## 1. Restricted 2D Semantics



Two-dimensional semantics was introduced to model the semantics of
context-sensitive expressions in natural language, like indexicals and
demonstratives. A similar 2D framework was developed to model
important aspects of tense and modal logic.


### 1.1 Indexicals


#### 1.1.1 The need for a 2D framework



Semantic theories explain how the truth or falsity of whole sentences
depends on the meanings of their parts by stating rules governing the
interpretation of subsentential expressions and their modes of
combination. A semantic framework provides a standard formalism for
stating such rules. The simplest (0-dimensional) semantic frameworks
work by assigning *extensions* as the semantic values of
particular expressions. Intuitively, the extension includes those
things in the actual world to which the expression applies: e.g., the
extension of the name 'Barack Obama' is the man Obama, the
extension of the predicate 'is cool' is the set of all the
actual cool things, and the extension of a two-place predicate
'is cooler than' is the set of pairs of actually existing
things the first of which is cooler than the second. A whole sentence
is assigned a truth-value (True or False) as its extension, which is
computed on the basis of the extensions of the component expressions:
e.g., the sentence, 'Barack Obama is cool', will have the
semantic value True just in case Obama is a member of the set of
actual cool things. A two-dimensional semantic framework is the result
of enriching this simple extensional framework in two distinct
ways.



The first enrichment, *standard possible worlds semantics*, is
introduced to explain the meaning of modal operators like
'possible' and 'necessary' and to distinguish
the intuitive subject matter represented by particular subsentential
expressions. Consider the expressions 'Roger Federer',
'the greatest tennis player of all time', and 'the
most famous Swiss citizen in 2020'. Let's assume all three
expressions happen to have exactly the same extension: a particular
individual RF. So a simple extensional semantics will assign exactly
the same semantic value to all three expressions. But clearly they
differ in meaning: if events had unfolded only slightly differently
than they actually did, the three expressions would pick out different
people. In general, definite descriptions like 'the greatest
tennis player' or 'the most famous Swiss citizen'
pick out different individuals depending on who happens to have the
relevant properties in counterfactual situations; whereas proper names
like 'Roger Federer' *rigidly* pick out the very
same individual in every possible
 situation.[1]
 [See entry on
 rigid designators.]
 Moreover, such differences in what expressions pick out in
counterfactual situations affect the truth of modal claims: e.g.,
'Federer is necessarily Federer' is true, but
'Federer is necessarily the greatest tennis player' is
false. So there is an aspect of meaning that is not captured in simple
extensional semantics. The basic idea behind possible world semantics
is to map out such differences in meaning by specifying what an
expression picks out *relative to every possible way the world
could be* (every "possible world").



In standard (1-dimensional) possible worlds semantics, the semantic
value of an expression is an *intension*, a function that
assigns an extension to the expression "at" every possible
world. For instance, the semantic value of a definite description like
'the best known Swiss citizen' is a function that takes as
input a possible world and yields as output whoever happens to satisfy
that description in that world, and the semantic value of a proper
name like 'Roger Federer' is a constant function that maps
every possible world to the very same individual, RF. Such intensions
reflect commonsense intuitions about the "modal profile"
of the objects, kinds, or properties picked out by our words -
i.e. different possible ways those features could be
 instantiated.[2]
 This framework is also used to explain the meaning of modal operators
like 'necessarily' and 'possibly': a sentence
is necessarily true just in case it is true at every possible world,
and it is possibly true just in case it is true at some possible
world. [See the entries on
 intensional logic
 and
 modal logic.]



The second enrichment of the basic extensional semantic
framework--the one that is distinctive of two-dimensional
semantics--requires us to take possible worlds into account in a
different way. To see why this might be necessary for an adequate
account of meaning, let's focus on context-sensitive expressions
like 'I', 'here' or 'this'. In one
respect, these terms function like names, picking out the very same
thing in every possible world. For instance, if Hillary Clinton says
'I could have been president', her word 'I'
refers rigidly to the same woman, HC, in every possible world and her
claim is true just in case there is a possible way the world could be
in which HC is president. In standard possible worlds semantics, then,
the intension o of 'I' is exactly the same as the
intension of the name 'Hillary Clinton': a function that
yields the individual HC for every possible world. But clearly the
English word 'I' is not synonymous with the name
'Hillary Clinton'--for John McCain might utter the
sentence 'I could have been president' and in his mouth
the word 'I' would refer rigidly to a different person,
JM, in every possible world. What's distinctive of
context-sensitive expressions like 'I' or
'this' is that they represent different things depending
on the context in which they are used. David Kaplan
 (1989a)[3]
 first brought widespread attention to this phenomenon of
context-dependence by proposing his influential two-dimensional
semantic theory to clarify the rules governing such expressions.



Kaplan distinguishes two different aspects of the meaning of
expressions in a public language. The first aspect, *content*,
reflects the modal profile of the object, kind or property
represented. This is the aspect of meaning that is modeled by standard
possible world semantics. The second aspect of meaning,
*character*, reflects semantic rules governing how the content
of an expression may vary from one context of use to the next. A
context-invariant expression like 'Hillary Clinton' has a
constant character, picking out the very same object in every context
in which it's used, whereas indexical expressions like
'I' or 'this' have variable character, picking
out different things in different contexts of use.



Formally, character is defined as a function that maps possible
contexts of use to contents, and content is defined as a function
mapping possible worlds to extensions. Thus, a character is a function
that takes as input a context and yields as output a function from
possible worlds to extensions. This is a *two-dimensional
intension*, since there are two distinct roles that possibilities
play here: as a context of use, and as a circumstance of evaluation (a
possible situation relative to which we evaluate whether the relevant
object exists or property is instantiated). Contexts of use can be
thought of as "centered" worlds: possible worlds with a
designated agent and time within that world, which serve to locate a
particular situation in which the expression is used. We can then
represent a context as an ordered triple, \(\langle w,a,t\rangle\) ,
of a possible world *w*, an agent *a*
within that world, and a time *t*
when the agent exists in that
 world.[4]
 So possible worlds play two distinct roles in Kaplan's
formalism: contexts of use determine which content is expressed and
circumstances of evaluation reflect the modal profile of that content.
The conventional semantic rules governing an expression like
'I' can be easily represented using Kaplan's 2D
framework: in any possible context, \(\langle w,a,t\rangle\), an
utterance of 'I' *rigidly designates* the agent of
that context, *a*, in all possible circumstances of evaluation.
[See the entry on
 indexicals
 for a more detailed discussion].


#### 1.1.2 Introducing 2D matrices



A useful way of visualizing the dual role played by possible worlds in
a 2D framework is to construct *a two-dimensional matrix*
(Stalnaker 1978). To represent Kaplan's theory of indexicals, we
array possible circumstances of evaluation along the horizontal axis
and possible contexts of utterance along the vertical axis. Each
horizontal row of the matrix represents the *content* the
target expression would have if used in the context specified for that
row. This content is (partially) represented by recording the
extension of the term at each possible circumstance arrayed along the
horizontal axis. This procedure is then repeated for each context
listed along the vertical axis.



For instance, consider a particular utterance of 'I' made
by Barack Obama during his inaugural presidential address. This
context of use can be represented as the world \(w\_1\), centered on
the man BO, at time \(t\_0\). We can (partially) represent the content
of 'I' in this centered world thus:



*Obama's use of 'I' in his inaugural
address:*


|  |  |  |  |
| --- | --- | --- | --- |
|  | \(w\_1\) | \(w\_2\) | \(w\_3\) |
| \(\langle w\_1, \BO, t\_0\rangle\) | BO | BO | BO |




This simple one-dimensional matrix reflects the fact that, when used
in this context, 'I' refers rigidly to Obama at every
possible circumstance of evaluation--even at the counterfactual
worlds \(w\_2\) and \(w\_3\), in which John McCain or Hillary Rodham
Clinton won the 2008 presidential election. The context-dependence of
the expression 'I' is revealed when we evaluate the use of
'I' with respect to different possible contexts of use.
Let's consider two other contexts: \(\langle w\_2, \JM,
t\_0\rangle\) is a world in which McCain won the election, centered on
him at his inaugural address; and \(\langle w\_3, \HC, t\_0\rangle\) is
a world in which Clinton won, centered on her at her inaugural
address. We then rely on our implicit understanding of the semantic
rules governing 'I' to generate two more rows for our
matrix:



*Kaplanian character of 'I':*


|  |  |  |  |
| --- | --- | --- | --- |
|  | \(w\_1\) | \(w\_2\) | \(w\_3\) |
| \(\langle w\_1, \BO, t\_0\rangle\) | BO | BO | BO |
| \(\langle w\_2, \JM, t\_0\rangle\) | JM | JM | JM |
| \(\langle w\_3, \HC, t\_0\rangle\) | HC | HC | HC |




What the matrix reveals is that the expression 'I' rigidly
designates different individuals, depending on the context in which it
is used. Thus the 2D matrix provides a graphic illustration of how
content of the expression 'I' varies, depending on the
context in which it is used.



Such 2D matrices can be used to represent the differences between the
semantic rules governing indexicals, definite descriptions, and names.
For instance, the definite description 'the inaugural speaker in
2009' will generate the following Kaplanian matrix:



*Kaplanian character of 'the inaugural speaker in
2009':*


|  |  |  |  |
| --- | --- | --- | --- |
|  | \(w\_1\) | \(w\_2\) | \(w\_3\) |
| \(\langle w\_1, \BO, t\_0\rangle\) | BO | JM | HC |
| \(\langle w\_2, \JM, t\_0\rangle\) | BO | JM | HC |
| \(\langle w\_3, \HC, t\_0\rangle\) | BO | JM | HC |




Unlike the matrix for 'I', the horizontal rows of this 2D
matrix are all exactly the same. This reflects the fact that the
expression 'the inaugural speaker in 2009' is not
context-sensitive: it always represents the very same property
irrespective of the context in which it is used--namely, the
property of being the person who delivers the inaugural US
presidential address in 2009. This property is exemplified by
different individuals at different possible worlds: the person who is
the inaugural speaker at \(w\_1\) is Obama, at \(w\_2\) it's
McCain, and at \(w\_3\) it's Clinton. In general, the sequence
arrayed along the rows of this matrix reflects the variety of
different individuals who could instantiate the property represented
by 'the inaugural speaker' in different circumstances. Of
course, no finite matrix can fully capture the range of variation, but
it can give a useful partial representation of the property in
question.



