S5-denying Approach to Relativized Metaphysical Modality

S5-denying Approach to RelativizedMetaphysical Modality

Fernando Furtado

https://orcid.org/0000-0001-6390-5745University of Lisbon, School of Arts and Humanities

Centre of Philosophy, LanCogLisbon, Lisbon

[email protected]

Article Info:CDD:Received:28.02.2020; Accepted: 09.03.2020https://doi.org/10.1590/0100-6045.2020.V32N1.FF

Keywords:metaphysical modalitymodal logicessentialismrelativized modality

Abstract: This paper is organised as follows: first, I presentSalmon’s theory of modality (which I call ‘S5-denying ap-proach to relativized metaphysical modality’) and compareit with the standard interpretation of modality: ‘the non-relativized S5-friendly interpretation of metaphysical modal-ity’. Second, I explain Murray andWilson’s ‘two-dimensionalS5-friendly interpretation of relativized metaphysical modal-ity’. In the third and last part, I put forward a few argu-ments against Murray and Wilson’s attempt to provide anessentialist S5-friendly theory for modality. In general, thispaper argues that if one wants to hold an essentialist the-ory for relativized (metaphysical) modality, then his best

Manuscrito – Rev. Int. Fil. Campinas, v. 43, n. 1, pp. 1-40,Jan-Mar. 2020.

Fernando Furtado 2

option in the market right now is to stick with Salmon’sproposal, which better represents a genuine essentialist in-terpretation of relativized metaphysical modality.

1 Introduction

Nathan Salmon (1989) argued (following Hugh Chan-dler (1976)) that things that are possible may be merelycontingently possible instead of necessarily possible. Em-ploying the possible worlds vocabulary, the claim isthat what is possible may vary depending on whichpossible world is being taken into consideration. Ac-cording to the relativist treatment of modality, it isnatural to suppose that some things that are impos-sible from the standpoint of a possible world w1 arepossible from the perspective of a different possibleworld w2. That is equivalent to say that things whichqualify as viable possibilities from a possible world w1

might be different from the viable possibilities froma possible world w2. In this sense, possibilities (andalso necessities) are relativized to which possible worldis being taken into consideration. Although Salmon’ssuggestion might be interesting and capable of dealingwith some intricate philosophical problems, it has beenunderappreciated in literature probably due (I specu-late) to its dismissal of features of modal logic regard-ing which significant consensus has been reached. Theconsensus concerns the idea according to which thelogic for metaphysical modality should not be weakerthan the S5 modal logic system. The idea is thatan adequate theory for metaphysical modality requiresmodels in which the accessibility relation between pos-sible worlds is an equivalence relation. Since Salmon’ssemantics puts to use an accessibility relation betweenpossible worlds weaker than equivalence (intransitive)(therefore, the consensus is not fulfilled), philosophers


S5-denying Approach to Relativized Modality 3

may have found themselves dissuaded to take Salmon’smodal theory as seriously as it deserved. However,more recently, AdamMurray and Jessica Wilson (2012)took Salmon’s central insights seriously and proposedan interpretation of it which, if successful, can eventu-ally deliver the benefits of a relativized description ofmetaphysical modality without having to pay the priceof dropping S5. Hence, if they successfully achievetheir goal, then they will be able to provide an essen-tialist S5-friendly theory for relativized metaphysicalmodality. Unfortunately, however, I shall argue thatthey do not achieve their goal; therefore, if I am right,no essentialist S5-friendly theory for relativized meta-physical modality is available.1

1‘Modal logic’ is used here to refer to what is knownas ‘normal modal logic’: extensions of classical proposi-tional logic by the addition of two elements i) the axiom K□(φ→ ψ) → (□φ→ □ψ) and ii) the necessitation rule; if φ isa theorem of K, then □φ is also a theorem of K. As a result ofthe addition of i) and ii) to classical logic, we have the simplestand weakest modal system K. The modal system T is reachedby adding □T ⌜□φ → φ⌝ (or ◇T ⌜φ → ◇φ⌝) to system K. Bis reached by adding □B ⌜φ → □◇φ⌝ (or ◇B ⌜◇□ φ → φ⌝) toT. By adding □4 ⌜□φ → □□ φ⌝ (or ◇4 ⌜◇◇ φ → ◇φ⌝) to T,we get S4. And, finally, the addition of □5 ⌜◇φ → □◇ φ⌝ (or◇5 ⌜◇□ φ → □φ⌝) to T gives us S5. Each of these systemsis proven to be sound and complete relative to its respectivesemantics. From the semantic point of view, the systems differ-entiate by constraints on accessibility relation between possibleworlds. The accessibility relation is a binary relation on the setof possible worlds. Intuitively, a possible world w holds accessi-bility relation with a possible world w′ iff w

′ is possible relativeto (is “accessible” from) w. In the case of the semantics for T,the accessibility relation is characterised as reflexive. RegardingB, a reflexive and symmetrical semantics is needed. Concern-ing S4, a reflexive and transitive semantics is required. And forS5 a reflexive, symmetrical and transitive semantics is neces-sary. Some interesting logical relations among the systems canbe shown. S5 is the strongest modal system (any theorem ofa weaker system, such as S4, is also an S5-theorem, but thereare S5-theorems which are not S4-theorems. And B and S4 areindependent and alternative systems and both entail T. From


Fernando Furtado 4

This paper is organised as follows: firstly, I presentSalmon’s theory of modality (which I call ‘S5-denyingapproach’ short for ‘S5-denying approach to relativizedmetaphysical modality’) and compare it with the stan-dard interpretation of modality; ‘the non-relativizedS5-friendly interpretation of metaphysical modality’.Secondly, I explain Murray and Wilson’s ‘two dimen-sional S5-friendly interpretation of relativized meta-physical modality’ or ‘2D-interpretation’ (I focus ononly one of its implementations: the one with non-overlapping subspaces). Thirdly and most importantly,I put forward a few arguments against Murray andWilson’s attempt to provide an essentialist S5-friendlytheory for relativized metaphysical modality. As mightbe clear up to now, this paper can be read as an indi-rect defence of Salmon’s account of relativized meta-physical modality. Since it argues that if one wants tohold an essentialist theory for relativized metaphysicalmodality, then his best option in the market right nowis to stick with Salmon’s proposal, which (I shall argue)better represents a genuine essentialist interpretationof relativized metaphysical modality.

