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1. There are two standard ways of interpreting rigid designation, bothof which are consistent with Saul Kripke’s original introductionof the term (1970 lectures, published 1980). On the firstinterpretation, a name rigidly designates a given objectat everyworld in which that object exists, and it fails to designateanything at worlds in which the object does not exist. On the secondinterpretation, a name rigidly designates a given object at everyworldregardless of whether the object exists in that world.For present expository purposes, we’ll rely on the latterinterpretation.

2. An object’smodal profile ornature, in thissense, should be distinguished from its essential properties. A modalprofile (or nature) captures all the possible combinations ofproperties the object might instantiate in different possible worlds.In contrast, the object’s essential properties are the specificsubset of properties (if any) that the object must have in every worldin which it exists. Standard possible world semantics can be used tocapture the modal profile associated with ‘Roger Federer’by first assigning an intension to the name which maps every possibleworld onto a particular individual, RF, and then providing theresources for determining, of any given predicate, ‘P’,whether there is a possible situation inwhich RF exemplifies the property picked out by that predicate. Thecomplete modal profile of RF can be made explicit by surveying all thedifferent combinations of properties RF exhibits in the possibleworlds in which he exists.

3. Kaplan’s essay is based on a series of lectures delivered in1971.

4. This way of representing possible contexts presupposes that the agentexists at the designated time. This is known as a “propercontext”, and it reflects the empirical facts available forinterpreting actual utterances. But some theorists work with a weakernotion of context, which allows for arbitrary combinations of agents,spatio-temporal locations, and worlds. This weaker notion of contextcan help represent how we interpret indexical utterances, like arecording of ‘I am not here now’ on an answering machine,when the agent is not located at the time and place of the utterances(Predelli 1998).

5. The sentence counts as a logical truth in Kaplan’s 2D frameworkbecause logical validity is defined only onproper contexts,in which the designated agent is located at designated time and placein the world of the context (1989a, 544: clause 10, 547: remark3).

6. What is the semantic role of the logical operator,‘Fixedly’ (‘\(\mathcal{F}\)’), considered onits own? Borrowing formalism introduced by (Davies and Humberstone1980), we can say that a sentence is always evaluated with respect toan ordered pair of possible worlds, \(\langle w_x, w_y\rangle\), wherethe first member of the pair represents the world designated as actualand the second represents a circumstance of evaluation. A sentenceSisfixedly true at \(w_n\), iffSis true with respect to every ordered pair \(\langlew_x, w_n\rangle\), where we allow the \(w_x\) to vary arbitrarily.(Graphically, a sentenceS will be Fixedly true at\(w_n\) just in case the entire (vertical) column corresponding to\(w_n\) in a 2D matrix contains ‘True’ all the way down.)In effect, the Fixedly operator tells us about what must be true withrespect to a specific world considered as counterfactual, no matterwhich world is designated as actual. When we combine the operators‘Fixedly’ and ‘Actually’(‘\(\mathcal{F}\mathcal{A}\)’), the complex operatorbehaves like anecessity operator ranging over possible worldsplaying the actual world role. A sentence is fixedly actuallytrue just in case it istrue at the designated world nomatter which world is so designated. So a sentence is fixedly actuallytrue iff it is true with respect to every ordered pair \(\langlew_x,w_x\rangle\), where the value of \(w_x\) is allowed to varyarbitrarily. (Graphically,S will be fixedlyactually true iff the diagonal of the 2D matrix contains‘True’ all the way down.)

7. Notice that, unlike Kaplan, Crossley and Humberstone do not think ofthe possible worlds ranged along the vertical axis of the 2D matrix aspossible contexts of use. On their 2D framework, the worlds“considered as actual” are justarbitrarilydesignated as the reference of the term ‘actually’for the purposes of modeling the behavior of ‘actually’within the scope of operators like ‘necessarily’ or‘possibly’. Thus, their 2D semantics does not explain howthe significance of the natural language expression‘actually’ is affected by the context in which it’sused, only how the term interacts with other modal operators.

8. The semantic rules governing ‘actually’ allow us to knowthe definitions given in (i) and (ii) are true no matter which worldis actual (i.e. they are fixedly actually true). According to (i),whoever invented the zip (whether it was Mary Anne Evans orWhitcomb Judson), that person is Julius. Since you know that (i) istrue no matter which world is actual, you can know that it’strue on the basis of apriori reasoning alone. But still, (i) does notreflect the modal profile of Julius: Whitcomb Judson (the realinventor of the zip) could have died in infancy. So the definition in(i) does not ensure that the claim is necessary in the sense of thestandard modal operator, ‘\(\Box\)’. In contrast, the trueidentity claim ‘Julius = Whitcomb Judson’ is necessary inthis sense, since both names rigidly designate the same individual inall possible worlds. But the semantic rule in (i) does not allow us toknow this identity is true: we need to conduct some empiricalinvestigation to find out who the inventor was. Similar observationsfollow from the definition of ‘water’ in (ii).

