Two-Dimensional Semantics
David J. Chalmers
Philosophy Program
Research School of Social Sciences
Australian National University
Two-dimensional approaches to semantics, broadly understood, recognize two“dimensions” of the meaning or content of linguistic items. On these approaches,
expressions and their utterances are associated with two different sorts of semantic
values, which play different explanatory roles. Typically, one semantic value isassociated with reference and ordinary truth-conditions, while the other is associatedwith the way that reference and truth-conditions depend on the external world. The
second sort of semantic value is often held to play a distinctive role in analyzing
matters of cognitive significance and/or context-dependence.
In this broad sense, even Frege’s theory of sense and reference might qualify as asort of two-dimensional approach. More commonly, two-dimensional approaches areunderstood more narrowly to be a species of possible-worlds semantics, on which
each dimension is understood in terms of possible worlds and related modal notions.
In possible-world semantics, linguistic expressions and/or their utterances are
first associated with an extension. The extension of a sentence is its truth-value: forexample, the extension of ‘Plato was a philosopher’ is true. The extension of asingular term is its referent: for example, the extension of ‘Don Bradman’ isBradman. The extension of a general term is the class of individuals that fall under theterm: for example, the extension of ‘cat’ is the class of cats. Other expressions work
similarly.
One can then associate expressions with an intension, which is a function frompossible worlds to extensions. The intension of a sentence is a function that is true at a
possible world if and only if the sentence is true there: the intension of ‘Plato was aphilosopher’ is true at all worlds where Plato was a philosopher. The intension of asingular term maps a possible world to the referent of a term in that possible world:

In (E. Lepore & B. Smith, eds) Oxford Handbook of the Philosophy of Language. Oxford University
Press, forthcoming.

the intension of ‘Don Bradman’ picks out whoever is Bradman in a world. The
intension of a general term maps a possible world to the class of individuals that fallunder the term in that world: the intension of ‘cat’ maps a possible world to the classof cats in that world.
It can easily happen that two expressions have the same extension but different
intensions. For example, Quine’s terms ‘cordate’ (creature with a heart) and ‘renate’(creature with a kidney) pick out the same class of individuals in the actual world, sothey have the same extension. But there are many possible worlds where they pick out
different classes (any possible world in which there are creatures with hearts but nokidneys, for example), so they have different intensions. When two expressions havethe same extension and a different intension in this way, the difference in intension
usually corresponds to an intuitive difference in meaning. So it is natural to suggestthat an expression’s intension is at least an aspect of its meaning.
Carnap (1947) suggested that an intension behaves in many respects like a
Fregean sense, the aspect of an expression’s meaning that corresponds to its cognitive
significance. For example, it is cognitively significant that all renates are cordates andvice versa (this was a nontrivial empirical discovery about the world), so that ‘renate’and ‘cordate’ should have different Fregean senses. One might naturally suggest that
this difference in sense is captured more concretely by a difference in intension, andthat this pattern generalizes. For example, one might suppose that when two singularterms are cognitively equivalent (so that ‘a=b’ is trivial or at least knowable a priori,
for example), then their extension will coincide in all possible worlds, so that they
will have the same intension. And one might suppose that when two such terms are
cognitively distinct (so that ‘a=b’ is knowable only empirically, for example), then
their extensions will differ in some possible world, so that they will have different
intensions. If this were the case, the distinction between intension and extension could
be seen as a sort of vindication of a Fregean distinction between sense and reference.
However, the work of Kripke (1980) is widely taken to show that no such
vindication is possible. According to Kripke, there are many statements that areknowable only empirically, but which are true in all possible worlds. For example, it
is an empirical discovery that Hesperus is Phosphorus, but there is no possible worldin which Hesperus is not Phosphorus (or vice versa), as both Hesperus andPhosphorus are identical to the planet Venus in all possible worlds. If so, then‘Hesperus’ and ‘Phosphorus’ have the same intension (one that picks out the planet

Venus in all possible worlds), even though the two terms are cognitively distinct. The
same goes for pairs of terms such as ‘water’ and ‘H2O’: it is an empirical discovery
that water is H2O, but according to Kripke, both ‘water’ and ‘H2O’ have the same
intension (picking out H2O in all possible worlds). Something similar even applies toterms such as ‘I’ and ‘David Chalmers’, at least as used by me on a specific occasion:‘I am David Chalmers’ expresses nontrivial empirical knowledge, but Kripke’s
analysis entails that I am David Chalmers in all worlds, so that my utterances of these
expressions have the same intension. If this is correct, then intensions are strongly
dissociated from cognitive significance.
Still, there is a strong intuition that the members of these pairs (‘Hesperus’ and‘Phosphorus’, ‘water’ and ‘H2O’, ‘I’ and ‘David Chalmers’) differ in some aspect of
meaning. Further, there remains a strong intuition that there is some way the world
could turn out so that these terms would refer to different things. For example, it
seems to be at least epistemically possible (in some broad sense) that these terms
might fail to corefer. On the face of it, cognitive differences between the terms isconnected in some fashion to the existence of these possibilities. So it is natural tocontinue to use an analysis in terms of possibility and necessity to capture aspects ofthese cognitive differences. This is perhaps the guiding idea behind two-dimensionalsemantics.
Two-dimensional approaches to semantics start from the observation that theextension and even the intension of many of our expressions depend in some fashionon the external world. As things have turned out, my terms ‘water’ and ‘H2O’ havethe same extension, and have the same (Kripkean) intension. But there are ways
things could have turned out so that the two terms could have had a different
extension, and a different intension. So there is a sense in which for a term like‘water’, the term’s extension and its Kripkean intension depend on the character of
our world. Given that this world is actual, it turns out that ‘water’ refers to H2O, and
its Kripkean intension picks out Venus in all possible worlds. But if another world
had been actual (e.g. Putnam’s Twin Earth world in which XYZ is the clear liquid in
the oceans), ‘Hesperus’ might have referred to something quite different (e.g. XYZ),
and it might have had an entirely different Kripkean intension (e.g. one that picks out
XYZ in all worlds).

This suggests a natural formalization. If an expression’s (Kripkean) intension
itself depends on the character of the world, then we can represent this dependence bya function from worlds to intensions. As intensions are themselves functions fromworlds to extensions, this naturally suggests a two-dimensional structure. We canrepresent this structure diagramatically as follows:

H2O-world

XYZ-world…

H2O -world

H2O

H2O

…

XYZ-world

XYZ

XYZ

…

…

…

…

…

This diagram expresses an aspect of the two-dimensional structure associatedwith the term ‘water’. It is intended to express the intuitive idea that if the H2O -world
turns out to be actual (as it has), then ‘water’ will have a Kripkean intension that picksout H2O in all worlds; but if the XYZ-world turns out to be actual (as it has not), then‘water’ will have a Kripkean intension that picks out XYZ in all worlds. Intuitively,
the worlds in the column on the left represent ways the actual world can turn out(these are sometimes thought of more precisely as possible contexts of utterances, andare sometimes thought of as epistemic possibilities), while the worlds across the top
reflect counterfactual ways that a world could have been (these are sometimes thoughtof more precisely as possible circumstances of evaluation, and sometimes thought ofas metaphysical possibilities). It is sometimes said that worlds on the left column (oneworld per row), making up the “first dimension” of the matrix, correspond to different
worlds considered as actual; while the worlds in the top row (one world per column),
making up the “second dimension” of the matrix, correspond to different worlds
considered as counterfactual.
This two-dimensional matrix can be seen as a two-dimensional intension: afunction from ordered pairs of worlds to extensions. Such a function is equivalent to afunction from worlds to intensions, and seen this way can be regarded as capturing theintuitive idea that a term’s intension depends on the character of the actual world. One
can also recover the intuitive idea that a term’s extension depends on the character of
the actual world by examining the “diagonal” of this matrix, i.e. the cells that
correspond to the same world considered as actual and as counterfactual. In theexample above: where the H2O -world is considered as actual and as counterfactual,then ‘water’ picks out H2O, while if the XYZ-world is considered as actual and as
counterfactual, then ‘water’ picks out XYZ. We can say that an expression’s“diagonal intension” is a function mapping a world w to the term’s extension when w
is taken as both actual and as counterfactual. So the diagonal intension of ‘water’maps the H2O -world to H2O, the XYZ-world to XYZ, and so on.
We can then see how pairs of terms with the same extension and the same
Kripkean intension might nevertheless have different two-dimensional intensions, anddifferent diagonal intensions. For example, ‘water’ and ‘H2O’ have the same
Kripkean intension, but it is plausible that if the XYZ-world had turned out to be
actual, they would have had different Kripkean intensions: ‘water’ would have had an
intension that picked out XYZ in all worlds, while ‘H2O’ still would have had an
intension that picked out H2O in all worlds. If so, then these terms have different twodimensional intensions and different diagonal intensions.
One can make a case that something similar applies with ‘Hesperus’ and‘Phosphorus’, and with ‘I’ and ‘David Chalmers’: the members of each pair have a
different two-dimensional intension and a different diagonal intension. If so, then thisbegins to suggest that there is some sort of connection between an expression’s twodimensional intension (or perhaps its diagonal intension) and its cognitive
significance. One might even speculate that an expression’s diagonal intension
behaves in some respects like a Fregean sense, in a way that might vindicate Carnap’s
project.
At this point it must be acknowledged that things are not so simple. A number ofdifferent two-dimensional approaches to semantics have been developed in theliterature, by Kaplan (1979, 1989), Stalnaker (1978), Chalmers (1996, 2002a, 2004),
and Jackson (1998), among others; and closely related two-dimensional analysis ofmodal notions have been put forward by Evans (1977) and by Davies andHumberstone (1981). These approaches differ greatly in the way that they make theintuitive ideas above precise. They differ, for example, in just what they take the“worlds” in the left column to be, and they differ in their analysis of how a term’s
intension and/or extension depends on the character of the actual world. As a result,different approaches associate these terms with quite different sorts of two-dimensional semantic values, and these semantic values have quite different
connections to cognitive significance.
In what follows, I will first go over the two-dimensional approaches pioneered in
the 1970s by Kaplan, Stalnaker, Evans, and Davies and Humberstone.1 Each of theseapproaches can be seen as sharing some of the formal structure described above, butwith quite different conceptual underpinnings. Each of the approaches asserts some
sort of connection between two-dimensional semantic values and apriority, but theconnection is usually limited in scope, applying to indexicals (Kaplan) and todescriptive names (Evans), and ‘actually’-involving expressions (Davies andHumberstone), while Stalnaker’s later work rejects a connection to apriority
altogether. I will then describe the more general two-dimensional approach tosemantics developed in the 1990s by Chalmers, Jackson, and others. This approachassociated two-dimensional semantic values with expressions of all kinds, and asserts
a strong general connection between these semantic values and the domain of
apriority and cognitive significance. I will close by briefly describing some
applications of the framework, and by considering and responding to a numberof objections.

