Two-dimensional (2D) semantics is a formal framework that is used tocharacterize the meaning of certain linguistic expressions and theentailment relations among sentences containing them. Two-dimensionalsemantics has also been applied to thought contents. In contrast withstandard possible worlds semantics, 2D semantics assigns extensionsand truth-values to expressions relative to two possible worldparameters, rather than just one. So a 2D semantic framework providesfiner-grained semantic values than those available within standardpossible world semantics, while using the same basic model-theoreticresources. The 2D framework itself is just a formal tool. To develop asemantic theory for someone’s language, a proponent of 2Dsemantics must do three things: (i) explain what exactly the twopossible world parameters represent, (ii) explain the rules forassigning 2D semantic values to a person’s words and sentences,and (iii) explain how 2D semantic values help in understanding themeanings of the person’s words and sentences.

The two-dimensional framework has been interpreted in different waysfor different explanatory purposes. The two most widely acceptedapplications of two-dimensional semantics target restricted classes ofexpressions. David Kaplan’s 2D semantic framework for indexicalsis widely used to explain conventional semantic rules governingcontext-dependent expressions like ‘I’,‘that’, or ‘here’, which pick out differentthings depending on the context in which the expression is used. Andlogicians working on tense and modal logic use 2D semantics tocharacterize the logical implications of operators like‘now’, ‘actually’, and‘necessarily’. Such restricted applications of 2Dsemantics are intended to systematize and explain uncontroversialaspects of linguistic understanding.

Two-dimensional semantics has also been used for more ambitiousphilosophical purposes. Influential theorists like David Lewis, FrankJackson and David Chalmers argue that a generalized 2D semanticframework can be used to isolate an apriori aspect of meaning.Roughly, the idea is that speakers always have apriori access to thetruth-conditions associated with their own sentences. On the face ofit, this apriority claim seems to conflict with the observation thatcertain necessary truths, such as ‘water =H₂O’, can be known only on the basis of empiricalinquiry. But proponents of generalized 2D semantics argue that the 2Dframework undercuts this objection, by showing how such aposteriorinecessities are consistent with apriori access to truth-conditions.The positive reasons to accept generalized 2D semantics, however, arebound up with larger (and partly disjoint) explanatory projects. As aconsequence, debates over the merits of generalized 2D semantics touchon broader controversies about apriority, modality, semantic theoryand philosophical methodology.

The two-dimensional framework can also figure in a theory ofadhoc language use, instead of a theory of literal meanings. RobertStalnaker’s influential 2D account of assertion falls in thiscategory. His “metasemantic” interpretation of the 2Dframework is intended to characterize what is communicated whenconversational partners are partially ignorant or mistaken about theliteral meaning of their own words. Although it is formally similar togeneralized 2D semantics, Stalnaker’s use of the 2D frameworkavoids apriori accessible truth-conditions of the sort posited bygeneralized 2D semantics.

• 1. Restricted 2D Semantics
  • 1.1 Indexicals
    • 1.1.1 The need for a 2D framework
    • 1.1.2 Introducing 2D matrices
    • 1.1.3 The logic of indexicals and diagonal intensions
  • 1.2. Modal operators
    • 1.2.1 “Actually” and “Necessarily”
    • 1.2.2 A 2D model
    • 1.2.3 Philosophical applications
• 2. Generalized 2D Semantics
  • 2.1 Vindicating the traditional approach to meaning
  • 2.2 The empiricist project
    • 2.2.1 Linguistic understanding and coordination
    • 2.2.2 The role of the 2D framework
    • 2.2.3 Apriori conceptual analysis
    • 2.2.4 The role of conceptual analysis in metaphysics
  • 2.3 The rationalist project
    • 2.3.1 Connecting meaning, apriority and possibility
      • Supplement: The 2D argument against materialism
    • 2.3.2 The epistemic interpretation of the 2D framework
    • 2.3.3 Interpreting the E2D framework
    • 2.3.4 Rationalist semantic values
  • 2.4 Objections to generalized 2D semantics
    • Supplement: Objections to generalized 2D semantics
• 3. The Metasemantic Interpretation
  • 3.1 2D semantics for externalists
  • 3.2 A 2D account of assertoric content
  • 3.3 Propositional attitudes
  • 3.4 Metasemantic vs. semantic
  • Supplement: Objections to the metasemantic interpretation
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Restricted 2D Semantics

Two-dimensional semantics was introduced to model the semantics ofcontext-sensitive expressions in natural language, like indexicals anddemonstratives. A similar 2D framework was developed to modelimportant 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 sentencesdepends on the meanings of their parts by stating rules governing theinterpretation of subsentential expressions and their modes ofcombination. A semantic framework provides a standard formalism forstating such rules. The simplest (0-dimensional) semantic frameworkswork by assigningextensions as the semantic values ofparticular expressions. Intuitively, the extension includes thosethings in the actual world to which the expression applies: e.g., theextension of the name ‘Barack Obama’ is the man Obama, theextension of the predicate ‘is cool’ is the set of all theactual cool things, and the extension of a two-place predicate‘is cooler than’ is the set of pairs of actually existingthings the first of which is cooler than the second. A whole sentenceis assigned a truth-value (True or False) as its extension, which iscomputed on the basis of the extensions of the component expressions:e.g., the sentence, ‘Barack Obama is cool’, will have thesemantic value True just in case Obama is a member of the set ofactual cool things. A two-dimensional semantic framework is the resultof enriching this simple extensional framework in two distinctways.

The first enrichment,standard possible worlds semantics, isintroduced to explain the meaning of modal operators like‘possible’ and ‘necessary’ and to distinguishthe intuitive subject matter represented by particular subsententialexpressions. Consider the expressions ‘Roger Federer’,‘the greatest tennis player of all time’, and ‘themost famous Swiss citizen in 2020’. Let’s assume all threeexpressions happen to have exactly the same extension: a particularindividual RF. So a simple extensional semantics will assign exactlythe same semantic value to all three expressions. But clearly theydiffer in meaning: if events had unfolded only slightly differentlythan they actually did, the three expressions would pick out differentpeople. In general, definite descriptions like ‘the greatesttennis player’ or ‘the most famous Swiss citizen’pick out different individuals depending on who happens to have therelevant properties in counterfactual situations; whereas proper nameslike ‘Roger Federer’rigidly pick out the verysame individual in every possible situation.^[1] [See entry onrigid designators.] Moreover, such differences in what expressions pick out incounterfactual situations affect the truth of modal claims: e.g.,‘Federer is necessarily Federer’ is true, but‘Federer is necessarily the greatest tennis player’ isfalse. So there is an aspect of meaning that is not captured in simpleextensional semantics. The basic idea behind possible world semanticsis to map out such differences in meaning by specifying what anexpression picks outrelative to every possible way the worldcould be (every “possible world”).

In standard (1-dimensional) possible worlds semantics, the semanticvalue of an expression is anintension, a function thatassigns an extension to the expression “at” every possibleworld. For instance, the semantic value of a definite description like‘the best known Swiss citizen’ is a function that takes asinput a possible world and yields as output whoever happens to satisfythat description in that world, and the semantic value of a propername like ‘Roger Federer’ is a constant function that mapsevery possible world to the very same individual, RF. Such intensionsreflect 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 operatorslike ‘necessarily’ and ‘possibly’: a sentenceis 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 possibleworld. [See the entries onintensional logic andmodal logic.]

The second enrichment of the basic extensional semanticframework—the one that is distinctive of two-dimensionalsemantics—requires us to take possible worlds into account in adifferent way. To see why this might be necessary for an adequateaccount of meaning, let’s focus on context-sensitive expressionslike ‘I’, ‘here’ or ‘this’. In onerespect, these terms function like names, picking out the very samething 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 herclaim is true just in case there is a possible way the world could bein which HC is president. In standard possible worlds semantics, then,the intension o of ‘I’ is exactly the same as theintension of the name ‘Hillary Clinton’: a function thatyields the individual HC for every possible world. But clearly theEnglish word ‘I’ is not synonymous with the name‘Hillary Clinton’—for John McCain might utter thesentence ‘I could have been president’ and in his mouththe word ‘I’ would refer rigidly to a different person,JM, in every possible world. What’s distinctive ofcontext-sensitive expressions like ‘I’ or‘this’ is that they represent different things dependingon the context in which they are used. David Kaplan (1989a)^[3] first brought widespread attention to this phenomenon ofcontext-dependence by proposing his influential two-dimensionalsemantic theory to clarify the rules governing such expressions.

Kaplan distinguishes two different aspects of the meaning ofexpressions in a public language. The first aspect,content,reflects the modal profile of the object, kind or propertyrepresented. This is the aspect of meaning that is modeled by standardpossible world semantics. The second aspect of meaning,character, reflects semantic rules governing how the contentof an expression may vary from one context of use to the next. Acontext-invariant expression like ‘Hillary Clinton’ has aconstant character, picking out the very same object in every contextin which it’s used, whereas indexical expressions like‘I’ or ‘this’ have variable character, pickingout different things in different contexts of use.

Formally, character is defined as a function that maps possiblecontexts of use to contents, and content is defined as a functionmapping possible worlds to extensions. Thus, a character is a functionthat takes as input a context and yields as output a function frompossible worlds to extensions. This is atwo-dimensionalintension, since there are two distinct roles that possibilitiesplay here: as a context of use, and as a circumstance of evaluation (apossible situation relative to which we evaluate whether the relevantobject exists or property is instantiated). Contexts of use can bethought of as “centered” worlds: possible worlds with adesignated agent and time within that world, which serve to locate aparticular situation in which the expression is used. We can thenrepresent a context as an ordered triple, \(\langle w,a,t\rangle\) ,of a possible worldw, an agentawithin that world, and a timetwhen the agent exists in that world.^[4] So possible worlds play two distinct roles in Kaplan’sformalism: contexts of use determine which content is expressed andcircumstances 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 2Dframework: in any possible context, \(\langle w,a,t\rangle\), anutterance of ‘I’rigidly designates the agent ofthat context,a, in all possible circumstances of evaluation.[See the entry onindexicals for a more detailed discussion].