2 Standard Interpretation vs. S5-denyingApproach

This section explains both the standard interpretationof modality and Salmon’s S5-denying approach. As Isaid above, Salmon’s theory of modality can be under-stood as a restriction on the standard interpretation.The standard interpretation is what I have been call-ing ‘non-relativized S5-friendly interpretation’, whileSalmon’s theory I call ‘the S5-denying approach to rel-ativized metaphysical modality’. These labels suggest

a philosophical standpoint, it is also worth noting that it is awidespread consensus that the logic for metaphysical modalityshould be somewhere between T and S5.



the differences between the two approaches: accordingto Salmon’s interpretation, metaphysical modality isto be understood in a relativized way. As a result, theS5 modal logic system is regarded as too strong. Ac-cording to the standard interpretation, metaphysicalmodality behaves in a non-relativized manner, whichmeans that S5 modal logic system is exactly as strongas it should be. These explanations are supposed tojustify the use of the aforementioned labels, but theexact difference between them may not be clear yet.Hopefully, this section will bring to light all the differ-ences and similarities between them.

Metaphysical modality has usually been understoodin a non-relativized fashion, which means that what ispossible (or necessary) is possible (or necessary) nomatter what. Using possible worlds discourse, what ispossible (or necessary) relative to one possible worldis possible (or necessary) for any other possible world.According to the non-relativized interpretation, anypossible world is possible relatively to every possibleworld. Similarly, if a proposition p is possible at worldw0 (p is true in some possible world accessible from(possible relative to) w0), then p is possible relativeto any possible world or, equivalently, p is necessarilypossible. If that is true, then the S5 modal logic sys-tem has to be assumed when we are working on themetaphysics of modality. Looking closer, we will seethat the non-relativized interpretation is supportingthe following general modal principle:

MP5 If φ is possible, then φ is necessarilypossible.

The role of MP5 is played in modal logic by the charac-teristic axiom of S5 ⌜◇φ→ □◇φ⌝. Interestingly, wedo not have a direct and definitive argument defend-ing MP5. Philosophers seem to take it as plausibleand start to work presupposing it without arguing in


Fernando Furtado 6

its defence. The following argument by Plantinga isan exception in this regard:

“Suppose we focus our attention on broadlylogical necessity. Are there propositionsthat are in fact necessary, but would havebeen merely contingent if things had beendifferent, if some other possible state ofaffairs had been actual? (13) All bache-lors are unmarried and (14) If all bache-lors are unmarried and Dirk is a bache-lor, then Dirk is unmarried are necessarytruths; could they have been merely con-tingent? If so, there must be some possiblestate of affairs S such that if S had beenactual, then (13) and/or (14) would havebeen contingent. But are there states ofaffairs that with any show of plausibilitycould be said to meet this condition? [...] Ithink we can see that (13) and (14) are notmerely necessary; they could not have beencontingent. [...] Are there propositionsthat in fact are possible, but would havebeen impossible had things been differentin some way? (15) Socrates never marriedand (16) Socrates was a carpenter are falsebut possible propositions; could they havebeen impossible? The answer, I think, isclear; (15) and (16) themselves could nothave been impossible.” (Plantinga, 1974,pp. 52-53)

One first point about Plantinga’s argument is to thinkabout what exactly he means by ‘broadly logical ne-cessity’. It is not obvious what he means by that. Itis not clear whether ‘broadly logical necessity’ meansthe same as ‘metaphysical necessity’ or something elsesuch as ‘conceptual necessity’ or ‘analytical necessity’,for instance. Even though one may want to equate



‘broadly logical necessity’ with ‘metaphysical neces-sity’, these expressions do not mean the same thing.One piece of evidence in this direction is Chandler’s ar-ticle ‘Plantinga and the Contingently Possible’, wherehe tries to make the point that there must be at leastsome metaphysical necessities only necessary relativeto some possible worlds but not to all of them. And aswe have seen, Plantinga’s argument defends the ideathat broadly logical necessity does not vary from onestate of affair (possible world) to another. Being thatthe case, if a proposition p is necessary in one state ofaffair, then p will also be necessary in any other stateof affair. Consequently, Plantinga’s argument can beunderstood a defence of MP5, at least with concern tothe broadly logical modality. I will not dive any deeperinto this discussion here because it would lead to prob-lems which are beyond the scope of this paper. It wasworth mentioning, though, since, as Salmon pointedout, Chandler’s paper is the source of the idea whichI am calling (following (Murray & Wilson, 2012)) ‘rel-ativized metaphysical modality’.

One different source of motivation for accepting thestandard interpretation of modality is the widespreadidea according to which metaphysical modality is thebroadest kind of alethic modality. As Burgess puts itand Murray and Wilson (2012) quote:

“[W]e may distinguish the species of physi-cal necessity, or what could not have beenotherwise so long as the laws of nature re-mained the same, frommetaphysical neces-sity, what could not have been otherwise nomatter what.” (Murray & Wilson, 2012, p.189)

The suggestion that metaphysical necessity correspondsto what could not have been otherwise no matter whatseems to be an additional motivation towards the stan-dard interpretation of metaphysical modality.


Fernando Furtado 8

“Very few metaphysicians have rejected theB axiom for metaphysical modality. If some-thing is so, how could it have been meta-physically impossible? Indeed, most meta-physicians accept S5 as the propositionalmodal logic of metaphysical modality, butin any case the most prominent objectionsto S5 in that role target the principle thatwhatever is necessary is necessarily neces-sary [or equivalently, whatever is possiblypossible is possible] rather than the B prin-ciple.” (Williamson, 2013, pp. 43-44)

The aim of this paper is not to argue against B or S5.2

In fact, what I am trying to do is to defend a specificinterpretation of modality supposing that we have in-dependent reasons to accept a relativized interpreta-tion of modality. Once one has accepted relativizedmetaphysical modality, the most straightforward wayto deal with it is to assume a modal logic to a sys-tem weaker than S5. Following the path indicated byWilliamson, what I am going to do is to present anobjection to S4 that is usually taken as motivatinga relativized understanding of metaphysical modality.But before going into that, I still have to explain thestandard or non-relativized interpretation of modality.

