an intension? Carnap's characterization suggests a natural definition: an intension is a function from possibilities to extensions.[*] The possibilities here correspond to different possible states of the world. Relative to any possibility, an expression has an extension: for example, a sentence (e.g. 'All renates are cordates') can be true or false relative to a possibility, and a singular term (e.g. 'the teacher of Aristotle') picks out an individual relative to a possibility. An expression's intension is the function that maps a possibility to the expression's extension relative to that possibility. When two expressions are necessarily co-extensive, they will pick out the same extension relative to all possibilities, so they will have the same intension. When two expressions are not necessarily co-extensive, they will not pick out the same extension relative to all possibilities, so they will have different intensions. So intensions behave just as Carnap suggests they should.

*[[This definition of an intension is often attributed to Carnap, but in Carnap 1947 it plays at most a minor role. He proposes something like this (in section 40, p. 181) as a way of understanding individual concepts, which are the intensions of names, but then moves to a slightly different understanding. Earlier in the book, he characterizes necessity ("L-truth") in terms of state-descriptions, which are akin to possible worlds. But state-descriptions soon drop out of the discussion, so that intensions are treated in effect as something of a primitive semantic value. This sort of construction is also discussed in Carnap 1963, pp. 892-94. (Thanks to Wolfgang Schwartz for pointers here.) A proposal close to the definition above is present in C.I. Lewis's suggestion that an intension "comprises whatever must be true of any possible world in order that the proposition should apply to it or be true of it" (Lewis 1944).]]

Seen this way, the notion of an intension provides a bridge between meaning and modality. Just as a sense can be seen as a sort of meaning that is constitutively tied to reason, an intension can be seen as a sort of meaning that is constitutively tied to modality. Furthermore, intensions seem to behave very much as senses are supposed to behave. Just as two expressions can have the same extension but different senses, two expressions can have the same extension but different intensions. And just as sense was supposed to determine extension, intension seems to determine extension, at least relative to a world.

One can make a direct connection by adding an additional claim connecting modality and reason. It has often been held that a proposition is necessary if and only if it is a priori (knowable independently of experience) or trivial (yields no substantive knowledge of the world). The notions of apriority and triviality are essentially rational notions, defined in epistemic terms. Carnap himself held a version of the thesis involving triviality, but it is more useful for our purposes to focus on the version involving apriority. In this form, the relevant thesis goes back at least to Kant, so we can call it:

Kantian Thesis: A sentence S is necessary iff S is a priori.

If we combine the Carnapian Thesis with the Kantian Thesis, we obtain the following:

Neo-Fregean Thesis: Two expressions 'A' and 'B' have the same intension iff 'A == B' is a priori.

If this claim is accepted, then one has recaptured something that is at least close to the Fregean thesis. For apriority is at least closely related to cognitive insignificance. When a proposition is cognitively insignificant, it is plausibly a priori. The reverse is not the case, on Frege's understanding of cognitive significance: many logical and mathematical propositions are cognitively significant, even though they are a priori. But in any case, apriority and cognitive insignificance are at least closely related rational notions. Typical cognitively significant identities, such as 'Hesperus is Phosphorus', 'Water is H₂O', and 'I am David Chalmers' are all a posteriori. If the Neo-Fregean Thesis is correct, it follows that 'Hesperus' and 'Phosphorus' have different intensions, as do 'water' and 'H₂O', and 'I' and 'David Chalmers'. So intensions behave quite like Fregean senses.

In effect, modality serves as a bridge in explicating the tie between meaning and reason. One constructs a notion of meaning using modal notions, combines this with the claim that modality is constitutively tied to reason, and ends with a link between all three. The central connection between meaning, reason, and modality is captured within the Neo-Fregean thesis: intension is a notion of meaning, defined in terms of modality, that is constitutively connected to reason.

This golden triangle was shattered by Kripke, who cut the connection between reason and modality. Kripke argued that the Kantian Thesis is false: there are many sentences that are necessarily true but whose truth is not knowable a priori. For example, Kripke argued that given that Hesperus is actually Phosphorus, it could not have been that Hesperus was not Phosphorus: Hesperus is necessarily the planet Venus, and so is Phosphorus. So although 'Hesperus is Phosphorus' is not knowable a priori, it is nevertheless necessary. More generally, Kripke argued that names and natural kind terms are rigid designators, picking out the same extension in all possible worlds. It follows that any true identity involving such terms is necessary. For example, 'Water is H₂O' is necessary, even though it is a posteriori. The same goes for claims involving indexicals: 'I am David Chalmers' (as used by me) is another a posteriori necessity.

If Kripke is right about the Kantian Thesis, then the Neo-Fregean Thesis is also false. Since 'Hesperus is Phosphorus' is necessary, 'Hesperus' and 'Phosphorus' have the same intension, picking out the planet Venus in all possibilities. But the equivalence between 'Hesperus' and 'Phosphorus' is nevertheless a posteriori and cognitively significant. So cognitively and rationally distinct pairs of expressions can have the same intension: witness 'Hesperus' and 'Phosphorus', 'water' and 'H₂O', 'I' and 'David Chalmers'. So the Neo-Fregean Thesis fails, and intensions no longer behave like Fregean senses.

In effect, Kripke leaves intact the Carnapian link between meaning and modality, but in severing the Kantian link between reason and modality, he also severs the Fregean link between meaning and reason. This is roughly the received view in contemporary analytic philosophy: meaning and modality are connected, but both are disconnected from reason.

Although most contemporary analytic philosophers accept Kripke's arguments against the Kantian thesis, many would still like to hold that Frege was right about something. There remains an intuition that 'Hesperus' and 'Phosphorus' (or 'water' and 'H₂O', or 'I' and 'David Chalmers') differ in at least some dimension of their meaning, corresponding to the difference in their cognitive and rational roles. One might try to do this by breaking the Carnapian connection between meaning and modality. Two-dimensional semantics takes another strategy: in effect, it finds another way of looking at modality that yields a Fregean aspect of meaning.

The core idea of two-dimensional semantics is that there are two different ways in which the extension of an expression depends on possible states of the world. First, the actual extension of an expression depends on the character of the actual world in which an expression is uttered. Second, the counterfactual extension of an expression depends on the character of the counterfactual world in which the expression is evaluated. Corresponding to these two sorts of dependence, expressions correspondingly have two sorts of intensions, associating possible states of the world with extensions in different ways. On the two-dimensional framework, these two intensions can be seen as capturing two dimensions of meaning.

These two intensions correspond to two different ways of thinking of possibilities. In the first case, one thinks of a possibility as representing a way the actual world might turn out to be: or as it is sometimes put, one considers a possibility as actual. In the second case, one acknowledges that the actual world is fixed, and thinks of a possibility as a way the world might have been but is not: or as it is sometimes put, one considers a possibility as counterfactual. When one evaluates an expression relative to a possible world, one may get different results, depending on whether one considers the possible world as actual or as counterfactual.

The second way of thinking about possibilities is the more familiar in contemporary philosophy. Kripke's arguments rely on viewing possibilities in this way. Take a possibility in which the bright object in the evening sky is a satellite around the earth, and in which Venus is visible and bright only in the morning. When we think of this possibility as a counterfactual way things might have been, we do not describe it as a possibility in which Hesperus is Mars, but as one in which Hesperus (and Phosphorus) is invisible in the evening. So relative to this possibility considered as counterfactual, 'Hesperus' picks out Venus. Correspondingly, the second-dimensional intensions of both 'Hesperus' and 'Phosphorus' both pick out Venus in this possibility, and in all possibilities in which Venus exists. It is this familiar sort of intension that yields the Kripkean gap between intension and cognitive significance.

The first way of thinking about possibilities is the less familiar in contemporary philosophy. If we take the possibility described above, and think of it as a way the world might actually be, we can say: if the world really is that way, then 'Hesperus' picks out a satellite. So relative to this possibility considered as actual, 'Hesperus' picks out not Venus but the satellite. Correspondingly, the first-dimensional intension of 'Hesperus' picks out the satellite in this possibility, while that of 'Phosphorus' picks out Venus. So 'Hesperus' and 'Phosphorus' have different first-dimensional intensions. This difference is tied to the fact that the actual-world reference of 'Hesperus' and 'Phosphorus' is fixed in quite different ways, although as things turn out, their referents coincide. Because of this, it seems that the first dimension may be better suited than the second for a link to reason and to cognitive significance.

The possibilities evaluated in the second dimension are usually thought of as possible worlds. The possibilities evaluated in the first dimension are a little different, as they reflect the nature of a world from the point of view of a speaker using an expression within a world. It is useful for many purposes to see these possibilities as centered worlds: worlds marked with a "center", which is an ordered pair of an individual and a time. We can think of the center of the world as representing the perspective of the speaker within the world.

I have been deliberately vague about just how the relevant intensions are to be defined, since as we will see, there are many different ways to define them. Because of this, giving detailed examples is tricky, because different frameworks treat cases differently. Nevertheless, it is useful to go through some examples, giving an intuitive analysis of the results that two-dimensional semantics might be expected to give if it is to yield something like a Fregean sense in the first dimension. We will later see how this can be cashed out in detail. For now, I will use "1-intension" as a generic name for a first-dimensional intension, and "2-intension" as a generic name for a second-dimensional intension.

First, 'Hesperus is Phosphorus'. In a centered world considered as actual, this is true roughly when the morning star visible from the center of that world is the same as the evening star. In a world considered as counterfactual, it is true when Venus is Venus. 'Hesperus' functions roughly to pick out the evening star in the actual world, so the 1-intension of 'Hesperus' picks out the evening star in a given centered world. Likewise, the 1-intension of 'Phosphorus' picks out the morning star in a centered world. Both of these terms behave rigidly in counterfactual evaluation, so their 2-intensions pick out their actual referents in all worlds. So the 2-intensions of both 'Hesperus' and 'Phosphorus' pick out Venus in all worlds.

Second, 'Water is H₂O'. In a centered world considered as actual, this is true roughly when the clear, drinkable liquid around the center of that world has a certain pattern of chemical structure. In a world considered as counterfactual, it is true when H₂O is H₂O. The reference of 'water' is fixed roughly by picking out the substance with certain superficial properties and a certain connection to the speaker in the actual world, so its 1-intension picks out roughly the substance with those properties connected to the center of a given world. Similarly, the 1-intension of 'H₂O' picks out the substance with the right sort of chemical structure in a centered world. As in the first case, both expressions behave rigidly in counterfactual evaluation, so their 2-intensions pick out H₂O in all worlds.

Third, 'I am a philosopher'. In a centered world considered as actual, this sentence is true when the being at the center of the world is a philosopher. In a world considered as counterfactual, this sentence (or at least my utterance of it) is true if David Chalmers is a philosopher in that world. The actual-world reference of 'I' is fixed by picking out the subject who utters the token; so the 1-intension of 'I' picks out the subject at the center of a given world. 'I' behaves as a rigid designator in counterfactual evaluation, so its 2-intension picks out the actual referent (in this case, David Chalmers) in all possible worlds. 'Philosopher', by contrast, is a broadly descriptive term: both its 1-intension and its 2-intension function to pick out beings with certain characteristic attributes.