The matrix for a proper name like 'Barack Obama' reveals
another very different pattern:



*Kaplanian character of 'Barack Obama':*


|  |  |  |  |
| --- | --- | --- | --- |
|  | \(w\_1\) | \(w\_2\) | \(w\_3\) |
| \(\langle w\_1, \BO, t\_0\rangle\) | BO | BO | BO |
| \(\langle w\_2, \JM, t\_0\rangle\) | BO | BO | BO |
| \(\langle w\_3, \HC, t\_0\rangle\) | BO | BO | BO |




According to Kaplan, proper names are context-invariant: they always
have the very same content irrespective of the context in which they
are used. Proper names are also rigid designators: they pick out a
single individual at every possible world. The upshot is that the 2D
matrix for a proper name will be completely uniform: the very same
individual appears in every cell of the matrix. This reflects the idea
that the semantic function of a name in a public language is simply to
pick out a particular individual, not to convey any information about
how to identify the individual in question. [For a different account
of proper names, see SS2.2 below.]


#### 1.1.3 The logic of indexicals and diagonal intensions



Kaplan's semantic rules for indexicals guarantee that certain
sentences will be true whenever they are uttered, and certain
inferences will be truth preserving. This account paved the way for
Kaplan's formal logic of indexicals (Kaplan 1989a). In this
system, logical validity is defined in terms of different possible
contexts of use: a sentence is valid iff it is true in every possible
context of use; and an inference is valid iff the truth of the
premises ensures the truth of the conclusion in every possible context
of use.



On Kaplan's account, sentences can be logically valid, even if
they express contingent propositions. For instance, the semantic rules
governing indexicals ensures that the sentence 'I am here'
will be true in any context of use. But the content expressed is
normally contingent: I could easily *not* have been here right
now, but at the beach instead.



To illustrate, we can construct a partial 2D matrix for the sentence
using our previous example. Suppose 'I am here now' is
uttered by the new president at the inauguration
(*t0*) in \(w\_1\) where Obama won, \(w\_2\), where
McCain won, and \(w\_3\) where Clinton won. Let's assume Obama
would attend the inaugural address of McCain but not of Clinton,
McCain would avoid the inauguration of anyone who defeated him, and
Clinton would attend Obama's inauguration but not
McCain's. This yields the following 2D matrix:



*Kaplanian character of 'I am here'*


|  |  |  |  |
| --- | --- | --- | --- |
|  | \(w\_1\) | \(w\_2\) | \(w\_3\) |
| \(\langle w\_1, \BO, t\_0\rangle\) | **T** | T | F |
| \(\langle w\_2, \JM, t\_0\rangle\) | F | **T** | F |
| \(\langle w\_3, \HC, t\_0\rangle\) | T | F | **T** |




The horizontal rows of the matrix represent the different propositions
expressed by the sentence in each context of use. Each utterance
expresses a different contingent proposition, as can be seen by the
fact that each contains both 'T's and 'F's,
and that the patterns differ. The 2D matrix also graphically
represents the fact that the sentence is guaranteed to be true
whenever uttered. Notice the diagonal of the matrix running from the
top left corner to the bottom right, which contains all
'T's. This reflects the fact that the sentence is
guaranteed to be true whenever uttered. With a nod to Stalnaker
(1978), we can call this the *diagonal intension* of the
sentence. In Kaplan's semantic framework, a necessary diagonal
intension indicates that a sentence is *logically valid* and
 *analytic*.[5]


### 1.2 Modal Operators



At around the same time that Kaplan began developing his account of
indexicals, logicians working on tense and modal logic had begun using
2D semantic frameworks to explain the behavior of sentential operators
like 'now' and 'actually' (Aqvist 1973;
Kamp 1971; Segerberg 1973; Vlach 1973). Unlike Kaplan, these logicians
were not primarily concerned with the semantic rules governing natural
languages. In particular, modal logicians were not focused on how the
context in which an expression is used can affect its reference.
Rather, they were interested in developing formal systems for
representing valid inferences about time and possibility. It turns out
that tense and modal logic are formally very similar and that both
require double-indexing for expressive adequacy. Thus, to fully
capture reasoning about what's necessary and possible, we need
to move from standard possible worlds semantics to a 2D semantic
framework.


#### 1.2.1 'Actually' and 'Necessarily'



Consider the following sentence:



(1)
 It is possible for
everything that is actually red to be shiny. 


Standard possible worlds semantics lacks the expressive power to
capture what is said by this sentence (Crossley and Humberstone 1977;
Hazen 1976, 1978). The claim is not that there is a possible world
such that all the things that are red in that world are also shiny in
that world (they're supposed to be red in the *actual*
world, not the counterfactual one). Nor is the claim that for each
object that is red, there is a possible world in which it is shiny
(the objects are all supposed to be shiny *together* within a
single possible world). So here is a relation among objects in
possible worlds that cannot be expressed in standard possible world
semantics. To capture the relation, we need to introduce an extra
element into the formal framework: we simply designate one world
within the model (the set of possible worlds) to play the role of the
actual world. We can then introduce a sentential operator
'\(\mathcal{A}\)' (read as 'Actually'), which
requires us to *evaluate any claim within its scope at the
designated world*, even when the operator is embedded within the
scope of other modal operators. Using this enriched possible worlds
framework, we can represent the truth-conditions of our sample
sentence in a straightforward way:



\((1')\)
 \(\Diamond \forall x(\mathcal{A}\Red(x) \rightarrow \Shiny(x))\) 


This sentence is true just in case there is some possible world, *w*,
in which everything that is red in the designated
world, \(w\_@\), is shiny in *w*.



One awkward consequence of this 2D semantic account of
'Actually' is the way this operator interacts with the
standard modal operator 'Necessarily'. Intuitively, what
the actual world is like seems logically and metaphysically
contingent. But according to the proposed semantics for
'Actually', any true sentence *S* will
yield a *necessary* *truth* when embedded within the
scope of the operator '\(\mathcal{A}\)'. For instance,
consider the following sentence:



(2)
Obama actually won the 2008 election. 


If Obama won in the designated world of our model, then it's
true at every possible world in that model that Obama won at its
designated world. So on the proposed 2D semantics, the sentence is
*necessarily true*. (When we embed (2) within the necessity
operator '\(\Box\)' we get a truth; and any claim of the
form \(\mathcal{A}S \rightarrow \Box \mathcal{A}S\) will be logically
valid.) But intuitively it's a *contingent* matter how
the 2008 elections turned out. To mitigate this counterintuitive
consequence, Crossley and Humberstone introduce a new logical
operator, 'Fixedly' ('\(\mathcal{F}\)') in
such a way that the complex operator 'Fixedly Actually'
('\(\mathcal{F}\mathcal{A}\)'), captures the sense of
necessity we have in mind when we deny that (2) is necessary. A
sentence is fixedly actually true just in case it is *true no
matter which world is designated as
 actual*.[6]


#### 1.2.2 A 2D model



Once again, 2D matrices can be used to graphically depict how the
semantic theory works. Let's take our universe of possible
worlds to contain just three worlds: \(w\_1\) is a world where Obama
won, \(w\_2\) a world where McCain won, and \(w\_3\) a world where
Clinton won. To explain the 'Fixedly Actually' operator,
we need to consider possible worlds playing two different roles: the
standard role as a circumstance of evaluation and the special role of
being designated as the actual world. To construct a 2D matrix, we
array possible worlds playing the standard role along the horizontal
axis, and along the vertical axis we array the same worlds playing the
role of being designated as actual. Each horizontal row of this matrix
represents a different model with a particular world designated as
 actual.[7]
 On this account, the truth of a sentence embedded within the
'Actually' operator depends entirely on what's true
in the world designated as actual in a given model. So we can fill in
the 2D matrix as follows:



*C&H matrix for 'Obama actually won.'*


|  |  |  |  |
| --- | --- | --- | --- |
|  | \(w\_1\) | \(w\_2\) | \(w\_3\) |
| \(w\_1\) as actual | **T** | T | T |
| \(w\_2\) as actual | F | **F** | F |
| \(w\_3\) as actual | F | F | **F** |




In any world in a model (a row in the matrix), 'Actually *S*'
is always evaluated by looking at the
designated world of that model. So such sentences are either
necessarily true (True at every world in the model) or necessarily
false (False at every world in the model). This is the sense of
necessity that corresponds to the standard modal operator
'\(\Box\)'. On this understanding of necessity, the target
sentence is necessarily true (since \(w\_1\) represents the
*actual* actual world). But intuitively there is a sense in
which the sentence seems contingent, since a different world could
have been actual: if \(w\_2\) or \(w\_3\) had been actual, the sentence
'Obama actually won' would have been false. This fact is
reflected in the 2D matrix by the diagonal intension, where the
sentence comes out true with respect to \(\langle w\_1, w\_1\rangle\),
but false with respect to \(\langle w\_2, w\_2\rangle\) and \(\langle
w\_3, w\_3\rangle\). The 'Fixedly Actually' operator is
sensitive to the necessity or contingency of the diagonal intension.
The sense in which the target sentence (2) is not necessary is that
it's not fixedly actually true.



While this double-indexing model has become standard in the
literature, as (Rabern 2012b) points out, the same expressive adequacy
considerations Crossley and Humberstone (1977) used to support two
indices for modal and temporal operators also support
*multiple-indexing* for those operators. For instance,
consider the following sentence:



(3)
If the Titanic had
not hit an iceberge on its maiden voyage, it would still have been
possible for everyone would then have survived the maiden voyage to
die on the maiden voyage. (Rabern 2012b: 119) 


To generate the correct analysis, we need to invoke three distinct
worlds: the actual world \(w\_1\), where the Titanic hit an iceberg, a
counterfactual world \(w\_2\) where it did not, and another
counterfactual world \(w\_3\) where all survivors of the voyage in
\(w\_2\) died. So double-indexing and 2D matrices cannot fully capture
the compositional semantics of modal sentences: we will need an
infinite sequence of possible world indices. For further work on
multiple-indexed semantics, see (Ninan 2010; Yalcin 2015).


#### 1.2.3 Philosophical applications



An influential paper by Martin Davies and Lloyd Humberstone
(1980) brought the formal tools developed in 2D modal logic to bear on
philosophical puzzles about modality. Following Gareth Evans (1979),
Davies and Humberstone suggest that there are two notions of
metaphysical necessity involved in ordinary modal thinking: deep and
superficial necessity. They argue that the two logical operators,
'\(\mathcal{F}\mathcal{A}\)' and '\(\Box\)',
respectively, provide a
clear formal elucidation of these two notions.



