Are two Dimensions Too Many? - A one-dimensional rival to two-dimensional semantics

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rival to two-dimensional semantics

Sacha Bourgeois-Gironde, Denis Bonnay

To cite this version:

Sacha Bourgeois-Gironde, Denis Bonnay. Are two Dimensions Too Many? - A one-dimensional rival to two-dimensional semantics. 2005. �ijn_00000659�

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Title: Are Two Dimensions Too Many? A One-dimensional Rival to Two-Dimensional Semantics Article Type: Original Research

Section/Category:

Keywords: Two Dimensional Semantics, Semantics for Knowledge and Belief, Modal logic, Possibility, Conceivability, Impossible worlds, Theory of reference, Transworld Identity

Corresponding Author: Dr Sacha Bourgeois-Gironde, PhD Corresponding Author's Institution: Institut Jean Nicod / ENS LSH First Author: Sacha Bourgeois-Gironde, PhD

Order of Authors: Sacha Bourgeois-Gironde, PhD; Denis Bonnay, MD Manuscript Region of Origin:

Abstract: We discuss two interpretations of two-dimensional semantics (2DMS) of modality due to D. Chalmers and R. Stalnaker. The main problem

with both interpretations of the formal framework is the relinquishing of rigidity for terms. They are in this sense unfaithful to an agent's beliefs. We present alternative

principles to capture what we take to be agents' beliefs. These are the principles of hyper-rigidity and backward reference to actuality. We propose, then, to go back to a one-dimensional semantics which affords a satisfactory model of beliefs' reports.

Our proposal, like 2DMS, addresses typical problems of

We discuss two interpretations of two-dimensional semantics (2DMS) of modality due to D. Chalmers and R. Stalnaker. The main problem with both interpretations of the formal framework is the

relinquishing of rigidity for terms. They are in this sense unfaithful to an agent's beliefs. We present alternative

principles to capture what we take to be agents’ beliefs. These are the principles of hyper-rigidity and backward reference to actuality. We propose, then, to go back to a one-dimensional semantics which affords a satisfactory model of beliefs’ reports.

Our proposal, like 2DMS, addresses typical problems of representation for beliefs and epistemological difficulties related to modal knowledge.

Click here to download abstract: abstract2DMS.txt

Are Two Dimensions Too Many? A One-dimensional

Rival to Two-dimensional Semantics

Abstract. Wediscusstwointerpretationsoftwo-dimensionalsemantics(2DMS)

of modality dueto D. Chalmers and R. Stalnaker. The mainproblem withboth

interpretationsof the formal framework is the relinquishing of rigidity for terms.

Theyare inthissenseunfaithfultoanagent'sbeliefs.Wepresentalternativeprin-

ciples to capture what we take to be agents beliefs. These are the principles of

hyper-rigidityandbackwardreferencetoactuality.Wepropose,then,togobackto

a one-dimensionalsemanticswhich aords a satisfactory modelofbeliefs reports.

Our proposal, like 2DMS, addresses typical problemsof representationfor beliefs

andepistemologicaldiÆcultiesrelated tomodalknowledge.

Introduction

There are two basic stories concerning how the reference of proper

names is xed through possible worlds. The Hintikkean account as-

sociates names with individual concepts: each speaker associate with

names some modes of identicationthat determine reference inbelief

worlds. Thus, "Cicero", as used by a speaker with a poor knowledge

of ancient Rome, does not rigidly refer to Cicero, but, for example,

to whomever shemight take to be the mostfamous oratorof the last

years of the Republic. The Kripkean account has it that individuals,

rigidlydesignatedbythenamesweuseforthem,arethebasicbuilding

blocks of possible worlds: there is in fact no problem as to how the

reference of "Cicero" is xed, because it is precisely used to refer to

Cicero, whateverhis occupationinRomearound 50B.C.,includingor

notbeing afamousorator.

At rst sight, these two accounts seem incompatible. But one can

make them compatible if one takes them to deal with two dierent

kindsofpossibility,namely,fortherstone,epistemicpossibility,and,

forthe second one,metaphysical possibility.Theidea isthusthat two

coreferential terms can fail to corefer, rigidity notwithstanding, in a

belief world, just in case the agent is mistaken as to the reference of

these terms and does not know they do corefer. Starting with this

idea,proponentsoftwodimensionalmodalsemantics(2DMSforshort)

suggest to merge the two accounts into one and the same theory. In

2DMS, a sentence can be associated withtwo kinds of intensions, the

rstonealongKripkeanpossibleworlds,whichwillaccountforthefact

Click here to download Manuscript: aretwodimensionstoomany.tex

thatthesentenceexpressesanecessaryorcontingentpropostition,and

the second one along Hintikkean possible worlds, which will account

for failure of substitutivity in intensional contexts and for epistemic

propertiesof sentences.