1.1.2 Introducing 2D matrices

A useful way of visualizing the dual role played by possible worlds ina 2D framework is to constructa two-dimensional matrix(Stalnaker 1978). To represent Kaplan’s theory of indexicals, wearray possible circumstances of evaluation along the horizontal axisand possible contexts of utterance along the vertical axis. Eachhorizontal row of the matrix represents thecontent thetarget expression would have if used in the context specified for thatrow. This content is (partially) represented by recording theextension of the term at each possible circumstance arrayed along thehorizontal axis. This procedure is then repeated for each contextlisted along the vertical axis.

For instance, consider a particular utterance of ‘I’ madeby Barack Obama during his inaugural presidential address. Thiscontext of use can be represented as the world \(w_1\), centered onthe man BO, at time \(t_0\). We can (partially) represent the contentof ‘I’ in this centered world thus:

This simple one-dimensional matrix reflects the fact that, when usedin this context, ‘I’ refers rigidly to Obama at everypossible circumstance of evaluation—even at the counterfactualworlds \(w_2\) and \(w_3\), in which John McCain or Hillary RodhamClinton won the 2008 presidential election. The context-dependence ofthe 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 onhim at his inaugural address; and \(\langle w_3, \HC, t_0\rangle\) isa world in which Clinton won, centered on her at her inauguraladdress. We then rely on our implicit understanding of the semanticrules governing ‘I’ to generate two more rows for ourmatrix:

What the matrix reveals is that the expression ‘I’ rigidlydesignates different individuals, depending on the context in which itis used. Thus the 2D matrix provides a graphic illustration of howcontent of the expression ‘I’ varies, depending on thecontext in which it is used.

Such 2D matrices can be used to represent the differences between thesemantic rules governing indexicals, definite descriptions, and names.For instance, the definite description ‘the inaugural speaker in2009’ will generate the following Kaplanian matrix:

Unlike the matrix for ‘I’, the horizontal rows of this 2Dmatrix are all exactly the same. This reflects the fact that theexpression ‘the inaugural speaker in 2009’ is notcontext-sensitive: it always represents the very same propertyirrespective of the context in which it is used—namely, theproperty of being the person who delivers the inaugural USpresidential address in 2009. This property is exemplified bydifferent individuals at different possible worlds: the person who isthe inaugural speaker at \(w_1\) is Obama, at \(w_2\) it’sMcCain, and at \(w_3\) it’s Clinton. In general, the sequencearrayed along the rows of this matrix reflects the variety ofdifferent individuals who could instantiate the property representedby ‘the inaugural speaker’ in different circumstances. Ofcourse, no finite matrix can fully capture the range of variation, butit can give a useful partial representation of the property inquestion.

The matrix for a proper name like ‘Barack Obama’ revealsanother very different pattern:

According to Kaplan, proper names are context-invariant: they alwayshave the very same content irrespective of the context in which theyare used. Proper names are also rigid designators: they pick out asingle individual at every possible world. The upshot is that the 2Dmatrix for a proper name will be completely uniform: the very sameindividual appears in every cell of the matrix. This reflects the ideathat the semantic function of a name in a public language is simply topick out a particular individual, not to convey any information abouthow to identify the individual in question. [For a different accountof proper names, see §2.2 below.]

1.1.3 The logic of indexicals and diagonal intensions

Kaplan’s semantic rules for indexicals guarantee that certainsentences will be true whenever they are uttered, and certaininferences will be truth preserving. This account paved the way forKaplan’s formal logic of indexicals (Kaplan 1989a). In thissystem, logical validity is defined in terms of different possiblecontexts of use: a sentence is valid iff it is true in every possiblecontext of use; and an inference is valid iff the truth of thepremises ensures the truth of the conclusion in every possible contextof use.

On Kaplan’s account, sentences can be logically valid, even ifthey express contingent propositions. For instance, the semantic rulesgoverning indexicals ensures that the sentence ‘I am here’will be true in any context of use. But the content expressed isnormally contingent: I could easilynot have been here rightnow, but at the beach instead.

To illustrate, we can construct a partial 2D matrix for the sentenceusing our previous example. Suppose ‘I am here now’ isuttered by the new president at the inauguration(t₀) in \(w_1\) where Obama won, \(w_2\), whereMcCain won, and \(w_3\) where Clinton won. Let’s assume Obamawould attend the inaugural address of McCain but not of Clinton,McCain would avoid the inauguration of anyone who defeated him, andClinton would attend Obama’s inauguration but notMcCain’s. This yields the following 2D matrix:

The horizontal rows of the matrix represent the different propositionsexpressed by the sentence in each context of use. Each utteranceexpresses a different contingent proposition, as can be seen by thefact that each contains both ‘T’s and ‘F’s,and that the patterns differ. The 2D matrix also graphicallyrepresents the fact that the sentence is guaranteed to be truewhenever uttered. Notice the diagonal of the matrix running from thetop left corner to the bottom right, which contains all‘T’s. This reflects the fact that the sentence isguaranteed to be true whenever uttered. With a nod to Stalnaker(1978), we can call this thediagonal intension of thesentence. In Kaplan’s semantic framework, a necessary diagonalintension indicates that a sentence islogically valid andanalytic.^[5]

1.2 Modal Operators

At around the same time that Kaplan began developing his account ofindexicals, logicians working on tense and modal logic had begun using2D semantic frameworks to explain the behavior of sentential operatorslike ‘now’ and ‘actually’ (Åqvist 1973;Kamp 1971; Segerberg 1973; Vlach 1973). Unlike Kaplan, these logicianswere not primarily concerned with the semantic rules governing naturallanguages. In particular, modal logicians were not focused on how thecontext in which an expression is used can affect its reference.Rather, they were interested in developing formal systems forrepresenting valid inferences about time and possibility. It turns outthat tense and modal logic are formally very similar and that bothrequire double-indexing for expressive adequacy. Thus, to fullycapture reasoning about what’s necessary and possible, we needto move from standard possible worlds semantics to a 2D semanticframework.

1.2.1 ‘Actually’ and ‘Necessarily’

Consider the following sentence:

(1)

It is possible foreverything that is actually red to be shiny.

Standard possible worlds semantics lacks the expressive power tocapture what is said by this sentence (Crossley and Humberstone 1977;Hazen 1976, 1978). The claim is not that there is a possible worldsuch that all the things that are red in that world are also shiny inthat world (they’re supposed to be red in theactualworld, not the counterfactual one). Nor is the claim that for eachobject that is red, there is a possible world in which it is shiny(the objects are all supposed to be shinytogether within asingle possible world). So here is a relation among objects inpossible worlds that cannot be expressed in standard possible worldsemantics. To capture the relation, we need to introduce an extraelement into the formal framework: we simply designate one worldwithin the model (the set of possible worlds) to play the role of theactual world. We can then introduce a sentential operator‘\(\mathcal{A}\)’ (read as ‘Actually’), whichrequires us toevaluate any claim within its scope at thedesignated world, even when the operator is embedded within thescope of other modal operators. Using this enriched possible worldsframework, we can represent the truth-conditions of our samplesentence 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 designatedworld, \(w_@\), is shiny inw.

One awkward consequence of this 2D semantic account of‘Actually’ is the way this operator interacts with thestandard modal operator ‘Necessarily’. Intuitively, whatthe actual world is like seems logically and metaphysicallycontingent. But according to the proposed semantics for‘Actually’, any true sentenceS willyield anecessarytruth when embedded within thescope 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’strue at every possible world in that model that Obama won at itsdesignated world. So on the proposed 2D semantics, the sentence isnecessarily true. (When we embed (2) within the necessityoperator ‘\(\Box\)’ we get a truth; and any claim of theform \(\mathcal{A}S \rightarrow \Box \mathcal{A}S\) will be logicallyvalid.) But intuitively it’s acontingent matter howthe 2008 elections turned out. To mitigate this counterintuitiveconsequence, Crossley and Humberstone introduce a new logicaloperator, ‘Fixedly’ (‘\(\mathcal{F}\)’) insuch a way that the complex operator ‘Fixedly Actually’(‘\(\mathcal{F}\mathcal{A}\)’), captures the sense ofnecessity we have in mind when we deny that (2) is necessary. Asentence is fixedly actually true just in case it istrue nomatter which world is designated as actual.^[6]

1.2.2 A 2D model

Once again, 2D matrices can be used to graphically depict how thesemantic theory works. Let’s take our universe of possibleworlds to contain just three worlds: \(w_1\) is a world where Obamawon, \(w_2\) a world where McCain won, and \(w_3\) a world whereClinton won. To explain the ‘Fixedly Actually’ operator,we need to consider possible worlds playing two different roles: thestandard role as a circumstance of evaluation and the special role ofbeing designated as the actual world. To construct a 2D matrix, wearray possible worlds playing the standard role along the horizontalaxis, and along the vertical axis we array the same worlds playing therole of being designated as actual. Each horizontal row of this matrixrepresents 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 truein the world designated as actual in a given model. So we can fill inthe 2D matrix as follows:

In any world in a model (a row in the matrix), ‘ActuallyS’is always evaluated by looking at thedesignated world of that model. So such sentences are eithernecessarily true (True at every world in the model) or necessarilyfalse (False at every world in the model). This is the sense ofnecessity that corresponds to the standard modal operator‘\(\Box\)’. On this understanding of necessity, the targetsentence is necessarily true (since \(w_1\) represents theactual actual world). But intuitively there is a sense inwhich the sentence seems contingent, since a different world couldhave been actual: if \(w_2\) or \(w_3\) had been actual, the sentence‘Obama actually won’ would have been false. This fact isreflected in the 2D matrix by the diagonal intension, where thesentence comes out true with respect to \(\langle w_1, w_1\rangle\),but false with respect to \(\langle w_2, w_2\rangle\) and \(\langlew_3, w_3\rangle\). The ‘Fixedly Actually’ operator issensitive to the necessity or contingency of the diagonal intension.The sense in which the target sentence (2) is not necessary is thatit’s not fixedly actually true.