These two notions of necessity, they argue, help explain some of Saul
Kripke's (1980) puzzling examples in which necessity and
apriority come apart. Using 2D modal logic, it's easy to
construct necessary aposteriori truths. The semantic rules governing
the modal operator '\(\mathcal{A}\)' guarantee that every
claim of the form \(\mathcal{A}S\) will be either necessarily true or
necessarily false in the sense of '\(\Box\)'. But when the
embedded sentence *S* is an ordinary empirical truth
like 'Obama won', \(\mathcal{A}S\) will be knowable only
aposteriori: so \(\mathcal{A}S\) will be a necessary aposteriori
truth. The 'Actually' operator can also be used to
construct contingent apriori truths. Any claim of the form
\((\mathcal{A}S \rightarrow S)\) is guaranteed by the semantic rules
governing 'Actually' to be true at the designated world no
matter which world is designated as actual (i.e., it's fixedly
actually true). But when *S* is an ordinary empirical
truth, the complex claim is not necessary in the sense of
'\(\Box\)': there will be some worlds in the model where
*S* is false while \(\mathcal{A}S\) is true. In such
cases, the complex sentence will be a contingent apriori truth.



Davies and Humberstone also suggest that the 2D modal operator
'Actually' might help analyze certain referring
expressions in natural language. In particular, they focus on
Evans' (1982) notion of a 'descriptive name' (a name
whose reference is fixed by a description) and on natural kind terms.
Suppose the following definitions capture the semantic rules governing
the relevant expressions in natural language:


1. Julius \(=\_{df}\) the actual inventor of the zip.
2. Water \(=\_{df}\) the actual chemical kind to which that liquid
belongs which falls from clouds, flows in rivers, is drinkable,
colorless, odorless, etc.



If such analyses are correct, then the semantics for
'actually' will allow us to explain why 'Julius
invented the zip' is contingent and apriori and 'water =
H2O' is necessary
 aposteriori.[8]
 [See the entries on
 names
 and
 descriptions.
 For a survey of other philosophical applications of the 2D framework
in modal logic see (Humberstone 2004).]



Davies and Humberstone themselves express reservations about the
adequacy of analyses using 'actually' for natural language
expressions, particularly in the case of proper names (1980,
17-21). As a consequence, they did not take 2D modal logic to
provide a complete response to Kripke's puzzles about necessary
aposteriori and contingent apriori truths. However, the use of the 2D
framework to explain these puzzles was subsequently taken up and
refined by proponents of generalized 2D semantics.


## 2. Generalized 2D Semantics


### 2.1 Vindicating the traditional approach to meaning



In the previous sections, we considered applications of the 2D
framework that seek to explain the meaning of specific types of
expression: indexicals and modal operators. In contrast, proponents of
generalized 2D semantics (G2D) believe that the 2D framework can be
used to explain an important aspect of the meaning of *all*
expressions. In particular, G2D is meant to vindicate the traditional
idea that we have apriori access to our own meanings through armchair
reflection.



According to the philosophical tradition, to know the meaning of a
subsentential expression like 'bachelor' is to implicitly
grasp a criterion that determines exactly which individuals count as
bachelors in any possible situation. (Accounts of meaning broadly
along these lines were advanced by Plato, Descartes, Locke, Hume,
Frege, Russell, Carnap, and many others.) On the traditional account,
speakers' implicit grasp of a criterion plays two key
theoretical roles:


1. Semantic competence: Two speakers (or one speaker on two
occasions) share the same meaning just in case they associate the very
same criterion with their expressions.
2. Reference determination: The criterion a speaker currently
associates with an expression determines which things fall into its
extension in every possible situation.



The first claim requires that speakers who share the same meaning must
share a criterion for identifying the reference; while the second
requires that this criterion be veridical. If this traditional account
of meaning is correct, then one can make one's own meanings
explicit by engaging in *apriori conceptual analysis*. Such
conceptual analysis allows you to determine what exactly it takes to
count as a bachelor in any possible world; and it allows you to
specify what exactly someone must be prepared to accept in order to
genuinely agree or disagree about bachelors.



G2D is a strategy for defending a variant of this traditional view of
meaning against a series of influential objections. In the
1970s and 80s, semantic externalists used a variety of persuasive
examples to argue that the traditional account of meaning yields an
unrealistic picture of (i) semantic competence, (ii) reference
determination, and (iii) epistemic access to modal facts. Proper names
and natural kind terms seem especially problematic for the traditional
 account.[9]
 By commonsense standards, you don't need to know a specific
rule for identifying Godel in any possible world in order to
count as competent with the name 'Godel'; and no such
knowledge seems required for your use of the name to pick out the
relevant man in every possible world (Donnellan 1970; Kripke
 1980).[10]
 Similarly, you don't need to know precisely what it takes for
something to count as water in any possible world to be competent with
the word 'water' or for your word to pick out the chemical
substance H2O in every possible world (Kripke 1980; Putnam
1970, 1972). Indeed, making room for the possibility of ignorance and
error about reference-conditions seems crucial to explaining empirical
inquiry into the nature of familiar things, and to vindicating the
commonsense realist idea that we can refer to things whose nature we
don't fully understand (Burge 1979, 1986; Putnam 1972, 1973). If
these critics are right, then the traditional account of meaning is
untenable. Implicit knowledge of reference-conditions is *not*
required either for linguistic competence or for determinate
reference. And apriori conceptual analysis cannot be trusted to reveal
what's genuinely possible--at best, it reveals one's
current fallible assumptions about the topic in question. [See the
entry on
 externalism about mental content.]



Proponents of G2D believe this pessimistic conclusion is unwarranted.
What critics' examples really show, they argue, is that the
traditional view of meaning should be refined, not junked. Moreover,
the 2D semantics developed for indexicals and modal operators suggests
a promising strategy for accommodating putative externalist
counterexamples within a broadly traditional account of meaning. In
the case of indexicals and rigidified definite descriptions, competent
speakers grasp a *reference-fixing criterion* without grasping
*the modal profile* of the object, kind, or property picked out
by the expression. For instance, you can know that 'I'
always refers to the speaker whenever it is uttered without knowing
the nature of the person who is actually picked (e.g., what it takes
to be Barack Obama in any possible world). Perhaps our understanding
of names and natural kind terms is structured in a similar way:
competent speakers always have apriori access to the reference-fixing
criterion for their own use of the name 'Barack Obama',
but they have only aposteriori access to the associated modal profile.
If this suggestion is on the right track, then a G2D framework could
be used to clarify the nature of this semantic understanding. Moreover
we may be able to explain certain epistemic operators, like 'it
is conceptually possible that' or 'it is apriori
that', as operating on such 2D semantic values.



The basic philosophical idea behind G2D--that subjects have
apriori access to reference-fixing criteria for their words but not to
the modal profile of the subject matter picked out--has been
suggested by a number of theorists. David Lewis, in particular, was a
powerful champion of the idea that we can give apriori definitions for
terms whose precise reference we do not understand. Lewis articulated
the 'analytic functionalist' approach to specifying the
meaning of mental predicates and of theoretical terms in science
(1966; 1970; 1972; 1979; 1980; 1994); and he was also an early
advocate of a generalized 2D approach to semantics (1981; 1994). Other
influential proponents of the idea that we can have implicit knowledge
of reference-fixing criteria without knowing the modal profile of the
reference include Michael Dummett (1973; 1981), Gareth Evans (1982),
and John Searle (1983). Early proponents of an explicitly
two-dimensional semantics for names and natural kind terms include
Harry Deutsch (1990, 1993), Ulrike Haas-Spohn (1995), and Kai-Yee Wong
(1996). However, it is two later theorists--Frank Jackson (1994;
1998a; 1998b; 2004) and David Chalmers (1996; 2002b; 2002c; 2004;
2006a)--who have most systematically developed and defended G2D
as a way of reconciling the lessons of semantic externalism with the
traditional apriori approach to meaning and modality.



It's worth noting that G2D has been motivated primarily by
epistemic, metasemantic, and metaphysical concerns, rather than by
issues in compositional semantics. In particular, G2D seeks to
vindicate the traditional idea that we can know the truth-conditions
of our own sentences via armchair reasoning about hypothetical cases.
The approach promises to explain why certain necessary truths can only
be known aposteriori by appealing to the structure of our implicit
semantic understanding. Proponents of G2D make claims about how the
two types of intension may interact with modal and epistemic
operators. However, working out the details of the compositional
semantics has been a relatively recent concern of proponents of G2D
(e.g. Chalmers 2011a, c; Chalmers and Rabern 2014; Johannesson and
Packalen 2016; Kipper 2017).



The 2D semantic frameworks proposed by Jackson and Chalmers are very
similar in their broad aims and formal structure, and commentators
often treat the two versions as interchangeable. However, the two
frameworks are developed in the service of two quite different
philosophical projects, emphasizing different aspects of the
traditional approach to meaning. Jackson takes up the traditional
empiricist project of explaining empirical facts about language use
and communication, while Chalmers pursues a broadly rationalist
project of explaining key structural interconnections between meaning,
apriority, and possibility. This difference in explanatory aims leads
to different interpretations of the 2D framework.


### 2.2 The empiricist project


#### 2.2.1 Linguistic understanding and coordination



An empiricist account of meaning is a high-level causal explanation of
uncontroversial facts about language use. In particular, the
empiricist seeks to characterize the psychological states that guide
individuals' application of an expression to particular cases,
and to explain how linguistic coordination within a linguistic
community is achieved.



Clearly individual speakers must have some implicit assumptions about
the reference of a word that guide their verdicts about whether it
applies to particular cases (Jackson 1998a, 29-42). Your
judgments about whether a particular Gettier case counts as knowledge,
for instance, are guided by your prior understanding of the term
'knowledge', and your answer is only justified insofar as
it reflects that prior understanding. An empiricist seeks to explain
these facts by positing a stable internal psychological
state--something like an *internal reference-fixing
template*--that guides your verdicts no matter what the
actual world turns out to be like.



It's equally clear that members of the same linguistic community
generally manage to use words to reliably coordinate their beliefs and
actions (Jackson 1998b, 2004). When I ask you to pass the salt, you
know roughly which white granular substance I'm asking
for--you know, for instance, that it would be inappropriate to
pass the sugar bowl or the pepper grinder. This sort of everyday
coordination requires speakers to have similar dispositions to
classify things as falling into the extension of words, and it
requires that these similarities in classificatory dispositions be
mutually obvious to all concerned: for it to make sense for me to say
'please pass the salt' in order to get salt, it must be
common knowledge between us that we're inclined to classify
roughly the same things as 'salt'. An empiricist explains
this common knowledge by positing *implicit conventions* that
require everyone to associate the very same reference-fixing template
with a given word (Jackson 1998b; Lewis 1969).