9. Some theorists push the argument further, arguing that the potentialfor ignorance and error highlighted by externalist examplesgeneralizes to other types of expression, such as artifact terms(‘pencil’, ‘sofa’), conventional kinds(‘arthritis’, ‘fortnight’), or even logicalexpressions (‘not’, ‘every’) (Putnam 1972;Burge 1979; Williamson 2007).

10. Kripke’s book is based on a series of lectures delivered in1970.

11. Jackson emphasizes that his account of meaning isnotintended to explain thought contents (1998b; 2004). The reason forthis is simple: the account posits implicit conventions that requireindividual speakers to associate a word with specific reference-fixingassumptions about the object, kind or property in question. Theaccount thus presupposes an independent explanation of the content ofthese thoughts. Moreover, it’s hard to see how anything likelinguistic conventions could apply to thought contents directly: sincewe don’t seem to have any way of focusing on our own mentalrepresentations independently of our understanding of their contents,it’s implausible that we could establish conventions requiringthinkers to associate some pattern of understanding with a givenmental representation.

12. It’s worth noting that Jackson is not committed to anyparticular interpretation of the possible worlds that define his 2Dframework. Jackson’s 2D framework is just a tool for testingone’s current understanding of how two different vocabulariesrelate to each other, and cataloguing the resulting information in aperspicuous way. The goal is to identify conditions, specified in somebase vocabulary, that one takes to suffice for the applicability of agiven target expression. For these purposes,any vocabularycan be used to describe hypothetical cases, as long as it providesenough information for the speaker to be confident in applying thetarget expression. The fact that Jackson sometimes takes thecommitments of a sophisticated proponent of physicalism as his primaryexample may obscure this point, suggesting that the possibilities thatdefine the 2D framework should always be couched in the language ofidealized physics. But it’s important to keep in mind that theempiricist theory of meaning is supposed to characterize actuallinguistic competence—a psychological and socialphenomenon—not to characterize the nature of reality. So, incontrast to the rationalist project, the empiricist projectneedn’t posit an idealized vocabulary that can exhaustivelyspecify all possible worlds.

13. It’s worth noting that indirect reference-fixing is alsopossible when the nature of the reference can be discovered throughapriori reflection—and in such cases the difference betweenreference-fixing criteria and theoretical criteria will not correspondto different A- and C-intentions. Jackson and Pettit’s‘moral functionalism’ is a case in point: the reference ofmoral terms is allegedly fixed by a variety of superficial properties(e.g. killing is normally wrong, people are normally motivated to dowhat they judge right, etc), but these properties do not constitutethe ultimate theoretical characterization of the properties picked outby moral terms. The fact that you don’t realize that your term‘morally right’ picks out a rule-utilitarian property(let’s say) may be due to a failure of apriori reflection ratherthan lack of empirical knowledge of your environment (Jackson 1998a,ch. 5 & 6; Jackson and Pettit 1995).

14. N.B., the rationalist is not committed to the idea that a 2D semanticframework accounts forall aspects of meaning: in particular,further aspects of meaning may be required to account for differencesin content between apriori equivalent claims. See the discussion ofsemantic pluralism in 2.3.3 below.

15. According to a modal rationalist like Chalmers, the primary job ofmodal notions likepossibility andnecessity is tokeep track of apriori consistency relations among semantic contents.To fulfill this explanatory role, the rationalist posits a“logically possible world” for every apriori conceivableway the world might be. Chalmers thinks that critics of modalrationalism should concede (i) that this notion of logical possibilityis an important and useful one for keeping track of consistency amongthought contents and (ii) thatat least some logicalpossibilities represent genuine metaphysical possibilities. The debateover the truth of modal rationalism would then hinge on whetherevery logically possible world is also metaphysicallypossible. Put graphically: is the space of metaphysically possibleworlds a proper subset of the space of logically possible worlds?Chalmers argues that any way of demarcating metaphysical from logicalpossibilities would constitute “anad hoc proliferationof modalities” which posited a second “brute andinexplicable” modal primitive. So we should accept a singlespace of possibility, in which every apriori coherent hypothesisrepresents a genuine metaphysical possibility (1996, 136­-8; 1999,488–91).