One quite direct way to explain the standard in-terpretation is talking about the modal principles en-coded in the axioms of the modal logic systems. Wecan speak of two pairs of modal principles that are partof the standard interpretation and will differ from therelativized one: first, the modal principle which statesthat i) what is possible is necessarily possible. ii) whatis possibly necessary is necessary. Given i), ii) followsa theorem (and vice-versa). And, second, the modal

2Nathan Salmon suggests a kind of agnosticism or scepticismregarding B, but he does not offer a counterexample to it. (1989,pp. 25-29)



principle which states that iii) what is possibly possibleis possible. iv) what is necessary is necessarily neces-sary. Given iii), iv) follows as a theorem (and vice-versa). Any of the of principles of the first pair corre-spond precisely to the axioms that should be added tothe system T of modal logic to get S5. And the secondpair corresponds precisely to the axioms that shouldbe added to the system T to get S4. The system S5includes S4, in the sense that all theorems of S4 aretheorems of S5. Therefore, rejecting any of the modalprinciples implies the rejection of S5: directly, by re-jecting either i) or ii); indirectly, by rejecting either iii)or iv).

The application of these modal principles to meta-physical modality results in the following: i) if some-thing is a metaphysical possibility, then it is necessar-ily a metaphysical possibility. And iii) if something ispossibly a metaphysical possibility, then it is a meta-physical possibility. Put as a slogan, all metaphysicalpossibilities are metaphysical possibilities in any situ-ation. The set of metaphysical possibilities is the samein any case. Denying any of these ideas is the same asaccepting some version of a relativized metaphysicalmodality. It might still not clear what exactly the dif-ferences between the standard interpretation of meta-physical modality and the relativized one are. Fromthis point onwards, our best bet to make things cleareris to appeal to Possible Worlds Semantics (PwS). Pos-sible worlds semantics is likely to help us to make intu-itive sense of the differences that we wish to establish.Possible worlds semantics is often used to understandmodality, and it indeed helps us make sense of the ax-iomatization of modal logic mainly when we need tounderstand better the iteration of modal operators.

Possible world semantics for propositional modallogic might be roughly speaking presented as a triple⟨W,R, v⟩, where W stands for a non-empty set of pos-


Fernando Furtado 10

sible worlds, R represents the accessibility relation thatranges over W, and v is a function which assigns truthvalues to each sentential variable regarding each pos-sible world. A definition for necessity operator, then,is what follows:

v(□p, w) = T iff for every world w′ in Wsuch that wRw′, v(p, w′) = T.

For the standard interpretation of modality, the clausesfor the truth of modal claims can be simplified. Sinceevery possible world is accessible (possible relative) toevery possible world, the binary relation (the acces-sibility relation) does not need to be mentioned. So,the following clauses for the modal operators are goodenough:

v(□p, w) = T iff for every world w′ in W,v(p, w′) = T.

v(◇p, w) = T iff for some world w′ in W,v(p, w′) = T.

If we grant that modal claims should be somehow rela-tivized, then the clauses just given are no longer enough.However, the possible world semantics gives us a quitestraightforward way to make room for an adequate in-terpretation of the relativized modality. All we needto do is to take advantage of the tools available inthe standard semantics. In this case, the binary rela-tion of accessibility between possible worlds will allowus to interpret the relativized modality properly. Theclauses for the truth of modal claims that make roomfor relativized modality are the following:


v(◇p, w) = T iff for some world w′ in Wsuch that wRw′, v(p, w′) = T.



For the truth of necessity claims regarding a specificpossible world w, only one subset of the set of possibleworlds, rather than the whole universe of them, will berelevant, but which is the relevant one? Only the oneincluding the possible worlds accessible from w. Thevery same idea, of course, applies for the evaluation ofpossibility claims.

The standard interpretation of modality understandsthe accessibility relation between possible worlds asan equivalence relation. As I understand it, strictlyspeaking, any theory that employs a notion of meta-physical modality that can be modelled by a seman-tics in which the accessibility relation between possibleworlds is weaker than an equivalence relation should beunderstood as a relativized theory of modality. Havingas framework the semantics for normal modal logic,we have a few alternatives, we can, for instance, ei-ther reject the symmetry or the transitivity or eventhe reflexivity of the accessibility relation between pos-sible worlds. No matter which direction one decidesto go, one will be supporting what I understand as arelativized modality. Since it is not easy to motivatethe rejection of symmetry of the accessibility relationbetween possible worlds and, regarding metaphysicalmodality, reflexivity has not risen any suspicious sofar (it is quite plausible to suppose that it should beat least reflexive), the best available option is to showthat the accessibility relation is intransitive. Salmon’sidea was to follow that path presenting what he thinksto be a counterexample to transitivity. Before movingon to Salmon’s counterexample, let us take a look at adiagram which is going to help to understand the ap-proaches. The following PwS-diagram represents thestandard interpretation of non-relativized metaphysi-cal modality:


Fernando Furtado 12

w2

w1

p

w3

Figure 1: Standard interpretation of non-relativized metaphysicalmodality. The accessibility relation is transitive, symmetrical andreflexive. Once p is true in w3, p is possible to every possible world.Therefore, p is necessarily possible (as it supposed to be in S5).

This diagram helps us to visualise the standard inter-pretation of metaphysical modality. As can be seen,the accessibility relation between possible worlds (in-dicated by the direction of the arrows) is reflexive,transitive and symmetrical. Therefore, a propositionpossibly possible p is truew1 iff p is true in at leastone the possible world accessible from w1 (w1, w2 orw3). Since, all possible worlds “access” all other pos-sible worlds, the very thing that makes the iteratedproposition possibly possible p true [p being true ei-ther in w1 or w2 or w3] makes possibly p true as well.All we need in order to both of them to be true is theproposition p to be true in w3 even if it is false in w1

and w2. The very same scenario that makes possiblypossible p true also makes possibly p true. Strictly,what validates the inference from possibly possible pto possibly p is the transitivity of the accessibility re-lationship between possible worlds [since w1 accessesw2 and w2 accesses w3, w1 accesses w3].