An empiricist use of the 2D framework is intended to show that this
core explanatory project is not undermined by the intuitions about
names and natural kind terms highlighted by semantic externalists
(Jackson 1998a;
 1998b).[11]


#### 2.2.2 The role of the 2D framework



Externalists argue that ordinary speakers are often ignorant or
mistaken about the precise nature (modal profile) of the objects,
kinds or properties their words pick out. But linguistic conventions
don't always fix the reference by specifying the nature of the
reference. Perhaps the conventions governing names and natural kind
terms are structured in a similar way to indexicals. For instance, we
might have an implicit semantic rule requiring us to take
'water' to pick out whatever chemical kind
*actually* explains a certain suite of superficial observable
properties: e.g., being a clear, potable, odorless liquid that fills
lakes and streams around here (Jackson 1998a, 1998b). On this
analysis, 'water' just is an implicitly indexical
expression, picking out different chemical kinds depending on which
world is actual. If this rule is what one must accept to count as
competent with the meaning of the expression type 'water',
then it is no surprise that competent speakers often fail to realize
that water = H2O.



Of course, it is an empirical question whether names and natural kind
terms are in fact governed by indirect reference-fixing rules of this
sort. But according to Jackson, you can test whether your implicit
understanding of 'water' is structured in this way by
considering possible situations in two different roles: as *your
actual environment* or as *a mere counterfactual
possibility* (Jackson 1998a, ch. 2). Consider two different
possible worlds based on Putnam's Twin Earth thought experiment
(Putnam 1972). In the first world, Earth, the clear potable stuff that
fills lakes and streams and is habitually called 'water'
by English speakers is H2O. The second world, Twin Earth,
is exactly the same except that the stuff that has these properties is
the complex chemical kind, XYZ. If your commonsense understanding of
'water' is governed by the proposed reference-fixing
convention, it would lead you to identify different chemical
substances as water depending on what your actual environment is like:
if your actual environment is Earth, then water is H2O; but
if your actual environment is Twin Earth, then water is XYZ. If you
assume that water is *actually* H2O, moreover, you
will judge that water is *essentially* H2O in all
counterfactual circumstances. And if you assume water is
*actually* XYZ, then you'll judge water is
*essentially* XYZ.



This pattern of dispositions to apply the term 'water' can
be depicted on a 2D matrix as follows:



*Empiricist 2D matrix for 'water':*


|  |  |  |
| --- | --- | --- |
|  | Earth | Twin Earth |
| \(\langle\)Earth, \(a, t\rangle\) | **H2O** | H2O |
| \(\langle\)Twin Earth, \(a, t\rangle\) | XYZ | **XYZ** |




Along the vertical axis are ranged centered possible worlds (a
possible world, with a designated agent *a* and time
*t* within that world) representing different ways
your actual environment could be like; and the same worlds are ranged
along the horizontal axis representing different counterfactual
circumstances of evaluation. This matrix reflects your commonsense
dispositions to apply the term 'water' to different
chemical kinds on the basis of whether it actually plays certain
superficial roles described in other commonsense terms
('clear', 'potable', 'liquid',
 etc).[12]



Semantic externalists take these sorts of judgments about Twin Earth
to militate against a traditional account of meaning--for they
suggest that your understanding does not fully determine the nature of
the reference. But according to Jackson, the only conclusion that is
warranted is that the meaning of your term is more complex than the
tradition suggests: your verdicts about possible worlds considered as
actual reflect your *naive reference-fixing criterion*,
and your verdicts about possible worlds considered as counterfactual
reflect the *theoretical criterion* you would accept after you
learned all the relevant empirical facts about your actual
environment. These two types of criterion can be modeled in possible
world semantics as intensions: an A-intension (for
'Actual') is a function from worlds considered as actual
to extensions, while a C-intension (for 'Counterfactual')
is a function from worlds considered as counterfactual to extensions
(Jackson 1998a, ch.2). The diagonal of our matrix corresponds to the
A-intension you associate with 'water'; and the first
horizontal row corresponds to the C-intension of your term
'water' (assuming that \(\langle\)Earth, \(a, t\rangle\)
represents your real-world environment).



Semantic externalists acknowledge only the C-intension as modeling an
expression's semantic content, but 2D empiricists insist that
both A- and C-intensions reflect important aspects of a competent
English speaker's understanding of a word like
'water'. In particular, they take A-intensions to reflect
what is understood and communicated by minimally competent English
speakers and what guides their everyday classifications. The
suggestion, then, is that A-intensions capture the shared,
conventionally entrenched understanding of reference-fixing conditions
posited by the empiricist approach to meaning.



By itself, this 2D framework offers no guarantee that the hypothetical
judgments recorded by an A-intension are produced by a stable
reference-fixing criterion. Nor does it guarantee that the very same
A-intension will be generated for all competent speakers in your
linguistic community. However, according to Jackson, we have solid
empirical reasons to think these conditions are satisfied in the case
of names and natural kind terms. First, the widespread acceptance of
the externalist thought experiments demonstrates that we do in fact
share similar reference-fixing criteria for terms like
'water' and 'Godel' (Jackson 1998a,
38-39). Second, the empiricist model of meaning provides the
best psychological explanation of how such linguistic coordination is
achieved (Jackson
 1998b).[13]


#### 2.2.3 Apriori conceptual analysis



In addition to clarifying the structure of our semantic understanding,
the 2D framework can help justify specific conceptual analyses. The
criteria that implicitly guide our everyday use of a term are often
embodied in recognitional or inferential dispositions rather than in
consciously accessible rules. Indeed, Jackson likens grasp of
reference-fixing criteria for particular expressions to our ability to
recognize the grammaticality of sentences in one's own language
(2000). Just as linguists can construct a grammar for your language on
the basis of your judgments about the acceptability of particular
sentences, you can construct an analysis of the meaning you associate
with an expression on the basis of your application of a term to
hypothetical cases. The correct analysis must capture the full range
of your confident judgments involving the target expression, while
abstracting away from performance errors and other interfering factors
(Jackson 1998a, 31-37).



This psychological model helps explain how you can come to know the
correct analysis of your term 'water' by noting which
properties you treat as "obvious and central" when filling
in a 2D matrix like the one above (Jackson 1998a, 31). The 2D
framework prompts you to consider possible cases in two different
ways-as actual or as counterfactual. This allows you to know
whether the content of your term varies depending on what your actual
environment is like (e.g. 'water') or whether it is stable
(e.g. 'bachelor'). Moreover, careful attention to your
reactions to these suppositions will allow you to make explicit which
superficial properties implicitly guide your application of the term.
For instance, you may discover that your implicit criterion for
applying 'water' is that water \(=\_{df}\) the actual
chemical kind in your environment that is a clear, potable, odorless
liquid that falls as rain and fills lakes and streams. Alternatively,
your use of the term 'water' may be guided by the types of
causal relations invoked in causal theories of reference: water
\(=\_{df}\) the actual natural kind that has been the dominant cause of
your community's past use of the term 'water'.
Indeed, Jackson suggests that standard causal theories of reference
are based on this method of conceptual analysis (1998a, 37-41).
[See the entry on
 causal theories of mental content.]



The conceptual analyses produced by this method count as apriori,
according to the 2D empiricist, because you can know them to be
correct "independently of knowing what the actual world is
like" (Jackson 1998a, 51). The evidence that supports such
analyses consists in purely hypothetical judgments: judgments about
how to classify cases *on the supposition* that your
environment is like *X*, or like *Y*.
Since such hypothetical judgments don't
require you to determine what your *real* environment is like,
your justification for accepting an analysis is not based on empirical
knowledge. And to change your judgment about a purely hypothetical
case would be to change the meaning of your term (Jackson 1998a,
44-46).


#### 2.2.4 The role of conceptual analysis in metaphysics



Jackson claims that apriori conceptual analysis plays a crucial role
in metaphysics (Jackson 1994; 1998a). Metaphysical reductions provide
a constitutive account of some target domain (e.g., beliefs, free
will, water, moral rightness) in terms of more basic features of the
world (e.g., the properties postulated by an idealized physics, ideas
in the mind of God, the mosaic of sense data). A physicalist about
mental states, for instance, is committed to there being specific
facts about the microphysical structure of the world that suffice for
the existence of beliefs, desires and sensory experiences. The
physicalist is thus committed to metaphysically necessary
"entailments" connecting claims about the two domains:
it's metaphysically necessary that if such-and-such physical
facts obtain, then such-and-such mental facts obtain. This
metaphysical entailment relation can arguably be cashed out in terms
of global supervenience (Jackson 1998a, 6-14). [See the entry on
 supervenience.]



The role of conceptual analysis is to show that a putative reduction
respects the original meaning of the target expression (Jackson 1998a,
28). A physicalist won't succeed in accounting for free will if
she identifies free will with having a temperature of 37.4o C
- such a "reduction" would simply change the subject
under discussion. A successful reduction must be answerable to our
original shared understanding of the target expression--and
elucidating this original understanding just is what conceptual
analysis does. So if conceptual analyses are knowable apriori, it
follows that metaphysical reductions must always be backed by apriori
entailments between the base-level claim (such as a physical
description of the world) and the target claim (such as the claim that
humans have free will).



On this empiricist account, conceptual analysis plays a modest
metaphysical role. Conceptual analysis captures apriori entailment
relations among your ideas; but it cannot tell you whether there
*are* any objects, kinds, or properties that satisfy your
current reference-fixing assumptions (Jackson 1998a, 42-4).
Moreover, the meaning you currently associate with a term may be
pragmatically deficient: e.g., it may not be determinate enough to
settle certain hard cases or it may not allow you to draw useful
distinctions in your actual environment. In such cases, you would have
good pragmatic reasons to change the meaning of your term (Jackson
1998a, 44-6, 53-4). What the empiricist denies is that
changing your current criteria for applying a term can ever get you
closer to the truth about what you meant all along.


### 2.3 The rationalist project


#### 2.3.1 Connecting meaning, apriority, and possibility



For an empiricist, an expression's meaning reflects the causal
mechanisms guiding everyday classification and communication. For a
rationalist, in contrast, an expression's meaning reflects what
is apriori accessible to the speaker on the basis of ideal reflection.
The empiricist is primarily concerned with causal explanation of
linguistic facts, while the rationalist is primarily concerned with
idealized apriori rationality and insights into objective possibility.
This difference in emphasis can have significant ramifications for
G2D.



David Chalmers has developed a detailed and influential rationalist
interpretation of the 2D framework. This semantic project is situated
within a broadly rationalist tradition that posits a "golden
triangle" of necessary constitutive relations between meaning,
apriority, and possibility (2004; 2006a; 2012).



Following Frege (1892), Chalmers is interested in capturing a notion
of meaning that is finer-grained than reference. Frege pointed out
that sentences containing co-referential expressions like
'Hesperus' and 'Phosphorus' can differ in
*cognitive significance*: someone who is competent with these
two names may not realize they are co-referential and may therefore
use them differently in making and justifying claims. Frege took
sameness of cognitive significance to be the mark of sameness of
meaning. According to a 2D rationalist, sameness of cognitive
significance can in most cases be elucidated in terms of *apriori
equivalence*: two expressions are associated with the same meaning
iff one can know that they pick out the very same things on the basis
of apriori reflection alone (Chalmers
 2002b).[14]
 This constitutive link between meaning and apriority constitutes the
first side of the "golden triangle".