While this double-indexing model has become standard in theliterature, as (Rabern 2012b) points out, the same expressive adequacyconsiderations Crossley and Humberstone (1977) used to support twoindices for modal and temporal operators also supportmultiple-indexing for those operators. For instance,consider the following sentence:

(3)

If the Titanic hadnot hit an iceberge on its maiden voyage, it would still have beenpossible for everyone would then have survived the maiden voyage todie on the maiden voyage. (Rabern 2012b: 119)

To generate the correct analysis, we need to invoke three distinctworlds: the actual world \(w_1\), where the Titanic hit an iceberg, acounterfactual world \(w_2\) where it did not, and anothercounterfactual world \(w_3\) where all survivors of the voyage in\(w_2\) died. So double-indexing and 2D matrices cannot fully capturethe compositional semantics of modal sentences: we will need aninfinite sequence of possible world indices. For further work onmultiple-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 onphilosophical puzzles about modality. Following Gareth Evans (1979),Davies and Humberstone suggest that there are two notions ofmetaphysical necessity involved in ordinary modal thinking: deep andsuperficial necessity. They argue that the two logical operators,‘\(\mathcal{F}\mathcal{A}\)’ and‘\(\Box\)’,respectively, provide aclear formal elucidation of these two notions.

These two notions of necessity, they argue, help explain some of SaulKripke’s (1980) puzzling examples in which necessity andapriority come apart. Using 2D modal logic, it’s easy toconstruct necessary aposteriori truths. The semantic rules governingthe modal operator ‘\(\mathcal{A}\)’ guarantee that everyclaim of the form \(\mathcal{A}S\) will be either necessarily true ornecessarily false in the sense of ‘\(\Box\)’. But when theembedded sentenceS is an ordinary empirical truthlike ‘Obama won’, \(\mathcal{A}S\) will be knowable onlyaposteriori: so \(\mathcal{A}S\) will be a necessary aposterioritruth. The ‘Actually’ operator can also be used toconstruct contingent apriori truths. Any claim of the form\((\mathcal{A}S \rightarrow S)\) is guaranteed by the semantic rulesgoverning ‘Actually’ to be true at the designated world nomatter which world is designated as actual (i.e., it’s fixedlyactually true). But whenS is an ordinary empiricaltruth, the complex claim is not necessary in the sense of‘\(\Box\)’: there will be some worlds in the model whereS is false while \(\mathcal{A}S\) is true. In suchcases, the complex sentence will be a contingent apriori truth.

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

1. Julius \(=_{df}\) the actual inventor of the zip.
2. Water \(=_{df}\) the actual chemical kind to which that liquidbelongs 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 ‘Juliusinvented the zip’ is contingent and apriori and ‘water =H₂O’ is necessary aposteriori.^[8] [See the entries onnames anddescriptions. For a survey of other philosophical applications of the 2D frameworkin modal logic see (Humberstone 2004).]

Davies and Humberstone themselves express reservations about theadequacy of analyses using ‘actually’ for natural languageexpressions, particularly in the case of proper names (1980,17–21). As a consequence, they did not take 2D modal logic toprovide a complete response to Kripke’s puzzles about necessaryaposteriori and contingent apriori truths. However, the use of the 2Dframework to explain these puzzles was subsequently taken up andrefined 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 2Dframework that seek to explain the meaning of specific types ofexpression: indexicals and modal operators. In contrast, proponents ofgeneralized 2D semantics (G2D) believe that the 2D framework can beused to explain an important aspect of the meaning ofallexpressions. In particular, G2D is meant to vindicate the traditionalidea that we have apriori access to our own meanings through armchairreflection.

According to the philosophical tradition, to know the meaning of asubsentential expression like ‘bachelor’ is to implicitlygrasp a criterion that determines exactly which individuals count asbachelors in any possible situation. (Accounts of meaning broadlyalong 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 keytheoretical roles:

1. Semantic competence: Two speakers (or one speaker on twooccasions) share the same meaning just in case they associate the verysame criterion with their expressions.
2. Reference determination: The criterion a speaker currentlyassociates with an expression determines which things fall into itsextension in every possible situation.

The first claim requires that speakers who share the same meaning mustshare a criterion for identifying the reference; while the secondrequires that this criterion be veridical. If this traditional accountof meaning is correct, then one can make one’s own meaningsexplicit by engaging inapriori conceptual analysis. Suchconceptual analysis allows you to determine what exactly it takes tocount as a bachelor in any possible world; and it allows you tospecify what exactly someone must be prepared to accept in order togenuinely agree or disagree about bachelors.

G2D is a strategy for defending a variant of this traditional view ofmeaning against a series of influential objections. In the1970s and 80s, semantic externalists used a variety of persuasiveexamples to argue that the traditional account of meaning yields anunrealistic picture of (i) semantic competence, (ii) referencedetermination, and (iii) epistemic access to modal facts. Proper namesand natural kind terms seem especially problematic for the traditional account.^[9] By commonsense standards, you don’t need to know a specificrule for identifying Gödel in any possible world in order tocount as competent with the name ‘Gödel’; and no suchknowledge seems required for your use of the name to pick out therelevant man in every possible world (Donnellan 1970; Kripke 1980).^[10] Similarly, you don’t need to know precisely what it takes forsomething to count as water in any possible world to be competent withthe word ‘water’ or for your word to pick out the chemicalsubstance H₂O in every possible world (Kripke 1980; Putnam1970, 1972). Indeed, making room for the possibility of ignorance anderror about reference-conditions seems crucial to explaining empiricalinquiry into the nature of familiar things, and to vindicating thecommonsense realist idea that we can refer to things whose nature wedon’t fully understand (Burge 1979, 1986; Putnam 1972, 1973). Ifthese critics are right, then the traditional account of meaning isuntenable. Implicit knowledge of reference-conditions isnotrequired either for linguistic competence or for determinatereference. And apriori conceptual analysis cannot be trusted to revealwhat’s genuinely possible—at best, it reveals one’scurrent fallible assumptions about the topic in question. [See theentry onexternalism about mental content.]

Proponents of G2D believe this pessimistic conclusion is unwarranted.What critics’ examples really show, they argue, is that thetraditional view of meaning should be refined, not junked. Moreover,the 2D semantics developed for indexicals and modal operators suggestsa promising strategy for accommodating putative externalistcounterexamples within a broadly traditional account of meaning. Inthe case of indexicals and rigidified definite descriptions, competentspeakers grasp areference-fixing criterion without graspingthe modal profile of the object, kind, or property picked outby the expression. For instance, you can know that ‘I’always refers to the speaker whenever it is uttered without knowingthe nature of the person who is actually picked (e.g., what it takesto be Barack Obama in any possible world). Perhaps our understandingof names and natural kind terms is structured in a similar way:competent speakers always have apriori access to the reference-fixingcriterion 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 couldbe used to clarify the nature of this semantic understanding. Moreoverwe may be able to explain certain epistemic operators, like ‘itis conceptually possible that’ or ‘it is apriorithat’, as operating on such 2D semantic values.

The basic philosophical idea behind G2D—that subjects haveapriori access to reference-fixing criteria for their words but not tothe modal profile of the subject matter picked out—has beensuggested by a number of theorists. David Lewis, in particular, was apowerful champion of the idea that we can give apriori definitions forterms whose precise reference we do not understand. Lewis articulatedthe ‘analytic functionalist’ approach to specifying themeaning of mental predicates and of theoretical terms in science(1966; 1970; 1972; 1979; 1980; 1994); and he was also an earlyadvocate of a generalized 2D approach to semantics (1981; 1994). Otherinfluential proponents of the idea that we can have implicit knowledgeof reference-fixing criteria without knowing the modal profile of thereference include Michael Dummett (1973; 1981), Gareth Evans (1982),and John Searle (1983). Early proponents of an explicitlytwo-dimensional semantics for names and natural kind terms includeHarry 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 G2Das a way of reconciling the lessons of semantic externalism with thetraditional apriori approach to meaning and modality.

It’s worth noting that G2D has been motivated primarily byepistemic, metasemantic, and metaphysical concerns, rather than byissues in compositional semantics. In particular, G2D seeks tovindicate the traditional idea that we can know the truth-conditionsof our own sentences via armchair reasoning about hypothetical cases.The approach promises to explain why certain necessary truths can onlybe known aposteriori by appealing to the structure of our implicitsemantic understanding. Proponents of G2D make claims about how thetwo types of intension may interact with modal and epistemicoperators. However, working out the details of the compositionalsemantics has been a relatively recent concern of proponents of G2D(e.g. Chalmers 2011a, c; Chalmers and Rabern 2014; Johannesson andPackalén 2016; Kipper 2017).

The 2D semantic frameworks proposed by Jackson and Chalmers are verysimilar in their broad aims and formal structure, and commentatorsoften treat the two versions as interchangeable. However, the twoframeworks are developed in the service of two quite differentphilosophical projects, emphasizing different aspects of thetraditional approach to meaning. Jackson takes up the traditionalempiricist project of explaining empirical facts about language useand communication, while Chalmers pursues a broadly rationalistproject of explaining key structural interconnections between meaning,apriority, and possibility. This difference in explanatory aims leadsto 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 ofuncontroversial facts about language use. In particular, theempiricist seeks to characterize the psychological states that guideindividuals’ application of an expression to particular cases,and to explain how linguistic coordination within a linguisticcommunity is achieved.

Clearly individual speakers must have some implicit assumptions aboutthe reference of a word that guide their verdicts about whether itapplies to particular cases (Jackson 1998a, 29–42). Yourjudgments 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 asit reflects that prior understanding. An empiricist seeks to explainthese facts by positing a stable internal psychologicalstate—something like aninternal reference-fixingtemplate—that guides your verdicts no matter what theactual world turns out to be like.