Salmon’s strategy is to show that some inferences ofthe kind that requires transitivity are fallacious. Howdoes he argue against transitivity? In order to moti-vate his idea, he appeals to an essentialist intuitionabout the material origin of an artefact; a wooden ta-ble, which he calls ‘Woody’.

“The case against S4 modal logic [transi-tivity of accessibility relation] stems fromthe intuition (which many of my opponentsshare) that a particular material artefact— say, a particular wooden table which wemay call ‘Woody’ — could have originatedfrom matter slightly different from its ac-tual original matter m (while retaining itsnumerical identity, or its haecceity) but notfrom entirely different matter.” (Salmon,1989, p. 130)

From these modal intuitions, we can formulate a ver-sion of Salmon’s argument with the following struc-ture:3

1. Woody originates from matter m.

2. It is possible that Woody originates from matterm′.

3. It is not possible that Woody originates frommatter m′′.

4. If Woody had originated from matter m′, then itwould have been possible for Woody to originatefrom matter m′′.

5. It is possibly possible that Woody originates frommatter m′′. (2, 4)

3Salmon’s version can be found at (Salmon, 1989, pp. 130-131). Murray and Wilson’s can be found at (Murray & Wilson,2012, p. 193).


Fernando Furtado 14

6. It is not possible that Woody originates fromm′′,

but it is possibly possible that Woody originatesfrom matter m′′. (3, 5)

7. It is not true that if it is possibly possible thatWoody originates from matter m′′, then it is pos-sible that Woody originates form matter m′′.

As we can see, the conclusion of the argument is sup-posed to be a counterexample to the S4 modal logicsystem, which states that if something is possibly pos-sible, then it is possible. The conclusion of the ar-gument says that although it is possibly possible forWoody originates from matter m′′, it is not possiblefor Woody originates from m

′′. In this case, there is nopossible world directly accessible to the actual world inwhich Woody originates fromm

′′. That model requiresthe accessibility relation between possible worlds to beintransitive, which means that “S4 modal logic is fal-lacious”. (Salmon, 1989, p. 131). The following PwS-diagram represents Salmon’s modal structure (the S5-denying approach to relativized metaphysical modal-ity):

p

w3

¬p

w2

¬p

w1

Figure 2: S5-denying approach: intransitive accessibility relation.Let p stand for: Woody originates from m

′′. According to this intran-sitive model, since w3 is not accessible from w1, p being true in w3 doesnot make ◇p true in w1. Therefore, even though v(◇◇ p,w1) = V ,v(◇p,w1) = F .



As can be noticed at Figure 2, the accessibility rela-tion between possible worlds is reflexive and also sym-metric. However, it is not transitive. Supposing thatWoody originates fromm in w1, the intransitive modelmakes room for the proposition that it is possible thatit is possible to Woody originates from m

′′ to be true,even though the proposition that it is possible thatWoody originates from m

′′ is not true. In other words,the possible world w3 (where Woody originates fromm′′) is not possible from w1 but merely possibly pos-

sible. One key feature of Salmon’s idea is that w3 isstill possible from w2 where Woody originates fromm′, what is shown in the following counterfactual con-

ditional: if Woody had originated from m′, it would

have been possible for Woody to originate from m′′.

that Salmon accepts the conditional.4

3 Two-dimensional Interpretation

Once we have got a clear understanding of the differ-ences between the standard interpretation of modalityand the S5-denying approach, we can now move onstraight to the two-dimensional S5-friendly interpreta-tion of relativized modality. To explain Murray andWilson’s approach, we can start by asking for the mo-tivations to their proposal. What are precisely thereasons to reject both the standard interpretation ofmodality and the S5-denying approach? As we haveseen, they want to make room for a sort of relativizedinterpretation of modality inspired by Salmon’s argu-ment concerning Woody. And, at the same time, theywant to keep the S5 modal logic system as the patternfor modal metaphysics; neither S5-denying approach

4Murray and Wilson have provided an alternative interpreta-tion of this conditional that avoids what they call ‘in situ shifts’,which, according to them, is one of the main sources of theirconcerns with Salmon’s proposal. (Murray & Wilson, 2012, p.197)


Fernando Furtado 16

nor the standard interpretation can meet these goals.

We motivate the alternative proposal byattention to discussions in Salmon (1989)[...]. In [this] discussion, the author can-vasses data which he takes to support a cer-tain thesis—that the transitivity of the ac-cessibility relation between possible worlds,and associated systems of modal logic S4and S5, should be rejected as characteristicof metaphysical modality. We argue thatthe data [about Woody] can be accommo-dated, compatible with transitive accessi-bility [...], if metaphysical necessities andpossibilities are relativized to indicative ac-tualities. (Murray & Wilson, 2012, p. 190)

As they make clear, their primary motivation is to pro-vide a theory capable of accommodating Woody’s data(what S5-denying account also can do) in a systemin which the accessibility relationship between possi-ble worlds is an equivalence relation (as it is in thestandard interpretation). In order achieve that goal,they provide a description in which the semantic struc-ture is relativized to which possible world is consideredas indicatively actual. Following that path, they endup adding a new parameter to the standard possibleworld semantics for modal logic. Although Murrayand Wilson and myself may be using the term ‘rela-tivized’ to apprehend, let us say, the samemetaphysicalphenomenon, the way how each of us implements theidea in the formal semantics is quite different. For me(following Salmon), it concerns the restriction on theaccessibility relation between possible worlds; what isenough to make room for being possible (or impossible)to be relative. According to their understanding, a newparameter needs to be added to the standard semanticsof modal logic to work as a function from indicativelyactual worlds to truth values. Strictly speaking, their



implementation is a kind of two-dimensional seman-tics, which they acknowledge by saying that their the-ory is “formally analogous epistemic interpretation ofthe two-dimensional 2D semantic framework.” (2012,p. 198). So, it might be a good idea to keep that inmind to help us to grasp their theory correctly. In whatfollows I will explain how it works, but for now I justwanted to highlight a potential difference in meaningconcerning the use of the term ‘relativized’. The fol-lowing complex PwS-diagram gives us a grasp of theirproposal:

¬pw1

¬p

w2

¬p

w3S1

¬pw1

¬p

w2

p

w3S2

¬pw1

¬p

w2

p

w3S3

Figure 3: Two-dimensional strategy. There are three distinct isolatedsubspaces (S1, S2 e S3) relativized to possible worlds taken as indica-tively actuals (indicated by the white fills regarding each subspace:[S1 −w1], [S2 −w2], [S3 −w3]).p: Woody originates from m

′′.