The second side of the "golden triangle" connects meaning
with possibility. Following Carnap (1947), Chalmers suggests that we
can use possible worlds semantics to individuate particular meanings
in terms of their representational properties. In standard possible
world semantics, the meaning of 'doctor' is identified
with an *intension* that maps possible worlds to extensions. An
expression's intension reflects the modal profile of the object,
kind, or property picked out. Identifying meanings with intensions
therefore establishes an important constitutive connection between
meanings and modal claims. If 'doctor' and
'physician' are associated with the same meaning, then
it's *true in all possible worlds* that all doctors are
physicians and all physicians are doctors. And conversely if two
expressions are *co-extensive in all possible worlds*, then
they have the same meaning.



The third side of the "golden triangle" connects
possibility with apriority. Following Kant (1787), a rationalist about
modality holds that what is necessary is always knowable apriori and
what is knowable apriori is always necessary. Thus ideal apriori
reflection can be trusted to reveal the space of possibility.



This "golden triangle" of constitutive relations generates
a distinctive rationalist account of meaning. To be competent with an
expression's meaning is to be in an internal cognitive state
that puts one in a position to identify its extension in any possible
world on the basis of apriori reflection alone. Apriori reflection
will also suffice to determine whether two expressions are associated
with the same or different meanings. This rationalist approach to
meaning contrasts with the empiricist one: whereas the empiricist uses
*causally efficacious cognitive mechanisms* to isolate the
reference-fixing criteria currently associated with an expression, the
rationalist uses the subject's *ideally rational
judgments* to isolate the complex cognitive states that would
ground those reflective judgments. As a consequence, the aspect of
understanding that corresponds to a rationalist meaning may turn out
to be more heterogeneous and less stable than the shared, causally
efficacious 'templates' postulated by the empiricist. The
"golden triangle" also involves a distinctive rationalist
account of modal epistemology, according to which ideal apriori
conceivability is a fail-safe guide to metaphysical possibility. This
modal rationalism affords a simple and attractive account of our
access to modal facts (Chalmers 1996, 136-8; 1999, 488-91;
 2002a).[15]



This simple rationalist picture of meaning and modality, however, was
undermined by externalist thought experiments. Kripke's (1980)
observation that certain necessary truths, like 'Hesperus =
Phosphorus', are only knowable aposteriori threatens both the
idea that linguistic competence affords apriori access to the truth-
and applicability-conditions of one's words and the idea that
necessary truths are always apriori knowable. The guiding idea behind
2D rationalism is that a rationalist can accommodate Kripke's
examples by moving to a 2D semantic framework. In particular, the 2D
framework can be used to isolate an aspect of meaning that satisfies
the "golden triangle" of constitutive relations among
meaning, apriority and modality.



Roughly, the idea is that 'Hesperus = Phosphorus' is
aposteriori because we associate distinct reference-fixing criteria
with the two names: e.g., being the brightest star visible in the
evening and the brightest star visible the morning. According to the
2D rationalist, these reference-fixing criteria are (i) an aspect of
meaning, (ii) which can be known apriori via conceptual analysis, and
(iii) which suffices to fix the applicability conditions for every
possible world considered as one's actual environment. If the 2D
framework can be used to isolate such an aspect of meaning for all
expressions, we will have vindicated the rationalist's
"golden triangle" connecting meaning, apriority and
possibility.



Vindicating this "golden triangle" constitutes a primary
theoretical constraint for a rationalist interpretation of 2D
semantics. A 2D semantics that meets this constraint would play a
wide-ranging role in philosophy. It would account for core semantic
roles associated with the Fregean notion of sense (Chalmers 2002b) and
the traditional notion of a proposition (Chalmers 2011a). In addition,
rationalist 2D semantics promises to define a versatile notion of
narrow thought content suited to playing key explanatory and
evaluative roles in commonsense psychology (Chalmers 2002c).
Furthermore, the rationalist approach to meaning and modality
underwrites a distinctive form of apriori reasoning about the nature
of the actual world:



> 
> There is a long tradition in philosophy of using apriori methods to
> draw conclusions about what is possible and what is necessary, and
> often in turn to draw conclusions about matters of substantive
> metaphysics. Arguments like this typically have three steps: first an
> epistemic claim (about what can be known or conceived), from there to
> a modal claim (about what is possible or necessary), and from there to
> a metaphysical claim (about the nature of things in the world).
> (Chalmers 2002a, 145)
> 



Chalmers has developed an influential anti-physicalist argument
along these lines, which relies on a rationalist 2D semantic framework
to establish that facts about phenomenal consciousness cannot be
reduced to physical or functional facts about the brain (1996; 2009).
See the supplementary document:



*The 2D argument against materialism.*



#### 2.3.2 The epistemic interpretation of the 2D framework



Any interpretation of the 2D framework must answer the following two
questions:


1. What is the nature of the worlds to be considered as actual?
2. What is the principle for determining an expression's
extension relative to worlds considered as actual?



But the rationalist project imposes specific constraints on how these
questions are answered. To vindicate the "golden
triangle", the rationalist must identify a way of mapping an
individual speaker's understanding of particular expressions
onto possible worlds that affords apriori access to the entire space
of possibility. This is not a trivial requirement: standard ways of
interpreting the 2D framework cannot vindicate the rationalist
project. However Chalmers has developed a distinctive
"epistemic" interpretation of the 2D framework that he
believes can establish the relevant constitutive links between
meaning, apriority and possibility (2004, 2006a).



A rationalist 2D semantics must vindicate the following principle:



Core Thesis: For any sentence \(S,\) \(S\) is apriori iff \(S\)
has a necessary 1-intension. (Chalmers 2004, 165)




A sentence's 1-intension is an intension that corresponds to the
diagonal of a 2D matrix. So the Core Thesis affirms that a token
sentence is apriori (for a subject at a particular time) just in case
there is no possible way the world might be that, if it actually
obtained, would make *S* false. In effect, the Core
Thesis sums up the "golden triangle" of constitutive
connections the rationalist hopes to establish between meaning,
apriority, and possibility: (i) it postulates a possible way the world
could be for every apriori coherent hypothesis, and vice versa; and
(ii) this tight connection between apriority and possibility is
reflected in an aspect of linguistic meaning, the
 1-intension.[16]



The major obstacle to vindicating the Core Thesis for standard
interpretations of the 2D framework is the assignment
principle--the way 2D theories assign extensions relative to
"worlds considered as actual". A natural way of
understanding the injunction to consider a possible world as actual is
to simply imagine a possible world, locate a person in it at a time,
and then rely on ordinary interpretive methods to decide what exactly
that person in those empirical circumstances is referring to when
using a given expression. Chalmers calls this strategy for assigning
1-intensions to expressions a "contextualist"
interpretation of the 2D framework. What's distinctive of a
contextualist approach is (i) that a token of the target expression
must be located within the world considered as actual, and (ii) that
the expression is assigned an extension on the basis of how it's
used in that world. On this approach, a 1-intension will be
*undefined* for possible worlds that do not contain a token of
the target expression: no extension can be assigned for such worlds,
not even an empty extension.



This contextualist approach to assigning 1-intensions is incompatible
with the Core Thesis (Chalmers 2004, 167-176). Consider
sentences like 'Language exists' or 'A sentient
being exists': the meaning of these sentences seems to guarantee
that they will be true in every possible context in which they are
used. So on the contextualist approach, these sentences should be
assigned necessary 1-intensions, mapping every possible context of use
to the truth-value True. But contrary to the Core Thesis, these
sentences are *not* apriori truths knowable independently of
any empirical evidence. There's no contradiction in the very
idea of a world without language or thought and we can easily imagine
what such a world would be like; it's just that our everyday
experience allows us to immediately rule out the possibility that our
actual environment is like that. The problem is that contextualist
1-intensions are undefined for worlds without thought or language,
even though they are both apriori coherent and metaphysically
possible. So a necessary contextual 1-intension does not track
apriority or metaphysical necessity. Contextualist 1-intensions,
therefore, cannot satisfy the rationalist's Core
 Thesis.[17]



This difficulty can be avoided, Chalmers argues, if we rely on a
notion of *epistemic possibility*--what *seems*
possible after ideal rational reflection--to interpret the 2D
framework. More specifically, he focuses on the notion of apriori
coherence: claims that could be true for all one can tell on the basis
of idealized apriori
 reasoning.[18]
 This notion of apriori coherence is used to answer the two
interpretive questions highlighted above: (i) apriori coherence is
used to characterize the possibilities relative to which 1-intensions
are defined, and (ii) apriori coherence is invoked to assign
1-intensions to a speaker's expressions.



First consider the possibilities that define 1-intensions. On the
epistemic interpretation, the possibilities are not metaphysically
possible contexts of use, but epistemically possible
"scenarios": maximally specific hypotheses about what
one's actual environment might be like that cannot be ruled out
through apriori reasoning alone. Scenarios provide a complete
characterization of the entire history of a universe, down to the last
microphysical detail. They also provide perspectival
information--a notional "center"--that indicates
the location from which the hypothetical universe is to be considered.
The crucial point is that scenarios are defined by their epistemic
role: they represent ways we can conceive of the actual world, within
which we can try to identify familiar objects, kinds or
 properties.[19]



The second distinctive element of the epistemic interpretation of 2D
semantics is the procedure for assigning 1-intensions to a
speaker's expressions. On the epistemic approach, 1-intensions
reflect relations of apriori coherence between descriptions of
possible scenarios and ordinary language sentences:



> 
> The epistemic 1-intension for a sentence *S* is True
> at a scenario *W* iff \((W\) & not\(-S)\) is
> apriori incoherent. (Chalmers 2004, 180-4)
> 



This principle for assigning 1-intensions relies on the
speaker's ordinary ability to engage in object-level reasoning
about combinations of hypotheses: given the assumption that the
scenario description *W* is true, you're asked
to decide whether *S* must be true as well. If
it's incoherent to accept \((W\) & not\(-S)\), your
epistemic intension for *S* maps *W*
to True, otherwise *W* is mapped to False. This
epistemic assignment principle contrasts sharply with the
contextualist principle. The contextualist approach requires us to
engage in explicit meta-linguistic reasoning to interpret the
expression '*S*' *as it's
used* within the possible world *W*. On the
epistemic approach, in contrast, an extension is assigned to
'*S*' on the basis of the subject's
own object-level reasoning using the expressions '*W*'
and '*S*'.