Certain patterns seem to emerge. The first two sentences are necessary (at least if Kripke is right), and both of them have a 2-intension that is true in all worlds. The third sentence is contingent, and its 2-intension is false in some worlds. So it seems that a sentence is necessary precisely when it has a necessary 2-intension. This corresponds directly to the Carnapian thesis: 2-intensions, in effect, are defined so that two expressions will have the same 2-intensions when they are necessarily equivalent.

On the other hand, all three of these sentences are a posteriori, and all of them appear to have a 1-intension that is false in some centered worlds. At the same time, a priori sentences such as (perhaps) 'All bachelors are unmarried males' or 'Hesperus (if it exists) has been visible in the evening' can plausibly be seen as having a 1-intension that is true in all centered worlds. So it is at least tempting to say that a sentence is a priori precisely when it has a necessary 1-intension. This corresponds to the neo-Fregean thesis: one might naturally suggest that two expressions have the same 1-intension precisely when they are a priori equivalent. To illustrate, one can note that the difference in the 1-intensions of 'Hesperus' and 'Phosphorus', or of 'water' and 'H₂O', seems to be closely tied to their a priori inequivalence. All this needs to be analyzed in more depth, but one might at least characterize the general sort of behavior suggested in the examples above, where differences in 1-intensions go along at least roughly with differences in cognitive significance, as quasi-Fregean.

Along with the 1-intension and the 2-intension of a given expression, one can also define a two-dimensional intension. In many cases, just as an expression's extension depends on how the actual world turns out, an expression's 2-intension depends on how the actual world turns out. The expression's two-dimensional intension captures this dependence: it can be seen as a function from centered worlds to 2-intensions, or equivalently as a function from pairs of centered worlds and worlds to truth-values. In the case of 'Hesperus', for example, the two-dimensional intension maps a centered world V to the 2-intension that picks out V's evening star (if it exists) in any worlds W. The actual 2-intension of an expression corresponds to the two-dimensional intension evaluated at the actual centered world of the speaker: given that Venus is the actual world's evening star, the 2-intension of 'Hesperus' picks out Venus in all worlds. The 1-intension of an expression can be reconstructed by "diagonalizing" the two-dimensional intension: one evaluates the two-dimensional intension at a centered world W, yielding a 2-intension, and then one evaluates this 2-intension at the same world (stripped of its center). One might think of the two-dimensional intension as representing the way that an expression can be used to evaluate counterfactual worlds, depending on which world turns out to be actual.

I will return to these themes later, but for now it must be acknowledged that the situation is much more complicated than I have made things sound. A number of different two-dimensional systems have been introduced, and many of these give different results. A partial list of proponents of these systems, along with the names they give to their two-dimensional notions, includes:

Kaplan (1978; 1989): character and content
Stalnaker (1978): diagonal proposition and propositional expressed
Evans (1979): deep necessity and superficial necessity
Davies/Humberstone (1981): "fixedly actual" truth and necessary truth
Chalmers (1996): primary intension and secondary intension
Jackson (1998): A-intension and C-intension

There are many differences between these systems, some on the surface, and some quite deep. Surface differences include the fact that where Chalmers and Jackson speak of two sorts of intensions, Evans and Davies and Humberstone speak of two sorts of necessity, while Kaplan and Stalnaker speak of propositions. This sort of difference is mostly intertranslatable. Given a notion of necessity and a corresponding way of evaluating possibilities (as with Evans and Davies and Humberstone), one can define a corresponding sort of intension, and vice versa. Stalnaker's propositional content is just a set of possible worlds, which is equivalent to the intension of a sentence, and Kaplan's content is closely related.[*] Kaplan's and Stalnaker's first-dimensional notions are defined over contexts (which are at least closely related to centered worlds), and initially involve a two-dimensional intension: a function from contexts to 2-intensions. Stalnaker diagonalizes this function, yielding a function from contexts to truth-values, or a 1-intension. Kaplan leaves his character as a two-dimensional function from contexts to 2-intensions, but a corresponding step could straightforwardly be taken. So in all these cases, there is a similar formal structure.

*[[Kaplan's content is strictly speaking a singular proposition rather than a set of worlds, but it immediately determines a set of worlds. For our purposes, the difference between singular propositions, other structured propositions, and sets of worlds in analyzing the second dimension of content will not be crucial, so for simplicity I will speak as if the relevant second-dimensional contents are intensions. The discussion can be straightforwardly adapted to other views.]]

At a conceptual level, the systems have something further in common. In each case, the first-dimensional notion is put forward at least in part as a way of better capturing the cognitive or rational significance of an expression than the second dimension. And in each case, at least some sort of link between the first-dimensional notion and apriority has been claimed. In Kaplan's and Stalnaker's original publications, it is held that character and diagonal propositions closely reflect matters of apriority, at least in some cases. For Evans and Davies and Humberstone, when a statement of a certain sort is knowable a priori, it is deeply necessary, or true fixedly actually. And for Chalmers and Jackson, whenever a sentence is a priori, it has a necessary primary intension or 1-intension.

But these similarities mask deep underlying conceptual differences. These systems are defined in quite different ways, and apply to quite different items of language, yielding quite different results. Correspondingly, proponents of these systems differ greatly in the scope and strength of their claims. Kaplan's analysis is restricted to just a few linguistic expressions: indexicals and demonstratives. He explicitly resists an extension of his system to other expressions, such as names and natural kind terms. Evans and Davies and Humberstone develop their analysis for a different narrow class of expressions: descriptive names, and perhaps (in the case of Davies and Humberstone) some natural kind terms. Stalnaker's analysis applies in principle to any sentence, but in more recent work, he has explicitly disavowed any strong connection with apriority, and has been skeptical about applications of two-dimensional semantics in that direction. By contrast, Chalmers and Jackson suggest that their notions are defined for a very wide class of expressions, and make strong claims about the connection between these notions and apriority. (The current paper might be viewed in part as a defense of these strong claims.)

These differences arise from different interpretations of the formal two-dimensional framework. The framework of worlds and intensions, taken alone, is simply an abstract structure in need of content. Different interpretations flesh out this content in different ways. The interpretations are not necessarily incompatible, although it is possible that some are ill-defined, or rest on false presuppositions. The relations between these interpretations, however, are not well-understood.

The main project of this paper is to explore the different ways in which a two-dimensional framework can be understood. What are the fundamental concepts underlying different interpretations of the framework? How are these related? How do the differences between these interpretations explain the differences in the scope and strength of the claims that are made for them? Which interpretations of the framework yield the strongest connections between the first dimension and the rational domain?

The central question on which I will focus is the following. Is there an interpretation of the two-dimensional framework that yields constitutive connections between meaning, reason, and modality? That is, is there an interpretation on which the first dimension is tied universally to the rational domain? On this way of thinking, the ideal form of the two-dimensional framework will recapture something like the neo-Fregean thesis: two terms will have the same 1-intension if and only if they are equivalent a priori. To get at this question, we can focus on the following core thesis:

Core Thesis: For any sentence S, S is a priori iff S has a necessary 1-intension.

Here, S should be understood as a sentence token (such as an utterance) rather than a sentence type, to accommodate the possibility that different tokens of the same expression type may have different 1-intensions. Correspondingly, we should understand apriority as a property of sentence tokens. I will say more about the relevant notion of apriority and the type/token distinction in section 3.8. But for now, to a first approximation, we can say that a sentence token S is a priori when S expresses a thought that can be justified independently of experience. And I will take it that the intuitive judgments about apriority above are correct: a typical utterance of 'Hesperus is Phosphorus' is not a priori, in this sense, while a typical utterance of 'All bachelors are unmarried' is a priori in this sense.

The Core Thesis links the rational notion of apriority, the modal notion of necessity, and the semantic notion of intension. If the Core Thesis is true, it restores a golden triangle of connections between meaning, reason, and possibility. It also immediately entails a version of the Neo-Fregean Thesis (given plausible principles about compositionality).

Neo-Fregean Thesis (2D Version): Two expressions 'A' and 'B' have the same 1-intension iff 'A == B' is a priori.

If the two-dimensional framework can be understood in such a way that the Core Thesis is true, it promises an account of a broadly Fregean aspect of meaning, tied constitutively to the epistemic domain. It also promises further rewards: perhaps an account of the contents of thought on which content is tied deeply to a thought's rational role (potentially yielding an account of so-called "narrow content" and "modes of presentation" in thought), and perhaps a view of modality on which there are deep links between the rational and modal domains (potentially grounding a connection between notions of conceivability and possibility). So the key question in what follows will be: can we define 1-intensions so that the Core Thesis is true?

To anticipate, my answer will be as follows. There are two quite different ways of understanding the two-dimensional framework: the contextual understanding and the epistemic understanding. The contextual understanding uses the first dimension to capture context-dependence. The epistemic understanding uses the first dimension to capture epistemic dependence. The contextual understanding is more familiar, but it cannot satisfy the Core Thesis. The epistemic understanding is less familiar, but it can satisfy the Core Thesis. The reason is that only on the epistemic understanding is the first dimension constitutively tied to the epistemic domain.

Within each of these general understandings of the framework, there are various possible specific interpretations. In what follows, I will first explore contextual interpretations (section 2), and then epistemic interpretations (section 3). Some of these interpretations are closely related to existing proposals, but rather than working directly with existing proposals, I will characterize these interpretations from first principles. This allows us to examine the properties of these interpretations in a clear light, free of problems of textual exegesis. Later in the paper, I will examine how existing proposals fit into this scheme.

A methodological note: in this paper I will adopt the approach of semantic pluralism, according to which expressions can be associated with semantic values in many different ways. Expression types and expression tokens can be associated (via different semantic relations) with extensions, various different sorts of intensions, and with many other entities (structured propositions, conventionally implied contents, and so on). On this approach, there is no claim that any given semantic value exhausts the meaning of an expression, and I will not claim that the semantic values that I focus on are exhaustive. (I think that such claims are almost always implausible.)

Likewise, this approach gives little weight to disputes over whether a given (purported) semantic value is "the" meaning of an expression, or even whether it is truly a "semantic" value at all. Such disputes will be largely terminological, depending on the criteria one takes to be crucial in one's prior notion of "meaning" or "semantics". On the pluralist approach, the substantive questions are: can expressions (whether types or tokens) be associated with values that have such-and-such properties? If so, what is the nature of the association and of the values? What aspects of language and thought can this association help us to analyze and explain?

My focus in this paper will be almost wholly on whether there is an association between expression tokens and 1-intensions that satisfies the Core Thesis, and on how this association can be understood. I will not say much more about the motivations for this sort of approach, about the broader shape of the resulting semantic theory, or about applications. Motivation and broader questions are discussed in "On Sense and Intension" (Chalmers 2002b), which gives a gentler introduction to these issues. Applications are discussed in "The Components of Content" (Chalmers 2002c) and in "Does Conceivability Entail Possibility" (Chalmers 2002a).