The general idea is to build a semantics in whichthe possible world structure is composed of multipleisolated subspaces of possible worlds instead of onlyone space of possible worlds. As can be noticed inFigure 3, there are three different subspaces to accom-modate the intended relativization to indicatively ac-tual possible worlds. Depending on which world isconsidered as indicatively actual, a different subspacecomes into play and, therefore, different truth valuesare assigned to the proposition in question. If we thinkof the proposition p (Woody originates from m

′′) andtake w1 (where Woody is made fromm) as indicativelyactual (subspace S1 represents it), then we will get thetruth value false for p (and also for ◇p and ◇◇p) in


Fernando Furtado 18

w1(S1). However, if we assume that the indicativelyactual world is the possible world w2 (where Woodyis made from m

′: subspace S2 represents this situa-tion), then, even though p will continue to be false (inw2(S2)), ◇p (and also ◇◇p) will turn out to be truein w2(S2). Something similar happens if w3 (whereWoody is made from m

′′) is taken as indicatively ac-tual. But in this case, p will be true in w3(S3) (thesame goes for ◇◇p and ◇p regardless whether or notp is true in either w1(S3) or w2(S3)).

It is worth noting that the only difference betweeneach world-token (for instance, w1 at the subspace S1and w1 at subspace S2), regards the relativization inplace, i.e. which world is considered as indicatively ac-tual. As it has shown, some true propositions concern-ing one relativization might come out false to another.That is precisely the intended result.

This section began by talking about Murray andWilson motivations for proposing an alternative inter-pretation for relativized metaphysical modality, which,as we now can see, are two primary motivations: first,to recognise Wood’s data as philosophically relevant;and second, to provide a solution to Woody’s case com-patible with the S5 modal logic system. Did they sat-isfy these motivations? It seems that they did meetthem. They accept the significance of Woody’s caseand propose a “new” semantics for the interpretationof modality, which allows us to deal with Wood’s caseand, in the meantime, preserve S5 features. If we lookat Figure 3 once again, we will see that, concerningeach isolated subspace, the accessibility relation be-tween possible worlds meets the S5 requirements, i.e.it is a transitive, symmetrical and reflexive binary re-lation. To be able to do that, what they did was topropose a semantics in which the truth value of a givenproposition is relativized to which world is being heldas indicatively actual. Even though Figure 3 gives us



a gentle visual grasp of their approach, we still need aprecise formulation of it. To get an accurate formula-tion, we can compare the formal features of their se-mantics with the standard semantics for propositionalmodal logic.As I said above, Murray and Wilson’s se-mantics increments the standard semantics for modallogic with an additional parameter. The semantics formodal logic is usually presented with a triple ⟨W,R, v⟩,where W stands for the set of possible worlds, R rep-resents the accessibility relation that ranges over W,and v is a function which assigns truth values to eachproposition regarding each possible world. Just as areminder, the clause for evaluation of a propositionlike □p is what follows:


Murray and Wilson’s semantics adds a new parame-ter to the semantic structure in such a way that it be-comes a quadruple ⟨W,R, v, w@⟩. This quadruple addsa function w@, which picks out one possible world ofthe set W to be considered as indicatively actual (any-thing else is kept equal). The new semantic clause fornecessity becomes:

v(□p, w,w@) = T iff taking w as w@, forevery world w′ in W such that wRw′, v(p, w′)= T.

Or, more naturally,

□p is truew1 iff taking w1 as indivative ac-tual, p is true in every possible world ac-cessible from w1.

As it may have been noticed, there is no constraintwhatsoever on R. R does indeed represent a reflexive,


Fernando Furtado 20

symmetrical and transitive relation. Hence, Murrayand Wilson’s semantics does indeed validate S5. How-ever, one may ask, does it handle Woody’s data prop-erly? Let us think about that. The general strategyof dealing with Woody’s data uses the relativizationas a function from indicatively actual worlds to truthvalues, which means that depending on which world isthe indicative actual, different truth values come out.

As it happens in the case of subspace S1, the world-type w1 is the indicative actual (w1 is possible relativeto any other possible world in S1; as S5 requires). InS1 where w1 is the indicative actual world, and Woodyoriginates from m, the proposition ◇p turns out to befalse (in w1(S1)). It happens because the propositionp is false in all possible worlds (in S1) (including w3

where, let us say, a table other than Woody originatesfrom m

′′). Since S5 is in place (thus, the accessibilityrelation is an equivalence relation), we can extrapolateand say that not only ◇p is false but also ◇◇p (inw1(S1)). Considering the subspace S2 (where w2 is theindicative actual world, and Woody originates fromm′), the proposition ◇p comes out true (in w2(S2)).

In this case, the table of w3(S2) that originates fromm′′ is Woody itself, which means (given that S5 is in

place) that not only◇p is true but also □◇p is true (inw2(S2)). In both cases, in the subspace S1 and in thesubspace S2, the S5 modal logic system is validated.Either it is impossible to Woody originates from m

′′,then it is necessarily so (as in S1). Or it is possible forWoody originates from m

′′, then it is necessarily so (asin S2). Either way, S5 is preserved.

Based on all we have discussed so far, we should (atleast for the sake of argument) grant that Murray andWilson have satisfied their motivations of making senseof the idea of preserving S5 and, at the same time,dealing with Woody’s data. From the next section on-wards, I argue that despite Murray and Wilson’s make



sense of their goals, their approach faces both techni-cal difficulties and philosophical challenges that mightgive us reasons to stick with the S5-denying approachto relativized metaphysical modality, if we are willingto understand modality in a relativized fashion.

4 Challenges for the 2D-interpretation of Rela-tivized Modality

In the second half of this paper, I will be presentingsome problems that the 2D-interpretation of the rela-tivized modality has to face. The potential problemswill be of two kinds: i) I will argue that such an ap-proach may have trouble trying to make sense of theconcepts used to formulate the theory and also thatthe possible world structure used in the account maynot be able to represent modality accurately. And ii)2D-interpretation may face serious challenges dealingwith the philosophical problems that it was initiallydesigned to address. It is worth noting that neitherthe first nor the second kind of challenges can under-mine the original project; that is not even the aim ofthis section. In fact, this section aims to argue thatif one wants to understand the relativized of modalityaccurately, the S5-denying approach is probably hismost conservative option.