Unlike the contextualist approach, therefore, the epistemic assignment
principle does not explicitly require that a scenario contain a token
of the relevant expression type in order to assign an extension
relative to that scenario. As a consequence, sentences like
'Language exists' seem to pose no special problem for
satisfying the Core Thesis. The sentence 'Language exists'
will have a contingent epistemic 1-intension, because there are
possible scenarios that are apriori consistent with both the truth and
falsity of that sentence. For instance, consider a scenario in which
the only object is a giant lump of salt. To test whether your sentence
'Language exists' is true at this scenario considered as
actual, you ask whether there is any incoherence in combining the
claim 'The only object that exists is a lump of salt' with
the claim 'It's not the case that language exists'.
Intuitively, this combination is coherent: there is no language in the
salt world. So the epistemic 1-intension for your sentence
'Language exists' yields the value False for that
scenario. Since there are other scenarios relative to which the
sentence 'Language exists' will have the value True, your
sentence will have a contingent epistemic 1-intension. This contingent
epistemic intension for your sentence 'Language exists'
reflects the fact that it's not apriori true that language
exists. So it seems the epistemic assignment principle will allow
apriority and necessity of the 1-intension to go hand in hand, as
required by the Core Thesis.


#### 2.3.3 Defining the E2D framework



There is further work to do in spelling out Chalmers' E2D
framework in such a way as to vindicate his rationalist project. One
way to think about the rationalist project is as a combination of the
following theses:


1. Semantic foundationalism holds that there is a canonical level of
description that affords transparent epistemic access to an objective
space of apriori possibilities - i.e. ways the world might be
for all that can be known independently of all experience.
2. Meaning rationalism holds that implicit conceptual competence puts
one in a position to conclusively know the applicability conditions of
one's words relative to every such scenario considered as
actual.
3. Modal rationalism holds that the space of apriori epistemic
possibility is a failsafe guide to the space of metaphysical
possibility: there are no modal illusions when scenarios or worlds are
described in canonical terms



Together, 1 and 2 constitute a sort of semantic reductionism: the
meaning of any ordinary language expression is reduced to the meanings
of the base vocabulary via the epistemic exercise of considering
scenarios as actual and worlds as counterfactual. And 3 ensures that
this epistemic exercise is an accurate guide to metaphysical
possibility. Chalmers' Core Thesis is meant to capture this
tight relationship between grasp of meaning, apriori reflection, and
metaphysical possibility.



However, simply rejecting C2D in favor of E2D does not yet provide any
positive account of what it is to entertain an epistemic scenario W,
and how we should update our beliefs in the light of the supposition
that W is actual. Without these details, it's impossible to
determine whether the Core Thesis is true. Perhaps it's
impossible to consider a scenario as actual without presupposing
one's own existence, or perhaps our best epistemic methods for
updating our beliefs presupposes the existence of those very beliefs;
or perhaps there is no way of thinking about the world that
doesn't rely on some further empirical assumptions about the
world.



Chalmers has entered into these interpretive questions in considerable
detail over many publications. The starting point for his approach is
outlined in (Chalmers and Jackson 2001), where he suggests that
scenarios can be understood as PQTI sentences: where P states
microphysical truths, Q states phenomenal truths, T is a
'that's all' clause indicating that P and Q provide
a complete description of a possible universe, and I indicates the
subject's notional location within that universe. P and Q employ
a canonical vocabulary that fully specifies the essential nature of
the fundamental properties upon which all other properties in a
possible world supervene. Thus, PQTI sentences provide an
epistemically transparent access to the space of epistemic and
metaphysical possibility (simply removing the self-locating
information from a PQTI sentence yields a complete description of a
corresponding possible world, PQT). The 1-intensions of one's
ordinary language expressions are then determined by the individual
subject's ideally reflective dispositions to judge sentences
true, assuming the truth of different PQTI-sentences. And 2-intensions
are fixed by one's reasoning about PQT sentences considered as
counterfactual (given assumptions about PQTI). (Chalmers 2006, 2011b)
further articulates how the space of epistemic possibility can be
understood, how scenarios are related to possible worlds, and how 1-
and 2-intensions are assigned to token representations.



More recently, in Constructing the World (2012), Chalmers has focused
squarely on the epistemic 'scrutability' relation that
connects our understanding of ordinary language expressions to
base-level descriptions of scenarios.



Apriori Scrutability: There is a compact class of basic truths D such
that for any truth S, one can conclusively know 'D \(\supset\)
S' apriori.



Chalmers still takes PQTI sentences to be a promising candidate for
specifying a scrutability base D, but he is open to the possibility
that the descriptive vocabulary in PQTI may need to be supplemented in
order to capture some truths, such as truths about causal relations or
quiddities. But while he can afford to be flexible about the exact
nature of the scrutability base, Chalmers' rationalist program
depends on vindicating Apriori Scrutability for any sentence that is
evaluable as possibly true or false (169-71). A good deal thus
hangs on whether he is right that ideal epistemic procedures allow for
apriori justification, given an exhaustive base-level description of a
scenario. Chalmers offers arguments to support the plausibility of
this view in (2012, ch. 4.).



According to the 'frontloading' argument, we can have
conclusive apriori knowledge of material conditionals of the form
(PQTI  \(\supset\)
S). Chalmers argues that all empirical information relevant to
justifying a verdict about S can be 'frontloaded' into the
antecedent of the conditional, so information about one's
real-world environment, E, cannot play any essential role in
justifying verdicts about the conditional. If E is itself apriori
entailed by PQTI, it is not needed to justify a verdict about the
conditional. And if E is not apriori entailed by PQTI, E will be
irrelevant to justifying a verdict about the conditional. So our
justification for the application conditionals that ground
1-intensions is wholly apriori, and immune to empirical
 defeat.[20]


#### 2.3.4 Rationalist semantic values



Epistemic 2D semantics differs in important respects from traditional
accounts of meaning. Semantic theories normally describe general
semantic rules governing expression *types*, whereas epistemic
2D semantics is based on a single individual's current
understanding of a *token* expression. Kaplan and Jackson, for
instance, use the 2D framework to characterize the implicit
conventions governing syntactically individuated expressions like
'I' or 'water' in our linguistic community. In
contrast, Chalmers uses the 2D framework to characterize your
potentially idiosyncratic understanding of a particular use of an
expression on a given occasion--e.g., the way you understood Al
Gore's fifth use of 'water' during a speech on
climate change. Moreover, on this account 2D semantic values depend on
the upshot of ideal rational reflection about apriori coherence
relations. Just what is involved in ideal rational reflection is an
open question. But it's plausible that it may depend on
substantive constructive theorizing about the empirical scenario in
question and on various non-obvious and idiosyncratic aspects of the
subject's initial cognitive state. In that case, identifying the
precise epistemic 1-intension associated with your understanding of
'water' will be a highly non-trivial matter, and it may be
far from obvious when your understanding of the term shifts so that
two tokens no longer share the same epistemic 1-intension. This is
why, in contrast with 2D empiricists like Jackson, a rationalist like
Chalmers denies that epistemic 1-intensions reflect the
subject's "implicit knowledge" of a reference-fixing
criterion (e.g., Chalmers 2002a, 185; 2006b, SS5).



Of course, it's possible that some epistemic intensions will
reflect stable reference-fixing rules that are entrenched by implicit
linguistic conventions. But it's also possible that some
epistemic intensions will reflect highly abstract, heterogeneous,
unstable, and idiosyncratic aspects of a speaker's understanding
at a given time. As a consequence, epistemic intensions are not
guaranteed to line up with conventional linguistic meanings (Chalmers
2002b). Given this divergence from standard semantic theories, one may
wonder whether epistemic intensions deserve to be considered a kind of
*meaning*.



However, according to the 2D rationalist, epistemic intensions play
the core semantic roles associated with Fregean senses (Chalmers
2002b). Like Fregean senses, epistemic 1-intensions lend themselves to
a compositional semantic theory: the epistemic intension of a sentence
is determined by the epistemic intensions of the component
expressions. Moreover, epistemic 1-intensions, like Fregean senses,
reflect the speaker's own rational perspective on what her words
represent. Two token names '*A*' and
'*B*' have the same Fregean senses iff
the identity '*A = B*' would strike the speaker as
*trivially true*. Similarly, a subject associates two token
expressions with the same epistemic intension iff they are *apriori
 equivalent*.[21]
 Finally, epistemic intensions may play a role similar to that of
Fregean senses in the semantics of attitude reports (Chalmers 2011a).
Overall, then, epistemic intensions seem to provide an attractive
theoretical refinement of the Fregean notion of sense.



In addition, epistemic intensions arguably carve out a well-defined
notion of narrow content suited to playing key roles in commonsense
psychology (Chalmers 2002c). Epistemic intensions reflect rational
relations among token mental states. Epistemic intensions can then be
used to specify representational state types that are relevant to
assessing a person's rationality and to explaining rational
thought processes.



It's important to note that epistemic 1-intensions are intended
to explain only *one aspect* of meaning. The 2D semantic
framework also posits 2-intensions ("counterfactual" or
"subjunctive" intensions), which reflect the modal profile
of the object, kind or property picked out by an expression. But
Chalmers emphasizes that his E2D does not exclude positing further
aspects of meaning:



> 
> 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 or the
> secondary intension is *the* content of an utterance. Both can
> be systematically associated 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 them play, 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. (Chalmers 2006a, SS3.5)
> 



In sum, Chalmers' highly idealized E2D framework is intended to
earn its semantic keep by defining a kind of meaning capable of
forging traditional rationalist connections between meaning,
rationality and possibility. But he is happy to allow that there may
be other types of semantic values or structures that are needed to
play other semantic roles. [See entry on
 propositions,
 and
 structured propositions.]


### 2.4 Objections to generalized 2D semantics



The G2D framework has attracted a wide variety of criticisms,
targeting its commitment to apriori conceptual analysis, its claim
that 1-intentions capture a type of meaning, and its internalist
approach to assigning contents. The specific rationalist and
empiricist applications of the G2D framework have also been
criticized. For a survey of these lines of criticism: see the
supplementary document:



> 
> *Objections to generalized 2D semantics*
> 


## 3. The Metasemantic Interpretation


### 3.1 2D semantics for externalists



Generalized 2D semantics seeks to vindicate a traditional internalist
conception of meaning: it posits an extra aspect of meaning for all
expressions (the intension corresponding to the diagonal of a 2D
matrix) that is fully determined by a subject's internal states,
and which in turn determines objective truth-conditions for their
sentences. By enriching compositional semantics in this way, G2D
promises a straightforward explanation of a variety of epistemic
properties of sentences: e.g., why a necessary sentence like
'Hesperus is Phosphorus' is not apriori knowable, what the
subject learns by accepting the sentence, or how the subject uses the
sentence in reasoning.