2

Early two-dimensional approaches

2.1 Kaplan: Character and content
Perhaps the best-known broadly two-dimensional approach is Kaplan’s analysisof the character and content of indexicals (Kaplan 1979, 1989). According to Kaplan,
his work is partly grounded in work in tense logic by Kamp (1971) and Vlach (1973),
which gives a sort of two-dimensional analysis of the behavior of ‘now’. Kaplan
applies his analysis to indexicals such as ‘I’, ‘here’, and ‘now’, as well as todemonstratives such as ‘this’ and ‘that’. Kaplan’s well-known analysis is described
elsewhere in this volume, so I will describe it only briefly here.

1

More detailed discussions of all of these two-dimensional frameworks and their interrelations can befound in two recent collections: the March 2004 special issue of Philosophical Studies on “The TwoDimensional Framework and its Applications”, and the book Two-Dimensional Semantics:
Foundations and Applications (Garcia-Carpintero and Macia, 2006). See especially Chalmers 2006,
Davies 2004, and Stalnaker 2004, and also the discussion in Soames 2005.

For Kaplan, the “worlds” involved in the first dimension are contexts ofutterance: these can be seen as at least involving the specification of a speaker and atime and place of utterance, within a world. The “worlds” involved on the second
dimension are circumstances of evaluation: these are ordinary possible worlds at
which the truth of an utterance is to be evaluated.
Consider an expression such as:(1) I am hungry now
According to Kaplan’s analysis, when this expression is uttered by Joe at time t1,
it expresses a proposition that is true if and only if Joe is hungry at t1. We can call this
proposition expressed the content of the utterance. This content can naturally be
represented as an intension that is true at all and only those worlds (thosecircumstances of evaluation) in which Joe is hungry. (Kaplan regards propositions as
structured entities rather than intensions, but the difference does not matter muchhere.) In a different context — say, a context with Diana speaking at t2 — an utteranceof the same expression will have a different content. This content will be a
proposition that is true at a world if and only if Diana is hungry at t2 in that world.
The character of an expression is a function from contexts to contents, mapping
a context of utterance to the content of that expression in that context. (If content isseen as an intension, then character is a sort of two-dimensional intension.) So thecharacter of ‘I am hungry’ maps the first context above to the proposition that Joe ishungry at t1, and the second context above to the proposition that Diana is hungry at
t2. Extending this idea to subsentential indexical terms, we can say that the characterof ‘I’ maps the first context to Joe and the second context to Diana; more generally, itmaps any context into the speaker in that context. Similarly, the character of ‘now’
maps any context into the time specified in that context.
The above definition of character is still somewhat imprecise, and many tricky
issues come up in giving a precise definition. But to a rough first approximation, one
can say that the character of an expression maps a context to the content that theexpression would have if uttered in that context. There is more to say than this(especially as Kaplan intends his analysis to apply even to contexts in which there is
no token of the relevant utterance), but this is enough for now. In general, character isassociated with an expression type rather than with an expression token, although thismatter is complicated somewhat by the case of demonstratives such as ‘this’ and‘that’, whose character may vary between different utterances.
On Kaplan’s analysis, the character of indexicals such as ‘I’, ‘now’, and ‘here’,as well as the character of demonstratives such as ‘this’ and ‘that’, reflects their
cognitive significance. For example, ‘I am here now’ has a propositional content thatis true in only some worlds, but its character yields a proposition that is true in allcontexts of utterance. (Kaplan does not “diagonalize” character into an intension, but
it would be easy enough to do so. If one did so, then ‘I am here now’ would be
associated with a diagonal intension that is necessarily true.) So the character rather
than the content seems to reflect the fact that the sentence can be known a priori (ornear enough). Likewise, when a true utterance of ‘this is that’ is cognitively
significant, the occurrences of ‘this’ and ‘that’ will refer to the same object, but theircharacters will differ. So at least in these domains, character behaves a little like a
Fregean sense.
This behavior does not extend to other expressions, however. For example,
Kaplan holds that names refer to the same individual in any context of utterance.
On this view, co-extensive names such as ‘Mark Twain’ and ‘Samuel Clemens’ will
have exactly the same character, and an identity such as ‘Mark Twain is Samuel
Clemens’ will have a character that yields a true proposition in every context, eventhough the identity appears to be a posteriori and cognitively significant. Somethingsimilar applies to natural kind terms such as ‘water’. So on Kaplan’s analysis, names
and natural kind terms have a “constant character” that is dissociated from their
cognitive roles.
One can diagnose the situation by noting that character is most closely tied to thepatterns of context-dependence associated with an expression, rather than to theexpression’s cognitive significance. In the case of indexicals, the patterns of contextdependence of an expression are themselves closely associated with the expression’s
cognitive significance. But for many other expressions, such as names and natural
kind terms, cognitive significance is strongly dissociated from patterns of contextdependence. (The same goes for numerous ordinary context-dependent expressions,such as ‘tall’.) As a result, in the general case, Kaplan’s framework is better suited tothe analysis of the context-dependence of expressions than to an analysis of their
cognitive significance.

2.2 Stalnaker: Diagonal proposition and proposition expressed
Stalnaker’s analysis starts with the idea that although sentences such as‘Hesperus is Phosphorus’ express necessary truths, they are sometimes used to convey
contingent information about the world. Stalnaker (1978) analyzes this contingent
information as the diagonal proposition associated with an utterance.
On Stalnaker’s analysis, the proposition expressed by an utterance is a standard
intension, or a set of possible worlds. So the proposition expressed by an ordinary
utterance of ‘Hesperus is Phosphorus’ is the set of worlds in which Hesperus is
Phosphorus, which is the set of all worlds (leaving aside questions about existence).
Stalnaker defines the propositional concept associated with an utterance as a function
from possible worlds to propositions, mapping a world to the proposition that thatutterance would express in that world. He then defines the diagonal proposition
associated with an utterance as a function that maps a possible world to the truth valueof that utterance when used in that possible world.
Stalnaker individuates utterances in such a way that a given utterance could havebeen used with an entirely different meaning. For example, an utterance of ‘Hesperus
is Phosphorus’ could have been used to express the proposition that Mark Twain is
George Bush, in a world w in which ‘Hesperus’ is used as a name for Twain and‘Phosphorus’ is used as a name for Bush. It follows that while the propositional
concept of my utterance maps the actual world to the proposition that Hesperus is
Phosphorus, it maps world w to the proposition that Twain is Bush (which is itself
presumably the empty set of worlds). The diagonal proposition of my utterance maps
the actual world to the truth-value of the former proposition in the actual world (true),and maps world w to the truth-value of the latter proposition in w (false). So although
my utterance of ‘Hesperus is Phosphorus’ expresses a necessary proposition in theordinary sense, it is associated with a contingent diagonal proposition.
Stalnaker’s propositional concept is a sort of two-dimensional intension, and his
diagonal proposition is the associated diagonal intension. Like Kaplan, Stalnaker’s
framework can be seen as capturing a certain way in which the content of an utterance
depends on the context in which it is uttered. But while Kaplan’s analysis is in effectrestricted to contexts in which the expression retains its original meaning, Stalnaker’s
analysis ranges over contexts in which the expression is used with entirely different
meanings. As a result, Stalnaker characterizes his use of the two-dimensionalframework as a “metasemantic” use: unlike Kaplan’s character, diagonal propositions

are not really part of the meaning of an utterance, but rather capture something about
how meaning is determined by the external world.
Stalnaker uses this framework mainly to analyze the information conveyed byassertions. In a context where the hearer knows the full meanings of the terms used inan utterance (e.g., if they know that ‘Hesperus’ and ‘Phosphorus’ both refer toVenus), and where this knowledge is common ground between speaker and hearer,then the utterance will convey its original propositional content. But if the hearer does
not know the meanings of the terms, then the utterance will convey a different
content. In particular, it will convey the diagonal proposition of the utterance: here,
the proposition that ‘Hesperus is Phosphorus’ expresses a truth. If the common
ground between speaker and hearer includes partial knowledge of meaning — say, theknowledge that ‘Hesperus’ is used to refer to the evening star and that ‘Phosphorus’ isused to refer to the morning star — then worlds outside this common ground are ineffect excluded by presuppositions, and the diagonal proposition will in effect beequivalent to the proposition that the morning star is the evening star (at least acrossthe relevant range of worlds). So in such a context, an assertion of ‘Hesperus is
Phosphorus’ will convey the information that the morning star is the evening star.
In his 1978 paper, Stalnaker says that if one defines an operator ‘†’ such that ‘†P’is true iff P has a necessary diagonal proposition, then ‘†’ is equivalent to the “a priori
truth” operator. In later work (e.g. Stalnaker 2004), however, he retracts that claim. It
is easy to see why. Even paradigmatic a priori claims such as ‘1+1=2’ do not have anecessary diagonal proposition: the diagonal proposition of ‘1+1=2’ is false at a world
where ‘1’ refers to 3 and ‘2’ refers to 7, for example. It is true that a statement such as‘Hesperus is the evening star’, which is arguably an a priori truth, will have a diagonal
proposition that is true in all worlds in a class that is restricted as in the previous
paragraph (by imposing the restriction that ‘Hesperus’ is used to refer to the eveningstar). But in this case, it is the restriction that is doing all the work in connecting the
diagonal proposition to a priori truth.
Because of this, there is no strong connection between diagonal propositions anda priori truth. There is sometimes a connection between an utterance’s diagonal
proposition and its cognitive significance, but this connection arises only in certain
contexts where certain special restrictions due to limited knowledge of meaning are inforce. Because of this, Stalnaker’s diagonal propositions cannot be used to ground a
two-dimensional approach to the cognitive significance of linguistic items in general.