On the contextual understanding of two-dimensional semantics, the possibilities involved in the first dimension represent possible contexts of utterance, and the intension involved in the first dependence represents the context-dependence of an expression's extension. There are many ways in which the extension of an expression can depend on the context in which it is uttered. On the contextual understanding, a 1-intension captures the way in which an expression's extension depends on its context. As we will see, this sort of context-dependence can itself be understood in a number of different ways.

To formalize this, we can start by focusing on expression tokens: spoken or written tokens of words, sentences, and other expressions. We can take it that any expression token has an extension. In cases where a token "aims" to have an extension but fails, as with an empty name, we can say that it has a null extension. If there are some expression tokens that do not even aim to have an extension (as perhaps with some exclamations), they are outside the scope of our discussion. A token of a sentence corresponds to an utterance; its extension is a truth-value.

Any expression token falls under a number of different expression types. A token may fall under an orthographic type (corresponding to its form), a semantic type (corresponding to its meaning), a linguistic type (corresponding to its identity within a language), and various other types. Different tokens of the same expression type will often have different extensions. When two tokens of the same expression type have different extensions, this reflects a difference in the context in which the tokens are embedded.

For our purposes, contexts can be modeled as centered worlds. The context in which an expression token is uttered will be a centered world containing the token. This can be modeled as a world centered on the speaker making the utterance, at the time of utterance. It is also possibly to model a context by a different sort of centered world with just an expression token marked at the center. The previous version will work for most purposes, however, as long as we assume that a subject makes at most one utterance at a given time.

One can now define the contextual intension of an expression type. This is a function from centered worlds to extensions. It is defined at worlds centered on a subject uttering a token of the expression type. At such a world, the contextual intension returns the extension of the expression token at the center.

One can also define the contextual intension of an expression token, relative to a type of which it is a token. This is also a function from centered worlds to extensions. It is defined at worlds centered on a token of the same type, and returns the extension of the token at the center. This contextual intension is the same as the contextual intension of the relevant expression type.

The first-dimensional intensions in the two-dimensional framework are often understood as contextual intensions of some sort. On this way of seeing things, a 1-intension mirrors the evaluation of certain metalinguistic subjunctive conditionals: if a token of the relevant type were uttered in the relevant context, what would its extension be? Of course, for every different way of classing expression tokens under types, there will be a different sort of contextual intension. In what follows I examine some of the relevant varieties of contextual intension.[*]

*[[Constructs akin to contextual intensions have been stressed by Robert Stalnaker in a number of writings (e.g. Stalnaker 1978, 1999). At the same time, Stalnaker and Ned Block have both been active critics of the overextension of this framework (e.g. Stalnaker 1990, 2001; Block 1991; Block and Stalnaker 1999). The discussion in this section owes a significant debt to Stalnaker and Block. Although I carve up the territory in a different way, a number of the varieties of contextual intension that I mention are touched on explicitly or implicitly by Stalnaker and Block at various points, and some of my critical points echo points made by them in criticizing certain applications of the two-dimensional framework.]]

We can say that two tokens are tokens of the same orthographic type when they have the same orthography. This holds roughly when they are made up of the same letters or sounds, regardless of their meaning, and regardless of the language in which they are uttered. The exact details of what counts as the same orthography can be understood in different ways, but these differences will not matter for our purposes.

The orthographic contextual intension of an expression token T is defined at centered worlds with a token of T's orthographic type at the center. It maps such a world to the extension of the relevant token in that world.

(The orthographic contextual intension of a sentence token is closely related to its diagonal proposition, as defined by Stalnaker (1978). I will return to this matter later.)

As an example, let S be Oscar's utterance of 'Water is H₂O'. Let W₁ be Oscar's world (Earth), centered on Oscar making this utterance. Oscar's utterance is true, so S's orthographic contextual intension is true at W₁. Let W₂ be a universe containing Twin Earth (where everything is just as on earth except that the watery liquid is XYZ), centered on Twin Oscar uttering 'Water is H₂O'. Twin Oscar's utterance is false (his word 'water' refers to XYZ), so S's orthographic contextual intension is false at W₂. Let W₃ be a universe containing Steel Earth, where the word 'water' refers to steel but chemical terms are the same, centered on Steel Oscar uttering 'Water is H₂O'. Steel Oscar's utterance is false, so S's orthographic contextual intension is false at W₃.

It is clear that orthographic contextual intensions do not satisfy the Core Thesis. For every orthographic type, some possible token of that type expresses a falsehood. For example, there are worlds in which the string 'bachelors are unmarried' means that horses are cows. In such a centered world, the orthographic contextual intension of 'bachelors are unmarried' is false. The same goes for any sentence. So no truth has a necessary contextual intension, and in particular no a priori truth has a necessary contextual intension. So if 1-intensions are understood as orthographic contextual intensions, the Core Thesis is obviously false.

We can say that two expression tokens are tokens of the same linguistic type when they are tokens of the same expression in a language. This assumes that expression tokens belong to languages, and that languages involve expressions such as words, phrases, and sentences. So any two tokens of the English word 'water' share a linguistic type, as do any two utterances of the French sentence 'C'est la vie'.

The linguistic contextual intension of an expression token T is defined at centered worlds with a token of T's linguistic type at the center. It maps such a world to the extension of the relevant token in that world.

(The linguistic contextual intension of an expression is in some respects like its character, as defined by Kaplan. I will return to this matter later.)

As before, let S be Oscar's utterance of 'Water is H₂O'. If W₁ is Oscar's own centered world (Earth): S's linguistic contextual intension is true at W₁. If W₂ is Twin Oscar's centered world (Twin Earth): it is arguable that Twin Oscar's word 'water' is a different word from Oscar's word 'water'. Certainly if the referent of 'water' is essential to the word, as many theorists hold, then Twin Oscar's 'water' is a different word. If so, S's linguistic contextual intension is not defined at W₂. If W₃ is Steel Oscar's centered world (where 'water' means steel): here it is reasonably clear that Steel Oscar's 'water' is a different word that has the same orthography. If so, S's linguistic contextual intension is not defined at W₃. Applying this sort of reasoning, one reaches the conclusion that S's contextual intension is true at every world in which at which it is defined, since the English word 'water' refers to H₂O in every world in which it exists, and so does the English expression 'H₂O'.

If this is right, then linguistic contextual intensions do not satisfy the core thesis. 'Water is H₂O' is a posteriori, but it seems to have a necessary contextual intension, true at every world at which it is defined. The same goes even more clearly for sentences involving names, such as 'Cicero is Tully'. It is widely held that names have their referents essentially; if so, the linguistic contextual intensions of true identities of this sort will be true at all worlds at which they are defined. As such, linguistic contextual intensions do not behave at all like Fregean senses. If 1-intensions are understood as linguistic contextual intensions, the Core Thesis is false.

There are some expressions for which linguistic contextual intensions behave more like Fregean senses. One such is 'I': setting certain odd cases aside, any token of the English word 'I' picks out the utterer of that token. So the linguistic contextual intension of 'I' picks out the speaker at the center of any centered world at which it is defined. In this way, it behaves much as we earlier suggested the 1-intension of 'I' should behave. Something similar applies to other indexicals, such as 'today', and to some broadly descriptive terms, such as 'philosopher'. It is in the case of names and natural kind terms that the fit seems to be worst.

We can say that two expression tokens are tokens of the same semantic type when they have the same semantic value. An expression token's semantic value is its meaning or content, or some aspect of its meaning or content. There are many different ways of assigning semantic values to expression tokens, so there are correspondingly many different ways of classing expression tokens under semantic types.

The semantic contextual intension of an expression token T is defined at centered worlds with a token of T's linguistic type at the center. It maps such a world to the extension of the relevant token in that world.

As before, let S be Oscar's utterance of 'Water is H₂O'. If W₁ is Oscar's own centered world (Earth): S's semantic contextual intension is true at W₁. If W₂ is Twin Oscar's centered world (Twin Earth): at least on many ways of assigning semantic value, Twin Oscar's term 'water' has a different semantic value from Oscar's, so S's semantic contextual intension (for this sort of semantic type) is undefined at W₂. If W₃ is Steel Oscar's centered world, then Steel Oscar's term 'water' clearly has a different semantic value from Oscar's, so S's semantic contextual intension is undefined at W₃. If W₄ is a world centered on French Oscar, a counterpart of Oscar who speaks French and is uttering 'eau est H₂O': then it is plausible that this utterance has the same semantic value as Oscar's, so S's semantic contextual intension is defined at W₁ and is true there.

Of course the behavior of a semantic contextual intension will depend on our choice of semantic value. For example, if we stipulate that the relevant semantic value of an expression is its extension, then any two co-extensive expressions will have the same semantic contextual intension, and there is no chance that the Core Thesis will be true. There are two choices of semantic value that are somewhat more interesting, however.

We might stipulate that the relevant semantic value of an expression is its standing meaning: roughly, the aspect of meaning that is common to all tokens of the expression's linguistic type. If we do this, then an expression's semantic contextual intension will be an extension of its linguistic contextual intension to a broader space of worlds. At worlds centered on a token of the same linguistic type, the intensions will give the same results. But the semantic contextual intensions will also be defined at other worlds, centered on synonyms and translations of the original expression. Nevertheless, if 1-intensions are understood as these semantic contextual intensions, the Core Thesis will be false for the same reasons as in the case of linguistic contextual intension. For example, if the extension of 'water' is essential to the word, then it is part of the word's standing meaning. So the semantic contextual of 'Water is H₂O' will be true at every world where it is defined, and the Core Thesis is false.

Alternatively, we might stipulate that the relevant semantic value of an expression token is its Fregean or descriptive content, corresponding roughly to the expression's cognitive significance for the subject. On this reading, the Core Thesis may be more plausible. For example, one might argue that Oscar's and Twin Oscar's terms 'water' have the same descriptive content. If so, then the semantic contextual intension of Oscar's utterance 'Water is H₂O' is defined at W₂ and is false there. On the other hand, Steel Oscar's term 'water' plausibly has a different descriptive content, so the semantic contextual intension of Oscar's utterance is not defined at W₃.

Understood this way, semantic contextual intensions behave as we might expect a Fregean 1-intension to behave, at least to some extent. One can argue that when a statement is a priori, any possible statement with the same descriptive content will be a priori and so will be true, so that the expressions semantic contextual intension will be necessary, as the Core Thesis requires. Correspondingly, one might suggest that when a statement is not a priori, then there will be possible statements with the same descriptive content that are false, so that the statement's semantic contextual intension will not be necessary, as the Core Thesis requires.