1

Two-dimensionalmodalsemantics is therefore an attractive frame-

work: it makes it possibleto give a possible world analysis of content

compatible with Kripke's analysis of rigidity. But this synthesis rests

on an equivocation. The mere gluing of the two approaches does not

accountforthespeaker'sintuitionasdescribedbyKripke.Ifrigiddes-

ignation,and individualspickedthroughit, isallwhat possibleworlds

are about, and if the content of the sentence expressed by a speaker

is to be a set of possibleworlds, these possibleworlds shouldbe built

on the very basis supplied by rigid designation, since it is the only

basis on which we have an intuitive access to possibleworlds. IfI can

consistently believe that Hesperus and Phosphorus are not the same

thing, it should be justbecause I can consistently of Hesperus and of

Phosporusthattheyarenotthesame thing.Unfortunately,thisisnot

allowedinthe2DMSframework,sinceitrepresentsthepossibilitythat

they are not the same by resorting to a world in which Hesperus or

Phosphorus or both do not refer to what they actually refer to. But

as far as it is possible for me to have de re beliefs about Hespherus

and Phosporus,anyaccountrelyingonthiskindoftrickmisrepresents

what happenswhen I entertain the possibilitythat they are dierent,

since it represents it by two other objects which are in fact dierent,

while Hesperus and Phosphorus on which the agent's thought bears

areactuallyidentical.Ouraimisthustorepresentonthecontrarythis

possibilitythatI might entertain asthe possibilitythat Hesperus and

Phosphorus { the two objects in the sky that I can rigidly refer to {

aredierent,evenifthispossibilityisnotametaphysicalpossibility(if

one grantsthatidentityisnecessary).

2DMS was preciselyconceived as a theory whose ideal was to rec-

oncile rst-person intuitionsaboutmeaningand content and "oÆcial"

semanticcontent. Sometimesthetwocoincide, butinmostinteresting

cases there is a conict between intuitions and actual content. It is

solvedby2-dimensionalistintermsoftheoreticallyprescribedKripkean

content onone side, andFregean content,on theother side,thelatter

meant to reect the agent's intuitions about meaning. However, the

agent may resist this explanation and be convinced of her mentally

having a rm grip on a Kripkean content. One basic critical point is

that2DMSisinsomeimportantrespectcognitivelyinadequatebecause

it too soon departs from possible Kripkean intuitions the agent may

want to preserve incontexts in which 2DMSintends to make senseof

the agent's beliefs by reinstatingFregean semantics.

One reasonwhywe wishintuitionsto supportoursemantic theory

isthatinonemajorareaof applicationsof2DMSwehavenothingbut

intuitions,namelymodalones,toaccountfor.Theagenthassomeintu-

itionaboutwhatcouldbethecase.Shemayhavesome furtherspecic

intuition about the way she uses the terms that shape up her modal

intuition.But infact ouronlyaccess to theagent's modalintuition is

throughthewayshethinkssheuses herterms andherreportthatshe

useditrigidlyordescriptively.Incontextsofthatsort,inordertomake

correct predictions about what modal judgements are expressed,it is

crucialthatourtheoryts thesesemantic intuitions. Unlike2DMSwe

claim that in most cases modal intuitions are supported by Kripkean

intuitionsconcerningthesemanticsoftherelevantterms,whichmeans

that theiruseisintendedto berigid.Fregean contentswerereinstated

by2DMS theoreticiansinorder to make senseof seeminglycontradic-

toryorirrationalmodalbeliefsliketheoneswhichimplyaviolationof

the identityof referredto individuals.

Wedefendanalternativewaytoconnectmodalityandrationalityby

keepingallalong onesingle kindofsemantic content andsubsequently

bydefusingthesystematictransitionbetweenintuition andpossibility

Fregeancontentsweresupposedtoallowbackinthehandsof2DMSists.