Instead, they are most useful for analyzing what is conveyed by utterances when there
is limited knowledge of meaning in place.
2.3 Evans: Deep necessity and superficial necessity
Evans’ analysis (Evans 1977) is focused on descriptive names: names whosereference is fixed by a description. His main example is the name ‘Julius’, which is
stipulated to be a name for whoever invented the zip, if anyone uniquely invented it (Iwill omit references to uniqueness in what follows, but they should be tacitly
understood). He considers the following sentence:(2) If anyone invented the zip, Julius invented the zip.
If one follows Kripke, then (2) expresses a contingent proposition. ‘Julius’ picks
out the actual inventor (William C. Whitworth) in all worlds, so the proposition isfalse in all worlds where someone other than Whitworth invented the zip.
According to Evans, however, this sort of contingency is superficial. (2) issuperficially contingent, in that the claim ‘It might have been the case that someone
other than Julius invented the zip’ is true. Superficial necessity and contingency of asentence turns on how it embeds within modal operators: S is superficially necessary
iff ‘It is necessary that S’ is true. But Evans suggests that in a deeper sense, (2) isnecessary. He holds that the sentence is necessary because it expresses a necessarycontent. On Evans’ view, there is a semantic rule connecting ‘Julius’ with theinvention of the zip, and this semantic rule makes it the case that the content of (2) isnecessarily true.
Evans’ framework has two modal operators, rather than two intensions. The
framework does have two semantic values: the proposition expressed by an sentence,
which is something like the familiar proposition that is true in all worlds where
Whitworth invented the zip, and the content of the sentence, which behaves ascharacterized above. Neither propositions nor contents are characterized as intensions,but it is easy enough to define intensions in the vicinity. We can say that thesuperficial intension of S is the set of worlds in which the proposition expressed by S
is true: roughly, the set of worlds w such that ‘if w had obtained, S would have been
the case’ is true. We can say that the deep intension of S is the set of worlds in which
the content of S is true. In these terms, (2) has a superficial intension that is false atsome worlds, but a deep intension that is true at all worlds.

In the case of descriptive names such as ‘Julius’, deep necessity (as opposed tosuperficial necessity) seems closely connected to apriority, and deep intensions areclosely connected to an expression’s cognitive role. It is tempting to extend thisconnection beyond the case of descriptive names, but Evans does not discuss otherexpressions, and it is not entirely clear how an extension would go. A more precise
analysis of Evans’ notion of deep necessity would require a more precise
understanding of his notion of “content”, which serves as something of an unanalyzed
primitive in his 1978 article.
From other work, it seems clear that Evans thinks in the case of ordinary proper
names (as opposed to descriptive names), there is a semantic rule that ties a name to
its referent, so that the referent is part of the content. Correspondingly, it seems that
Evans held that identities involving ordinary proper names have a content that isnecessary, so that an identity such as ‘Mark Twain is Samuel Clemens’ is not only
superficially necessary but deeply necessary. If this is right, then the two names
involved will have the same deep intension. So in these cases (and probably inanalogous cases involving natural kind terms), deep necessity and deep intensions are
not as strongly connected to cognitive significance or to apriority as in the case of
descriptive names.
2.4 Davies and Humberstone: ‘Fixedly Actually’ and ‘Necessarily’
The two-dimensional framework of Davies and Humberstone (1981) is based on
an analysis of the operator ‘actually’ (A). AP is true in a world w iff P is true in theactual world. Davies and Humberstone note that ‘P iff AP’ is contingent but knowablea priori. They suggest that although the sentence is contingent, there is an intuitive
sense in which it is necessary: intuitively, no matter which world turns out to be theactual world, ‘P iff AP’ will be true. Likewise, for a contingent empirical truth P, AP
will be necessary, but there is an intuitive sense in which it is contingent: intuitively,
there are some worlds such that if those worlds had been actual, then AP would havebeen false.
This intuition can be formalized by introducing a “floating” actual world into a
possible-worlds model. Instead of simply designating a fixed world as the actual
world, we take actuality to be a feature that can attach to different worlds. We canthen evaluate sentences in a world w, where a world w’ is taken to be actual(“considered as actual”). Or equivalently, we can evaluate sentences at pairs of worlds

(w’, w), where the first world represents the world that is designated as actual, and thesecond world represents the world in which the sentence is evaluated (relative to thedesignation of the first world as actual).2
Doubly-indexed evaluation behaves as follows. A sentence P without modal
operators is true at (w’, w) iff P is true at w according to ordinary singly-indexedevaluation. P is true at (w’, w) iff P is true at (w’, v) for all v (i.e. iff P is true at all
worlds relative to w’ considered as actual). AP is true at (w’, w) iff P is true at (w’, w)(i.e. iff P is true at w’ when w’ is considered as actual). In conjunction with theobvious semantics for truth-functional logical operators, this suffices to recursively
define doubly-indexed evaluation of sentences in modal propositional logic (including‘’ and ‘A’) in terms of standard singly-indexed evaluation of atomic sentences.
Davies and Humberstone then introduce the further operator “fixedly” (F), which
can be defined as follows: FP is true at (w’, w) iff P is true at (v, w) for all v (i.e. iff P
is true at w relative to all worlds considered as actual). The “fixedly actually” operator
FA is consequently such that FAP is true at (w’, w) iff AP is true at (v, w) for all v,i.e., iff P is true at (v, v) for all v. So FAP is true iff P is true at all worlds w when w
itself is considered as actual.
The two crucial modal operators here are  and FA. We can say that P isnecessary when P is true (i.e. when P is true at all worlds when our world isconsidered as actual), and that P is FA-necessary when FAP is true (i.e. when P istrue at all worlds w when w is considered as actual). Let us say that P is A-involving
iff P contains an instance of A or of F. It is easy to see that when P is not Ainvolving, P will be FA-necessary iff it is necessary. But when P is A-involving, thetwo may come apart. In particular, the sentence ‘P iff AP’ is not necessary, but it is
FA-necessary. Likewise, for a contingent atomic truth P, AP is necessary, but it is
FA-contingent. So Davies and Humberstone suggest that FA-necessity captures theintuitive sense in which these two sentences are necessary and contingent
respectively.
Davies and Humberstone also extend the discussion to Evans’ case of descriptive
names. They observe that descriptive names such as ‘Julius’ behave very much likeA-involving descriptions of the form ‘The actual inventor of the zip’. For example,2

For simplicity of presentation, I depart from Davies and Humberstone’s own formalization, but theformalization here gives equivalent results.

just as ‘Julius invented the zip’ seems contingent and a priori, ‘The actual inventorof the zip invented the zip’ seems contingent and a priori. Furthermore, it is easy to
see that when formalized in modal predicate logic, sentences of the latter form are notnecessary but are FA-necessary. This mirrors Evans’ claim that ‘Julius (if he exists)
invented the zip’ is not superficially necessary but is deeply necessary.
Davies and Humberstone suggest the natural hypothesis that descriptive names arein fact abbreviated A-involving descriptions, and that Evans’ deep necessity is justFA-necessity.
Davies and Humberstone speculate that all contingent a priori sentences may be(perhaps tacitly) A-involving sentences that are contingent and FA-necessary.3 They
also suggest that some necessary a posteriori sentences are A-involving sentences thatare necessary and FA-contingent: for example, ‘The actual F is G’ (where ‘the F is G’
is contingent) and analogous claims involving descriptive names. They speculatetentatively that natural kind terms (such as ‘water’) might be seen as abbreviated Ainvolving descriptions (such as ‘the actual waterish stuff around here’), in which casenecessary a posteriori identities such as ‘water is H2O’ may also benecessary and FA-contingent. However, they do not extend the claim to all necessary
a posteriori sentences. In particular, they hold that ordinary proper names are not Ainvolving, so that identities involving ordinary proper names (such as‘John is Tom’) are FA-necessary iff they are necessary. It follows from this thatnecessary a posteriori identities involving these names are FA-necessary, rather thanFA-contingent.
Davies and Humberstone do not posit two semantic values to go along with their
two modal operators, but one could naturally do so. We can say that the standard
intension of P is true at w iff P is true at w when our world is considered as actual (i.e.
iff P is true at (a, w), where a is the actual world), and that the FA-intension of P istrue w iff P is true at w when w is considered as actual (i.e. iff P is true at (w, w)). We
can also define the two-dimensional intension of P in the obvious way; then the FAintension will be equivalent to the “diagonal” of the two-dimensional intension.
As defined here, FA-intensions are closely tied to apriority for some sentences:
especially for A-involving sentences, and for tacitly A-involving sentences such as

3This claim will be true only if all contingent a priori sentences are A-involving. For some reasons for
doubt about this (involving indexical contingent a priori sentences, for example), see Chalmers 2006.

those involving descriptive names and perhaps natural kind terms (if these are indeed
tacitly A-involving). If the “actually” operator were the only source of the necessary aposteriori and the contingent a priori, then there would be a strong general tie between
FA-intensions and apriority. But if there are other sources of the necessary a posteriori
and the contingent a priori (such as ordinary proper names and indexicals), then inthese cases, FA-intensions will not be closely tied to apriority at all.

3

Two-Dimensionalism
The two-dimensional approaches discussed above all introduce “first-

dimensional” semantic values or modal notions that are more strongly connected toapriority and to cognitive significance than are the more familiar “seconddimensional” semantic values and modal notions. But in each of these approaches, theconnection is somewhat attenuated. In the case of Kaplan’s character, the connection
only applies in the case of indexicals. In the case of Evans it is asserted only fordescriptive names. In the case of Davies and Humberstone, it holds only for Ainvolving expressions and tacitly A-involving expressions such as descriptive names
and perhaps some natural kind terms. In the case of Stalnaker, it applies only undercertain strong restrictions on the domain of a diagonal proposition, or not at all.
In recent years, a number of philosophers (e.g. Chalmers 1996, 2002, 2004 andJackson 1998, 2004; see also Braddon-Mitchell 2004, Lewis 1993, and Wong 1996)
have advocated a two-dimensional approach on which first-dimensional semantic
values are connected to apriority and cognitive significance in a much stronger andmore general way. On this approach, the framework applies not just to indexicals anddescriptive names, but to expressions of all sorts. Proponents hold that any expression(or at least, any expression token of the sort that is a candidate for having an
extension) can be associated with an intension that is strongly tied to the role of theexpression in reasoning and in thought. The term two-dimensionalism is usually usedfor views of this sort.