But using the 2D framework to characterize the subject's
epistemic perspective is not beholden to this internalist project.
Semantic externalists reject the traditional view that our purely
internal states afford apriori access to reference-fixing conditions
for our words and thoughts. According to externalists, the basic
assignments in a compositional semantics relate the subject's
words and thoughts to objective features of her
environment--objects, kinds and properties whose nature is
captured by standard (1D) possible world semantics. Even externalists,
however, can define 2D matrices that reflect the subject's
epistemic perspective on the reference of her words and thoughts. For
the externalist, however, these 2D matrices will not represent
*meanings*--a specific aspect of understanding that is
required for linguistic or conceptual competence and which figures in
a compositional semantic theory that determines truth-conditions for
sentences. On an externalist interpretation, 2D matrices merely
reflect one aspect of a subject's *partial semantic
understanding* of what her words and thoughts represent. Because
externalist 2D matrices don't represent meanings, moreover, the
externalist is free to use the 2D framework strategically to focus on
different aspects of the subject's understanding for different
explanatory purposes.



Robert Stalnaker has articulated such an externalist interpretation of
the 2D framework in a series of influential papers spanning some
thirty years. He was the first to introduce 2D matrices to specify
what is communicated in situations where conversational partners are
partly ignorant or mistaken about the nature of the objects, kinds or
properties their words pick out (1978), and he later extended his 2D
framework to characterize the content of certain thoughts and attitude
attributions (1981; 1987; 1988). In both cases, the 2D framework is
used to define "diagonal" intensions that reflect the
subject's partial understanding of which objects, kinds or
properties her words and thoughts represent. These diagonal intensions
are not meanings or semantic values, since they do not figure in a
compositional semantic theory and they do not reflect conditions for
conceptual or linguistic competence. The only meaning of an expression
on this account is its ordinary "horizontal" intension. In
effect, Stalnaker's 2D matrices represent different meanings
that an expression *could have had* if it had occurred in
different empirical circumstances. This "metasemantic"
interpretation of the 2D framework contrasts sharply the
"semantic" interpretations favored by G2D theorists like
Jackson and Chalmers (Stalnaker 2001, 2004).



Proponents of G2D were influenced by Stalnaker's early papers
developing the 2D framework, and their views are often presented as
continuous in motivation and form. But there are important theoretical
consequences that flow from the choice between 2D metasemantics and
generalized 2D semantics. Indeed, Stalnaker himself is a vocal critic
of generalized 2D semantics, rejecting its commitment to the semantic
status of 2D matrices, its commitment to apriori conceptual analysis,
and its internalist approach to reference determination.


### 3.2 A 2D account of assertoric content



The metasemantic interpretation of the 2D framework was originally
developed as a way of explaining how the propositions conveyed by the
assertion of a sentence can vary depending on the conversational
context (Stalnaker 1978). In this seminal paper, Stalnaker proposes an
attractive theoretical account of the role of assertion in a
conversation, which is then used to explain how the assertoric use of
a necessary sentence like 'Hesperus = Phosphorus', can
convey a specific empirical proposition within a given conversation.
In particular, Stalnaker argues that our commitment to construing such
sentences as making felicitous and informative assertions will lead us
to *reinterpret* their content in ways that can be modeled
using the 2D framework.



The guiding idea is that in making an assertion the speaker is trying
to get the audience to rule out certain possibilities. In asserting
'It's cold today', for instance, I may be trying to
get you to rule out possibilities in which today's temperature
in Melbourne is over 10deg C. We can model what my assertion
conveys, then, as a function that maps possible worlds in which
today's temperature is under 10deg C to True and all other
worlds to False. However, the precise truth-conditions communicated by
an assertoric use of a sentence depend in part on the conversational
context in which it takes place. Just which temperatures count as
cold, for instance, depends on shared background assumptions in a
particular conversational context: what's cold in Melbourne is
mild in Manitoba.



A second guiding idea is that the proposition actually conveyed by the
assertion of a particular sentence depends on *presuppositions*
shared by the participants in the conversation--including
presuppositions about what particular words represent and
presuppositions about actual empirical circumstances. If you're
a Chinese speaker who doesn't understand anything at all about
what the term 'cold' represents in English, then I cannot
use 'It's cold today' to convey facts about
Melbourne's weather. And if you're a Canadian who
doesn't understand anything about Australian weather conditions,
you won't understand precisely what I am saying to my fellow
Melburnians when I assert that sentence. Stalnaker calls the set of
presuppositions that conversational partners treat as common knowledge
that they can rely on to get their point across the "context
set"--which he models as the set of possible worlds that
satisfy all of these mutual presuppositions. The context set will
encode shared assumptions about the meaning of words, about general
empirical facts, about the what's happened so far in the
conversation, and so on.



The goal of assertion, Stalnaker suggests, is to shrink the context
set. In making an assertion, the speaker tries to get the audience to
accept a new proposition as one of their shared presuppositions,
thereby shrinking the set of possible worlds that are considered live
options. For instance, in asserting 'It'll be very cold
today' to a group of Melburnians, I exploit background knowledge
of local weather conditions in June to get my audience to accept that
the temperature outside is somewhere between 5-10deg C, ruling
out live possibilities that it might be in the 15-20deg C
range. If all goes well, further planning will proceed on the basis of
a smaller and more accurate range of possibilities. In contrast, if I
were to assert 'It's cold' to the monolingual
Chinese speaker or to the parochial Canadian, my assertion would be
defective, since my audience wouldn't be able to figure out
which temperatures are ruled out by my assertion.



Identity claims, however, do not seem to fit this simple model of
assertion. As Kripke (1980) argued, identities are either necessarily
true or necessarily false. So accepting an identity will either leave
the context set unchanged or it will eliminate it altogether. Either
way, asserting an identity would be pointless. But clearly it is not.
Asserting an identity such as 'Lloyd is I.L. Humberstone'
can be genuinely informative, ruling out empirical possibilities
previously taken to be live options. According to the metasemantic
account, (i) the goal of assertion can explain why the assertion of a
necessary sentence will lead to a reinterpretation of the content of
the asserted sentence, and (ii) the 2D framework helps to specify just
which proposition will be conveyed by the sentence within a given
conversation.



In general, an identity claim is appropriate when one of the parties
to a conversation is (partially) ignorant about which object is picked
out by a name like 'Lloyd'. For an externalist like
Stalnaker, this is a case of *semantic ignorance*. If
O'Leary doesn't know that 'Lloyd' is
co-referential with 'I.L. Humberstone', then he does not
fully understand the semantic rules governing these names: i.e., that
both names are associated with a constant function from any possible
world to a specific individual. But O'Leary isn't utterly
incompetent with the meaning of these terms: he implicitly understands
both names as rigid designators, and he has some substantive
understanding of the object each name picks out. For instance, he may
understand that 'Lloyd' refers to the person to whom
he's just been introduced and that 'I.L.
Humberstone' refers to the author of 'Direction of
Fit'. O'Leary's semantic deficiency--his
failure to fully understand the meaning of these names in a
contextually appropriate way--is grounded in his ignorance of the
ordinary empirical fact that the man to whom he has been introduced is
the author of 'Direction of Fit'.



2D matrices can be used to represent this sort of partial semantic
understanding. O'Leary knows that \(if\) the man in front of him
is the author of the famous article, then 'Lloyd = I.L.
Humberstone' expresses a necessary truth; and he knows that
\(if\) the man in front of him isn't the author, the sentence
expresses a necessary falsehood. What O'Leary doesn't know
is which of these two possibilities corresponds to his actual
situation. Call the first possibility *i* and the
second *j*. O'Leary's epistemic situation
can then be summed up in a 2D matrix:



O'Leary: 'Lloyd = I.L. Humberstone'



|  |  |  |
| --- | --- | --- |
|  | *i* | *j* |
| *i* | T | T |
| *j* | F | F |




The matrix is defined only with respect to a specific set of relevant
alternative possibilities, *i* and *j*,
chosen in such a way as to reflect the
subjects' semantic understanding and our own explanatory
interests. The vertical axis represents these possible worlds in their
role as *contexts of use*, which determine the literal semantic
content of the expressions used in them. The horizontal axis
represents those same possible worlds as *circumstances of
evaluation*, relative to which we evaluate the truth or falsity of
the proposition expressed. Each row of the matrix thus represents a
different proposition that might be literally expressed by the
sentence. Stalnaker calls such matrices *propositional
concepts*, since they reflect the subject's current
imperfect conception of the meaning of the
 sentence.[22]
 This particular matrix reflects the fact that O'Leary's
current epistemic state is compatible with the identity sentence
expressing either a necessary truth or a necessary falsehood,
depending on empirical facts about the actual context of use.



What does O'Leary learn when he comes to accept Daniels'
assertion of 'Lloyd is I.L. Humberstone'? Since the actual
world is like *i*, the literal semantic content of
the asserted sentence is a necessary truth. But necessary truths rule
out no empirical possibilities whatsoever, so this cannot be the
informative proposition that is conveyed by Daniels' assertion.
Moreover, O'Leary is not in a position to recognize that this is
the literal semantic content of the sentence, since he doesn't
know whether the actual world is like *i* or *j*.
The natural suggestion is that the information
conveyed by Daniels' assertion is that the real world is like
*i* and not *j*. When O'Leary
accepts 'Lloyd is I.L. Humberstone', he will no longer
treat the possibility that the man in front of him is *not* the
author of the famous article as a live option: this empirical
possibility will be eliminated from his context set. Thus, the
proposition that seems to be conveyed by Daniels' assertion
corresponds to the diagonal intension determined by our 2D matrix for
that assertion. Moreover, this observation generalizes: when subjects
are partially ignorant of the semantic values of their words, the
diagonal proposition determined by the propositional concept can
capture the empirical information conveyed by the assertion.



But why is this so? To explain why assertions sometimes express the
diagonal proposition, the metasemantic account appeals to rational
maxims governing conversational cooperation. The following maxim seems
to govern the practice of assertion:



> 
> The very same proposition should be expressed relative to every
> possible world in the context set. (Stalnaker 1978, 88)
> 



Speakers should conform to this maxim, because assertion involves an
intention to get one's audience to eliminate worlds from the
context set in accordance with the proposition expressed--and in
order for this intention to succeed the audience must be in a position
to figure out just which worlds they are being asked to eliminate.
When this sort of rational maxim governing the communication of
information is flouted, the audience will look for a non-standard
interpretation of the utterance that would bring it back into
conformity with the maxims (Grice 1989). [See the entries on
 pragmatics
 and on
 implicature.]