It’s equally clear that members of the same linguistic communitygenerally manage to use words to reliably coordinate their beliefs andactions (Jackson 1998b, 2004). When I ask you to pass the salt, youknow roughly which white granular substance I’m askingfor—you know, for instance, that it would be inappropriate topass the sugar bowl or the pepper grinder. This sort of everydaycoordination requires speakers to have similar dispositions toclassify things as falling into the extension of words, and itrequires that these similarities in classificatory dispositions bemutually obvious to all concerned: for it to make sense for me to say‘please pass the salt’ in order to get salt, it must becommon knowledge between us that we’re inclined to classifyroughly the same things as ‘salt’. An empiricist explainsthis common knowledge by positingimplicit conventions thatrequire everyone to associate the very same reference-fixing templatewith a given word (Jackson 1998b; Lewis 1969).

An empiricist use of the 2D framework is intended to show that thiscore explanatory project is not undermined by the intuitions aboutnames 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 ormistaken about the precise nature (modal profile) of the objects,kinds or properties their words pick out. But linguistic conventionsdon’t always fix the reference by specifying the nature of thereference. Perhaps the conventions governing names and natural kindterms are structured in a similar way to indexicals. For instance, wemight have an implicit semantic rule requiring us to take‘water’ to pick out whatever chemical kindactually explains a certain suite of superficial observableproperties: e.g., being a clear, potable, odorless liquid that fillslakes and streams around here (Jackson 1998a, 1998b). On thisanalysis, ‘water’ just is an implicitly indexicalexpression, picking out different chemical kinds depending on whichworld is actual. If this rule is what one must accept to count ascompetent with the meaning of the expression type ‘water’,then it is no surprise that competent speakers often fail to realizethat water = H₂O.

Of course, it is an empirical question whether names and natural kindterms are in fact governed by indirect reference-fixing rules of thissort. But according to Jackson, you can test whether your implicitunderstanding of ‘water’ is structured in this way byconsidering possible situations in two different roles: asyouractual environment or asa mere counterfactualpossibility (Jackson 1998a, ch. 2). Consider two differentpossible worlds based on Putnam’s Twin Earth thought experiment(Putnam 1972). In the first world, Earth, the clear potable stuff thatfills 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 isthe complex chemical kind, XYZ. If your commonsense understanding of‘water’ is governed by the proposed reference-fixingconvention, it would lead you to identify different chemicalsubstances as water depending on what your actual environment is like:if your actual environment is Earth, then water is H₂O; butif your actual environment is Twin Earth, then water is XYZ. If youassume that water isactually H₂O, moreover, youwill judge that water isessentially H₂O in allcounterfactual circumstances. And if you assume water isactually XYZ, then you’ll judge water isessentially XYZ.

This pattern of dispositions to apply the term ‘water’ canbe depicted on a 2D matrix as follows:

Along the vertical axis are ranged centered possible worlds (apossible world, with a designated agenta and timet within that world) representing different waysyour actual environment could be like; and the same worlds are rangedalong the horizontal axis representing different counterfactualcircumstances of evaluation. This matrix reflects your commonsensedispositions to apply the term ‘water’ to differentchemical kinds on the basis of whether it actually plays certainsuperficial roles described in other commonsense terms(‘clear’, ‘potable’, ‘liquid’, etc).^[12]

Semantic externalists take these sorts of judgments about Twin Earthto militate against a traditional account of meaning—for theysuggest that your understanding does not fully determine the nature ofthe reference. But according to Jackson, the only conclusion that iswarranted is that the meaning of your term is more complex than thetradition suggests: your verdicts about possible worlds considered asactual reflect yournaïve reference-fixing criterion,and your verdicts about possible worlds considered as counterfactualreflect thetheoretical criterion you would accept after youlearned all the relevant empirical facts about your actualenvironment. These two types of criterion can be modeled in possibleworld semantics as intensions: an A-intension (for‘Actual’) is a function from worlds considered as actualto 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 theA-intension you associate with ‘water’; and the firsthorizontal 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 anexpression’s semantic content, but 2D empiricists insist thatboth A- and C-intensions reflect important aspects of a competentEnglish speaker’s understanding of a word like‘water’. In particular, they take A-intensions to reflectwhat is understood and communicated by minimally competent Englishspeakers and what guides their everyday classifications. Thesuggestion, then, is that A-intensions capture the shared,conventionally entrenched understanding of reference-fixing conditionsposited by the empiricist approach to meaning.

By itself, this 2D framework offers no guarantee that the hypotheticaljudgments recorded by an A-intension are produced by a stablereference-fixing criterion. Nor does it guarantee that the very sameA-intension will be generated for all competent speakers in yourlinguistic community. However, according to Jackson, we have solidempirical reasons to think these conditions are satisfied in the caseof names and natural kind terms. First, the widespread acceptance ofthe externalist thought experiments demonstrates that we do in factshare similar reference-fixing criteria for terms like‘water’ and ‘Gödel’ (Jackson 1998a,38–39). Second, the empiricist model of meaning provides thebest psychological explanation of how such linguistic coordination isachieved (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. Thecriteria that implicitly guide our everyday use of a term are oftenembodied in recognitional or inferential dispositions rather than inconsciously accessible rules. Indeed, Jackson likens grasp ofreference-fixing criteria for particular expressions to our ability torecognize the grammaticality of sentences in one’s own language(2000). Just as linguists can construct a grammar for your language onthe basis of your judgments about the acceptability of particularsentences, you can construct an analysis of the meaning you associatewith an expression on the basis of your application of a term tohypothetical cases. The correct analysis must capture the full rangeof your confident judgments involving the target expression, whileabstracting away from performance errors and other interfering factors(Jackson 1998a, 31–37).

This psychological model helps explain how you can come to know thecorrect analysis of your term ‘water’ by noting whichproperties you treat as “obvious and central” when fillingin a 2D matrix like the one above (Jackson 1998a, 31). The 2Dframework prompts you to consider possible cases in two differentways–as actual or as counterfactual. This allows you to knowwhether the content of your term varies depending on what your actualenvironment is like (e.g. ‘water’) or whether it is stable(e.g. ‘bachelor’). Moreover, careful attention to yourreactions to these suppositions will allow you to make explicit whichsuperficial properties implicitly guide your application of the term.For instance, you may discover that your implicit criterion forapplying ‘water’ is that water \(=_{df}\) the actualchemical kind in your environment that is a clear, potable, odorlessliquid that falls as rain and fills lakes and streams. Alternatively,your use of the term ‘water’ may be guided by the types ofcausal relations invoked in causal theories of reference: water\(=_{df}\) the actual natural kind that has been the dominant cause ofyour community’s past use of the term ‘water’.Indeed, Jackson suggests that standard causal theories of referenceare based on this method of conceptual analysis (1998a, 37–41).[See the entry oncausal 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 becorrect “independently of knowing what the actual world islike” (Jackson 1998a, 51). The evidence that supports suchanalyses consists in purely hypothetical judgments: judgments abouthow to classify caseson the supposition that yourenvironment is likeX, or likeY.Since such hypothetical judgments don’trequire you to determine what yourreal environment is like,your justification for accepting an analysis is not based on empiricalknowledge. And to change your judgment about a purely hypotheticalcase 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 rolein metaphysics (Jackson 1994; 1998a). Metaphysical reductions providea constitutive account of some target domain (e.g., beliefs, freewill, water, moral rightness) in terms of more basic features of theworld (e.g., the properties postulated by an idealized physics, ideasin the mind of God, the mosaic of sense data). A physicalist aboutmental states, for instance, is committed to there being specificfacts about the microphysical structure of the world that suffice forthe existence of beliefs, desires and sensory experiences. Thephysicalist is thus committed to metaphysically necessary“entailments” connecting claims about the two domains:it’s metaphysically necessary that if such-and-such physicalfacts obtain, then such-and-such mental facts obtain. Thismetaphysical entailment relation can arguably be cashed out in termsof global supervenience (Jackson 1998a, 6–14). [See the entry onsupervenience.]

The role of conceptual analysis is to show that a putative reductionrespects the original meaning of the target expression (Jackson 1998a,28). A physicalist won’t succeed in accounting for free will ifshe identifies free will with having a temperature of 37.4º C– such a “reduction” would simply change the subjectunder discussion. A successful reduction must be answerable to ouroriginal shared understanding of the target expression—andelucidating this original understanding just is what conceptualanalysis does. So if conceptual analyses are knowable apriori, itfollows that metaphysical reductions must always be backed by apriorientailments between the base-level claim (such as a physicaldescription of the world) and the target claim (such as the claim thathumans have free will).

On this empiricist account, conceptual analysis plays a modestmetaphysical role. Conceptual analysis captures apriori entailmentrelations among your ideas; but it cannot tell you whether thereare any objects, kinds, or properties that satisfy yourcurrent reference-fixing assumptions (Jackson 1998a, 42–4).Moreover, the meaning you currently associate with a term may bepragmatically deficient: e.g., it may not be determinate enough tosettle certain hard cases or it may not allow you to draw usefuldistinctions in your actual environment. In such cases, you would havegood pragmatic reasons to change the meaning of your term (Jackson1998a, 44–6, 53–4). What the empiricist denies is thatchanging your current criteria for applying a term can ever get youcloser 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 causalmechanisms guiding everyday classification and communication. For arationalist, in contrast, an expression’s meaning reflects whatis apriori accessible to the speaker on the basis of ideal reflection.The empiricist is primarily concerned with causal explanation oflinguistic facts, while the rationalist is primarily concerned withidealized apriori rationality and insights into objective possibility.This difference in emphasis can have significant ramifications forG2D.

David Chalmers has developed a detailed and influential rationalistinterpretation of the 2D framework. This semantic project is situatedwithin a broadly rationalist tradition that posits a “goldentriangle” of necessary constitutive relations between meaning,apriority, and possibility (2004; 2006a; 2012).