4.1 Extensional Inadequacy

The 2D-interpretation is supposed to be a theory ofmodality based on a type of possible worlds seman-tics. As such, it also aims to represent possibilities(necessities, and other potentially derived modal con-cepts, e.g., essence) in terms of a possible worlds struc-ture. This section argues that 2D-interpretation maynot be able to represent the modal discourse. I willargue that the 2D-interpretation provides an exten-sionally inadequate description of modality. On the


Fernando Furtado 22

one hand, it does not represent some possibilities thatit should be able to represent. On the other hand,it acknowledges too many possible worlds: there aremore possible worlds than there should be.

4.1.1 Too Many Possibilities to Handle

The strategy is to show that some general featuresof the underlying concept of possible worlds used tobuild the semantic structure of the 2D-interpretationprevents it to represent all the facts (or propositions,state of affairs, etc.) that it was supposed to represent.Therefore, the notion of possible world employed doesnot, in fact, represent the concept of possible world (asusually understood).

The inspiration for what follows is David Kaplan’schallenge for the foundation of possible worlds seman-tics, which Kaplan relates to Russell’s paradox. Kaplanargues for a paradox that shows up in possible worldsemantics. As he puts it, the challenge “shows a seri-ous difficulty in the naive foundation of PwS [possibleworld semantics].” (Kaplan, 1995, p. 41). As it wouldbe expected, there are many different versions of theargument for many different purposes. What follows isa simplified version of Kaplan’s case for the purposesthis paper.5

The argument aims to show that there cannot beentities such as possible worlds or any other kind ofrepresentation of the set of all the propositions (facts,states of affairs, etc.). To be more precise, the problemin question here relates to the notion of maximality. Ingeneral, the idea is that for any supposed arrangementof all propositions that one considers, there will alwaysbe some propositions left out. The argument attacks

5For a more detailed and precise discussion about the case,I would suggest both Kaplan’s paper (1995) and the sectionon possible worlds paradoxes in John Divers’ Possible Worlds(2002).



the notion of maximal consistent sets of propositionsarguing for the incoherence of such a concept. Sincepossible worlds, under some perspectives, are charac-terised as maximal consistent sets of propositions, anargument demonstrating the inconsistency of that con-cept will also show the inconsistency of the concept ofpossible worlds so described. The notion of a maximalset of propositions is characterised as follows:6

S is a maximal set of propositions iff forany proposition p, either p or its negationis a member of S.

To make sense of the idea, let us begin to suppose pos-sible world w and a proposition p which represents theconjunction of all true propositions concerning w. Theproblem shows up when a proposition refers to p. Forexample, let us think about a person A and a futuretime t, and suppose that it is possible that A thinksof p at t. At this point, the reader may have alreadyfigured out the strategy. Suppose that q is the proposi-tion that represents the possibility just described. Theproblem, then, is that q cannot be part of p. Neverthe-less, p was supposed to represent all the propositions.Therefore, we conclude either p did not represent allthe propositions in the first place or propositions suchas q cannot constitute a genuine proposition. Sincethere is no reasonable ground (at least in the naivefoundation of PwS) to argue that q does not representa real proposition, one should conclude that p doesnot represent all the propositions. This conclusion ex-trapolates for any representation of all propositions.And that should be enough for us to reach a contra-diction, given that we began for supposing that p does

6It does not need to be about sets of propositions. In fact,many philosophers understand the concept of possible worldsdifferently. What matters is whether the concept employed usesor not the notions of (naive) sets and maximality. If that is thecase, then the argument is going to follow regardless.


Fernando Furtado 24

represent all propositions. Hence, the additional sup-position that a possible world is defined as a maximalset of propositions will allow us to conclude that w isnot a possible world.

As might be clear, this kind of argument will be, asfar as I can see, an issue for any treatment of modalitywhich uses possible worlds understood as a maximalset of propositions (or some other favoured notion). Infact, the problem will potentially arise for any attemptof providing a semantics for modality which uses naiveset theory as the base on the top of which the modalsemantic is built and tries to give a coherent notionof maximal consist set, which supposedly would rep-resent all propositions. So, this particular argumentis not an issue specifically to the 2D-interpretation.However, as I said before, inspired by the argumentjust discussed, we can make a case for a problem thatthe 2D-interpretation of relativized modality will needto face. That is what we are going to do now.

It is not complicated to come up with an examplethat shows up only in the 2D-interpretation. One ofthese cases considers the proposition:

p: Woodyw1(S1) is identical to Woodyw1(S2) .

Just to remember, S1 and S2 refers to different sub-spaces in the 2D semantic structure. What is the prob-lem with p? There are a couple of issues regarding p.The first of them being: is p a member of either w1(S1)or w1(S2)? In fact, there seem to be reasons to thinkthat p is neither a member of w1(S1) nor w1(S2). Basedon what we have seen so far and considering the con-cept of possible world used in the semantics proposedby the 2D-interpretation is indeed the familiar conceptof possible world, if p is not a member of w1(S1), thennot−p should be. However, for precisely the same rea-sons that p is not a member of w1(S1), not − p cannotbe a member of w1(S1) as well. So, w1(S1) is not max-imal. Therefore, w1(S1) is not a possible world. The



very same reasoning can be developed to show thatw1(S2) is also not a possible world. Extrapolating theresult, we conclude that the concept of possible worldsused in the 2D-interpretation is not the familiar con-cept of possible worlds. This case is only an issue forthe 2D-interpretation not for any standard semantics.

The last case presents, let’s say, a more “technical”issue, but there are some other philosophical problemspotentially even more complicated. One of the philo-sophical difficulties that I want to discuss here concernsthe concept of identity showing up in p. A straight-forward way of making the first problem evident iscomparing it with an analogous case in the standardsemantics.

p′: Woodyw1

identical to Woodyw2.

What is the notion of identity in p′? The standard

semantics can say that the notion of identity in issue isthe notion of transworld identity ; the ordinary notionof identity in the standard philosophical interpretationof the semantics for modal logic.