I will argue shortly that this is not quite right. But even if it were right, it is clear that this sort of 1-intension cannot underwrite the full ambitions of the Fregean two-dimensionalists. The Fregean two-dimensionalist, as sketched previously, intends to use the two-dimensional framework to ground an aspect of meaning that is constitutively tied to meaning. But semantic contextual intensions as defined here presuppose such a Fregean semantic value, and so cannot independently ground such an account. If this is the best a two-dimensionalist can do, then if someone is independently doubtful about a Fregean aspect of meaning, two-dimensionalism cannot help. At best, two-dimensionalism will be a helpful tool in analyzing such a notion of meaning, given an independent grounding for the notion.[*]

*[[This sort of point in made quite clearly, in the context of discussing narrow content, by Stalnaker 1991, Block 1991, and Block and Stalnaker 1999.]]

We have seen that orthographic contextual intensions are far from satisfying the Core Thesis, while linguistic contextual intensions are closer at least in some cases, and some sort of semantic contextual intensions may be closer still. But there is a further problem that arises for any sort of linguistic or semantic contextual intension, suggesting that no such contextual intension can satisfy the Core Thesis.

Let S be a token of 'A sentence token exists' (where a sentence token is understood to be a concrete entity produced by speech, writing, or a similar process). Then S is true. Furthermore, any token of the linguistic item 'A sentence token exists' is true. Any token that means the same thing as 'A sentence token exists' is true. So it seems that S will have a necessary linguistic contextual intension, and a necessary semantic contextual intension, under any reasonable way of classifying linguistic and semantic types. But S is clearly a posteriori: it expresses empirical knowledge of the world, which could not be justified independently of experience. So S is a counterexample to the Core Thesis. So the Core Thesis is false for any sort of semantic or linguistic contextual intension.

The same goes for a number of other sentences. If S₁ is 'Language exists' (where a language is understood to be a spoken or written language, not just an abstract language), then any utterance of the same expression or with the same meaning will be true. So S₁ has a necessary linguistic and contextual intension, despite being a posteriori. If S₂ is 'I exist', then any utterance of the same expression with the same meaning will be true, so S₂ has a necessary linguistic and semantic contextual intension. But (somewhat controversially) S₂ is a posteriori, justifiable only on the basis of experience. If S3 is 'I am uttering now', then any utterance of the same expression or with the same meaning will be true. S3 is clearly a posteriori, but has a necessary linguistic and semantic contextual intension.

All these cases are counterexamples to the Core Thesis. All of them are a posteriori and cognitively significant, and many of them seem to be as cognitively significant as paradigmatic expressions of empirical knowledge. But all have necessary semantic and linguistic contextual intensions. So the Core Thesis is false for all such intensions.

The trouble is that apriority and being true whenever uttered are fundamentally different notions. The first builds in an epistemic or rational element, but the second builds in no such element. The second build in a metalinguistic element, but the first builds in no such element. It is possible to understand the second in a way that makes it coincide with the first in many cases, in effect by building in an epistemic element into the individuation of the relevant linguistic types. But it is impossible to do so in all such cases, since the second has an ineliminable metalinguistic element that goes beyond the epistemic domain.

I think the moral is that to satisfy the Core Thesis, we must understand the two-dimensional framework in a quite different, non-contextual way. But before doing so, I will more briefly examine some further ways in which one might define a contextual intension.

Given orthographic, linguistic, and semantic types for expression tokens, it is possible to define hybrid types corresponding to conjunctions of two or more of these types. One can then define corresponding hybrid contextual intensions.

For example, one might say that two expressions share the same orthographic/semantic type when they share the same orthographic type and the same semantic type. One can then define the orthographic/semantic contextual intension of an expression as the function that maps a world centered on a token of the appropriate orthogaphic/semantic type to the extension of that token.

Hybrid contextual intensions may be useful for some purposes, but it is clear that they will not satisfy the Core Thesis any better than non-hybrid contextual intensions. So I will set them aside here.

It is possible to define a slightly different sort of contextual intension for an expression token by focusing not on the types that the token falls under, but on the token itself. Let us assume that an expression tokens are not tied to their context essentially: a given token might have been uttered in another context. Then we can say that the token-reflexive contextual intension of an expression token T is a function that maps a centered world containing T to the extension of T in that world.

The precise behavior of a token-reflexive contextual intension will depend on what properties an expression token has necessarily. It is plausible that such a token has any properties necessarily, it has its orthographic properties necessarily. If so, its token-reflexive contextual intension will be a restriction of its orthographic contextual intension, obtained by eliminating worlds centered on a different token of the same orthographic type. One might also hold that a token has some semantic value necessarily, or that it has its linguistic type necessarily. If so, its token-reflexive contextual intension will be a restriction of its semantic or linguistic contextual intension. If an expression has more than one of these things necessarily, its token-reflexive contextual intension will be a restriction of a hybrid contextual intension. If it has further properties necessarily (e.g. its speaker), it will be a further restriction of the relevant contextual intension.

It is not obvious how to decide exactly which properties an expression token has necessarily. But however we do this, it is clear that token-reflexive contextual intensions cannot satisfy the Core Thesis. The counterexamples discussed above, such as 'I am uttering now', will apply equally to token-reflexive contextual intensions. Furthermore: insofar as tokens have any properties necessarily, one can likely construct sentence tokens attributing these properties that are true whenever uttered, but not a priori (e.g. 'This token has four words'; 'David Chalmers is speaking now'). And insofar as tokens have few properties necessarily, one can likely construct sentences that are a priori but that are not true whenever uttered (e.g. 'All bachelors are unmarried'). So if 1-intensions are understood as token-reflexive contextual intensions, the Core Thesis is false.

In an attempt to get around the problems posed by sentences such as 'I am uttering now', one might attempt to construct contextual intensions that are defined at centered worlds that do not contain a token of the relevant expression type. The most obvious way to do this is to appeal to certain counterfactual conditionals. Let us say that the extended contextual intension is defined at any centered world, independently of whether a token of the type is present there. At a given centered world, the extended contextual intension returns what the extension of a token of that type would be, if it were uttered at the center of that world.

One can then say that the extended contextual intension of an expression token (relative to a type) is maps a centered world to what the extension of a token of that type would be, if it were uttered at the center of the world. So in principle, one might have extended linguistic contextual intensions, extended semantic contextual intensions, and so on. One could define an extended token-reflexive contextual intension in an analogous way.

An obvious problem here is that in many cases, it is unclear how to evaluate the counterfactual. It may be reasonably straightforward in some cases, such as 'I am a philosopher': true just when an utterance of 'I am a philosopher' by the subject at the center would be true, so true just when the person at the center is a philosopher. But how is one to evaluate what a token of 'water' would refer to if it were used in a world where there is no liquid, and in which nobody speaks a language? How does one evaluate whether an utterance of 'I am speaking loudly' would be true if it were uttered, in a world where the subject at the center is not in fact speaking? In some cases, it seems impossible for a token of the relevant type to be uttered in the relevant context. In other cases, it may be possible, but it is possible in many different ways, yielding many different results. So the truth of the relevant counterfactuals seems to be underdetermined, and an expression's extended contextual intensions seems to be ill-defined.[*]

*[[A point of this sort is made by Stalnaker 1990.]]

Another problem: even if extended contextual intensions behave coherently, they give results that are different from what we need. For example, let S = 'I am uttering now'. S is a posteriori, so the Core Thesis requires that its 1-intension be false at some worlds. For example, it is desirable that S's 1-intension be false at an utterance-free world. Let W be such an utterance-free centered world. To evaluate S's contextual intension at W, we ask: if S were uttered at the center of W, what would its extension be? It is clear that if S were uttered in W, it would be true. So S's extended contextual intension is true at W, and indeed is true at all worlds. So the Core Thesis is still false for extended contextual intensions.

To get anything like the result that is needed, we would need to evaluate S's extension in W without S being present in W. But it is very hard to do that on the contextual model. On the contextual understanding, 1-intensions are derivative on facts about the extensions of various possible tokens, as uttered in various possible contexts. It seems clear that on such an understanding, the 1-intension of a sentence such as 'There are sentence tokens' will never be false.

I think that the idea of an extended contextual intension is getting at something important: that we need to be able to evaluate an expression's 1-intension in centered worlds that do not contain a token of the expression. But this is the wrong way to achieve the goal. To do this properly, I think we need to go beyond the contextual understanding of 1-intensions.

One might suggest that to capture a token's cognitive significance, we should not focus on a token's broadly linguistic properties, such as its orthography, its semantic value, and its language. Instead, we need to focus on its cognitive properties, which correspond to mental features of the subject that produces the token. Some such features include: the concept or belief that the token expresses; the cognitive role associated with the token; and the intentions associated with the token. Assuming that we have a way of individuating the mental types in question, we can then classify expression tokens under corresponding cognitive types.

For a given scheme of cognitive typing, one can then define the cognitive contextual intension of an expression token as the intension that maps a world centered on a token of the same cognitive type to the extension of that token. In the three cases above: a conceptual contextual intension will be defined at worlds centered on a token expressing the same concept or belief; a cognitive-role contextual intension will be defined at worlds centered on a token associated with the same cognitive role; and an intention-based contextual intension will be defined at worlds centered on a token associated with the same intentions.

Assuming that one can make sense of the relevant typing, there is a natural extension of this idea. One could define a sort of extended cognitive contextual intension, defined at worlds that do not contain the token at all, but merely contain the relevant mental feature. For example, the extended conceptual contextual intension will be defined at any world that contain the relevant concept at its center, irrespective of whether it contains any token, and will return the extension of the concept. (This assumes that concepts have extensions, which seems reasonable enough.) The extended cognitive-role contextual intension might be defined at any world centered on a concept that plays the relevant cognitive role, returning the concept's extension; and the extended intention-based contextual intension will be defined at any world centered on a concept is associated with the same intentions.

This sort of intension has some promise of dealing with the central problems raised so far. In the case of 'A sentence token exists': one can make a case that the extended conceptual contextual intension of this expression is false at some centered worlds: those in which a subject has the relevant concepts and the relevant thought, but in which there are no sentence tokens. So the intension is not necessary, reflecting the aposteriority of the sentence. The same goes for 'Language exists', and for 'I am uttering now'. By allowing intensions to be evaluated without relying on language, the metalinguistic element of contextual intensions has been reduced or eliminated.

Still, analogous problems arise. 'I am thinking now' will plausibly have a necessary conceptual contextual intension, but it is plausibly a posteriori: the thought itself is justified only by experience, albeit by introspective experience. The same goes for 'I exist'. And the same will apply to specific attributions of mental features: a thought such as 'I have the concept concept' will be true whenever it is thought, but it is not justifiable a priori. Something similar applies to thoughts attributing certain cognitive roles or certain intentions. So even here, some a posteriori sentences and thoughts will have a necessary 1-intension.

As for the other main sort of problem discussed so far, that associated with 'Water is H₂O': a proponent might hold that although Oscar and Twin Oscar do not have the same word 'water', their words express the same concept, at least under one reasonable way of individuating concept types. If so, then the conceptual contextual intension of Oscar's token 'Water is H₂O' will be false at the world centered on Twin Oscar, as the Fregean conception requires. At the same time, it might be undefined at the world centered on Steel Oscar (since he seems to have a different concept), as required.