Thisneedsnotbetakenasanempiricalpointaboutthenatureofmodal

intuitions but as an important conceptual possibility that may make

empiricaland intuitivesense.To putitbluntly:IfIbelievethat water

couldhaveachemicalstructuredierentfromH2O,whatImaysimply

meanisthatIbelieveofH2Othatitcouldbesomethingdierentfrom

H2O.I maywellbelieve animpossibilityaboutH2O and water rather

thanapossibilityaboutsomethingIdidnotobviouslyintendedtorefer

to.Wepreferatheorywhichpredictsthiskindofsemanticlucidityand

modal(really modal) illusions to one which attributes to the subjects

an unintendedchangeintheuseoftheirterms.We tryto presentsuch

a theoryand its comparative advantages andcosts with2DMS.

Werstpresentthe2DMSsetting,emphasizinghowtheadditionof

a newdimensioncan helptosolvetheproblemof puttingtogetherthe

Kripkean account of proper names and the possible world account of

content.Thispointisabittrickybecauseof thevariousinterpretation

oftheformalsettingof2DMS:Stalnaker'smetasemanticinterpretation

seems immuneto someof thecriticismsaimedat Chalmers'ssemantic

interpretation. But we see in the second section that the objections

we direct at 2DMS bear on both interpretations. Our main point is

that one should remain faithful in hostile contexts to Kripkean intu-

itionsofrigiditywhichwethinkcontinuetoadequatelycharacterizethe

semantic behavior of singular terms and natural substances terms in

suchcontexts. When we dealwith thesame problems that2DMS was

introducedtosolve,wedonotsaythatthesubjectisnotmistakeninbe-

lievingwhatshebelieves,butwe atleastsupposethatsheknowswhat

she believes. This epistemic candor carries over to the semantic level

by keeping aoat intuitions of rigidity in those places where Fregean

contentswere reinstatedby2DMS.We areconsequentlyledto extend

the principle of transworld identity to worlds guring in the content

of beliefs deemed contradictory; hence an enrichment of modal space

withaspeciesofimpossibleworlds,whichweprefertoitsduplicationby

2DMS.Theformalismisintroducedinthethirdsection.Wespellouta

fewsemanticrequirementsandourvisionofmodalspaceinparagraphs

3.2and3.3.Itisremarkablethatourone-dimensionalmodalsemantics

(1DMS) both accommodates traditional objections (paragraphs 2.1)

thatwereformulatedagainst2DMSaswellasourown.Aswesaid,our

particularobjections{inparagraphs2.2and2.3{hintbacktotheidea

that semantic predictions must be subjectively adequate to the sub-

ject. Wethinkthat ordinarymodalintuitionsare better dealt withby

incorporatingin oursemanticsso-called requirementsof hyperrigidity

andbackwardreferencetoactuality.Thoseprinciplesstaterespectively

that thesubjectkeeps thereferenceof herterms,oncerigidlyxed,in

every context and that sheremains lucidaboutthe kindof departure

fromactualityherintuitionsmayimpart.Wecaneasilytranslatethose

requirements in our model, which in one case (hyperrigidity) 2DMS

refuses to do by principle, so to speak, and in the other (backward

reference to actuality) it simply cannot do, as we show below. This

mono-dimensionalist approach is not free lunch. The mainhypothesis

supporting its implementation is that propernames are directly used

to refer to individuals in possible situations. Each name comes with

a witness for it, and though it will generally be the case that such a

witness is identical in a certain world with certain objects, no such

thing is necessary from the point of view of the subject. This implies

thatweallowforimpossiblevariationoftheidentityrelationsincertain

worlds, i.e. we allow for some impossible worlds in order to complete

thepictureof themodalspace.Butitwillbearguedinparagraph3.4.

that our reliance on impossible worlds is relatively immune to usual

criticismsagainst impossibleworlds.

1. Aims and scope of two-dimensional modal semantics

1.1. The formal apparatus of 2DMS

Two-dimensionalmodal semantics is a modal semantics according to

whichexpressionsareevaluatedwithrespectto pairsofworldsinstead

of worldssimpliciter.The ideais thatis trueat (w

1

;w

2

) i itwould

be true in w

2

, considered as counterfactual, had w

1

been the actual

world.Let'sconsiderthefollowingstatements:

() Hesperusappears intheeveningsky.

( ) Hesperus isPhosphorus.