3.1 The core claims of two-dimensionalism
Five core claims of two-dimensionalism are as follows.
(T1) Every expression token (of the sort that is a candidate to have an
extension) is associated with a primary intension, a secondary intension,and a two-dimensional intension. A primary intension is a function fromscenarios to extensions. A secondary intension is a function frompossible worlds to extensions. A two-dimensional intension is a function
from ordered pairs of scenarios and worlds to extensions.
(T2) When the extension of a complex expression token depends
compositionally on the extensions of its part, the value of each of its
intensions at an index (world, scenario, or ordered pair) depends in thesame way on the values of the corresponding intensions of its parts atthat index.
(T3) The extension of an expression token coincides with the value of its
primary intension at the scenario of utterance and with the value of thesecondary intension at the world of utterance.
(T4) A sentence token S is metaphysically necessary iff the secondary
intension of S is true at all worlds.
(T5) A sentence token S is a priori (epistemically necessary) iff the
primary intension of S is true at all scenarios.
In what follows I will first clarify and motivate these principles, without
precisely defining all of the key notions or making a case for their truth. In later
sections, I will discuss how the relevant notions (especially the notion of a primary
intension) can be defined, in such a way that the principles might be true. These
principles should not be taken to provide an exhaustive characterization of twodimensionalism, but they lie at the core of the view.
Start with claim (T1). Here, a scenario is something akin to a possible world, but
it need not be a possible world. In the most common two-dimensionalist treatments, ascenario is a centered world: an ordered triple of a possible world along with anindividual and a time in that world. Other treatments of scenarios are possible (seeChalmers 2004), but I will use this understanding here.

An expression’s secondary intension (or what Jackson calls its C-intension) is
just its familiar post-Kripkean intension, picking out the extension of the expressionin counterfactual worlds. For example, the secondary intension of a token of ‘I’ as
used by speaker S picks out S in all worlds. The secondary intension of ‘water’ picksout H2O in all worlds. The secondary intension of ‘Julius’ picks out William C.
Whitworth in all worlds. And so on.
An expression’s primary intension works quite differently. I will defer a fullcharacterization, but some examples will give a rough idea. The primary intension ofa token of ‘I’, evaluated at a centered world, picks out the designated individual at the“center” of that world. (So the primary intension of my use of ‘I’, evaluated at a world
centered on Napoleon, picks out Napoleon, rather than David Chalmers.) The primary
intension of a token of ‘water’, very roughly, picks out the clear, drinkable liquid withwhich the individual at the center is acquainted. (So the primary intension of my useof ‘I’, evaluated at a “Twin Earth” world centered on a subject surrounded by XYZ in
the oceans and lakes, picks out XYZ, rather than H2O.) The primary intension of atoken of ‘Julius’ picks out whoever invented the zip in a given world. (So the primary
intension of ‘Julius’, evaluated at a world where Tiny Tim invented the zip, picks out
Tiny Tim, rather than William C. Whitworth.) And so on.
Thesis (T1) also holds that expression tokens can be associated with a twodimensional intension: roughly, a function from (scenario, world) pairs to extensions.
We can then say that at least on the centered worlds understanding, the primary
intension coincides with the “diagonal” of the two-dimensional intension (i.e. thevalue of S’s primary intension at a centered world w coincides with the value of S’s
two-dimensional intension at the pair (w, w*), where w*is the possible-world elementof w). Likewise, the secondary intension coincides with the “row” of the twodimensional intension determined by the scenario of an utterance (i.e. the value of S’s
secondary intension at a world w coincides with the value of S’s two-dimensional
intension at (a, w), where a is the scenario of utterance). However, for most purposes
the two-dimensional intension of an expression is somewhat less important than its
primary and secondary intension, and the two-dimensionalist need not hold that an
expression’s primary and secondary intension are derivative from its two-dimensional
intension.

Thesis (T2) says that the primary and secondary intensions of a complex
expression depend on the primary and secondary intensions of its parts according to
the natural compositional semantics. For example, the primary intension of ‘I am
Julius’ will be true at a scenario if the individual at the center of that scenario is theinventor of the zip in that scenario.
Thesis (T3) states a natural connection between the intensions and the extensionof an expression token. This thesis requires that for every utterance, just as there is
one world that is the world of the utterance, there is also one scenario that is the
scenario of the utterance. If scenarios are understood as centered worlds, this will be aworld centered on the speaker and the time of the utterance. When evaluated at the
scenario and world of utterance, the primary and secondary intensions (respectively)of an expression token will coincide with the extension of the expression token. At
other worlds and scenarios, however, the values of these intensions may diverge fromthe original extension, and from each other.
Turning to claims (T4) and (T5): Here, we can say that S is a priori when it
expresses a thought that can be justified independently of experiences. S is
metaphysically necessary when it is true with respect to all counterfactual worlds(under the standard Kripkean evaluation). Thesis (T4) is a consequence of thestandard understanding of metaphysical necessity and the corresponding intensions.
Thesis (T5) is intended to be an analog of thesis (T4) in the epistemic domain.
Thesis (T5) is the distinctive claim of two-dimensionalism. It asserts a verystrong and general connection between primary intensions and apriority, one muchstronger than obtains with the other two-dimensional frameworks discussed earlier. It
is possible that a two-dimensionalist might grant some limited exceptions to thesis(T5) (say, for certain complex mathematical statements that are true but unknowable)
while still remaining recognizably two-dimensionalist. But it is crucial to the twodimensionalist position that typical a posteriori identities involving proper names or
natural kind terms, such as ‘Mark Twain is Samuel Clemens’ or ‘water is H2O’, havea primary intension that is false in some scenario.
Consequences of the previous theses include the following:(T6) A sentence token S is necessary a posteriori iff the secondary
intension of S is true at all worlds but the primary intension of S is false atsome scenario.

(T7) A sentence token S is contingent a priori iff the primary intension of
S is true at all scenarios but the secondary intension of S is false at someworld.
So two-dimensionalism proposes a unified analysis of the necessary a posteriori:
all such sentences have a necessary secondary intension but a contingent primary
intension. Likewise, it proposes a unified analysis of the contingent a priori: all suchsentences have a contingent primary intension but a necessary secondary intension.
From the previous theses, one can also draw the following conclusions about the
primary and secondary intensions of both sentential and subsentential expressions.
Here ‘A’ and ‘B’ are arbitrary expressions of the same type, and ‘A≡B’ is a sentencethat is true iff ‘A’ and ‘B’ have the same extension. For example, if A and B are
singular terms, ‘A≡B’ is just the identity statement ‘A≡B’, while if A and B aresentences, ‘A≡B’ is the biconditional ‘A iff B’.
(T8) ‘A≡B’ is metaphysically necessary iff A and B have the same
secondary intension.
(T9) ‘A≡B’ is a priori (epistemically necessary) iff A and B have the same
primary intension.
It follows that for a posteriori necessary identities involving proper names, suchas ‘Mark Twain is Samuel Clemens’, the two names involved will have the same
secondary intensions, but different primary intensions. Something similar applies tokind identities such as ‘water is H2O’. If this is correct, then primary intensions
behave in these cases in a manner somewhat reminiscent of a Fregean sense.
3.2 Epistemic two-dimensionalism
For these claims, especially claim (T5), to be grounded, we need to have a betteridea of what primary intensions are. Clearly, they must differ from characters,
diagonal propositions, deep intensions, and FA-intensions, at least as these areunderstood by their proponents. Here, I will outline one approach (the approach I
favor) to understanding primary intensions. This approach, which we might call
epistemic two-dimensionalism, is elaborated in much greater detail in other works(Chalmers 2002a, 2002b, 2004, 2006; Chalmers and Jackson 2001).