16. Chalmers introduces the term ‘1-intension’ as a way oftalking about the various different kinds of intension that correspondto the diagonal of a 2D framework. Strictly speaking, therefore, itmakes no sense to talk aboutthe 1-intension of a particularexpression without further ado, since the notion of a 1-intension isdefined purely in terms of the formalism. To obtain a specific1-intension for an expression, we need tointerpret the 2Dframework by specifying the possibilities relative to which extensionsare assigned and explaining the principles for assigning extensions.Chalmers’ Core Thesis doesn’t appeal to any specificinterpretation of the 2D formalism, instead it is intended as aconstraint on how the framework should be interpreted if it is tovindicate the rationalist conception of meaning. (Similarly, Chalmersintroduces the term ‘2-intension’ to pick outthefirst horizontal row of a 2D matrix, regardless of how exactlythe 2D formalism is interpreted. In any 2D framework, the first rowassigns an intension relative to the actual world considered asactual. So an expression’s 2-intension will correspond to thekind of intension posited by standard possible world semantics.)

17. Chalmers notes a further dialectical problem in appealing tocontextualist 2D semantics to vindicate rationalism (2004, 2006a). Inorder to generate a 1-intension, a contextualist must specify whichproperties of an utterance are held fixed in every possible context ofuse. Very different contextual 1-intensions will be generated for yourterm ‘I’, for instance, depending on whether we requirethat the target utterance of ‘I’ bespelled acertain way in every context or if we instead require that it beunderstood a certain way by the agent of the context. Acontextualist 1-intension based on spelling will not capture anyinteresting notion of meaning: depending on how it’s used andunderstood in a context, the string ‘I’ may pick outapples, Antarctica, arguments, or anything else. So if contextualist1-intensions are to capture a kind of meaning, we will need someindependent way of identifying those properties of an expression that,if held fixed, guarantee sameness of meaning. In the case ofindexicals like ‘I’ this requirement is not too onerous,since there are conventional linguistic rules for identifying theextension in a context that all competent speakers readilyacknowledge. But in the case of names or natural kind terms, it ishighly controversial whether there are any conventional linguisticrules, except the aposteriori rules that link these expressionsdirectly to their 2-intensions. So contextualist 2D semantics byitself does nothing to allay skepticism about the existence ofrationalist meanings for expressions like names and natural kindterms.

18. Chalmers leaves the door open for weaker ways of understandingepistemic possibility. For instance, he suggests that one might defineepistemic possibilities in terms of whether a sentence (or set ofsentences) seems coherent after a specified amount of “cognitivework”. However, Chalmers takes the notion of ideal aprioricoherence to be the most principled sense of epistemic possibility,and the one that underwrites our access to metaphysical modality(2002b).

19. Chalmers is neutral about a more substantive analysis of scenarios.He sketches three different approaches: (i) scenarios might beidentified with centered possible worlds, (ii) scenarios might beidentified with sets of apriori coherent sentences in an idealizedlanguage, or (iii) scenarios might be taken as epistemic primitives.As a modal rationalist, he takes these analyses to be extensionallyequivalent, but he argues that the different options have importantconsequences for how the 2D framework is elaborated (2004, 2006a).

20. It’s worth noting that there are other ways of interpreting anE2D framework defined on epistemic and metaphysical possibilities,which do not support Chalmers’ rationalist program. Someadvocates of E2D have suggested that we havedefeasible,rather thanconclusive apriori access to applicationconditionals (Kipper 2012, 39–40; Biggs and Wilson 2020, 69). But asWittmer (2013) points out, defeasible judgments will not suffice tofix the semantic facts about 1-intensions. So these interpretationsmust be construed as accounts of our epistemic access to independentlydetermined semantic facts. Other theorists take scenarios to be aspace of objective metaphysical possibilities, which need not beepistemically transparent to individuals. Wolfgang Spohn, forinstance, likens scenarios to concrete universes, and he takescounterfactual metaphysical possibilities to be constructed fromdescriptions of such universes (Spohn 2016). The result is apragmatist approach to G2D that is ill suited to Chalmers’rationalist E2D project.

21. The Fregean triviality test is not exactly the same as aprioriequivalence: cognitive significance marks what’s obvious to asubject at a given time, whereas apriori equivalence is often far fromobvious. Mathematical identities like ‘\(12^3 = 1728\)’are a clear case where the two notions come apart. Still, Chalmersargues that apriori equivalence is a well-behaved theoreticalrefinement of the Fregean notion. He also suggests that the 2Dframework might capture something close to the Fregean notion oftriviality if we substitute a weaker epistemic relation for‘apriori coherence’ in defining the epistemic framework(Chalmers 2002b).