The 2D-theorist can only use transworld identitywithin each subspace, not outside. Outside of eachsubspace, one would have come up with somethingdifferent such as trans-subspace identity. The problemwith this notion is that it is a brand-new concept whichnobody is familiar with and it is not explicitly defined.Additionally, it is not clear the role that such a con-cept would be playing in theory as a whole. Would itbe used elsewhere, or it would just be postulated tomake sense of p? Even though transworld identity isreplaceable (for a counterpart relation, for instance),it still is a crucial notion in the standard semantics.Is trans-subspace identity also an important notion?I cannot see how. At least not without making 2D-interpretation, for many purposes, indistinguishablefrom standard approach.


Fernando Furtado 26

I cannot say, however, that Murray and Wilson didnot expect some related issues. As they say in thefootnote 12:

“Even though worlds in different subspacesare not identical on this view, worlds in dif-ferent partitions may be taken to be ‘basic’or ‘canonical’ counterparts of each other,[...] worlds in different spaces may be basi-cally alike; such similarity may serve as thebasis for loose (as opposed to strict) iden-tification of worlds across subspaces. So,for example, distinct subspaces might eachcontain worlds that are basically, canoni-cally, or qualitatively similar, in contain-ing, e.g., a table-shaped hunk of matterm′′; such worlds, we might say, are of type

w3. In a world of type w3, is it true or falsethat Woody originates from m

′′? That de-pends. In a subspace where the indica-tively actual world is (of type) w1, this isfalse; but in a subspace where the indica-tively actual world is (of type) w2, this istrue.” (Murray & Wilson, 2012, p. 207)

Even though they did give some indication regardingthe identity (or identification) of worlds across sub-spaces, they clearly avoided talking about the identityof objects across subspace. That’s evidenced by theuse of the indefinite article ‘a’ at ‘a table-shaped [...]’instead of a proper name or even a definite description.I do not know whether they think that this is not im-portant, or they are planning to give an account of thenotion in the future, but, from my point of view, it isa crucial point of the theory and precision about it isfundamental to make sense of their approach.

Transworld identity plays a vital role in possibleworlds semantics, as employed by the standard inter-pretation. In fact, transworld identity is what enables



the standard possible world semantics to represent, forinstance, a modal concept of essence or to make senseof the notion of de re possibility. Transworld identityis precisely the function that maps objects in differentpossible worlds to answer questions about possibilitiesfor a given object. In order to answer whether or notis possible for Woody to be originally made of m′ (apiece of wood slightly different from the matter thatWoody was actually made), we need be able to identifyWoody in different possible worlds correctly. Withouttransworld identity, possible worlds semantics wouldnot be suited to model metaphysical modality theoriesaccurately.7

As it was said above, 2D-interpretation does notgive a straight answer whether or not Woody could beoriginally made from m

′. To answer questions aboutpossibilities regarding Woody, we need first to pick outa subspace and then, depending on which subspace weend up selecting, the questions will be answered posi-tively or negatively.8 As long as the first stage of theevaluation (the picking out subspace stage) has beencompleted, the theory will need to guarantee that, inthe end, we were talking about Woody itself and notany other ‘table-shaped such and such’. To be ade-quate to provide an account to metaphysical modalityand to make room for essentialism (or, more generally,to be able to answer questions about de re possibility),the 2D-interpretation of modality would need to give

7As said before, transworld identity might be replaced in thesemantic structure by some other function such as a counter-part relation to perform the role of mapping objects in differentpossible worlds. Being that the case, a precise definition of thelogical proprieties of the function employed is going to be nec-essary. Whereas transworld identity is an equivalence relation,counterpart relation is usually intentionally weaker than that(intransitive, for instance). Regardless, the logical properties ofthe counterpart relation that one uses need to be made explicit.

8It is worth remembering that transworld identity withineach subspace works just like in the 1D-semantic framework.


Fernando Furtado 28

us precise procedures to evaluate modal claims. Thoseprocedures would probably require some method toidentify objects in different subspace in addition to thetransworld identity, which would still play within eachsubspace. If that is correct, then the 2D-interpretationof relativized metaphysical modality needs to makesense of a concept of trans-subspace identity. Sincethey have not provided such a concept, we can tryto figure out how it could work. Regarding the stan-dard interpretation and also the S5-denying approach,transworld identity is defined as an equivalence rela-tion.

As we have seen, the S5-denying approach takesadvantage of the intransitivity in the accessibility re-lation between possible worlds to deal with Woody’scase. One suggestion of a notion of a world’s counter-part and similarity that can be seen in the followingquotation:

“Even though worlds in different subspacesare not identical on this view, worlds in dif-ferent partitions may be taken to be ‘basic’or ‘canonical’ counterparts of each other,[...] worlds in different spaces may be ba-sically alike; such similarity may serve asthe basis for loose (as opposed to strict)identification of worlds across subspaces.”(Murray & Wilson, 2012, p. 207)9

9This is unusual employment of the word ‘counterpart’, whichusually refers to the attempt of placing transitivity elsewhere.‘Counterpart theories’ typically refer to some strategy thatchanges the function which maps the objects in different possibleworlds. The most natural of those strategies would be to weakenthe mapping function making it intransitive. In that case, in-stead of transworld identity, a notion of counterpart (intention-ally designed to behave as a intransitive concept) would playthe role of picking out the objects in different possible worlds.David Lewis (1986) has famously followed that path for reasonsindependent of the present discussion.



If taken seriously, the last quotation suggests that Mur-ray and Wilson might prefer to place a sort of intransi-tivity on the concept of trans-subspace identity. Beingthat the case, they could say that a far way subspacewhere ‘a table shaped such and such’ exist and is suf-ficiently different from Woody is not Woody anymore.

Another way to achieve similar results would imple-ment a supplementary notion of accessibility relationbetween subspaces. Case in which, a subspace whereWoody is sufficiently different for some reason does notexist or is not a possible being. At this point, we seemto be either coming up with semi-technical ad hoc solu-tions or making the 2D-interpretation more and moresimilar to the S5-denying approach. Either way, it doesnot help 2D-interpretaion.