According to epistemic two-dimensionalism, the connection between primary
intension and epistemic notions such as apriority requires that primary intensions
should be characterized in epistemic terms from the start. On this approach, thescenarios that are in the domain of a primary intension do not represent contexts ofutterance. Rather, they represent epistemic possibilities: highly specific hypotheses
about the character of our world that are not ruled out a priori. The value of an
expression’s primary intension at a scenario reflects a speaker’s rational judgments
involving the expression, under the hypothesis that the epistemic possibility inquestion actually obtains.
For example, ‘water is not H2O’ is epistemically possible, in the sense that its
truth is not ruled out a priori. Correspondingly, it is epistemically possible that ourworld is the XYZ-world (or at least, that it is qualitatively just like the XYZ-world).
And if we suppose that our world is the XYZ-world (that is, that the liquid in theoceans and lakes is XYZ, and so on), then we should rationally endorse the claim‘water is XYZ’, and we should rationally reject the claim ‘water is H2O’. So theprimary intension of ‘water is H2O’ is false at the XYZ-world, and the primary
intension of ‘water is XYZ’ is true there.
Likewise, ‘Mark Twain is not Samuel Clemens’ is epistemically possible, in thesense that it is not ruled out a priori. Correspondingly, it is epistemically possible that
our world is a world w where one person wrote the books such as Tom Sawyer that we
associate with the name ‘Mark Twain’, and a quite distinct person is causallyconnected to our use of the term ‘Samuel Clemens’. If we suppose that w is our world,
then we should rationally endorse the claim ‘Mark Twain is not Samuel Clemens’. So
the primary intension of ‘Mark Twain is Samuel Clemens’ is false at w.
According to two-dimensionalism, something similar applies to any Kripkean aposteriori necessity. For any such sentence S, the negation of S is epistemicallypossible. And it is plausible that for any such S, there is a world w such that if we
suppose that our world is qualitatively like w, we should rationally reject S. If so, thenthe primary intension of S is false at w. If this pattern generalizes to all a posteriori
necessary sentences, then any such sentence has a primary intension that is false atsome scenario, as thesis (T6) above suggests.
Here, primary intensions are characterized in thoroughly epistemic terms. It
should be noted that the claims above are in no tension with the Kripkean claims that‘water is H2O’ is metaphysically necessary, or that ‘water’ picks out H2O in all
worlds. Even Kripke allows that ‘water is not H2O’ is epistemically possible. And it isa familiar Kripkean point that there can be an epistemic necessitation between two
statements A and B even when there is no metaphysical necessitation between them(witness ‘X is the source of heat sensations’ and ‘X is heat’). We simply have tostrongly distinguish this sort of epistemic evaluation of sentences in worlds (whichturns on epistemic necessitation) from the usual sort of counterfactual evaluation(which turns on metaphysical necessitation). Primary intensions are grounded in theformer; secondary intensions are grounded in the latter.
3.3 Defining primary intensions
It remains to define primary intensions more precisely. To generalize from theabove, we might suggest that the primary intension of a sentence S is true at a
scenario w iff the hypothesis that w is actual should lead us to rationally endorse S.
Somewhat more carefully, we can say that the primary intension of S is true at a
scenario w iff D epistemically necessitates S, where D is a canonical specification of
w. It remains to clarify the notion of a scenario, a canonical specification, andepistemic necessitation.
Scenarios are highly specific epistemic possibilities. On the centered-worlds
version of epistemic two-dimensionalism, scenarios are identified with centeredworlds. It is also possible to develop a version of epistemic two-dimensionalism
where scenarios are more strongly dissociated from ordinary possible worlds (seeChalmers 2004; forthcoming a), and instead are characterized in more purely
epistemic terms (for example, as maximal epistemically consistent sets of sentences in
an idealized language). But I will focus on the centered-worlds understanding here.
For any possible world w, it is epistemically possible that w is actual; or at least,
it is epistemically possible that a world qualitatively identical to w is actual. (Moreprecisely: it is epistemically possible that D is the case, where D is a complete
qualitative characterization of w. More on this notion shortly.) But epistemic
possibilities are more fine-grained than possible worlds. For example, the informationthat the actual world is qualitatively like a possible world w is epistemically consistent
with various different epistemic possible claims about one’s self-location: forexample, it is consistent with the claims ‘It is now 2004’ and ‘It is now 2005’.

To handle these claims about self-location, we model epistemic possibilities
using centered worlds. The individual and the time marked at the “center” of acentered world serve as a “you are here” marker, which serves to settle these claims
about self-location. For a given thinker, the hypothesis that a given centered world w
is actual can be seen as the hypothesis: ‘D is the case, I am F, and the current time isG’, where D is a complete qualitative characterization of w, and F and G are
qualitative descriptions that pick out the individual and the time at the center of w. We
can think of this conjunctive claim as a canonical specification of the centered worldin question.
In the foregoing, a qualitative vocabulary is, to a first approximation, avocabulary that is free of terms (such as names and natural kind terms) that give riseto Kripkean a posteriori necessities and a priori contingencies. (Restricting worlddescriptions to a vocabulary of this sort avoids obvious problems that would arise if
we allowed, for example, ‘water is H2O’ into the description of the XYZ-world. For
more on the characterization of qualitative, or “semantically neutral”, vocabulary, see
Chalmers 2004.) A complete qualitative characterization of w is a qualitative
statement D such that (i) D is true of w, and (ii) if E is a qualitative statement that istrue of w, then D necessitates E.
We also need to define epistemic necessitation. To a first approximation, we cansay that D epistemically necessitates S iff accepting D should lead one to rationallyendorse S (without needing further empirical information, given idealized reflection).
On a refined definition, we can say that D epistemically necessitates S iff a
conditional of the form ‘D ⊃ S’ is a priori. The refined definition is arguably better insome difficult cases, but for many purposes, the first approximation will suffice.
Because they are defined in epistemic terms, there is an inbuilt connection
between primary intensions and the epistemic domain. In particular, there will be astrong connection to apriority. When a sentence token S is a priori, then it will be
epistemically necessitated by any sentence whatsoever (this is especially clear for thesecond understanding of epistemic necessitation above), so its primary intension willbe true in all scenarios. When a sentence token S is not a priori, then its negation willbe epistemically possible, and S will be false relative to some highly specificepistemic possibility. As long as there is a scenario for every epistemic possibility,
then the primary intension of S will be false in some scenario. (On the centered worlds

understanding of scenarios, the existence of a scenario for every epistemic possibility
is a substantive but plausible claim; see Chalmers 2002c and 2004.) If so, then thesis(T5) will be correct.
One can define the secondary intension of a sentence in a similar, if morefamiliar, way. The secondary intension of S is true at a world w iff D metaphysicallynecessitates S, where D is a canonical specification of w. Here a canonical specificant
can be characterized much as before as a complete specification, although here it is
not necessary to impose the restriction to qualitative specifications. Metaphysicalnecessitation could be taken as basic, or perhaps better, we can define it in terms ofsubjunctive conditionals: D metaphysically necessitates S when a subjunctive
conditional of the form ‘if D had been the case, S would have been the case’ is true.
One can likewise define the two-dimensional intension of a sentence. The twodimensional intension of S is true at (v, w) iff D epistemically necessitates that D’metaphysically necessitates S, where D is a canonical specification of the scenario v
and D’ is a canonical specification of the world w. If we understand epistemicnecessitation in terms of a priori material conditionals and metaphysical necessitationin terms of subjunctive conditionals, this will be the case iff ‘D ⊃ (D’ ⇒ S)’ is apriori, where the outer conditional is material and the inner conditional is subjunctive.
This discussion of the intensions of sentences can be extended to the intensionsof subsentential expressions in a reasonably straightforward way. For details, see
Chalmers 2004.
3.4 The roots of epistemic two-dimensionalism
The epistemic two-dimensional framework is grounded in a thesis about the
scrutability of reference and truth: once a subject is given enough information about
the character of the actual world, then they are in a position to make rational
judgments about what their expressions refer to and whether their utterances are true.
For example, once we are given enough information about the appearance, behavior,composition, and distribution of various substances in our environment, as well as
about their relations to ourselves, then we are in a position to conclude (without
needing further empirical information) that water is H2O. And if instead we were
given quite different information, characterizing our environment as a “Twin Earth”
environment, then we would be in a position to conclude that water is XYZ.

Of course, if we allow the “enough information” to include arbitrary truths, suchas ‘water is H2O’, the scrutability claim will be trivial. But we can impose significant
restrictions on the information without compromising the plausibility of the thesis.
For example, one can argue that even if we restrict ourselves to truths that do not use
the term ‘water’ or cognates, it remains the case that given enough truths of this kind,
we are in a position to know the truth of ‘water is H2O’ (see Chalmers and Jackson
2001). The same goes for many or most other terms, plausibly including most names
or natural kind terms.
The upshot is that there is some reasonably restricted vocabulary V, such that forarbitrary statements T, then once we know enough V-truths we will be in a position toknow (without needing further empirical information) the truth-value of T. Just how
restricted such a vocabulary can be is an open question. Chalmers and Jackson (2001)
argue that PQTI, a conjunction of microphysical, phenomenal, and indexical truths
along with a “that’s all” truth, can serve as a basis. But this claim is not required here.
All that is required for present purposes is that some qualitative vocabulary, conjoinedwith indexical terms such as ‘I’ and ‘now’, is sufficient.4
This suggests that for any true sentence token S, there is a V-truth D such that D
epistemically necessitates S, in that a subject given the information that D will be in aposition to rationally endorse S (given ideal rational reflection). Furthermore, itappears that in principle, no further empirical information is needed to make thisjudgment; if such information were required, we could simply include it (orequivalent qualitative information) in D to start with. This strongly suggests that there
is a non-empirical warrant for the transition from D to S. In particular, one can makethe case that in these cases, the material conditional ‘D ⊃ S’ will be a priori. (Thiscase is made at length by Chalmers and Jackson 2001). If this is correct, then D
epistemically necessitates S in the second, stronger sense given above.

4

As before, a qualitative vocabulary is one that excludes terms, such as names and natural kind terms
that give rise to Kripkean a posteriori necessities. A qualitative vocabulary may include all sorts ofhigh-level expressions: ‘friend’, ‘philosopher’, ‘action’, ‘believe’, and ‘square’, for example. It will notdesignate individuals by using names: instead it will make existential claims of the form ‘there exist
such-and-such individuals with such-and-such qualitative properties’. Some theoretical terms (perhaps
including microphysical terms) may be excluded, but information conveyed using these terms caninstead be conveyed by the familiar Ramsey-sentence method, characterizing a network of entities andproperties with appropriate causal/nomic connections to each other and to the observational and thephenomenal. For familiar reasons, no important information is lost by doing this.

The scrutability claim does not apply only to the actual world. It is plausible thatfor all sorts of scenarios, if we are given the information that the scenario is actual,then we are in a position to make a rational judgment about the truth-value ofarbitrary sentences. For example, if we are given a complete qualitative
characterization of the bodies visible in the sky at various times, with the feature thatno body is visible both in the morning sky and the evening sky, then we should
rationally reject the claim ‘Hesperus is Phosphorus’. This sort of judgment is part ofthe inferential role associated with our use of the terms ‘Hesperus’ and ‘Phosphorus’.
The point is general: for any expression that we use, then given sufficient information
about the actual world, certain judgments using the expression will be irrational, andcertain other judgments using the expression will be rational. It is arguable that theexpressions of any language user will have this sort of normative inferential role. Thisis just part of what being a language user involves.
It is this sort of inferential role that grounds the primary intension of an arbitrary
expression (as used by an arbitrary speaker). A given sentence token will be
associated with a raft of conditional rational judgments, across a wide variety of
scenarios. This raft of conditional judgments corresponds to the sentence’s primary
intension. Something very similar applies to subsentential expressions: for a singularterm, for example, there will be a raft of conditional rational judgments using theexpression across a wide variety of scenarios, and these can be used to define theextension of the expression relative to those scenarios (see Chalmers 2004). So wewill have substantial primary intensions for a wide range of sentential and subsential
expression tokens.
It should be noted that nothing here requires that the expressions in question be
definable in simpler terms (such as in qualitative terms), or that they be equivalent todescriptions (even to rigidified or “actually”-involving descriptions). The inferential
roles in question will exist whether or not the term is definable and whether or not it isequivalent to a description (for more on this, see Chalmers and Jackson 2001 andChalmers 2002a).
These claims are quite compatible with Kripke’s epistemological argument thatterms such as ‘Gödel’ are not equivalent to descriptions. In effect, Kripke describes a
scenario w where someone called ‘Schmidt’ proved the incompleteness of arithmetic,
and then it was stolen by someone called ‘Gödel’ who moved to Princeton, and so on.
Kripke’s argument might be put by saying that (i) w is not ruled out a priori, and (ii) if