22. The word ‘concept’ here is not meant to imply that thesematrices reflect stable semantic units of thought—it is merelymeant to suggest that they reflect some aspect of the subject’scurrent understanding. Stalnaker rejects concepts as theoreticalentities that reflect stable units of cognitive significance inthought (1984).

23. On Stalnaker’s account, to determine just which proposition isascribed by a particular belief report, we must answer threequestions. First, how should we characterize the possible worlds thatdefine propositional concepts: which objects, kinds, or propertiesshould we invoke in describing the different ways the world might be?Second, which possible worlds so characterized should we treat asmembers of the context set for the belief ascription? Third, whichaspects of theattributor’s understanding should wehold constant in evaluating her report in those different worldsconsidered as contexts of use? These factors determine the content ofthe diagonal proposition attributed to the believer.

24. There is a further way in which the interests of an interpreter aretaken into account on Stalnaker’s approach. Whereas arationalist 2D theorist like Chalmers will invoke a single space ofapriori possible scenarios to define 2D matrices, Stalnaker’sexternalist commitments lead him to deny that we can exhaustivelycharacterize the space of metaphysical possibility in a descriptivevocabulary that’s devoid of any empirical presuppositions. Sincethere is no privileged way of describing the space of possibility, theexplanatory interests and presuppositions of the theorist will affectjust what objects, kinds or properties she uses to individuate thepossible worlds that define 2D matrices:

It might be nice if we had a neutral language with an internallygrounded semantics, a language that required no factual assumptionsfor its interpretation and that could provide a complete descriptionof the world, and of all possible worlds. It might be nice if therewere a pure epistemic space to which we had apriori access and interms of which we could locate our disagreements about what the actualworld is like. But I don’t think these things are possible. Theonly way we can describe the world is to use the materials that theactual world offers us—the things, properties and relations thatwe find there. (Stalnaker 2004, 319)

25. Chalmers argues that establishing the stability of phenomenal termslike ‘pain’ is not really essential to establishing hisanti-materialist conclusion (1996, 2009). His idea is that theepistemic 1-intension for ‘pain’ itselfcorresponds to a property—call it ‘diagonalpain’—that does not supervene on the physical propertiesof the actual world. Diagonal pain \(is\) instantiated in the actualworld and it isnot instantiated in the scenario that, ifactual, would make \((P \amp \neg Q)\) true. Given that there is ametaphysically possible world for every coherent scenario (premise 4),there will be a possible world just like ours physically but withoutdiagonal pain. Since the actual world has a property—diagonalpain—that doesn’t supervene on the physical, it followsthat materialism is false.

26. Chalmers in fact believes that phenomenal terms are stable, but he isless confident of the semantic stability of microphysical predicateslike ‘is a quark’: perhaps different intrinsic propertiesare picked out depending on what the actual world is like. To allowfor this possibility, Chalmers weakens the conclusion of his argument:either materialism is false or phenomenal properties are determined bythe intrinsic nature of basic physical properties (Chalmers 2009). AsChalmers emphasizes, most materialists will reject even this weakerconclusion, since it denies that phenomenal properties can beexplained by the structural and causal properties studied by thephysical sciences.

27. Chalmers (2009, §5) replies that the criticism misfires becauseit targets the apriori coherence of an explicitly modal claim,\(\Diamond(P \amp \neg Q)\), rather than the apriori coherence of \((P\amp \neg Q)\).

28. It’s worth noting that a 2D empiricist isnotcommitted to the rationalist’s Core Thesis. The empiricistproject is to explain how speakers manage to coordinate theirreference-fixing criteria through implicit conventions. For thesepurposes, there is no need to insist that 1-intensions must be definedfor worlds with no thought or language: we may have no interest indeciding how to apply terms in situations in which we ourselvesdon’t exist. It is the rationalist project of linking meaning tothe space of apriori coherent scenarios that gives rise to the CoreThesis.

29. Stalnaker’s initial response to proponents of finer-grainedpropositional structure is to insist that possible world semanticsprovides aminimal kind of content that all theorists shouldaccept (1984, 1999b). Stalnaker’s context-sensitive use of the2D framework is intended to provide a flexible framework forexplaining commonsense intuitions about logical equivalence anddere thought. And self-locating aspects of thought, according toStalnaker, should not be considered an aspect of propositionalcontents, but rather part of one’s epistemicrelationto a propositional content (Stalnaker 2006).