According to Stalnaker, this is precisely what is going on in the case
of identity claims like the one we have been considering.
Daniels' assertion of 'Lloyd is I.L. Humberstone'
clearly flouts the proposed maxim. We can assume that Daniels is aware
that O'Leary doesn't know whether he is in a world like
*i*, where the man to whom he's been introduced
is the famous author, or a world like j where they are distinct. Yet
Daniels utters a sentence that expresses a different proposition
depending on whether the actual world is like *i* or
like *j*. In such circumstances, the audience should
look for an alternative interpretation of the assertion.
Daniels' assertion can be brought back into conformity with the
maxim by re-interpreting it as conveying the proposition expressed by
the diagonal of the matrix. At a rough intuitive level, we can say
that Daniels is trying to get O'Leary to accept that the
sentence 'Lloyd is I.L. Humberstone' expresses a truth.
But the 2D framework also allows us to specify more precisely just
what empirical information is conveyed within a given conversational
context. Given O'Leary's and Daniels' common
presuppositions about what the two names represent, Daniels'
assertion also expresses the proposition *that the man to whom
O'Leary has just been introduced is the author of
'Direction of Fit'*.



It's worth emphasizing, however, that the very same sentence
asserted in a different conversational context could express an
entirely different empirical proposition: just which proposition is
expressed, on this metasemantic account, depends on what the
individual parties to the conversation are currently presupposing
about the meanings of the expressions used.



In recent work, Stalnaker has enriched his notion of a context. In
addition to a sets of possible worlds, he introduces (i) multiple
centers within those worlds representing the participants in a
conversation, and (ii) accessibility relations among the centers
representing interlocutors' ways of identifying each other
(Stalnaker 2008, 2014). This added structure is intended to capture
self-locating presuppositions that can help explain what's
communicated by indexical expressions like 'I',
'you' and 'now'. For a helpful overview of the
motivation for such an approach, see (Ninan 2010a) and for further
development of multi-centered accounts of propositional content see
(Torre 2010, Ninan 2013).


### 3.3 Propositional attitudes



The metasemantic 2D framework was originally developed to explain
communication, but the framework can also be used to specify the
content of certain beliefs and the content of assertions that
attribute beliefs.



Stalnaker (1984) defends a coarse-grained account of belief contents,
which individuates particular belief states in terms of a set of
possible worlds that would make them true. If this project is to
succeed, it must be possible to fully specify beliefs without invoking
anything like Fregean senses or conceptual structure. But there is an
important class of beliefs that seem to pose insuperable problems for
a simple possible worlds account of their content: beliefs in
necessary truths. The problem for a standard possible worlds analysis
is that all necessary truths have precisely the same content (the
function mapping every world to True). So the belief *that Hesperus
= Phosphorus* will have exactly the same content as a belief
*that Hesperus = Hesperus & Fermat's last theorem is
true*. But these are clearly distinct belief states. Beliefs in
identities have been particularly important in motivating theories of
finer-grained thought contents.



But fine-grained Fregean senses or conceptual structures are not
strictly required to distinguish beliefs in identities. Stalnaker
(1981; 1987) argues that the metasemantic 2D framework he developed to
explain what is communicated by an *assertion* of an identity
sentence can also explain the content of the *belief states*
attributed using an identity sentence to specify its content. If
O'Leary were to notice the pole star and think to himself
*that's Mars*, for instance, the truth-conditions of his
thought can be captured by a judiciously defined diagonal proposition
(Stalnaker 1987, 125). In this case, the worlds we include in the
context set may involve facts about which object is the target of
O'Leary's visual attention and facts about salient
empirical properties he associates with the name 'Mars'.
On the metasemantic approach, then, the proposition we attribute in
saying O'Leary believes that *that is Mars* is the
proposition that the visually salient object is the object that has
those Martian properties.



A further complication arises in specifying the content of attitude
attributions. On the metasemantic account of assertion, the content
conveyed by a sentence depends on the shared presuppositions of the
speaker and audience. But sometimes the parties to a discussion are
better informed than the person they are describing. In philosophical
discussions, for instance, it is standardly presupposed that
'Hesperus' and 'Phosphorus' are
co-referential. So the diagonal intension associated with the sentence
'Hesperus \(\ne\) Phosphorus' will be *necessarily
false* when it's asserted in philosophical contexts (i.e.,
it will be false when uttered in any situation compatible with what is
being presupposed in the philosophical conversation). And yet when a
philosopher says that O'Leary doesn't know that Hesperus
is Phosphorus, she still manages to communicate that O'Leary
fails to grasp some contingent empirical proposition. On the face of
it, the metasemantic account of assertion cannot explain how this is
possible, since every cell of the metasemantic matrix for the identity
claim in this philosophical conversation will be assigned the value
True.



Stalnaker's response to this problem is to suggest that the
context set for a belief report must be expanded so as to include
worlds that correspond to the way that the believer himself (i.e.
O'Leary) takes things to be. The diagonal proposition of the
philosopher's sentence is thus determined by considering what
she would be saying if her sentence were asserted in contexts
compatible with O'Leary's beliefs (1987; 1988). However,
there is no general rule for choosing which worlds are the relevant
ones:



> 
> The procedure I am proposing for extending propositional concepts so
> that the diagonalization strategy can be applied to problematic belief
> attributions takes examples case by case. It is not, as yet, very
> satisfactory if we are looking for a systematic way to explain why the
> complements of belief attributions denote the propositions that they
> seem to denote. But if, using this procedure, we can find a possible
> worlds proposition that is a plausible candidate to be the object of
> belief being attributed in the various problematic examples, then
> [...] it will not be completely mysterious how these propositions
> can be expressed by the sentences that seem to express them.
> (Stalnaker 1987, 129)
> 



Thus, the metasemantic 2D framework provides adequate descriptive
resources for characterizing mental states and our discourse about
them, without invoking fine-grained Fregean senses, concepts, or
syntactic structures. However, the metasemantic theory used to
construct the relevant 2D matrices relies on unsystematic norms of
charitable interpretation to identify the precise contents of
particular attitudes and attitude reports (Stalnaker 1999b,
 18-19).[23]


### 3.4 Metasemantic vs. semantic



The semantic vs. metasemantic distinction was first drawn by Kaplan
(1989b, 573-6). A semantic theory for a language assigns
semantic values (meanings) to particular expressions in the language.
In contrast, a metasemantic theory explains *why* expressions
have those semantic values: i.e., what facts about an expression make
it the case that it has a certain meaning. A semantic theory, for
instance, might tell us that the semantic value for 'Barack
Obama' is a 1D intension that maps any possible world to a
specific individual, while the semantic value for 'I' is a
2D intension that maps contexts of use (centered worlds) to a constant
function from any possible world to the individual who is the speaker
in the context. A metasemantic theory will tell us what makes it the
case that these are the correct interpretations of their meaning:
e.g., the metasemantic theory might appeal to the speaker's
dispositions and history, how she is causally connected to her social
or physical context, the linguistic conventions of the local
linguistic community, details about the conversational context, and so
on.



A *semantic interpretation* of the 2D framework takes the 2D
framework to specify a semantic value of an expression. Kaplan's
theory of indexicals is a semantic interpretation of the framework, as
are Jackson's and Chalmers' generalized 2D semantics.
Stalnaker calls his own interpretation of the 2D framework a
*metasemantic* one because his 2D matrices reflect general
principles for assigning semantic values (horizontal intensions) to
expressions on the basis of empirical facts about their use.
Stalnaker's 2D matrices thus reflect metasemantic facts about
interpretation, not semantic facts about the meanings of specific
expressions.



The metasemantic interpretation of the 2D framework is structurally
different from semantic interpretations like Jackson's and
Chalmers' in a number of important respects. First, on the
metasemantic approach, 2D matrices are defined in terms of a
restricted set of possible worlds. Metasemantic 2D matrices are
defined only with respect to those worlds that are consistent with
some relevant set of the subject's background assumptions. On a
semantic interpretation, in contrast, 2D matrices are defined on the
set of *all* centered possible worlds.



Second, just which set of possible worlds is used to construct a
metasemantic 2D matrix depends in part on the
*theorist's* contingent explanatory interests. For
instance, Daniels' current internal state of understanding the
expression 'Mars' may be associated with different 2D
matrices, depending on whether we are interested in characterizing his
conversation with O'Leary or in characterizing his thought
identifying a speck visible in the night sky as Mars. In many
contexts, moreover, we can characterize the content associated with
Daniels' internal state without resorting to the 2D apparatus at
all: e.g., when Daniels asserts 'Mars is a planet' we can
usually model what he says and thinks in terms of a simple horizontal
intension. Semantic interpretations of the 2D framework treat the
intension corresponding to the diagonal of a 2D matrix as fully
determined by the subject's internal states, whereas on a
metasemantic approach which diagonal intension is assigned--or
whether a diagonal intension is invoked at all--depends on the
explanatory interests of an
 interpreter.[24]



Third, the assignment principle used to fill in a metasemantic 2D
matrix for a subject's words is not beholden to an internalist
account of reference-fixing. On a semantic interpretation, 2D matrices
are filled in on the basis of the subject's idealized judgments
about hypothetical scenarios. But the metasemantic approach is not
committed to relying on the subject's epistemic judgments to
fill in 2D matrices: it's the *theorist* who must be able
to assign horizontal intensions to a subject's words and
thoughts on the basis of the totality of empirical facts about that
subject. Nor should we think of reference-fixing as somehow determined
by the internal states of the theorist, since even theorists are prone
to mistakes on externalist accounts of reference-fixing.



Fourth, on the metasemantic approach, the basic semantic assignments
are always horizontal intensions, which reflect the nature of the
objects, kinds and properties the subject is thinking and talking
about. On a metasemantic account, diagonal intensions represent *ad
hoc reinterpretations* that we resort to when the normal
horizontal intension for a subject's sentences or thoughts is
necessarily true or necessarily false, and hence makes no substantive
claim about the empirical world. Such diagonal intensions, moreover,
are theoretical abstractions *derived* from a set of horizontal
intensions that are compatible with the subject's partial
ignorance about nature of the objects, kinds, and properties she is
representing (i.e., her partial ignorance of the horizontal intensions
for her words and thoughts). Semantic interpretations of the 2D
framework, in contrast, treat the intensions picked out by the
diagonal of a 2D matrix as basic semantic values in their own
right.



As a consequence of these structural features of the metasemantic
interpretation, metasemantic diagonal intensions play none of the
basic explanatory roles attributed to 1-intensions by proponents of
generalized 2D semantics. In particular, (i) diagonal intensions do
not figure in a compositional semantic theory, (ii) they do not
reflect stable aspects of linguistic or conceptual competence, and
(iii) they do not represent apriori accessible reference-fixing
conditions for the subject's words or thoughts (Stalnaker 1999b,
2004 , 2007).



The metasemantic interpretation use of the 2D framework has come in
for criticism on a number of fronts. For a brief survey, see the
supplementary document:



> 
> *Objections to the metasemantic interpretation*
>