we accept that w obtains, we should reject the claim ‘Gödel proved theincompleteness of arithmetic’, so (iii) ‘Gödel proved the incompleteness ofarithmetic’ is not a priori. A two-dimensionalist will put this by saying that the
primary intension of ‘Gödel proved the incompleteness of arithmetic’ is false at w, sothat the primary intension of ‘Gödel’ differs from that of ‘the prover of theincompleteness of arithmetic’. If Kripke’s argument generalizes to other descriptions,
it will follow that the primary intension of Gödel is not equivalent to the primary
intension of any such description. But nothing here begins to suggest that ‘Gödel’lacks a primary intension.
Although the primary intension of an expression may not be equivalent to that ofa description, one can often at least approximately characterize an expression’s
primary intension using a description. For example, one might roughly characterize
the primary intension of a typical use of ‘water’ by saying that in a centered world w,
it picks out the dominant clear, drinkable liquid with which the individual at thecenter of w is acquainted. And one might roughly characterize the primary intensionof ‘Gödel’ by saying that it picks out that individual who was called ‘Gödel’ by thosefrom whom the individual at the center acquired the name. But these characterizations
will usually be imperfect, and it will be possible to find Kripke-style counterexamples
to them. Ultimately a primary intension is not grounded in any description, but rather
is grounded in an expression’s inferential role.
3.5 Two-dimensionalism and semantic pluralism
Two-dimensionalism is naturally combined with a semantic pluralism, according
to which expressions and utterances can be associated with many different semantic(or quasi-semantic) values, by many different semantic (or quasi-semantic) relations.
On this view, there should be no question about whether the primary intension of thesecondary intension is the content of an utterance. Both can be systematicallyassociated with utterances, and both can play some of the roles that we want contents
to play. Furthermore, there will certainly be explanatory roles that neither of themplay, so two-dimensionalism should not be seen as offering an exhaustive account of
the content of an utterance. Rather, it is characterizing some aspects of utterance
content, aspects that can play a useful role in the epistemic and modal domains.
Likewise, there should be no question about which of the two-dimensionalframeworks described in this paper is the “correct” framework. Each frameworkoffers a different quasi-semantic relation that associates expressions with twodimensional semantic values, and each of these may play an explanatory role indifferent domains. Each has different properties. Most obviously, primary intensions
have a stronger connection to apriority and cognitive significance than the semantic
values described earlier. Unlike characters, deep intensions, and FA-intensions, theprimary intension associated with an a posteriori identity such as ‘Mark Twain isSamuel Clemens’ will be contingent. Unlike diagonal propositions, the primary
intension of an a priori sentence such as ‘2+2=4’ will be necessary.
These differences arise from the differences in the way the semantic relations aredefined. Unlike characters and diagonal propositions, primary intensions are notdefined in terms of context-dependence. Unlike deep intensions, they are not definedin terms of a prior notion of content. Unlike FA-intensions; they are not defined interms of the behavior of an ‘actually’ operator. Rather, they are defined in epistemic
terms.
Because they are defined in epistemic terms, primary intensions can often vary
between tokens of an expression type. This will happen most obviously for contextdependent terms such as ‘tall’, for which tokens in different contexts will be
associated with different inferential roles. Primary intensions may also vary between
different tokens of the same name (especially by different speakers), for different
tokens of the same demonstrative (e.g. ‘this’ or ‘that’), and perhaps also for different
tokens of the same natural kind term. It follows that in these cases, a primary
intension does not constitute an expression’s linguistic meaning, where this isunderstood as what is common to all tokens of an expression type, or as what isrequired for any competent use of the expression. Instead, a primary intension can beseen as a kind of utterance content.
Even if they are not always part of linguistic meaning, primary intensions arenevertheless a sort of truth-conditional content. The primary intension of an utterance
yields a condition under which the utterance will be true. For example, the primary
intension of ‘there is water in the glass’ will be true at some scenarios and false atothers, and the utterance will be true iff the primary intension is true at the scenario of
the utterance (roughly, if the glass picked out by the individual at the center of the
scenario contains the dominant watery stuff in the environment around the center).
This can be seen as an epistemic truth-condition for the utterance, specifying how thetruth of the utterance depends (epistemically) on which epistemically possible

scenario turns out to be actual. This contrasts with the “metaphysical” truth-condition
corresponding to the secondary intension, which might be seen as specifying how thetruth of the utterance depends (metaphysically) on which metaphysically possible
world is actual. Again, there is no need to decide the question of which of these is thetruth-condition associated with an utterance.
Are primary intensions a sort of semantic content? This depends on how weunderstand the notion of semantic content. If we stipulate that the semantic content ofan utterance is truth-conditional content, then primary intensions are a variety ofsemantic content. On the other hand, if we stipulate that semantic content is linguistic
meaning in the sense above, or that semantic content is always associated withexpression types and not tokens, then primary intensions are not in general part ofsemantic content (though they may be part of semantic content for some expressions,such as some indexicals and qualitative expressions). In any case, once we are clear
on the various properties of these intensions, nothing important to the frameworkturns on the terminological question of whether they count as “semantic”.
A semantic pluralist can allow that for some explanatory purposes, it may be
useful to modify two-dimensionalist semantic values in some respects. For example,
one might define the structured primary intension of a complex expression as astructured entity involving the primary intensions of the simple expressions involvedin the expression’s logical form. One might likewise define structured secondary andtwo-dimensional intensions. Given compositionality, a structured primary intension
will determine an unstructured primary intension (and likewise for the otherintensions), but the reverse need not be the case. This means that structured primary
intensions are more fine-grained than unstructured primary intensions: for example,
all a priori truths will have the same unstructured primary intension (one that is true atall scenarios), but they will have different structured primary intensions. The finegrainedness of structured intensions makes a difference for certain purposes,
described below.
What are propositions, according to two-dimensionalism? Some twodimensionalists (e.g. Jackson 1998) hold that propositions are sets of possible worlds,
in which case a given utterance expresses two propositions (a primary proposition anda secondary proposition). This view is naturally combined with the view that there are
no necessary a posteriori propositions: necessary a posteriori sentences have a
primary proposition that is contingent and knowable only a posteriori, and asecondary proposition that is necessary and knowable a priori. Other twodimensionalists may hold that propositions have more structure than this. For
example, one could hold that propositions are structured entities involving both theprimary and secondary intensions (and/or perhaps the two-dimensional intension) of
the simple expressions involved. A two-dimensionalist of this sort may allow that
there are necessary a posteriori propositions.
A semantic pluralist view tends to suggest that there are numerous entities which
can play some of the explanatory roles that propositions are supposed to play, and that
there is no need to settle which of these best deserves the label ‘proposition’. My own
view is that if one were forced to identify propositions with one sort of entity that canbe modeled in the framework, there would be a good case for choosing structured
two-dimensional entities of some sort (perhaps those discussed as candidates forFregean senses, below). But one might also allow that at least for some purposes,
propositions should be seen as entities more fine-grained than any two-dimensional
objects, so that propositions can be associated with intensions without themselves
being intensions. In any case, core two-dimensionalism as characterized above iscompatible with a wide range of views here.