It is controversial, however, whether concepts (or roles or intensions) can be individuated in a way that yields these results. Many theorists hold that even a token concept expressed by 'water' has its extension essentially, and that all concepts of the same type have the same extension. If so, then a statement such as 'Water is H₂O' will have a necessary intension. They might concede that concepts or thoughts can also be individuated syntactically or formally; but on this way of doing things, 'All bachelors are unmarried' will have a contingent intension. So either way, the Core Thesis is false.

One might argue that there is an intermediate way of individuating concept types that yields the right results. But many will deny this. It might be objected that this requires individuating concepts by their narrow content (that aspect of their content that is determined by a subject's intrinsic properties), and it is highly controversial whether narrow content exists. Some think that the two-dimensional framework can be used to give an account of narrow content; but in this context, it seems illegitimate for the framework to presuppose narrow content. This is a precise analog of the problem that arose for the Fregean version of semantic contextual intensions above.

I think that the situation here is not entirely clear. One could argue with some plausibility that there is an intuitive sense in which Oscar and Twin Oscar have the same concept, where there is no corresponding intuitive sense that they have the same word. If so, one could appeal to this intuitive sort of concept individuation to ground some sort of conceptual contextual intension here. One might arguably be able to do the same sort of thing with cognitive roles, or intentions. But the intuitions in question are likely to be disputed by many, so this approach will be at best weakly-grounded, unless one can give some sort of independent account of the relevant concept types.

On my view, (extended) cognitive contextual intensions are the sort of contextual intensions that are closest to satisfying the Core Thesis. But ultimately, the central problems arise for them too. One might try appealing to related notions that carry features of the subject across worlds: for example, an evidential contextual intension, requiring sameness of evidence; a fixing contextual intension, requiring sameness of reference-fixing procedures or intentions; a physical contextual intension, requiring that subjects be physical duplicates; functional, phenomenal, physical-phenomenal contextual intensions, which require that subjects be functional, phenomenal, and physical-phenomenal duplicates; and so on. But it is not hard to see that all of these suggestions are subject to versions of the problems mentioned above. So we still need an account of the relevant intensions.

[One can also define other sorts of contextual intensions. One can define hybrid contextual intensions, which type sentences according to combinations of their orthographic, semantic, or linguistic properties. One can define token-reflexive contextual intensions, which turn on the extension of the original sentence token (not type-identical tokens) in other contexts. One can define extended contextual intensions, which turn on the extension that a sentence of the relevant type would have had if it had been uttered in a context. And one can define cognitive contextual intensions, which turn on holding fixed a speaker's cognitive properties (as opposed to a speaker's utterances) across centered worlds. I discuss intensions of these sorts in Chalmers (2004). Some of them have interesting properties, but a detailed examination suggests that none of them satisfy the Core Thesis.]

Overall, it seems that there is no way to define contextual intensions so that they satisfy the Core Thesis. Two central problems have arisen repeatedly. First, by building in a token of the relevant mental or linguistic type into the world of evaluation, the constitutive connection with the a priori is lost. Second, for a contextual intension to behave in a quasi-Fregean manner, we need to antecedently classify tokens under some sort of quasi-Fregean type, so that the framework cannot independently ground quasi-Fregean notions, as was originally hoped.

Contextual intensions may still be useful for many purposes. But they do not yield any restoration of the golden triangle, and in particular they do not deliver a notion of meaning that is deeply tied to reason. The fundamental problem is that although some contextual intensions yield a reasonably strong correlation with the epistemic domain, none are constitutively connected to the epistemic domain. To restore the connection between meaning and reason, we need to approach the two-dimensional framework in epistemic terms.

On the epistemic understanding of two-dimensional semantics, the possibilities involved in the first dimension are understood as epistemic possibilities, and the intensions involved in the first dimension represent the epistemic dependence of the extension of our expressions on the state of the world.

There are two key ideas here. The first is the idea of epistemic space: there are many ways the world might turn out to be, and there is a corresponding space of epistemic possibilities. The second is the idea of scrutability: once we know how the world has turned out, or once we know which epistemic possibility is actual, we are in a position to determine the extensions of our expressions. Together, these two ideas suggest than at expression can be associated with a function from epistemic possibilities to extensions: an epistemic intension.

Take the first idea first. There are many ways the world might be, for all we know. And there are even more ways the world might be, for all we know a priori. The oceans might contain H₂O or they might contain XYZ; the evening star might be identical to the morning star or it might not. These ways the world might be correspond to epistemically possible hypotheses, in a broad sense. Let us say that a claim is epistemically possible (in the broad sense) when it is not ruled out a priori. Then it is epistemically possible that water is H₂O, and it is epistemically possible that water is XYZ. It is epistemically possible that Hesperus is Phosphorus, and epistemically possible that Hesperus is not Phosphorus.

Just as one can think of metaphysically possible hypotheses as corresponding to an overarching space of metaphysical possibilities, one can think of epistemically possible hypotheses as corresponding to an overarching space of epistemic possibilities. Some possibilities in the space of metaphysical possibilities are maximally specific: these can be thought of as maximal metaphysical possibilities, or as they are often known, possible worlds. In a similar way, some possibilities in the space of epistemic possibilities are maximally specific: these can be thought of as maximal epistemic possibilities, or as I will call them, scenarios.

A scenario corresponds, intuitively, to a maximally specific way the world might be, for all one can know a priori. Scenarios stand to epistemic possibility as possible worlds stand to metaphysical possibility. Indeed, it is natural to think of a scenario as a sort of possible world, or better, as a centered possible world. There are some complications here, but for the moment it is helpful to think of scenarios intuitively in such terms.

For any scenario, it is epistemically possible that the scenario is actual. Intuitively speaking, for any world W, it is epistemically possible that W is actual. And for any centered world W, it is epistemically possible that W is actual. Here the center represents a hypothesis about my own location within the world. In entertaining the hypothesis that W is actual, I entertain the hypothesis that the actual world is qualitatively just like W, that I am the subject at the center of W, and that now is the time at the center of W.

For example, let the XYZ-world be a specific centered "Twin Earth" world, in which the subject at the center is surrounded by XYZ in the oceans and lakes. Then no amount of a priori reasoning can rule out the hypothesis that the XYZ-world is my actual world: i.e., that I am in fact living in such a world, where the liquid in the oceans and lakes around me is XYZ. So the XYZ-world represents a highly specific epistemic possibility.

When we think of a world as an epistemic possibility in this way, we are considering it as actual. On the epistemic understanding, to consider a world W as actual is to consider the hypothesis that W is one's own world. When one considers such a hypothesis, in effect considers the hypothesis that D is the case, where D is a statement giving an appropriate description of W. One can think of D, intuitively, as a description of W in neutral qualitative terms, along with a specification in indexical terms of a center's location in W. I will return to this matter later.

The second key idea is that of scrutability: the idea that there is a strong epistemic dependence of an expression's extension on the state of the world. If we come to know that the world has a certain character, we are in a position to conclude that the expression has a certain extension. And if we were to learn that the world has a different character, we would be in a position to conclude the expression has a different extension. That is: we are in a position to come to know the extension of an expression, depending on which epistemic possibility turns out to be actual.

If we take the case of 'Water is H₂O': we can say that given that the world turns out as it actually has, with H₂O in the oceans and lakes, then it turns out that water is H₂O. So if the H₂O-world is actual, water is H₂O. But if we were to discover that the oceans and lakes in the actual world contained XYZ, we would judge that water is XYZ. And even now, we can judge: if it turns out that the liquid in the oceans and lakes is XYZ, it will turn out that water is XYZ. Or we can simply say: if the XYZ-world is actual, then water is XYZ.

The same goes more generally. If W₁ is a specific scenario in which the morning and evening star are the same, and W₂ is a scenario in which the morning and evening star are different, then we can say: if W₁ is actual, then Hesperus is Phosphorus; if W₂ is actual, then Hesperus is not Phosphorus. The same goes, in principle, for a very wide range of scenarios and statements. Given a statement S, and given enough information about an epistemically possible state of the world, we are in a position to judge whether, if that state of the world obtains, S is the case.

All this is reflected in the way we use language to describe and evaluate epistemic possibilities. It is epistemically possible that water is XYZ. It is also epistemically possible that the XYZ-world is actual. And intuitively speaking, the epistemic possibility that the XYZ-world is actual is an instance of the epistemic possibility that water is XYZ. We can say as above: if the XYZ-world turns out to be actual, it will turn out that water is XYZ. We might also use a straightforward indicative conditional: if the XYZ-world is actual, then water is XYZ. Or we can use the Ramsey test, commonly used to evaluate indicative conditionals: if I hypothetically accept that the XYZ-world is actual, I should hypothetically conclude that water is XYZ.

We can put all this by saying that the XYZ-world verifies 'Water is XYZ', where verification is a way of expressing the intuitive relation between scenarios and sentences described above.[*] Intuitively, a scenario W verifies a sentence S when the epistemic possibility that W is actual is an instance of the epistemic possibility that S is the case; or when we judge that if W turns out to be actual, it will turn out that S is the case; or if the indicative conditional 'if W is actual, then S is the case' is rationally assertible, or if hypothetically accepting that W is actual leads to hypothetically concluding that S is the case. We can also say that when W verifies S, W makes S true when it is considered as actual. Verification captures the way that we use language to describe and evaluate epistemic possibilities.

*[[The term 'verify' is used for a related idea in Evans 1979. See also Yablo 1999.]]

This dependence can be represented by the epistemic intension of a sentence S. This is a function from scenarios to truth-values. If a scenario W verifies S, then S's epistemic intension is true at W; if W verifies ~S, then S's epistemic intension is false at W; otherwise, S's epistemic intension is indeterminate at W. So the epistemic intension of 'Water is XYZ' is true at the XYZ-world.

Given this intuitive conception of epistemic intensions, there is a strong prima facie case that they satisfy the Core Thesis. When S is a priori, we would expect that every scenario verifies S. And when S is not a priori, ~S is epistemically possible, so we would expect that there is a scenario that verifies ~S. If these claims hold true, then S is a priori iff S has a necessary epistemic intension (one that is true at all scenarios).

Epistemic intensions resemble contextual intensions in some superficial respects, but they are fundamentally quite different. The central difference, as we will see, is that epistemic intensions are defined in epistemic terms. From what we have seen so far, epistemic intensions behave at least somewhat as one would like a quasi-Fregean 1-intension to behave. But to investigate this matter, we must define the relevant notions more precisely.