Nowletw

0

betheactualworld,andw

1

apossibleworldinwhichthe

morningstarisstillVenusbutinwhichtheobjectappearinginthesky

on theevening isSiriusinsteadofVenus. In(w

0 ,w

0

), referenceisxed

intheactualworld,and sentencesareevaluatedintheactualworld as

well, so that and are both true in (w

0 ,w

0

). In (w

0 ,w

1

), reference

isstillxedintheactualworld,sothat "Hesperus" and"Phosphorus"

bothrefer to Venus, butthecircumstances ofevaluationsare given by

thecounterfactualworld w

1

.Becausereferencexinghasnotchanged,

is still true, and because facts are dierent, is false. In (w

1 ,w

1 ),

reference is xed in w

1

, therefore "Hesperus" refersto Sirius,whereas

"Phosphorus" refers to Venus as before. As a consequence, will be

false,butasthecircumstancesofevaluation aregiven byw

1

,istrue,

sincew

1

ispossibleworld inwhich Siriusappears intheevening sky.

Statementsare thusinterpretedbya functionwith two arguments,

this isStalnaker'snotionof propositional concept (Stalnaker1978). A

propositionalconcept f(x

1

;x

2

)can expresstwo distinguished proposi-

tions:

i) rst the horizontal one, f(w

0

;x), corresponding to the propo-

sition usually associated to the statement, all reference xing takes

place considering the actual world as actual. To x terminology, this

is Chalmers's secondary intension (Chalmers 1996) and Jackson's C-

intension(Jackson 1998).

ii) The other propositionof special interest is f(x;x), the diagonal

proposition, which gives the truth value of the statement in a world,

that world being considered as actual. This is Chalmers's primary

intensionand Jackson's A-intension.

1.2. Expected Advantages

This formaldevice hopesto solve two kindsof intricate problems.

representation problem Rigiddesignationgeneratesvariousseman-

ticpuzzles.Moreprecisely,theproblemistomakeKripkeanrigid-

ityand facts aboutcontent compatiblewith interpreting proposi-

tionsassets ofpossibleworlds.Onespecialaspectofthisproblem

concerns identity statements: if Kripke is right, all true identity

statements 'a=b' are necessarily true. They have therefore the

same truth value in every possible world, and hence the same

propositional content, and this content is trivial. The notion of

propositional content does not seem to give then an adequate

representationof theinformative content of these statements.

Another side of the problem concerns beliefs: as long as 'a' and

'b' have thesame referent,the propositionalcontent of '(a)'and

'(b)'willbethesame,nevertheless,itseemsperfectlyrationalfor

anagenttobelieveboth(a)and:(b)incasehedoesnotbelieve

that a=b.Howthen can we representthiskindof belief?

epistemological problem Thisis is thelooser problemof providing

abetterunderstandingofthepost-Kripkeaninterplaybetweenthe

distinctions a priori / a posteriori (epistemic) and necessary /

contingent (metaphysical). On the traditional view of epistemic

logic,epistemicalternativesforanagent arerepresentedassetsof

possible worlds, and possible worlds are the metaphysically pos-

sible worlds ( what else?). How then could it be that something

necessary isnota prioriknownorat least knowable?

In anutshell,2DMSsolvestherepresentationproblem by resorting

to primary intensions. "Hesperus is Phosphorus" does not lead to a

necessary diagonal proposition, just because in (w

1 ,w

1

), the object

which is the evening star and the object which is the morning star

according to w

1

are dierent. The same idea works with belief: "Hes-

perus is a planet" and "Phosphorus is a planet" are interpreted by

dierent diagonal propositions. In the world w

1

discussed above, the

morning star happens to be Venus; 'Hesperus is a planet' is false in

(w

1 ,w

1

) whereas'Phosphorusisa planet' turnsout to be true.

Given that a single statement expresses two dierent propositions,

horizontal and diagonal ones, we have two notions of being true ev-

erywhere.Necessityisinterpretedasusualasnecessityofthehorizon-

tal proposition. Epistemically, one must rst grant that the diagonal

proposition is determined by the descriptive content of terms - rigid

ornot. Theidea isthatone determinesthereferent of'Hesperus'inw

consideredasactualbyaskingwhichobjectistheeveningstarinw.The

presuppositionis that 'Hesperus' has as descriptive content 'the rst

heavenlybodyappearingintheeveningsky' andthat'swhy'Hesperus

appears intheevening sky' istrue inevery (w,w). 'Hesperusappears

in the evening sky' is a priori. More generally, always true diagonal

propositionsare true invirtue of thedescriptive content, ormeanings

of theterms.Therefore, these propositionsare apriori.