4

Applications of two-dimensionalism
I will briefly sketch some applications of the two-dimensionalism outlined in theprevious section.
(i) Fregean sense (Chalmers 2002a): Thesis (T9) above says that two expressions
A and B have the same primary intensions iff ‘A≡B’ is epistemically necessary. This isreminiscent of the Fregean claim that two singular terms A and B have the same sense
iff ‘A=B’ is cognitively insignificant. It suggests that primary intensions can play atleast some of the roles of a Fregean sense, individuating expressions by theirepistemic role. Of course there are some differences. For example, primary intensions
are not as fine-grained as Fregean senses: a priori equivalent expressions (such as‘7+3’ and ‘10’) will have different Fregean senses, but they have the same primary
intension (though they will usually have different structured primary intensions).
Further, there are differences between primary intensions and Fregean senses in thecase of indexicals: for example, uses of ‘I’ by different speakers have the same
primary intension, whereas Frege held that they have different senses. Relatedly,where Frege held that sense determines reference, primary intensions do not
determine extensions in a strong sense (although they may still determine extension
relative to context), as two expressions may have the same primary intensions anddifferent extensions. Still, may nevertheless think of primary intensions as a broadly
Fregean aspect of an expression’s content.
One can also use the two-dimensional framework to define semantic values thatbehave even more like Fregean senses. (Here I go beyond the discussion in Chalmers
2002a.) We might stipulate that the sense of a simple expression token is an orderedpair of its primary intension and its extension, and that the sense of a complex
expression token is a structured complex made up of the senses of its parts. Now,
most pairs of a priori equivalent expressions, such as ‘7+3’ and ‘10’ will havedifferent senses. (The only potential exceptions will arise if there are a priori
equivalent but cognitively distinct simple expressions, which is not obvious.)
Furthermore, uses of ‘I’ by different speakers will have different senses. And now,sense determines reference in the strong sense. So entities of this sort might be seen asvery much akin to Fregean senses, and we might think of the structured entity
associated with a sentence token as akin to a Fregean thought.
(ii) Contents of thoughts (Chalmers 2002b). One can extend the framework above
so that primary and secondary intensions are not just associated with sentences butwith thoughts, where these are understood as occurrent mental states. For example,
my thought water is H2O will have a contingent primary intension (false in the XYZscenario) but a necessary secondary intension. One can then argue that a thought’s
primary intension is a sort of narrow content: content that is shared between
intrinsically identical thinkers. For example, when Oscar on Earth and Twin Oscar on
Twin Earth say ‘water is wet’, the thoughts they express will have different secondary
intensions (so secondary intensions are a sort of “wide content”), but they will havethe same primary intension.
(iii) Belief ascriptions (Chalmers 2002b): One can use this framework to analyze
ascriptions of belief and other propositional attitudes. As a first attempt, one mightsuggest that an ascription ‘S believes that P’ is true iff the referent of S has a belief
with the primary intension of ‘P’ (in the mouth of the ascriber), or a belief with thesecondary intension of ‘P’ (in the mouth of the ascriber). Neither of these suggestions
works: the first is falsified by cases such as ‘John believes that I am hungry’, whilethe second is falsified by cases such as ‘Lois believes that Clark Kent can fly’.
However, more sophisticated analyses are possible. For example, Chalmers 2002b
suggests
An utterance of ‘S believes that P’ is true iff the referent of S has a belief
with the structured secondary intension of ‘P’ (in the mouth of theascriber) and with an appropriate structured primary intension.
Here, “appropriate” functions to pick out a range of primary intensions (allowing,for example, that the Pierre can satisfy a ‘London’-involving ascription even if heuses the term with a different primary intension), where this range may depend on thecontext of utterance. (Structured intensions are needed in order that independent“appropriateness” constraints may be imposed separately on each element of a belief.)
This analysis is closely related to “hidden-indexical” analyses of belief ascriptions,with primary intensions playing the role of “modes of presentation”.
One can also use primary intensions to give an analysis of de re attitude
ascriptions, in the style of Kaplan 1968.
A de re attitude ascription ‘S believes of X that it is F’ is true iff S has abelief with the secondary intension of ‘X is F’, and which picks out thereferent of X under a de re-appropriate primary intension.
Here, the conditions on a de re-appropriate primary intension may again becontext-independent, but to a first approximation we can think of such an intension as
one that is acquaintance-entailing: necessarily, if a subject S has a state with a de reappropriate primary intension that picks out extension E, then S will be acquainted
with E.
(iv) Indicative conditionals (Weatherson 2001): We can also use epistemic twodimensionalism to give a possible-worlds-style analysis of the intuitive acceptabilityconditions of indicative conditionals that is analogous to the familiar Lewis-Stalnaker
analysis of subjunctive conditionals.
A token of an indicative conditional ‘If P, then Q’ is acceptable iff theepistemically closest scenario satisfying the primary intension of ‘P’ (inthe mouth of the speaker) also satisfies the primary intension of ‘Q’.
Of course an elaboration of this account requires an elaboration of what
epistemic closeness amounts to. But given that the familiar Ramsey Test for theacceptability of an indicative conditionals is defined in epistemic terms (if one
conditionally accepts P, should one rationally conclude Q?), and given that primary
intensions are defined in very similar terms, it is not surprising that there is a close
relation.
(v) Conceivability and possibility (Chalmers 2002): If thesis (T5) is correct, itlicenses a certain sort of move from conceivability to possibility. Let us say that S isconceivable when it is epistemically possible: that is, when S is not ruled out a priori.
If T5 is correct, then when S is conceivable, the primary intension of S will be true atsome scenario. If scenarios are centered worlds, then there will be some centered(metaphysically possible) world w satisfying the primary intension of S. This does notentail that S is metaphysically possible, but it nevertheless allows us to draw
conclusions about metaphysical possible worlds from premises about conceivability.
Reasoning of this sort is central to some uses of conceivability arguments in thephilosophy of mind (e.g. in Chalmers 1996).

5Objections to two-dimensionalism
A number of objections to two-dimensionalism have been made in the literature.

Some objections (the first nine considered here) rest on the attribution of views towhich two-dimensionalism is not committed. They might be considered objections tocertain versions of two-dimensionalism, but they do not apply to the epistemic twodimensionalism that I have outlined. Other objections (the next two considered here)
show that the claims of two-dimensionalism must be restricted in certain respects.
Still others (the last three considered here) raise substantive issues whose adjudication
is an ongoing project.
What is held constant? (Block and Stalnaker 1999): Evaluation of primary
intensions turns on claims about what a term such as ‘water’ would have picked out incounterfactual circumstances. But this raises the question of what is held constantacross worlds in counting an expression as a token of ‘water’. If only orthography isheld constant, then many tokens of ‘water is watery’ will be false; if reference is held
constant, then no token of ‘water is H2O’ will be false. So to yield the desired results,
a two-dimensionalist must hold constant some intermediate sort of content, such as

Fregean or descriptive or narrow content. But it is question-begging for a twodimensionalist to presuppose such a notion of content.
Response: Evaluation of primary intensions does not turn on metalinguistic
claims about what a term would have picked out in counterfactual circumstances. One
could define an expression’s contextual intension as a mapping from worlds
containing a token of the expression to the extension of that token in that world. The
question of what is held constant would then become relevant: one would obtain
different sorts of contextual intensions depending on just what one counts as arelevant token. But primary intensions are not like this. They simply turn on the
epistemic properties of an expression in the actual world. For example, it isepistemically possible (not ruled out a priori) that there are no utterances, and so theprimary intension of ‘There are no utterances’ will be true in an utterance-free world(whereas the contextual intension of ‘There are no utterances’ will not be defined
there). Because properties of counterfactual tokens are irrelevant to the evaluation ofprimary intensions (except in some special cases), the problem of “what is held
constant” does not arise.
Twin Earth intuitions are irrelevant (Soames 2005). Intuitions about thereference of ‘water’ as used on Twin Earth are irrelevant to the meaning of our term
‘water’, as the term ‘water’ on Twin Earth has a different meaning.
Response: Again, evaluation of primary intensions does not depend on thereferents of homonymous terms in counterfactual worlds. Rather, it depends on
certain epistemic properties associated with uses of ‘water’ in our world. For example,
if we are given the information that the liquid in the oceans and lakes is and has
always been XYZ, we should conclude that water is XYZ. This is a fact about theinferential role associated with uses of our term ‘water’. Epistemic twodimensionalism uses this inferential role to analyze an aspect of the content of theseuses of the term.
Names and natural-kind terms are not indexicals (Nimtz and Beckermann
forthcoming; Soames 2005): Two-dimensionalism entails that terms such as ‘water’are really disguised indexicals that can pick out different referents in different
contexts. But such terms are not indexicals. Any utterance of the English term ‘water’,in any context, picks out H2O.

Response: Epistemic two-dimensionalism does not entail that names and natural
kind terms are disguised indexicals, and it is consistent with the claim that any
utterance of the English term ‘water’ refers to H2O. If primary intensions were
Kaplanian characters or contextual intensions, then the claim that ‘water’ refers to
H2O in any context would be inconsistent with the two-dimensionalist claim that the
primary intension of ‘water’ picks out XYZ in the Twin Earth world. But primary
intensions are not Kaplanian characters or contextual intensions. To ground thedesired behavior of primary intensions, the two-dimensionalist simply requires theplausible claim that it is epistemically possible (i.e. not ruled out a priori) that water is
XYZ. This claim is consistent with the claim that (given that ‘water’ actually refers toH2O), all metaphysically possible tokens of the English term ‘water’ refer to H2O.
Names are not rigidified descriptions (Soames 2005). Two-dimensionalism
entails that names and natural kind terms are disguised rigidified descriptions (of theform ‘the actual φ’, for some φ). But Kripke’s epistemic arguments show that names
are not rigidified descriptions, as do considerations about the way that names anddescriptions behave in belief ascriptions.
Response: Two-dimensionalism does not entail that names and natural kind terms
are rigidified descriptions. We have noted already that Kripke’s epistemic arguments
are accommodated by the observation that primary intensions cannot always be
encapsulated into a description. Furthermore, as noted above, it is consistent with
two-dimensionalism to hold that names and natural kind terms, unlike rigidified
descriptions, have the same referent in any context of utterance. It is also consistent
with two-dimensionalism to hold that the primary intension of a name or natural kindterm may vary between speakers. The account of belief ascriptions given above does
not entail that names will behave like rigidified descriptions in belief contexts, andhandles the relevant data straightforwardly.
Speakers lack identifying knowledge (Byrne and Pryor 2005; Schiffer 2003).
Two-dimensionalism requires that every name N (at least as used by a speaker) be
associated with a “uniqueness property” φ (such that at most one individual has φ),and requires that the speaker have a priori “identifying knowledge” of the form ‘N is

φ’. But speakers in general lack this sort of knowledge.
Response: Two-dimensionalism does not require that speakers possess
identifying knowledge. It is true that primary intensions can be associated with

uniqueness properties (or better, uniqueness relations, because of the role ofcentering). But speakers need not have beliefs about these uniqueness properties(expressible in the form ‘N is φ’). Epistemic two-dimensionalism simply requires thatspeakers have a conditional ability to determine the referent of N (or better, todetermine the truth-value of claims using N), given relevant information about thecharacter of the actual world and given idealized rational reflection. This conditional
ability need not be grounded in the possession of identifying knowledge. Furthermore,
the invocation of rational reflection makes this a normative claim that idealizes awayfrom cognitive limitations of the speaker. For example, even if a child cannot actually
identify a referent across all circumstances, there may still be idealized inferential
norms on how they should update their relevant beliefs given relevant information
about the world. These norms are all that is required.
Ordinary expressions are not ambiguous (Bealer 2002; Marconi 2005): Twodimensionalism explains the difference in truth-value between(3) It is metaphysically necessary that water is H2O.
(4) It is epistemically necessary that water is H2O.
by saying that ‘water’ expresses its primary intension in the first context and its
secondary intension in the second context. But this entails implausibly that ‘water’ isambiguous. Further, this view cannot handle combined contexts, such as ‘It ismetaphysically necessary but not epistemically necessary that water is H2O’.
Response: Two-dimensionalism does not hold that ordinary expressions areambiguous. ‘Water’ has exactly the same content in both (3) and (4) above: in bothcontexts (and in all contexts) it has both a primary intension and a secondary intension(or equivalently, it has a complex semantic value involving both a primary and asecondary intension). This does not entail that ‘water’ is ambiguous, any more thanthe distinction between character and content entails that indexicals are ambiguous.
The distinction between (3) and (4) is handled instead by the difference between themodal operators. The semantics of these operators are such that ‘It is metaphysicallynecessary that S’ is true when S has a necessary secondary intension, while ‘It isepistemically necessary that S’ is true when S has a necessary primary intension.
Combined contexts are handled in the obvious combined way.