Following Frege (1892), Chalmers is interested in capturing a notionof meaning that is finer-grained than reference. Frege pointed outthat sentences containing co-referential expressions like‘Hesperus’ and ‘Phosphorus’ can differ incognitive significance: someone who is competent with thesetwo names may not realize they are co-referential and may thereforeuse them differently in making and justifying claims. Frege tooksameness of cognitive significance to be the mark of sameness ofmeaning. According to a 2D rationalist, sameness of cognitivesignificance can in most cases be elucidated in terms ofaprioriequivalence: two expressions are associated with the same meaningiff one can know that they pick out the very same things on the basisof apriori reflection alone (Chalmers 2002b).^[14] This constitutive link between meaning and apriority constitutes thefirst side of the “golden triangle”.

The second side of the “golden triangle” connects meaningwith possibility. Following Carnap (1947), Chalmers suggests that wecan use possible worlds semantics to individuate particular meaningsin terms of their representational properties. In standard possibleworld semantics, the meaning of ‘doctor’ is identifiedwith anintension that maps possible worlds to extensions. Anexpression’s intension reflects the modal profile of the object,kind, or property picked out. Identifying meanings with intensionstherefore establishes an important constitutive connection betweenmeanings and modal claims. If ‘doctor’ and‘physician’ are associated with the same meaning, thenit’strue in all possible worlds that all doctors arephysicians and all physicians are doctors. And conversely if twoexpressions areco-extensive in all possible worlds, thenthey have the same meaning.

The third side of the “golden triangle” connectspossibility with apriority. Following Kant (1787), a rationalist aboutmodality holds that what is necessary is always knowable apriori andwhat is knowable apriori is always necessary. Thus ideal apriorireflection can be trusted to reveal the space of possibility.

This “golden triangle” of constitutive relations generatesa distinctive rationalist account of meaning. To be competent with anexpression’s meaning is to be in an internal cognitive statethat puts one in a position to identify its extension in any possibleworld on the basis of apriori reflection alone. Apriori reflectionwill also suffice to determine whether two expressions are associatedwith the same or different meanings. This rationalist approach tomeaning contrasts with the empiricist one: whereas the empiricist usescausally efficacious cognitive mechanisms to isolate thereference-fixing criteria currently associated with an expression, therationalist uses the subject’sideally rationaljudgments to isolate the complex cognitive states that wouldground those reflective judgments. As a consequence, the aspect ofunderstanding that corresponds to a rationalist meaning may turn outto be more heterogeneous and less stable than the shared, causallyefficacious ‘templates’ postulated by the empiricist. The“golden triangle” also involves a distinctive rationalistaccount of modal epistemology, according to which ideal aprioriconceivability is a fail-safe guide to metaphysical possibility. Thismodal rationalism affords a simple and attractive account of ouraccess to modal facts (Chalmers 1996, 136–8; 1999, 488–91; 2002a).^[15]

This simple rationalist picture of meaning and modality, however, wasundermined by externalist thought experiments. Kripke’s (1980)observation that certain necessary truths, like ‘Hesperus =Phosphorus’, are only knowable aposteriori threatens both theidea that linguistic competence affords apriori access to the truth-and applicability-conditions of one’s words and the idea thatnecessary truths are always apriori knowable. The guiding idea behind2D rationalism is that a rationalist can accommodate Kripke’sexamples by moving to a 2D semantic framework. In particular, the 2Dframework can be used to isolate an aspect of meaning that satisfiesthe “golden triangle” of constitutive relations amongmeaning, apriority and modality.

Roughly, the idea is that ‘Hesperus = Phosphorus’ isaposteriori because we associate distinct reference-fixing criteriawith the two names: e.g., being the brightest star visible in theevening and the brightest star visible the morning. According to the2D rationalist, these reference-fixing criteria are (i) an aspect ofmeaning, (ii) which can be known apriori via conceptual analysis, and(iii) which suffices to fix the applicability conditions for everypossible world considered as one’s actual environment. If the 2Dframework can be used to isolate such an aspect of meaning for allexpressions, we will have vindicated the rationalist’s“golden triangle” connecting meaning, apriority andpossibility.

Vindicating this “golden triangle” constitutes a primarytheoretical constraint for a rationalist interpretation of 2Dsemantics. A 2D semantics that meets this constraint would play awide-ranging role in philosophy. It would account for core semanticroles associated with the Fregean notion of sense (Chalmers 2002b) andthe traditional notion of a proposition (Chalmers 2011a). In addition,rationalist 2D semantics promises to define a versatile notion ofnarrow thought content suited to playing key explanatory andevaluative roles in commonsense psychology (Chalmers 2002c).Furthermore, the rationalist approach to meaning and modalityunderwrites a distinctive form of apriori reasoning about the natureof the actual world:

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

Chalmers has developed an influential anti-physicalist argumentalong these lines, which relies on a rationalist 2D semantic frameworkto establish that facts about phenomenal consciousness cannot bereduced 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 twoquestions:

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

But the rationalist project imposes specific constraints on how thesequestions are answered. To vindicate the “goldentriangle”, the rationalist must identify a way of mapping anindividual speaker’s understanding of particular expressionsonto possible worlds that affords apriori access to the entire spaceof possibility. This is not a trivial requirement: standard ways ofinterpreting the 2D framework cannot vindicate the rationalistproject. However Chalmers has developed a distinctive“epistemic” interpretation of the 2D framework that hebelieves can establish the relevant constitutive links betweenmeaning, 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 thediagonal of a 2D matrix. So the Core Thesis affirms that a tokensentence is apriori (for a subject at a particular time) just in casethere is no possible way the world might be that, if it actuallyobtained, would makeS false. In effect, the CoreThesis sums up the “golden triangle” of constitutiveconnections the rationalist hopes to establish between meaning,apriority, and possibility: (i) it postulates a possible way the worldcould be for every apriori coherent hypothesis, and vice versa; and(ii) this tight connection between apriority and possibility isreflected in an aspect of linguistic meaning, the 1-intension.^[16]

The major obstacle to vindicating the Core Thesis for standardinterpretations of the 2D framework is the assignmentprinciple—the way 2D theories assign extensions relative to“worlds considered as actual”. A natural way ofunderstanding the injunction to consider a possible world as actual isto simply imagine a possible world, locate a person in it at a time,and then rely on ordinary interpretive methods to decide what exactlythat person in those empirical circumstances is referring to whenusing a given expression. Chalmers calls this strategy for assigning1-intensions to expressions a “contextualist”interpretation of the 2D framework. What’s distinctive of acontextualist approach is (i) that a token of the target expressionmust be located within the world considered as actual, and (ii) thatthe expression is assigned an extension on the basis of how it’sused in that world. On this approach, a 1-intension will beundefined for possible worlds that do not contain a token ofthe target expression: no extension can be assigned for such worlds,not even an empty extension.

This contextualist approach to assigning 1-intensions is incompatiblewith the Core Thesis (Chalmers 2004, 167­–176). Considersentences like ‘Language exists’ or ‘A sentientbeing exists’: the meaning of these sentences seems to guaranteethat they will be true in every possible context in which they areused. So on the contextualist approach, these sentences should beassigned necessary 1-intensions, mapping every possible context of useto the truth-value True. But contrary to the Core Thesis, thesesentences arenot apriori truths knowable independently ofany empirical evidence. There’s no contradiction in the veryidea of a world without language or thought and we can easily imaginewhat such a world would be like; it’s just that our everydayexperience allows us to immediately rule out the possibility that ouractual environment is like that. The problem is that contextualist1-intensions are undefined for worlds without thought or language,even though they are both apriori coherent and metaphysicallypossible. So a necessary contextual 1-intension does not trackapriority 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 anotion ofepistemic possibility—whatseemspossible after ideal rational reflection—to interpret the 2Dframework. More specifically, he focuses on the notion of aprioricoherence: claims that could be true for all one can tell on the basisof idealized apriori reasoning.^[18] This notion of apriori coherence is used to answer the twointerpretive questions highlighted above: (i) apriori coherence isused to characterize the possibilities relative to which 1-intensionsare defined, and (ii) apriori coherence is invoked to assign1-intensions to a speaker’s expressions.

First consider the possibilities that define 1-intensions. On theepistemic interpretation, the possibilities are not metaphysicallypossible contexts of use, but epistemically possible“scenarios”: maximally specific hypotheses about whatone’s actual environment might be like that cannot be ruled outthrough apriori reasoning alone. Scenarios provide a completecharacterization of the entire history of a universe, down to the lastmicrophysical detail. They also provide perspectivalinformation—a notional “center”—that indicatesthe location from which the hypothetical universe is to be considered.The crucial point is that scenarios are defined by their epistemicrole: they represent ways we can conceive of the actual world, withinwhich we can try to identify familiar objects, kinds or properties.^[19]

The second distinctive element of the epistemic interpretation of 2Dsemantics is the procedure for assigning 1-intensions to aspeaker’s expressions. On the epistemic approach, 1-intensionsreflect relations of apriori coherence between descriptions ofpossible scenarios and ordinary language sentences:

The epistemic 1-intension for a sentenceS is Trueat a scenarioW iff \((W\) & not\(-S)\) isapriori incoherent. (Chalmers 2004, 180–4)

This principle for assigning 1-intensions relies on thespeaker’s ordinary ability to engage in object-level reasoningabout combinations of hypotheses: given the assumption that thescenario descriptionW is true, you’re askedto decide whetherS must be true as well. Ifit’s incoherent to accept \((W\) & not\(-S)\), yourepistemic intension forS mapsWto True, otherwiseW is mapped to False. Thisepistemic assignment principle contrasts sharply with thecontextualist principle. The contextualist approach requires us toengage in explicit meta-linguistic reasoning to interpret theexpression ‘S’as it’sused within the possible worldW. On theepistemic approach, in contrast, an extension is assigned to‘S’ on the basis of the subject’sown object-level reasoning using the expressions ‘W’and ‘S’.