2

1.3. Semantic vs metasemanticinterpretations

Even though 2DMS is a valuable tool in providing the formal setting

to solve these problems, it must be recognized that the validity of its

solution is highly dependent upon non-trivial theses about meaning

and belief. Theformalisminand of itself isbyno meansan argument

for these theses. In particular,there is a persistingdisagreement over

theclaimsof2DMStomodelandexplainaprioriknowledge:Stalnaker

(Stalnaker 2001) has proposed to label semantic the interpretation of

2DMS according to which the diagonal propositionis a component of

meaning and metasemantic the interpretation according to which the

diagonalpropositiononlyrepresentsthepossiblevariationsofmeanings

dependingon thefacts.

The semantic view, defendedbyChalmersand Jackson, rests upon

a commitment to descriptivism: to determine the value of 'Hespe-

rus' along the diagonal line, one appeals to the descriptive content

of thename."Descriptivecontent" hereistantamount tosome kindof

Fregean sense. The existence of such a sense is a necessary condition

for truth everywhere along the diagonal line to be interpretable as a

priori truth. This new form of descriptivismhas recently come under

heavyattack:thecoreargumentisthattheotherkindofintensionsthe

semantic interpretation presupposes simply do not exist, because the

kindofknowledgetheyimplycannotbetiedtoknowledgeofmeanings.

Argumentsherearethesame oldonesasKripke's againstthedescrip-

tivetheoryofpropernames.Thecompetentuserofanamedoesnotin

general possess substantial identifying knowledge. The speaker might

fail to have such knowledge, by ignorance (many people do not know

enoughaboutCicerotobe ableto identifyhimuniquely)andbyerror

(while our beliefs about G

'odel's actions might be false, they would

not prevent us from referring to him). Therefore, mastering the use

of a name doesnot presuppose knowledge of some kindof identifying

meaning.

The metasemantic view isn't committed to such descriptivism. No

linkbetweenaprioritruthandtrutheverywherealongthediagonalline

is suggested. Moreover, there are no a priori truths, since if semantic

facts had been dierent, meanings could have been radicallydierent

sothat"Themeterstickisonemeterlong"couldhavemeant"Michael

Jacksonisvefeetteninchestall".Thesolutionoftheepistemological

problem is partial: on the metasemantic view 2DMS handles well a

posteriorinecessary statements, butitfallsshortof explainingapriori

contingentones.

2. Objections to two-dimensionalism

Werstconsidercriticismsof2DMSfoundintheliterature.Theybear

mainlyuponthesemanticinterpretation,butwewillarguethattheway

out suggested by the metasemantic interpretation is not as safe as it

seems.We proposesomenewcriticismofwhat's wrongingeneralwith

the 2DMS strategy: we introducetwo principles governing ourmodal

intuitions,hyperrigidityandbackwardreference toactuality.Weshow

then how 2DMSviolates them.

2.1. Descriptivism strikes back

At rst sight, objections against neo-descriptivism are aimed only at

the semantic interpretation. The specicity of the metasemantic in-

terpretation is that, though it uses the diagonal "line" to solve the

representation problem,it doesnotpresupposethat lineto be partof

meaning.But, ifthediagonal propositionis notpart of meaning,how

can it play a role in solving representation problems? This question

is twofold:if the diagonal propositionis not ingeneral the content of

thesentence,howcanitbesometimesthepropositionexpressed?How

canweattributetospeakersthenecessaryknowledgeto determinethis

horizontalline, iftheknowledge inquestionisn't semantical?

Stalnaker'sanswerto therstquestion(Stalnaker1978)isGricean.

"Undercertainconditions,thecontentofanassertionisnotthepropo-

sition determined by the ordinary semantic rule, but instead the di-

agonal propositionofthepropositionalconceptdetermined."Andthis

happenswhensome principlesaboutassertions areviolated:

3

Non-triviality: A proposition asserted is always true in some but

notall of thepossibleworldsinthecontext set.

4

Context-independence:Thesame propositionisexpressedrelative

to each possibleworldin thecontext set.

The diagonal propositioncomes into play because because some con-

versationalmaxims wouldbe violated,had thecontent beentheusual

one,sothatthecontenthastobereinterpretedalongthemetasemantic

line.

As to the second point, the determination of the diagonal propo-

sition seems to involve reexive semantic competence on behalf of