The intuitive picture of the epistemic understanding above can be regarded as capturing what is essential to an epistemic understanding. To fill in the picture, however, a more precise analysis is required. What follows is one way to flesh out these details. Not all of the details that follow are essential to an epistemic account per se, but they provide a natural way of elaborating such an account.[*]

*[[Note that some of these details are necessarily complex, and some readers may prefer to skim the remainder of this section or skip ahead to section 4 on a first reading. Some other papers cover some of this material in more depth: notably, "The Nature of Epistemic Space" (Chalmers forthcoming), which covers the issues in 3.4 in more detail; "Conceptual Analysis and Reductive Explanation" (Chalmers and Jackson 2001), which is especially relevant to the issues in 3.6; and "Does Conceivability Entail Possibility?" and "On Sense and Intension" (Chalmers 2002a and 2002b), which discuss a number of aspects of these issues that are not discussed here.]]

Starting with the intuitive picture, we can say that the epistemic intension of a sentence token is a function from a space of scenarios to the set of truth-values, such that:

The epistemic intension of a sentence token S is true at a scenario W iff the hypothesis that W is actual epistemically necessitates S.

When the conditions specified here obtain, we can also say that W verifies S. The epistemic intension of S will be false at W when W verifies ~S, and it will be indeterminate at W when W verifies neither S nor ~S.

Rather than leaving the notion of "the hypothesis that W is actual" as primitive, it is useful (although not mandatory) to invoke the notion of a canonical description of a scenario. We can then characterize an epistemic intension as follows.

The epistemic intension of a sentence token S is true at a scenario W iff D epistemically necessitates S, where D is a canonical description of W.

It remains to clarify three notions: the notion of a scenario, that of a canonical description, and that of epistemic necessitation. I investigate each of these in what follows.

First, we need to say more about epistemic possibility and necessity. The epistemic understanding of two-dimensional semantics is grounded in a notion of deep epistemic possibility, or equivalently, of deep epistemic necessity. In the ordinary sense, we say that S is epistemically possible roughly when S may be the case for all we know, and that S is epistemically necessary roughly when we are in a position to know that S is the case. A notion of deep epistemic necessity goes beyond this sort of dependence on the shifting state of an individual's knowledge, to capture some sort of rational must: a statement is deeply epistemically necessary when in some sense, it rationally must be true.

Such a notion can be understood in various ways, but for our purposes there is a natural candidate. We can say that S is deeply epistemically necessary when it is a priori: that is, when the thought expressed by S can be justified independently of experience, yielding a priori knowledge. (I say more about the notion of apriority in section 3.9) Then S is deeply epistemically possible when the negation of S is not epistemically necessary: that is, when the thought that S expresses cannot be ruled out a priori. Henceforth, I will usually drop the modifiers "deep" and "deeply", and speak simply of epistemic possibility and necessity.

In this sense, 'Water is XYZ' is epistemically possible: one cannot know a priori that water is not XYZ. In the same way, 'Hesperus is not Phosphorus' is epistemically possible, as is 'I am not a philosopher'. On the other hand, 'Some bachelors are married' is not epistemically possible, and 'All bachelors are married' is epistemically necessary. Similarly, one can argue that 'Hesperus has never been visible in the evening sky' is epistemically impossible, and that its negation is epistemically necessary.

A claim is deeply epistemically possible, intuitively speaking, when it expresses a rationally coherent hypothesis about the actual world. The standards of rational coherence here are in one sense weaker than usual: if a hypothesis conflicts with empirical knowledge, it may still be deeply epistemically possible. The standards are in another sense stronger than usual: if a hypothesis can be ruled out only by a great amount of a priori reasoning, it is nevertheless deeply epistemically impossible. It is possible to define notions of possibility that meet different standards, but the current standards are best for our current purposes.

The epistemic necessity operator applies to both sentence types and sentence tokens. We require this as the sentences S whose epistemic intensions we are defining are tokens, and it is possible for two sentence tokens of the same linguistic type to have different epistemic properties (for the reasons, see section 3.8). The canonical descriptions D of scenarios, on the other hand, are sentence types, using expressions whose epistemic properties are fixed by the language. We also need an epistemic necessitation operator between sentence types of this sort and sentence tokens.

An epistemic necessity operator of this sort can be seen as a primitive of the system I am developing. On the picture where epistemic necessity corresponds to apriority, we can characterize its properties intuitively as follows. Let us say that thoughts are the sort of occurrent propositional attitudes expressed by assertive sentences. Then a sentence token S is epistemically necessary when the thought expressed by S can be justified independently of experience, yielding a priori knowledge. A sentence type D is a priori when it is possible for a token of S to be epistemically necessary. A sentence type D epistemically necessitates a sentence token S when a material conditional 'D ⊃ S' is epistemically necessary, where this is understood as a possible token material conditional whose constituent token of S expresses the same thought as the original token. I will say more about the characterization of epistemic necessity in section 3.9, but this understanding will suffice for present purposes.

We can now say that a scenario W verifies a sentence token S when a material conditional 'D ⊃ S' is epistemically necessary, where D is a canonical description of W. If epistemic necessity is understood as apriority, then on this model a scenario W verifies a sentence S when one could in principle rule out a priori the hypothesis that W is actual but S is not the case.

This definition works naturally with the characterizations we will give of scenarios and of canonical descriptions, but it should be noted that this is not the only possible definition. There are various ways in which an epistemic framework might characterize the required relationship between D and S in other terms, which need not appeal directly to notions such as apriority.

For example, one might appeal to the intuitive heuristics described earlier. One could say that W verifies S when the epistemic possibility that W is actual is an instance of the epistemic possibility that S is the case. Or appealing to canonical descriptions, one could say that W verifies S when the epistemic possibility that D is the case is an instance of the epistemic possibility that S is the case. Here one might leave this intuitive evaluation of epistemic possibilities as a primitive, much as the intuitive evaluation of counterfactual possibilities is often taken as a primitive.

Alternatively, one could ground epistemic necessitation in indicative conditionals: D epistemically necessitates S when the indicative conditional "if D is the case, then S is the case" is intuitively acceptable on rational reflection. (See Chalmers 1998 for a discussion of this approach.) In a closely related idea, one could ground epistemic necessitation in the Ramsey test: D epistemically necessitates S (relative to a subject) when if the subject hypothetically accepts that D is the case, the subject should rationally conclude that S is the case. The latter approach yields what we might call the Ramsey intension of an expression: the Ramsey intension of a subject's expression S is true at W when if the subject hypothetically accepts that D is the case (where D is a canonical description of W), the subject should rationally conclude that S is the case.

Ramsey intensions behave very much like epistemic intensions as defined above. It is plausible they often yield the same results: for example, both the epistemic intension and the Ramsey intension of "water is H₂O" are plausibly false at the XYZ-world. There are arguably some cases where they yield different results. For example, Yablo (2002) has argued that the indicative conditional "if 'tail' means leg, then tails are legs" is acceptable. If so, then the Ramsey intension of 'tails are legs' may be true in a world where 'tail' means legs, but the epistemic intension will not. (See Chalmers 2002a for discussion.) Likewise, if I accept that I have recently been given a drug that corrupts my adding abilities, then I should arguably suspend judgment about whether 57 plus 46 is 103. If so, the Ramsey intension of "57+46=103" will plausibly be indeterminate in a scenario where the subject at the center has been given such a drug, but the epistemic intension will not. It may that the Ramsey test can be understood in a way that handles the cases above differently, so that Ramsey intensions behave in the way that a Fregean intension should, but the matter is not entirely clear.

Ramsey intensions are a sort of epistemic intension in the general sense, as they are defined in epistemic terms. But where epistemic intensions as defined above are grounded in the notion of apriority, Ramsey intensions are grounded in the notion of rational inference. This has certain advantages: for example, those who are skeptical about apriority usually still accept that there is a coherent notion of rational inference. In what follows I will usually stay with epistemic intensions grounded in a notion of apriority, but the possibility of alternative understandings should be kept in mind.

These alternative understandings suggest that the epistemic understanding of the two-dimensional framework is not entirely beholden to the notion of apriority. Even if one rejects apriority, or if one rejects the application of apriority in this context, one should not reject the epistemic understanding. It is a prima facie datum that there is an epistemic dependence between epistemic possibilities and sentences, of the sort that was intuitively characterized earlier. One who rejects apriority will simply need to capture this dependence in other ways. My own view is that the understanding in terms of apriority runs the deepest, but the alternatives deserve exploration.

We can here note a fundamental difference between all of these sorts of epistemic evaluation and contextual evaluation. To evaluate a sentence S in a scenario W, there is no requirement that W contain a token of S. Even if W contains such a token, the definition gives it no special role to play. All that matters is the first-order epistemic relation between D and S, not whether D says something metalinguistic about a token of S. More generally, metalinguistic facts about how a token of S would behave in certain possible circumstances play no role in defining epistemic intensions. We will see shortly how this enables us to deal straightforwardly with the problem cases for contextual intensions.

Scenarios are intended to stand to epistemic possibility as possible worlds stand to metaphysical possibility. This claim can be expressed by the following:

Plenitude Principle: For all S, S is epistemically possible if and only if there is a scenario that verifies S.

In effect, the Plenitude Principle says that there are enough scenarios to verify every epistemically possible claim, and that no scenario verifies an epistemically impossible claim. It is easy to see that if we understand epistemic necessity as apriority, the Plenitude Principle is equivalent to the Core Thesis. (I give it a different name to leave open the option of understanding epistemic necessity in different terms.) So the only question is whether we can understand scenarios and verification so that the Plenitude Principle is true.

Intuitively, a scenario should correspond to a maximally specific epistemically possible hypothesis, or (for short) a maximal hypothesis: a hypothesis such that if one knew that it were true, one would be in a position to know any truth by reasoning alone. (Note that talk of "hypotheses" here is intuitive; formalizations of the relevant notions will follow.) We might say that a hypothesis H1 leaves another hypothesis H2 open if the conjunctions of H1 with both H2 and its negation are epistemically possible. A maximal hypothesis is one that leaves no possible hypothesis open. To every scenario, there should correspond a maximal hypothesis, and vice versa.

There are two concrete ways in which we might understand scenarios. The first is the way we have already sketched: as centered possible worlds. The uncentered part of the world corresponds to a hypothesis about the objective character of one's world. The centered part is needed to handle indexical claims, such as "I am in Australia". If we are given only a full objective description of a world, numerous indexical hypotheses will be left open, so such a description does not correspond to a maximal hypothesis. Correspondingly, there are numerous epistemically possible (but incompatible) objective-indexical claims: e.g. "the world is objectively thus and I am a philosopher" and "the world is objectively thus and I am not a philosopher". We need distinct scenarios to verify these claims: hence centered worlds.

There is good reason to believe that for every centered world, there is a corresponding maximal hypothesis, at least if we describe worlds under the right sort of canonical description. (It is arguable that for certain indexical hypotheses involving demonstratives, one may need further information in the center of the world: marked experiences, as well as a marked subject and time. But I will leave this matter to one side.) And one can easily make the case that an epistemically impossible sentence will be verified by no centered world (if it were so verified, it would not be epistemically impossible). The residual question is whether there are enough centered worlds to correspond to all maximal hypotheses, and to verify all epistemically possible statements. This matters turns on the following thesis:

Metaphysical Plenitude: For all S, if S is epistemically possible, there is a centered metaphysically possible world that verifies S.