Unlike the contextualist approach, therefore, the epistemic assignmentprinciple does not explicitly require that a scenario contain a tokenof the relevant expression type in order to assign an extensionrelative to that scenario. As a consequence, sentences like‘Language exists’ seem to pose no special problem forsatisfying the Core Thesis. The sentence ‘Language exists’will have a contingent epistemic 1-intension, because there arepossible scenarios that are apriori consistent with both the truth andfalsity of that sentence. For instance, consider a scenario in whichthe only object is a giant lump of salt. To test whether your sentence‘Language exists’ is true at this scenario considered asactual, you ask whether there is any incoherence in combining theclaim ‘The only object that exists is a lump of salt’ withthe claim ‘It’s not the case that language exists’.Intuitively, this combination is coherent: there is no language in thesalt world. So the epistemic 1-intension for your sentence‘Language exists’ yields the value False for thatscenario. Since there are other scenarios relative to which thesentence ‘Language exists’ will have the value True, yoursentence will have a contingent epistemic 1-intension. This contingentepistemic intension for your sentence ‘Language exists’reflects the fact that it’s not apriori true that languageexists. So it seems the epistemic assignment principle will allowapriority and necessity of the 1-intension to go hand in hand, asrequired by the Core Thesis.

2.3.3 Defining the E2D framework

There is further work to do in spelling out Chalmers’ E2Dframework in such a way as to vindicate his rationalist project. Oneway to think about the rationalist project is as a combination of thefollowing theses:

1. Semantic foundationalism holds that there is a canonical level ofdescription that affords transparent epistemic access to an objectivespace of apriori possibilities – i.e. ways the world might befor all that can be known independently of all experience.
2. Meaning rationalism holds that implicit conceptual competence putsone in a position to conclusively know the applicability conditions ofone’s words relative to every such scenario considered asactual.
3. Modal rationalism holds that the space of apriori epistemicpossibility is a failsafe guide to the space of metaphysicalpossibility: there are no modal illusions when scenarios or worlds aredescribed in canonical terms

Together, 1 and 2 constitute a sort of semantic reductionism: themeaning of any ordinary language expression is reduced to the meaningsof the base vocabulary via the epistemic exercise of consideringscenarios as actual and worlds as counterfactual. And 3 ensures thatthis epistemic exercise is an accurate guide to metaphysicalpossibility. Chalmers’ Core Thesis is meant to capture thistight relationship between grasp of meaning, apriori reflection, andmetaphysical possibility.

However, simply rejecting C2D in favor of E2D does not yet provide anypositive account of what it is to entertain an epistemic scenario W,and how we should update our beliefs in the light of the suppositionthat W is actual. Without these details, it’s impossible todetermine whether the Core Thesis is true. Perhaps it’simpossible to consider a scenario as actual without presupposingone’s own existence, or perhaps our best epistemic methods forupdating our beliefs presupposes the existence of those very beliefs;or perhaps there is no way of thinking about the world thatdoesn’t rely on some further empirical assumptions about theworld.

Chalmers has entered into these interpretive questions in considerabledetail over many publications. The starting point for his approach isoutlined in (Chalmers and Jackson 2001), where he suggests thatscenarios can be understood as PQTI sentences: where P statesmicrophysical truths, Q states phenomenal truths, T is a‘that’s all’ clause indicating that P and Q providea complete description of a possible universe, and I indicates thesubject’s notional location within that universe. P and Q employa canonical vocabulary that fully specifies the essential nature ofthe fundamental properties upon which all other properties in apossible world supervene. Thus, PQTI sentences provide anepistemically transparent access to the space of epistemic andmetaphysical possibility (simply removing the self-locatinginformation from a PQTI sentence yields a complete description of acorresponding possible world, PQT). The 1-intensions of one’sordinary language expressions are then determined by the individualsubject’s ideally reflective dispositions to judge sentencestrue, assuming the truth of different PQTI-sentences. And 2-intensionsare fixed by one’s reasoning about PQT sentences considered ascounterfactual (given assumptions about PQTI). (Chalmers 2006, 2011b)further articulates how the space of epistemic possibility can beunderstood, 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 focusedsquarely on the epistemic ‘scrutability’ relation thatconnects our understanding of ordinary language expressions tobase-level descriptions of scenarios.

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

Chalmers still takes PQTI sentences to be a promising candidate forspecifying a scrutability base D, but he is open to the possibilitythat the descriptive vocabulary in PQTI may need to be supplemented inorder to capture some truths, such as truths about causal relations orquiddities. But while he can afford to be flexible about the exactnature of the scrutability base, Chalmers’ rationalist programdepends on vindicating Apriori Scrutability for any sentence that isevaluable as possibly true or false (169–71). A good deal thushangs on whether he is right that ideal epistemic procedures allow forapriori justification, given an exhaustive base-level description of ascenario. Chalmers offers arguments to support the plausibility ofthis view in (2012, ch. 4.).

According to the ‘frontloading’ argument, we can haveconclusive apriori knowledge of material conditionals of the form(PQTI \(\supset\)S). Chalmers argues that all empirical information relevant tojustifying a verdict about S can be ‘frontloaded’ into theantecedent of the conditional, so information about one’sreal-world environment, E, cannot play any essential role injustifying verdicts about the conditional. If E is itself apriorientailed by PQTI, it is not needed to justify a verdict about theconditional. And if E is not apriori entailed by PQTI, E will beirrelevant to justifying a verdict about the conditional. So ourjustification for the application conditionals that ground1-intensions is wholly apriori, and immune to empirical defeat.^[20]

2.3.4 Rationalist semantic values

Epistemic 2D semantics differs in important respects from traditionalaccounts of meaning. Semantic theories normally describe generalsemantic rules governing expressiontypes, whereas epistemic2D semantics is based on a single individual’s currentunderstanding of atoken expression. Kaplan and Jackson, forinstance, use the 2D framework to characterize the implicitconventions governing syntactically individuated expressions like‘I’ or ‘water’ in our linguistic community. Incontrast, Chalmers uses the 2D framework to characterize yourpotentially idiosyncratic understanding of a particular use of anexpression on a given occasion—e.g., the way you understood AlGore’s fifth use of ‘water’ during a speech onclimate change. Moreover, on this account 2D semantic values depend onthe upshot of ideal rational reflection about apriori coherencerelations. Just what is involved in ideal rational reflection is anopen question. But it’s plausible that it may depend onsubstantive constructive theorizing about the empirical scenario inquestion and on various non-obvious and idiosyncratic aspects of thesubject’s initial cognitive state. In that case, identifying theprecise epistemic 1-intension associated with your understanding of‘water’ will be a highly non-trivial matter, and it may befar from obvious when your understanding of the term shifts so thattwo tokens no longer share the same epistemic 1-intension. This iswhy, in contrast with 2D empiricists like Jackson, a rationalist likeChalmers denies that epistemic 1-intensions reflect thesubject’s “implicit knowledge” of a reference-fixingcriterion (e.g., Chalmers 2002a, 185; 2006b, §5).

Of course, it’s possible that some epistemic intensions willreflect stable reference-fixing rules that are entrenched by implicitlinguistic conventions. But it’s also possible that someepistemic intensions will reflect highly abstract, heterogeneous,unstable, and idiosyncratic aspects of a speaker’s understandingat a given time. As a consequence, epistemic intensions are notguaranteed to line up with conventional linguistic meanings (Chalmers2002b). Given this divergence from standard semantic theories, one maywonder whether epistemic intensions deserve to be considered a kind ofmeaning.

However, according to the 2D rationalist, epistemic intensions playthe core semantic roles associated with Fregean senses (Chalmers2002b). Like Fregean senses, epistemic 1-intensions lend themselves toa compositional semantic theory: the epistemic intension of a sentenceis determined by the epistemic intensions of the componentexpressions. Moreover, epistemic 1-intensions, like Fregean senses,reflect the speaker’s own rational perspective on what her wordsrepresent. Two token names ‘A’ and‘B’ have the same Fregean senses iffthe identity ‘A = B’ would strike the speaker astrivially true. Similarly, a subject associates two tokenexpressions with the same epistemic intension iff they areapriori equivalent.^[21] Finally, epistemic intensions may play a role similar to that ofFregean senses in the semantics of attitude reports (Chalmers 2011a).Overall, then, epistemic intensions seem to provide an attractivetheoretical refinement of the Fregean notion of sense.

In addition, epistemic intensions arguably carve out a well-definednotion of narrow content suited to playing key roles in commonsensepsychology (Chalmers 2002c). Epistemic intensions reflect rationalrelations among token mental states. Epistemic intensions can then beused to specify representational state types that are relevant toassessing a person’s rationality and to explaining rationalthought processes.

It’s important to note that epistemic 1-intensions are intendedto explain onlyone aspect of meaning. The 2D semanticframework also posits 2-intensions (“counterfactual” or“subjunctive” intensions), which reflect the modal profileof the object, kind or property picked out by an expression. ButChalmers emphasizes that his E2D does not exclude positing furtheraspects of meaning:

Two-dimensionalism is naturally combined with asemanticpluralism, according to which expressions and utterances can beassociated with many different semantic (or quasi-semantic) values, bymany different semantic (or quasi-semantic) relations. On this viewthere should be no question about whether the primary intension or thesecondary intension isthe content of an utterance. Both canbe systematically associated with utterances, and both can play someof the roles that we want contents to play. Furthermore, there willcertainly be explanatory roles that neither of them play, sotwo-dimensionalism should not be seen as offering an exhaustiveaccount of the content of an utterance. Rather it is characterizingsome aspects of utterance content, aspects that can play a useful rolein the epistemic and modal domains. (Chalmers 2006a, §3.5)

In sum, Chalmers’ highly idealized E2D framework is intended toearn its semantic keep by defining a kind of meaning capable offorging traditional rationalist connections between meaning,rationality and possibility. But he is happy to allow that there maybe other types of semantic values or structures that are needed toplay other semantic roles. [See entry onpropositions, andstructured 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 claimthat 1-intentions capture a type of meaning, and its internalistapproach to assigning contents. The specific rationalist andempiricist applications of the G2D framework have also beencriticized. For a survey of these lines of criticism: see thesupplementary document:

Objections to generalized 2D semantics

3. The Metasemantic Interpretation

3.1 2D semantics for externalists

Generalized 2D semantics seeks to vindicate a traditional internalistconception of meaning: it posits an extra aspect of meaning for allexpressions (the intension corresponding to the diagonal of a 2Dmatrix) that is fully determined by a subject’s internal states,and which in turn determines objective truth-conditions for theirsentences. By enriching compositional semantics in this way, G2Dpromises a straightforward explanation of a variety of epistemicproperties of sentences: e.g., why a necessary sentence like‘Hesperus is Phosphorus’ is not apriori knowable, what thesubject learns by accepting the sentence, or how the subject uses thesentence in reasoning.