Two-dimensionalism cannot handle belief ascriptions (Soames 2005): It isnatural for two-dimensionalists to hold that ‘x believes that S’ is true when the subject
has a belief whose primary intension is the primary intension of S. But this view gives
the wrong result in a number of cases, and no better two-dimensionalist treatment ofbelief ascriptions is available.
Response: The view of belief ascriptions mentioned above is considered andrejected in Chalmers (1995; 2002), and to the best of my knowledge no twodimensionalist endorses the view. The account of belief ascriptions described in
section 4, straightforwardly handles most of the puzzle cases developed by Soames(see Chalmers 2004 for details). Soames raises some further puzzle cases for thisaccount involving the relationship between ordinary belief ascriptions and de re belief
ascriptions, but the account of de re belief ascriptions sketched above (and also given
in Chalmers 2002) handles these cases straightforwardly.
Two-dimensionalism requires global descriptivism (Stalnaker 2003, 2004): Twodimensionalism holds that the primary intension of an utterance or a belief isdetermined by the internal state of the speaker or believer. This requires an internalist“metasemantic” theory, showing how intentional content is determined by internal
state. The main candidate for such a theory is the “global descriptivism” of Lewis
1984, holding that the content of our utterances and beliefs is determined by whateverassignment of content yields the “best fit” between the beliefs and the world. But
global descriptivism is false.
Response: Two-dimensionalism does not require global descriptivism. Of course
there is not yet any satisfactory theory of the basis of intentionality, but there aremany possible internalist alternatives. For example, one might hold that the primary
intension of a mental state is determined in part by its internal functional role, and inpart by associated phenomenal states (where the latter may be especially relevant forphenomenal and perceptual concepts).
The wrong sentences are a priori: Two-dimensionalism requires the claim that
sentences such as ‘Hesperus (if it exists) is Phosphorus’ are not a priori, while
sentences such as ‘Julius (if he exists) invented the zip’ are a priori. But these claims
are incorrect: the former expresses a trivial singular proposition that can be justified apriori, while the latter expresses a nontrivial singular proposition that cannot bejustified a priori.

Response: If one stipulated that apriority of a name-involving sentence is to beunderstood in terms of the apriori knowability of an associated singular proposition,these (controversial and counterintuitive) claims would be correct. But the twodimensionalist takes this as good reason to reject the stipulation, or at least stipulates adifferent understanding of apriority for the purposes of the framework. For these
purposes, an utterance can be said to be a priori when it expresses a belief (or at leastan occurrent thought) that can be justified non-empirically, yielding a priori
knowledge. There is an obvious epistemic difference between beliefs expressed bytypical occurrences of ‘Hesperus is Hesperus’ and ‘Hesperus is Phosphorus’: noamount of non-empirical reasoning can convert the latter belief into a priori
knowledge, but the former is easily justified a priori. (Note that on this definition of
apriority, two different beliefs might be related to the same singular propositional
content while differing in their epistemic status: the epistemic status attaches
primarily to belief tokens, not to belief types or to propositional contents.) Thisepistemic difference at the level of thought can be used to ground the relevant claims
about the apriority of utterances. More generally, the primary intensions of utterances
are grounded in the (normative) cognitive role of associated thoughts.
Primary intensions are not linguistic meaning: Different speakers can use thesame name (‘Fred’) or natural kind term (‘water’) with quite different cognitive roles,
and with distinct patterns of epistemic evaluation. If so, the same expression will havedifferent primary intensions for different speakers. So an expression’s primary
intension is not part of its linguistic meaning, where this is understood as meaningthat is associated with an expression type simply by virtue of the conventions of alanguage.
Response: This point is correct: primary intensions are not always part of
linguistic meaning. For example, it can happen that an identity statement (e.g. ‘Bill
Smith is William Smith’) can be cognitively insignificant for one speaker (e.g. hiswife, who uses the two names interchangeably) but not for another (e.g. a colleague
who uses the names in quite different domains without knowing that they are
coextensive). If so, then the primary intensions of the names will coincide for one
speaker but not for another, so that the primary intension of at least one of them mustvary across speakers. Primary intensions can also vary for context-dependent terms
such as ‘tall’ and ‘heavy’. The moral is that for maximal generality, primaryintensions should be associated with expression tokens (or with utterances of
expression types) rather than with expression types.
Primary intensions are insufficiently fine-grained. Cognitively distinct
expressions may have the same primary intensions. When expressions are equivalenta priori, their primary intensions will coincide. For example, logical and mathematical
truths all have the same primary intension (true in all scenarios), and have the same
secondary intension too. But these clearly differ in meaning and in cognitive
significance. So two-dimensional semantic values do not exhaust meaning (orutterance content), and are not as fine-grained as Fregean senses.
Response: A two-dimensionalist can accommodate many of the relevant cases
here by invoking structured intensions. This will distinguish between different logicaland mathematical truths, for example. The only residual problem will arise if there arepairs of simple expressions that are equivalent a priori but that are cognitively distinct.
It is not obvious that there are such pairs, but if there are, there is more to meaning
than primary intensions. We might say that primary intensions individuate
expressions by their idealized cognitive significance, and so do not capture
differences in nonidealized cognitive significance. One might try to capture thesedifferences by moving to intensions that are defined over a space of finer-grained
epistemic possibilities. Or a two-dimensionalist might simply allow that in addition tointensions, expressions are associated with finer-grained semantic values that lie
behind and determine these intensions. But in any case, this point is no threat to thetwo-dimensionalist who is a semantic pluralist. Primary and secondary intensions are
not all there is to meaning, but nevertheless utterances can be associated with primaryand secondary intensions, in a way that can play the various explanatory roles
described above.
There are epistemic possibilities that correspond to no centered world (Yablo
1999, 2002). A key two-dimensionalist claim holds that when S is not ruled out apriori, then there is some centered world at which the primary intension of S is true.
This may be so for typical Kripkean a posteriori necessities such as ‘water is not
H2O’, but there are other sentences for which the claim false. For example, it may bethat the existence (or nonexistence) of a god is necessary without being a priori. If so,‘There is no god’ (or ‘There is a god’) is not ruled out a priori, but it is necessarily
false. There appears to be no relevant difference between primary and secondaryintensions here, so the primary intension is true in no possible world. Somethingsimilar applies if the laws of nature in our world are the laws of all possible worlds. If
these views are correct, then the space of epistemic possibilities outstrips the space of
metaphysical possibilities in a way that falsifies the two-dimensionalist claim.
Response: All of these purported counterexamples rest on controversial claims
about modality or apriority, and I have argued (Chalmers 1999; 2004) that none ofthem succeed. Furthermore, there is good reason to believe that the concept of
metaphysical modality itself has roots in the epistemic domain, so that there cannot be“strong necessities” that exhibit this sort of disconnect between epistemic andmetaphysical modalities. Still, the existence or nonexistence of strong necessities is adelicate and controversial issue. An alternative version of two-dimensionalism
remains neutral on this issue by understanding scenarios not as centeredmetaphysically possible worlds, but instead as maximal epistemic possibilities(corresponding roughly to maximal epistemically consistent sets of sentences). Theneven if no metaphysically possible world verifies ‘There is no god’, some maximal
epistemic possibility will verify ‘There is no god’, so there will be a scenario at which
the primary intension of this sentence will be true. Understood in this neutral way,
two-dimensionalism does not ground inferences from conceivability to metaphysicalpossibility (those inferences will turn on a further claim about the relationshipbetween scenarios and metaphysically possible worlds), but it can still play much thesame role as before in the epistemic and semantic domains.
Complete canonical descriptions are not available (Schroeter 2004): Epistemic
two-dimensionalism requires that there be qualitative descriptions of a given scenariothat are complete in that they epistemically determine the truth-value of arbitrary
judgments. But there may be some features of the world, such as intrinsic physicalfeatures, which cannot be captured in a qualitative description.
Response: It is not clear whether there are intrinsic properties that cannot be
captured in a qualitative description, but if there are, this will be irrelevant toepistemically determining the truth-value of any of our sentences. When information
about these features is needed to epistemically determine the truth-value of a sentence
in a scenario, a qualitative characterization of the features (e.g. an existential or aRamsey-sentence characterization) will suffice. (Such a characterization may not
suffice for metaphysical determination, and for evaluating truth-values of sentences incounterfactual worlds according to their secondary intensions. But qualitativedescriptions are only needed for primary intensions.) The minimal size of a
vocabulary that can epistemically determine the truth of all sentences is an important
open question, but there is good reason to believe that some qualitative (andindexical) vocabulary suffices. It should also be noted that if we take the purely
epistemic approach to scenarios described in the previous response, a restriction toqualitative vocabulary is not needed, and so the issue here does not arise.
Objections to the role of apriority (Block and Stalnaker 1999, Yablo 2002). It istrue that there is an epistemic relation between information about the world andclaims about reference: for example, given the information that we are in the H2O world (appropriately characterized) we should conclude that water is H2O, and given
the information that we are in the XYZ-world, we should conclude that water is XYZ.
And it is true that we can makes these conditional inferences from the armchair,
without needing to perform further investigation of the environment. But nevertheless,these inferences are not justified a priori. The inferences are justified in part bybackground empirical knowledge of the world (Block and Stalnaker) or by “peeking”
at our own judgments (Yablo). As a result, primary intensions are not connected toapriority as strongly as the two-dimensionalist supposes.
Response: Chalmers and Jackson (2001) argue that these connections are in fact apriori: although empirical facts about the world can play a causal role in determining
the relevant patterns of inference, there is good reason to believe that they do not playa justifying role. (Chalmers (2002) responds to Yablo.) It is also worth noting thateven a skeptic about apriority can use the epistemic two-dimensional framework.
Even if the relevant inferential connections are not a priori, one can still use them todefine primary intensions, and the resulting primary intensions will still behave muchas they are supposed to (assigning a necessary intension to ‘Hesperus is Hesperus’ but
not to ‘Hesperus is Phosphorus’, for example. The connection between primary
intensions and apriority will be lost, but primary intensions will still be strongly
connected to the epistemic domain.

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