The standard Kripkean cases of statements that are epistemically possible but metaphysically impossible are straightforwardly compatible with this thesis. For each such statement S, there is some way the world could turn out such that if things turn out that way, it will turn out that S is the case; and each of these ways the world could turn out can be seen as a centered world. In the case of 'Water is XYZ', the XYZ-world is such a world; something similar applies to other cases. One might worry about how a metaphysically possible world (the XYZ-world) can verify a metaphysically impossible statement ('Water is XYZ')? But two-dimensional evaluation makes this straightforward: 'Water is XYZ' is true at the XYZ-world considered as actual, but false at the XYZ-world considered as counterfactual. The metaphysical impossibility of 'Water is XYZ' reflects the fact that it is false at all worlds considered as counterfactual. But this is quite compatible with its being true at some worlds considered as actual.

Are there any counterexamples to the Metaphysical Plenitude thesis? I have argued elsewhere (Chalmers 2002a) that there are no such counterexamples. Certainly, there are no clear cases of epistemically possible claims that are verified by no centered world. Still, some controversial philosophical views entail that there are such cases. For example, some theists hold that it is necessary that an omniscient being exists, while also holding that it is not a priori that an omniscient being exists. If so, "No omniscient being exists" will be a counterexample to Metaphysical Plenitude: it will be an epistemically possible statement that is verified by no possible world. In effect, on this view the space of metaphysical possibilities is smaller in some respects than the space of epistemic possibilities.

The same goes for some other philosophical views. On some views on which the laws of nature of our world are the laws of all worlds, for example, the negation of a law of nature will be a counterexample to Metaphysical Plenitude. On views on which a mathematical claim (such as the Continuum Hypothesis) can be necessarily true but not knowable a priori, the negation of such a claim will be a counterexample to Metaphysical Plenitude. On some versions of the epistemic theory of vagueness, some claims involving vague terms (e.g. the statement that someone of a certain height is tall) may be a counterexample to Metaphysical Plenitude. On some materialist views about consciousness, the claim that there are zombies (unconscious physical duplicates of conscious beings) may be a counterexample to Metaphysical Plenitude. If these views are correct, there will be epistemically possible claims that are not verified by any centered metaphysically possible worlds. If so, Metaphysical Plenitude (and the Core Thesis for epistemic intensions over centered metaphysically possible worlds) will be false.

All of these views are highly controversial, and I have argued elsewhere (Chalmers 2002a) that all of them are incorrect. One can plausibly argue in reverse: the Metaphysical Plenitude thesis, which appears to fit all standard cases, gives us reason to reject these controversial views. More deeply, one can argue that these views rest on a mistaken conception of metaphysical possibility and necessity. My own view is that a careful analysis of the roots of our modal concepts supports constitutive links between epistemic and metaphysical modal notions, and thereby grounds the Metaphysical Plenitude thesis. If this is correct, then understanding scenarios in terms of centered worlds yields epistemic intensions that satisfy the Core Thesis.

It is nevertheless useful to have an approach to the space of epistemic possibilities that is neutral on these substantive questions about metaphysical possibility. This allows even those philosophers who deny Metaphysical Plenitude to make use of the notion of an epistemic intension, and allows a maximally general defense of the epistemic understanding of two-dimensional semantics.

The alternative is to understand scenarios in purely epistemic terms from the start. One might reasonably hold that since we want epistemic intensions to be constitutively connected to the epistemic realm, we need not invoke the metaphysical modality at all. Instead, we can do things wholly in terms of the epistemic modality. There are a couple of ways one might proceed here. One could introduce the notion of a scenario (a maximal epistemic possibility) as a modal primitive, in the same way that some philosophers introduce the notion of a world (a maximal metaphysical possibility) as a modal primitive. Or one could try to construct scenarios directly out of materials that are already at hand.

I take the second course in Chalmers (forthcoming), examining a detailed construction. I do not have space to do that here, but I can give a brief idea of how one might proceed. The idea I will outline is a linguistic construction of scenarios, constructed out of linguistic expressions in an idealized language, along with a basic operator of epistemic possibility.

Let us say that a sentence D of a language L is epistemically complete when (i) D is epistemically possible, and (ii) there is no sentence S of L such that both D&S and D&~S are epistemically possible. When D is epistemically complete, it is in effect as specific as an epistemically possible sentence can be. Let us say that D is compatible with H when D&H is epistemically possible, and D implies H when D&~H is epistemically impossible (that is, when there is an a priori entailment from D to H). Then if D is epistemically incomplete, it leaves questions open: there will be H such that D is compatible with H but D does not imply H. If D is epistemically complete, D leaves no questions open: if D is compatible with H, D implies H. Note that D need not explicitly include every such hypothesis as a conjunct; these hypotheses need only be implied.

Intuitively, scenarios should correspond to epistemically complete hypotheses, whether or not they are expressible in a language such as English. It is likely that actual languages do not have the expressive resources to express an epistemically complete hypothesis, as they are restricted to finite sentences and have a limited lexicon. So for the purposes of this construction, we need to presuppose an idealized language that can express arbitrary hypotheses. In particular, our language L should allow infinitary sentences (at least infinitary conjunctions) and should have terms that express every possible concept, or at least every concept of a certain sort. It is also important that expressions in L are epistemically invariant, so that there cannot be two tokens S₁ and S₂ of the same sentence type (used with full competence) such that S₁ is epistemically necessary and S₂ is not. The exact requirements for L raise subtle issues, but we can pass over them here.

We can then focus on epistemically complete sentences of L. By the idealization, every such sentence will express a maximally specific hypothesis, and vice versa. So scenarios should correspond to epistemically complete sentences in L, although perhaps with more than one such sentence per scenario. We can say that two sentences S and T are equivalent when S implies T and T implies S (that is, when S&~T and T&~S are epistemically impossible). Any epistemically complete sentences in L will then fall into an equivalence class. We can now identify scenarios with equivalence classes of epistemically complete sentences in L. To anticipate the definition of verification: we can also say that a scenario verifies a sentence S (of an arbitrary language) when D implies S, where D is an epistemically complete sentence of L in the scenario's equivalence class.

Defined this way, scenarios are tailor-made to satisfy the Plenitude Principle. This principle requires the following:

Epistemic Plenitude: For all S, if S is epistemically possible, then some epistemically complete sentence of L implies S.

Here S may be a sentence token in any language (not necessarily in L). To see the plausibility of this thesis, first note that because L has unlimited expressive power, some epistemically possible sentence S₁ of L will imply S. Second, it is plausible that any epistemically possible sentence S₁ of L is implied by some epistemically complete sentence D of L. Intuitively, to obtain D from S₁, one simply conjoins arbitrary sentences that are epistemically compatible with S₁ (and other conjoined sentences) until one can conjoin no more. The issue is not completely trivial, there might be endless infinitary conjunction with no maximal point, but under certain reasonable assumptions, such a sentence will exist. If so, then every epistemically possible sentence is verified by some scenario. In reverse, it is clear that any sentence verified by a scenario is epistemically possible. So the corresponding version of the Plenitude Principle is plausibly true.

In effect, this construction formalizes the intuitive idea of a maximal hypothesis: a maximal hypothesis is equivalent to an equivalence class of epistemically complete sentences in an idealized language. We might say that where the first-approach takes a metaphysical approach to scenarios, on which they correspond to centered metaphysically possible worlds, the second approach takes an epistemic approach to scenarios, on which they correspond to maximal hypotheses.

What is the relationship between the two constructions? My own view is there is a close correspondence: every centered world corresponds to a maximal hypothesis, and every maximal hypothesis corresponds to a centered world. (Not quite one-to-one: in certain cases there may be more than one centered world per maximal hypothesis, for example when there are symmetrical worlds with symmetrically corresponding centers.[*]) If so, then the Plenitude Principle will plausibly be satisfied either way. But philosophers who deny Metaphysical Plenitude will deny the close correspondence, holding that there are maximal hypotheses that correspond to no centered world. For example, a philosopher who holds that 'There is an omniscient being' is necessary but not a priori will hold that there is a maximal hypothesis that verifies the sentence in question, but that there is no centered metaphysically possible world in the vicinity. Such a philosopher should embrace the epistemic approach to scenarios.

*[[See Chalmers forthcoming, section 4(4) for more on ways in which there could be more than one centered world per maximal hypothesis. Schroeter (2004) raises the possibility that there are intrinsic properties for which there is no semantically neutral conception. If there are such properties, then this is another source of a many-to-one correspondence. If such properties exist, then an epistemically complete description of a centered world may not need to specify their precise distribution. If so, then an epistemically complete description need not be ontologically complete, and more than one centered world (with different but isomorphic distributions of intrinsic properties) may correspond to the same maximal hypothesis.]]

The epistemic approach to scenarios is grounded more purely in the epistemic realm, and its central theses require fewer commitments than the metaphysical approach. For this reason, one can argue that the epistemic approach to scenarios is more basic. Centered worlds are more familiar and are useful for various applications, however, so I will use both understandings of scenarios in what follows.

On either understanding, one scenario will be privileged with respect to any statement token as the actualized scenario at that token. On the world-based view, this will be the world centered on the speaker and the time of utterance. On the epistemic view, this will correspond to the maximal hypothesis that is true of the world from the speaker's perspective at the time of utterance. In general, we expect that when an expression token's epistemic intension is evaluated at the scenario that is actualized at that token, the result will be the token's extension.

When we consider a scenario as actual, in order to evaluate an expression, we always grasp it under a description. This raises an issue. A scenario can be described in multiple ways, and it is not obvious that all such descriptions will give equivalent results. So we have to isolate a special class of canonical descriptions of scenarios under which they must be considered.

If we take the epistemic approach to scenarios by the second construction above, the choice will be straightforward. A scenario will correspond to an equivalence class of epistemically complete sentences. Here, we can say that a canonical description of the scenario is any sentence in the corresponding equivalence class. Because all of these sentences are equivalent under implication, they will all give the same results under verification.

If we take the metaphysical approach to scenarios, things are more complicated. Here, we require that a canonical description be a complete neutral description of the world. Both neutrality and completeness need explanation.

First, neutrality. To describe a world, we must choose sentences that are true of it. But will these be sentences true of the world considered as actual, or of the world considered as counterfactual? If we choose the first, there is a danger of circularity: evaluation of a world considered as actual will be defined in terms of canonical descriptions, which will be defined in terms of evaluation of worlds considered as actual. If we choose the second, there is a danger of incoherence: the framework requires that the XYZ-world verifies 'water is not H₂O', but 'Water is H₂O' is true of the XYZ-world considered as counterfactual. Either way, we need to ensure that sentences such as 'Water is H₂O' are not present within canonical descriptions of the XYZ-world.