But using the 2D framework to characterize the subject’sepistemic perspective is not beholden to this internalist project.Semantic externalists reject the traditional view that our purelyinternal states afford apriori access to reference-fixing conditionsfor our words and thoughts. According to externalists, the basicassignments in a compositional semantics relate the subject’swords and thoughts to objective features of herenvironment—objects, kinds and properties whose nature iscaptured by standard (1D) possible world semantics. Even externalists,however, can define 2D matrices that reflect the subject’sepistemic perspective on the reference of her words and thoughts. Forthe externalist, however, these 2D matrices will not representmeanings—a specific aspect of understanding that isrequired for linguistic or conceptual competence and which figures ina compositional semantic theory that determines truth-conditions forsentences. On an externalist interpretation, 2D matrices merelyreflect one aspect of a subject’spartial semanticunderstanding of what her words and thoughts represent. Becauseexternalist 2D matrices don’t represent meanings, moreover, theexternalist is free to use the 2D framework strategically to focus ondifferent aspects of the subject’s understanding for differentexplanatory purposes.

Robert Stalnaker has articulated such an externalist interpretation ofthe 2D framework in a series of influential papers spanning somethirty years. He was the first to introduce 2D matrices to specifywhat is communicated in situations where conversational partners arepartly ignorant or mistaken about the nature of the objects, kinds orproperties their words pick out (1978), and he later extended his 2Dframework to characterize the content of certain thoughts and attitudeattributions (1981; 1987; 1988). In both cases, the 2D framework isused to define “diagonal” intensions that reflect thesubject’s partial understanding of which objects, kinds orproperties her words and thoughts represent. These diagonal intensionsare not meanings or semantic values, since they do not figure in acompositional semantic theory and they do not reflect conditions forconceptual or linguistic competence. The only meaning of an expressionon this account is its ordinary “horizontal” intension. Ineffect, Stalnaker’s 2D matrices represent different meaningsthat an expressioncould have had if it had occurred indifferent empirical circumstances. This “metasemantic”interpretation of the 2D framework contrasts sharply the“semantic” interpretations favored by G2D theorists likeJackson and Chalmers (Stalnaker 2001, 2004).

Proponents of G2D were influenced by Stalnaker’s early papersdeveloping the 2D framework, and their views are often presented ascontinuous in motivation and form. But there are important theoreticalconsequences that flow from the choice between 2D metasemantics andgeneralized 2D semantics. Indeed, Stalnaker himself is a vocal criticof generalized 2D semantics, rejecting its commitment to the semanticstatus 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 originallydeveloped as a way of explaining how the propositions conveyed by theassertion of a sentence can vary depending on the conversationalcontext (Stalnaker 1978). In this seminal paper, Stalnaker proposes anattractive theoretical account of the role of assertion in aconversation, which is then used to explain how the assertoric use ofa necessary sentence like ‘Hesperus = Phosphorus’, canconvey a specific empirical proposition within a given conversation.In particular, Stalnaker argues that our commitment to construing suchsentences as making felicitous and informative assertions will lead ustoreinterpret their content in ways that can be modeledusing the 2D framework.

The guiding idea is that in making an assertion the speaker is tryingto get the audience to rule out certain possibilities. In asserting‘It’s cold today’, for instance, I may be trying toget you to rule out possibilities in which today’s temperaturein Melbourne is over 10° C. We can model what my assertionconveys, then, as a function that maps possible worlds in whichtoday’s temperature is under 10° C to True and all otherworlds to False. However, the precise truth-conditions communicated byan assertoric use of a sentence depend in part on the conversationalcontext in which it takes place. Just which temperatures count ascold, for instance, depends on shared background assumptions in aparticular conversational context: what’s cold in Melbourne ismild in Manitoba.

A second guiding idea is that the proposition actually conveyed by theassertion of a particular sentence depends onpresuppositionsshared by the participants in the conversation—includingpresuppositions about what particular words represent andpresuppositions about actual empirical circumstances. If you’rea Chinese speaker who doesn’t understand anything at all aboutwhat the term ‘cold’ represents in English, then I cannotuse ‘It’s cold today‘ to convey facts aboutMelbourne’s weather. And if you’re a Canadian whodoesn’t understand anything about Australian weather conditions,you won’t understand precisely what I am saying to my fellowMelburnians when I assert that sentence. Stalnaker calls the set ofpresuppositions that conversational partners treat as common knowledgethat they can rely on to get their point across the “contextset”—which he models as the set of possible worlds thatsatisfy all of these mutual presuppositions. The context set willencode shared assumptions about the meaning of words, about generalempirical facts, about the what’s happened so far in theconversation, and so on.

The goal of assertion, Stalnaker suggests, is to shrink the contextset. In making an assertion, the speaker tries to get the audience toaccept a new proposition as one of their shared presuppositions,thereby shrinking the set of possible worlds that are considered liveoptions. For instance, in asserting ‘It’ll be very coldtoday’ to a group of Melburnians, I exploit background knowledgeof local weather conditions in June to get my audience to accept thatthe temperature outside is somewhere between 5–10° C, rulingout live possibilities that it might be in the 15–20° Crange. If all goes well, further planning will proceed on the basis ofa smaller and more accurate range of possibilities. In contrast, if Iwere to assert ‘It’s cold’ to the monolingualChinese speaker or to the parochial Canadian, my assertion would bedefective, since my audience wouldn’t be able to figure outwhich temperatures are ruled out by my assertion.

Identity claims, however, do not seem to fit this simple model ofassertion. As Kripke (1980) argued, identities are either necessarilytrue or necessarily false. So accepting an identity will either leavethe context set unchanged or it will eliminate it altogether. Eitherway, 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 possibilitiespreviously taken to be live options. According to the metasemanticaccount, (i) the goal of assertion can explain why the assertion of anecessary sentence will lead to a reinterpretation of the content ofthe asserted sentence, and (ii) the 2D framework helps to specify justwhich proposition will be conveyed by the sentence within a givenconversation.

In general, an identity claim is appropriate when one of the partiesto a conversation is (partially) ignorant about which object is pickedout by a name like ‘Lloyd’. For an externalist likeStalnaker, this is a case ofsemantic ignorance. IfO’Leary doesn’t know that ‘Lloyd’ isco-referential with ‘I.L. Humberstone’, then he does notfully understand the semantic rules governing these names: i.e., thatboth names are associated with a constant function from any possibleworld to a specific individual. But O’Leary isn’t utterlyincompetent with the meaning of these terms: he implicitly understandsboth names as rigid designators, and he has some substantiveunderstanding of the object each name picks out. For instance, he mayunderstand that ‘Lloyd’ refers to the person to whomhe’s just been introduced and that ‘I.L.Humberstone’ refers to the author of ‘Direction ofFit’. O’Leary’s semantic deficiency—hisfailure to fully understand the meaning of these names in acontextually appropriate way—is grounded in his ignorance of theordinary empirical fact that the man to whom he has been introduced isthe author of ‘Direction of Fit’.

2D matrices can be used to represent this sort of partial semanticunderstanding. O’Leary knows that \(if\) the man in front of himis 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 sentenceexpresses a necessary falsehood. What O’Leary doesn’t knowis which of these two possibilities corresponds to his actualsituation. Call the first possibilityi and thesecondj. O’Leary’s epistemic situationcan then be summed up in a 2D matrix:

The matrix is defined only with respect to a specific set of relevantalternative possibilities,i andj,chosen in such a way as to reflect thesubjects’ semantic understanding and our own explanatoryinterests. The vertical axis represents these possible worlds in theirrole ascontexts of use, which determine the literal semanticcontent of the expressions used in them. The horizontal axisrepresents those same possible worlds ascircumstances ofevaluation, relative to which we evaluate the truth or falsity ofthe proposition expressed. Each row of the matrix thus represents adifferent proposition that might be literally expressed by thesentence. Stalnaker calls such matricespropositionalconcepts, since they reflect the subject’s currentimperfect conception of the meaning of the sentence.^[22] This particular matrix reflects the fact that O’Leary’scurrent epistemic state is compatible with the identity sentenceexpressing 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 actualworld is likei, the literal semantic content ofthe asserted sentence is a necessary truth. But necessary truths ruleout no empirical possibilities whatsoever, so this cannot be theinformative proposition that is conveyed by Daniels’ assertion.Moreover, O’Leary is not in a position to recognize that this isthe literal semantic content of the sentence, since he doesn’tknow whether the actual world is likei orj.The natural suggestion is that the informationconveyed by Daniels’ assertion is that the real world is likei and notj. When O’Learyaccepts ‘Lloyd is I.L. Humberstone’, he will no longertreat the possibility that the man in front of him isnot theauthor of the famous article as a live option: this empiricalpossibility will be eliminated from his context set. Thus, theproposition that seems to be conveyed by Daniels’ assertioncorresponds to the diagonal intension determined by our 2D matrix forthat assertion. Moreover, this observation generalizes: when subjectsare partially ignorant of the semantic values of their words, thediagonal proposition determined by the propositional concept cancapture the empirical information conveyed by the assertion.

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

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

Speakers should conform to this maxim, because assertion involves anintention to get one’s audience to eliminate worlds from thecontext set in accordance with the proposition expressed—and inorder for this intention to succeed the audience must be in a positionto figure out just which worlds they are being asked to eliminate.When this sort of rational maxim governing the communication ofinformation is flouted, the audience will look for a non-standardinterpretation of the utterance that would bring it back intoconformity with the maxims (Grice 1989). [See the entries onpragmatics and onimplicature.]

According to Stalnaker, this is precisely what is going on in the caseof 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 awarethat O’Leary doesn’t know whether he is in a world likei, where the man to whom he’s been introducedis the famous author, or a world like j where they are distinct. YetDaniels utters a sentence that expresses a different propositiondepending on whether the actual world is likei orlikej. In such circumstances, the audience shouldlook for an alternative interpretation of the assertion.Daniels’ assertion can be brought back into conformity with themaxim by re-interpreting it as conveying the proposition expressed bythe diagonal of the matrix. At a rough intuitive level, we can saythat Daniels is trying to get O’Leary to accept that thesentence ‘Lloyd is I.L. Humberstone’ expresses a truth.But the 2D framework also allows us to specify more precisely justwhat empirical information is conveyed within a given conversationalcontext. Given O’Leary’s and Daniels’ commonpresuppositions about what the two names represent, Daniels’assertion also expresses the propositionthat the man to whomO’Leary has just been introduced is the author of‘Direction of Fit’.

It’s worth emphasizing, however, that the very same sentenceasserted in a different conversational context could express anentirely different empirical proposition: just which proposition isexpressed, on this metasemantic account, depends on what theindividual parties to the conversation are currently presupposingabout the meanings of the expressions used.

In recent work, Stalnaker has enriched his notion of a context. Inaddition to a sets of possible worlds, he introduces (i) multiplecenters within those worlds representing the participants in aconversation, and (ii) accessibility relations among the centersrepresenting interlocutors’ ways of identifying each other(Stalnaker 2008, 2014). This added structure is intended to captureself-locating presuppositions that can help explain what’scommunicated by indexical expressions like ‘I’,‘you’ and ‘now’. For a helpful overview of themotivation for such an approach, see (Ninan 2010a) and for furtherdevelopment of multi-centered accounts of propositional content see(Torre 2010, Ninan 2013).

3.3 Propositional attitudes

The metasemantic 2D framework was originally developed to explaincommunication, but the framework can also be used to specify thecontent of certain beliefs and the content of assertions thatattribute beliefs.

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

But fine-grained Fregean senses or conceptual structures are notstrictly required to distinguish beliefs in identities. Stalnaker(1981; 1987) argues that the metasemantic 2D framework he developed toexplain what is communicated by anassertion of an identitysentence can also explain the content of thebelief statesattributed using an identity sentence to specify its content. IfO’Leary were to notice the pole star and think to himselfthat’s Mars, for instance, the truth-conditions of histhought can be captured by a judiciously defined diagonal proposition(Stalnaker 1987, 125). In this case, the worlds we include in thecontext set may involve facts about which object is the target ofO’Leary’s visual attention and facts about salientempirical properties he associates with the name ‘Mars’.On the metasemantic approach, then, the proposition we attribute insaying O’Leary believes thatthat is Mars is theproposition that the visually salient object is the object that hasthose Martian properties.

A further complication arises in specifying the content of attitudeattributions. On the metasemantic account of assertion, the contentconveyed by a sentence depends on the shared presuppositions of thespeaker and audience. But sometimes the parties to a discussion arebetter informed than the person they are describing. In philosophicaldiscussions, for instance, it is standardly presupposed that‘Hesperus’ and ‘Phosphorus’ areco-referential. So the diagonal intension associated with the sentence‘Hesperus \(\ne\) Phosphorus’ will benecessarilyfalse when it’s asserted in philosophical contexts (i.e.,it will be false when uttered in any situation compatible with what isbeing presupposed in the philosophical conversation). And yet when aphilosopher says that O’Leary doesn’t know that Hesperusis Phosphorus, she still manages to communicate that O’Learyfails to grasp some contingent empirical proposition. On the face ofit, the metasemantic account of assertion cannot explain how this ispossible, since every cell of the metasemantic matrix for the identityclaim in this philosophical conversation will be assigned the valueTrue.

Stalnaker’s response to this problem is to suggest that thecontext set for a belief report must be expanded so as to includeworlds that correspond to the way that the believer himself (i.e.O’Leary) takes things to be. The diagonal proposition of thephilosopher’s sentence is thus determined by considering whatshe would be saying if her sentence were asserted in contextscompatible with O’Leary’s beliefs (1987; 1988). However,there is no general rule for choosing which worlds are the relevantones:

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

Thus, the metasemantic 2D framework provides adequate descriptiveresources for characterizing mental states and our discourse aboutthem, without invoking fine-grained Fregean senses, concepts, orsyntactic structures. However, the metasemantic theory used toconstruct the relevant 2D matrices relies on unsystematic norms ofcharitable interpretation to identify the precise contents ofparticular 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 assignssemantic values (meanings) to particular expressions in the language.In contrast, a metasemantic theory explainswhy expressionshave those semantic values: i.e., what facts about an expression makeit the case that it has a certain meaning. A semantic theory, forinstance, might tell us that the semantic value for ‘BarackObama’ is a 1D intension that maps any possible world to aspecific individual, while the semantic value for ‘I’ is a2D intension that maps contexts of use (centered worlds) to a constantfunction from any possible world to the individual who is the speakerin the context. A metasemantic theory will tell us what makes it thecase that these are the correct interpretations of their meaning:e.g., the metasemantic theory might appeal to the speaker’sdispositions and history, how she is causally connected to her socialor physical context, the linguistic conventions of the locallinguistic community, details about the conversational context, and soon.

Asemantic interpretation of the 2D framework takes the 2Dframework to specify a semantic value of an expression. Kaplan’stheory of indexicals is a semantic interpretation of the framework, asare Jackson’s and Chalmers’ generalized 2D semantics.Stalnaker calls his own interpretation of the 2D framework ametasemantic one because his 2D matrices reflect generalprinciples for assigning semantic values (horizontal intensions) toexpressions on the basis of empirical facts about their use.Stalnaker’s 2D matrices thus reflect metasemantic facts aboutinterpretation, not semantic facts about the meanings of specificexpressions.

The metasemantic interpretation of the 2D framework is structurallydifferent from semantic interpretations like Jackson’s andChalmers’ in a number of important respects. First, on themetasemantic approach, 2D matrices are defined in terms of arestricted set of possible worlds. Metasemantic 2D matrices aredefined only with respect to those worlds that are consistent withsome relevant set of the subject’s background assumptions. On asemantic interpretation, in contrast, 2D matrices are defined on theset ofall centered possible worlds.

Second, just which set of possible worlds is used to construct ametasemantic 2D matrix depends in part on thetheorist’s contingent explanatory interests. Forinstance, Daniels’ current internal state of understanding theexpression ‘Mars’ may be associated with different 2Dmatrices, depending on whether we are interested in characterizing hisconversation with O’Leary or in characterizing his thoughtidentifying a speck visible in the night sky as Mars. In manycontexts, moreover, we can characterize the content associated withDaniels’ internal state without resorting to the 2D apparatus atall: e.g., when Daniels asserts ‘Mars is a planet’ we canusually model what he says and thinks in terms of a simple horizontalintension. Semantic interpretations of the 2D framework treat theintension corresponding to the diagonal of a 2D matrix as fullydetermined by the subject’s internal states, whereas on ametasemantic approach which diagonal intension is assigned—orwhether a diagonal intension is invoked at all—depends on theexplanatory interests of an interpreter.^[24]

Third, the assignment principle used to fill in a metasemantic 2Dmatrix for a subject’s words is not beholden to an internalistaccount of reference-fixing. On a semantic interpretation, 2D matricesare filled in on the basis of the subject’s idealized judgmentsabout hypothetical scenarios. But the metasemantic approach is notcommitted to relying on the subject’s epistemic judgments tofill in 2D matrices: it’s thetheorist who must be ableto assign horizontal intensions to a subject’s words andthoughts on the basis of the totality of empirical facts about thatsubject. Nor should we think of reference-fixing as somehow determinedby the internal states of the theorist, since even theorists are proneto mistakes on externalist accounts of reference-fixing.

Fourth, on the metasemantic approach, the basic semantic assignmentsare always horizontal intensions, which reflect the nature of theobjects, kinds and properties the subject is thinking and talkingabout. On a metasemantic account, diagonal intensions representadhoc reinterpretations that we resort to when the normalhorizontal intension for a subject’s sentences or thoughts isnecessarily true or necessarily false, and hence makes no substantiveclaim about the empirical world. Such diagonal intensions, moreover,are theoretical abstractionsderived from a set of horizontalintensions that are compatible with the subject’s partialignorance about nature of the objects, kinds, and properties she isrepresenting (i.e., her partial ignorance of the horizontal intensionsfor her words and thoughts). Semantic interpretations of the 2Dframework, in contrast, treat the intensions picked out by thediagonal of a 2D matrix as basic semantic values in their ownright.

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

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

Objections to the metasemantic interpretation

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Other Internet Resources

• “Two-Dimensional Semantics”, 2018, Jens Kipper,Internet Encyclopedia of Philosophy,includes an annotated bibliography.
• “Two-Dimensional Semantics”, 2006, David Chalmers (manuscript version of Chalmers 2006b): comparesthe most influential interpretations of two-dimensionalism, andoutlines and defends the epistemic interpretation.
• “Two-Dimensional Semantics—the Basics”, 2007, Christian Nimtz: a concise overview that compares Kripke,Kaplan, Stalnaker and Chalmers.
• “Varieties of Two-Dimensionalism”, 2003, Jim Pryor: a somewhat more technical discussion exploresdifferent ways of interpreting 2D matrices.

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Acknowledgments

Many thanks to David Chalmers, Lloyd Humberstone, Frank Jackson,Robert Stalnaker and an anonymous referee for very helpful feedback onthis entry.