The solution is to restrict canonical descriptions to semantically neutral expressions. Intuitively, a semantically neutral expression is one that behaves the same whether one considers a world as actual or as counterfactual. We cannot simply define a semantically neutral expression in this way, since the definition presupposes evaluation in a world considered as actual, and this evaluation (as developed here) presupposes the notion of a canonical description. But nevertheless we have a good grasp on the notion. For example, 'water' and 'Hesperus' are not semantically neutral; but 'and', 'philosopher', 'friend', 'consciousness', and 'cause' plausibly are. One could rely on our intuitive grasp of this notion for current purposes, or one could seek to define it.

One promising approach is to define such an expression as one that is not "Twin-Earthable". We can say that two possible individuals (at times) are twins if they are physical and phenomenal duplicates; we can say that two possible expression tokens are twins if they are produced by corresponding acts of twin speakers. Then a token is Twin-Earthable if it has a twin with a different 2-intension. This test works for many purposes. A semantically neutral term (in the intuitive sense) is never Twin-Earthable. But the reverse is not quite the case. For example, let L be an expression that functions to rigidly designate the speaker's height. Then any twin of L will have the same 2-intension (since a twin speaker will have the same height), but L is not semantically neutral. One might respond by watering down the requirements of physical and phenomenal duplication (perhaps to some sort of mental duplication), but similar cases will still arise: e.g. if M is an expression that rigidly picks out 1 if the speaker has visual experience, and 0 if not, then M will be Twin-Earthable even by this sort of standard, but not semantically neutral.[*]

*[[Non-Twin-Earthability is related to Bealer's (1996) notion of semantic stability: "an expression is semantically stable iff, necessarily, in any language group in an epistemic situation qualitatively identical to ours, the expression would mean the same thing" (Bealer 1996, p. 134). It is clear that semantic stability cannot be used to characterize semantic neutrality, for same reasons as in the case of non-Twin-Earthability. For example, the expression M in the text is semantically stable but not semantically neutral.]]

A better characterization might be as follows: a semantically neutral expression is one whose extension in counterfactual worlds does not depend on how the actual world turns out (that is, on which epistemically possible scenario turns out to be actual). This is an intuitive characterization rather than a formal characterization: it invokes the intuitive idea of dependence of counterfactual extensions on the actual world, and formalizing this idea would require something equivalent to the two-dimensional framework (with ensuing circularity). But nevertheless, we have a good grip on the notion. In this sense, it is clear that most names, natural kind terms, and indexicals are not semantically neutral (and neither are L or M above), while numerous other terms (such as those listed above) are semantically neutral.

A precise formal characterization of semantic neutrality remains an open question for future research. One might try a characterization wholly in terms of our modal operators of epistemic and metaphysical necessity (i.e., apriority and necessity), but it is not entirely clear how this would work. In the meantime, the intuitive characterization suffices for our purposes. It is also useful to stipulate that terms with context-dependent behavior, such as "heavy", are not semantically neutral. This allows us to describe worlds using expression types and not just expression tokens.

To characterize a centered world, semantically neutral terms must be supplemented by some indexical terms, to characterize the location of a center. The best way to do this is the following. We can say that a statement is in canonical form when it has the form D & 'I am D₁' & 'now is D₂', where D, D₁, and D₂ are all semantically neutral, and D₁ and D₂ are identifying predicates relative to the information in D (that is: D implies 'Exactly one individual is D₁' and 'Exactly one time is D₂'). We can say that a neutral description of a centered world is a statement in canonical form such that D is true of the world, D₁ is true of the subject at the center, and D₂ is true of the time at the center. (If the center of a centered world includes entities other than an individual and a time, then one can extend similar treatment to these entities.)

In a few cases involving completely symmetrical worlds, there may be no identifying predicates available: that is, there may be no semantically neutral predicates true only of the individual (or time) at the center. In that case, one can invoke a maximally specific predicate instead: a predicate D₁ such that for all D₂ true of the center, D entails 'everything that is D₁ is D₂'. Here, two centered worlds that differ only in symmetrical placement of the center may yield the same canonical description. This is reasonable, as intuitively both worlds correspond to the same maximal hypothesis.

Second, completeness. We require that a canonical description be a complete neutral description of a centered world. There are two possibilities here. First, we can appeal to a criterion in terms of (metaphysical) necessity. Let us say that a semantically neutral description of a world is ontologically full when it (metaphysically) necessitates all semantically neutral truths about that world, and is minimal among the class of descriptions with this property. For example, if physicalism is true, a full semantically neutral specification of fundamental physical truths will be ontologically full. Then an ontologically complete neutral description of a centered world is a neutral description where the first (non-indexical) component of the description is ontologically full.

Alternatively, we can appeal to epistemic completeness. In this sense, a complete neutral description of a centered world is simply a neutral description that is epistemically complete. This requires the claim that for any centered world, there exists an epistemically complete neutral description. This claim is nontrivial, but there are good grounds to accept it. One can argue that although non-neutral terms are modally distinctive, they do not add fundamentally new epistemic power to a language, so that neutral terms constitute what I call an epistemic basis (see section 3.6) for the space of epistemic possibilities.

It is not hard to see that if Metaphysical Plenitude is correct, then an ontologically complete neutral description will also be a epistemically complete neutral description.[*] If so, we can then use either criterion for a canonical description. There will arguably be more explanatory power, however, in using a complete description in the ontological sense, and then allowing this description to epistemically determine all truths about a world considered as actual.

*[[The statement of Metaphysical Plenitude uses the notion of verification, which in turn requires the notion of a canonical description. For the purposes of interpreting Metaphysical Plenitude, we can assume that the canonical descriptions are required to be epistemically complete. If Metaphysical Plenitude formulated this way is correct, ontologically complete descriptions will give the same results as epistemically complete descriptions.]]

If Metaphysical Plenitude is false, then the two criteria will not coincide. An ontologically complete neutral description will not be epistemically complete, and it will leave some hypotheses unsettled (e.g. the complete physical truth about the world may leave the Continuum Hypothesis unsettled, even if it is necessarily true). If we require that canonical descriptions be ontologically complete, the epistemic intensions of these hypotheses will have an indeterminate truth-value. A consequence may be that when an expression's epistemic intension is evaluated at the actual centered world of the expression, it does not yield the expression's extension (e.g., the epistemic intension of CH may be indeterminate at the actual world, even if CH is true). If, on the other hand, we require epistemic completeness, then the epistemic intensions of the relevant claims will have a determinate truth-value (e.g. the epistemic intension of CH will be true or false at the world according to whether CH itself is true or false there). One might do things either way, depending on one's purposes, although perhaps the second is best overall. In any case, this situation will not matter much for our purposes, since we already know that if Metaphysical Plenitude is false, then the Core Thesis will be false when scenarios are understood as centered worlds.

(A third alternative is to require "qualitative completeness", where this is characterized as in Chalmers (2002a) in terms of a notion of positive conceivability. This yields a notion that is usefully intermediate between epistemic completeness and ontological completeness. But I will leave this option aside here.)

It is clear that if scenarios are understood as centered worlds, the characterization of canonical descriptions is significantly more complicated than if scenarios are understood in wholly epistemic terms. This may be another point in favor of the purely epistemic understanding of scenarios.

Given the epistemic understanding of scenarios, one might have the following worry: the epistemic intension of a sentence may be well-defined, but it is trivial. The triviality comes from the requirement that descriptions be epistemically complete. In may worry that in order for a description to be epistemically complete, it will need to specify the truth or falsity of most sentences S explicitly. For example, 'Water is H₂O' will be true precisely in those scenarios that have 'Water is H₂O' in their canonical description, and it will be false precisely in those scenarios that have 'Water is not H₂O' in their canonical description. If this sort of thing is typical, then epistemic evaluation as defined will have an uninteresting structure.

A related worry arises on the metaphysical understanding of scenarios. Here, the issue concerns the thesis (mentioned in the previous section) that there is an epistemically complete neutral description of any centered world. If one had the worry just mentioned about 'Water is H₂O', one might worry that an epistemically complete description of a centered world requires non-neutral terms, such as 'water'. The key question is whether the truth-value of all sentences S is epistemically necessitated by a description of a centered world in terms of semantically neutral expressions plus indexicals. If this is not the case, then as defined, the epistemic intension of the relevant sentences will be indeterminate at the relevant centered worlds.

These worries are reasonable enough, but I think that they are ultimately unfounded. In what follows, I will concentrate on the worry that applies to the epistemic understanding, but similar considerations also apply to the metaphysical understanding. To answer the worry, one needs to make the case that epistemically complete descriptions do not need to specify the truth or falsity of most statements explicitly, so that epistemic evaluation does not have a trivial structure. To see this, it is useful to focus on the actual world, and consider what an epistemically complete description of this world must contain. The sort of argument I give here is presented in much more depth by Chalmers and Jackson (2001) and Chalmers (2002a); but here I will give the basic idea.[*]

*[[Chalmers and Jackson (2001) can be seen as be seen as providing a crucial part of the foundation for the two-dimensional framework as it is understood here, even though the framework is hardly mentioned in the paper (which is packaged as a response to Block and Stalnaker on the explanatory gap). Section 3 of the paper in effect argues for the scrutability thesis in a general form, and sections 4 and 5 defend a specific version of the thesis. The reply to objection 6 in section 5 is particularly important in defending the a priori entailment version of the scrutability thesis. Sections 8 and 9 of Chalmers (2002a) provide a further defence of a version of the thesis.]]

The second principle underlying the epistemic understanding of the two-dimensional framework was what we might call the scrutability of truth. This can be put informally as the thesis that once we know enough about the state of the world, we are in a position to know the truth-values of our sentences. Furthermore, we usually need not be informed about a sentence explicitly in order to know whether it is true. We could put this somewhat more precisely as follows:

Scrutability of Truth: For most terms T used by a speaker, then for any truth S involving T, there exists a truth D such that D is independent of T, and such that knowing that D is the case puts the speaker in a position to know (without further empirical information, on idealized rational reflection) that S is the case.

Here, we can say that D is independent of T when D does not contain T or any close cognates. Of course this notion is somewhat vague, as is the notion of "most" above, but this does not matter for our purposes. To save breath, we can abbreviate "knowing that D is the case puts the speaker in a position to know (without further empirical information, on idealized rational reflection) that S is the case" as "D is epistemically sufficient for S".

Take the case of 'water'. Here, we can let D be a truth specifying an appropriate amount of information about the appearance, behavior, composition, and distribution of objects and substances in one's environment, as well as information about their relationship to oneself. D need not contain the term 'water' at any point: appearance can be specified in phenomenal terms, behavior and distribution in spatiotemporal terms, composition in microphysical or chemical terms. Then D is epistemically sufficient for 'Water is H₂O'. When one knows that D is the case, one will be in a position to know all about the chemical makeup of various liquids with various superficial properties in one's environment, and will thereby be able to infer that water is H₂O. After all, this informati