There is one more point that a 2D-theorist canmake, though. He could simply argue that he does notneed a notion of trans-subspace identity at all. In thatcase, I would answer saying that 2D-interpretation nolonger make room for essentialism. As a matter of fact,one would be embracing a radical anti-essentialismsince there would not be any restriction on the way howthings could have been in different subspaces. Onceagain, a quite strange result considering that the 2D-interpretation was motivated in the first place, amongother things, by the wish of accommodating Woody’sdata.

4.1.2 Too Many Possible Worlds

In the latest section, I argued to demonstrate that thesemantic structure provided by the 2D-interpretationof relativized modality is not extensionally adequatebecause it does not represent all the possibilities. Nev-ertheless, we can also argue in the opposite directiondemonstrating its extensional inadequacy by showingthat its semantic structure represents things that arenot possibilities. The semantics is designed in such a


Fernando Furtado 30

way that there will often be more than only one pos-sible world to represent each possibility.

The general idea of ‘extensional adequacy’ that I amusing here is that the semantic structure used to rep-resent modality should hold an isomorphic one-to-onerelation with the modal concepts which it is supposedto represent (in the current case, the modal conceptof possibility). That is to say that for each possibil-ity (regarding the kind the modality in issue; in thiscase: metaphysical modality), there must be one, andonly one, corresponding possible world.10 Arguably,that is the case considering both the standard interpre-tation of modality and also the S5-denying approachto relativized modality. However, concerning the 2D-interpretation things seem to be quite different.

Understanding possible worlds as maximal consis-tent sets of propositions (or something similar), theisomorphic relation can be quite straightforwardly demon-

10That is is strictly speaking false. In fact, for any possibilityP , there will be a family of possible worlds. For sake of simpli-fication, let us consider only two possibilities: suppose that Pstand for the possibility of myself being an alien and Q representthe possibility of the snow being white. Technically, regardingP (and supposing that the possible worlds are composed of onlytwo possibilities (P and Q)), there will be two possible worldscorresponding to the possibility P : the world where snow iswhite and the world where it is not white. To properly speak ofthe one-to-one relation between possible worlds and possibilities,we should talk about the notion of a family of possible worldsinstead of merely possible world. To get the one-to-one relationfrom the idea of family of possible worlds, what we need to dois arbitrarily keep all the other possibilities stable (unchanged).For example, if we keep Q stable (snow is white), there will beonly one possible world for corresponding to P . The complexitygrows really fast (2n, in semantics with only two truth values)according to the number of possibilities in consideration. So,for the sake of simplicity, I will continue using ‘possible worlds’instead of ‘family of possible worlds, supposing that these con-siderations have made things clear. Even if these clarificationsare not persuasive enough, I hope they will help to see that2D-interpretation overgenerates possible worlds.



strated.

w is a possible world iff for any propositionp, either p or its negation is a member ofw.

Let us suppose that w represents the possibility P inthe semantic theory. The substitution of propositionsthat are members of w by their respective negationsgives us a brand-new possible world w

∗ which repre-sents an also brand-new possibility P

∗. Despite be-ing quite straightforward, this reasoning allows us toshow that a concept of possible worlds characterisedas above, holds the isomorphic relation with the modalconcept of possibility, making it suited to play the roleof representing the modal concepts. Analogous rea-soning, though, cannot be pursued if one adopts the2D-interpretation of relativized modality. At least insome context (namely, those in which a relativized ap-proach is arguably well suited (e.g., Woody’s case)),the one-to-one isomorphic relation between possibilityand possible worlds cannot be established. In reality,regarding each possibility, there will be countless cor-responding possible worlds.

One willing to defend 2D-interpretation of relativizedmodality could argue that even though the semanticscannot establish a one-to-one relation between possi-ble worlds/possibility, it can establish a one-to-one re-lationship between possible world-type and possibility.

I recognise that approach as the correct way to go,but I still waiting for a precise explanation of why weneed possible world-type instead of the “simple” possi-ble world concept. And, what seems to be even morecomplicated, we need a clear explanation of what isprecisely the relationship between those two concepts– possible world and possible world-type. They aresomehow related, even though it is not obvious howand it is also not clear if it still makes sense to refer topossible world-type by the expression ‘possible world’.


Fernando Furtado 32

Strictly speaking, possible world, as used in formal se-mantics, can be virtually anything. But the question isthat, when applied to philosophical theories, possibleworlds need to correspond to some relevant philosoph-ical concept which they are supposed to clarify. Oth-erwise, instead of making philosophical theories moreprecise and easier to work with, the notion of possi-ble worlds will end up making them more obscure andharder to grasp. I do not want to argue that that isthe case with the 2D-interpretation (I would not gothat far). However, I do believe that the precise re-lationship between possibility and possible world-typestill needs to be made explicit. If it is not a one-to-one isomorphic relation, then what is it? As far as Iunderstand, this question remains unanswered. WhatI take as a downside of the proposal.

4.2 2D-interpretation and Essentialism

As we have seen in the first part of this paper, the 2D-interpretation of relativized modality was motivatedby, among other things, the ambition to accommodateWoody’s data in a reasonable understanding of meta-physical modality. Woody’s case shows up in a specifictheory of what is possible or not for an object, in thiscase, Woody. The consideration of something impos-sible for an object is often called ‘essentialism’. Anessentialist is going to say that, for instance, Woodycould not have been made from m

′′. That is the sameas saying that it is impossible to Woody to be (origi-nally) made ofm′′. What this does is to impose restric-tions on the possibilities available for Woody. Someessentialists are willing to hold a modal principle thatstates that, even though Woody could not have beenmade from m

′′, it could have been made from m′, sup-

posing that Woody was actually made from m. Assaid before, one of the main the motivations for 2D-interpretation is to accommodate essentialist cases like



Woody’s within a theory that does not drop S5 modallogic system. The question now is: can they really dowhat they intend? Can they provide such theory? Cantheir theory impose restrictions on what is possible ornot for Woody? It does not look like they can. Let ussee why.

How does S5-denying approach handle Woody’s case?Well, the explanation is quite straightforward. Sup-posing that Woody was actually made from m, it isnot possible for it to be made from m

′′. And why isthat so? The general idea is that there is a thresholdaccording to which, if the amount of the overlappingmatter used in the manufacturing process of makingWoody is smaller than x, then the table which eventu-ally comes out of this process cannot be Woody. Thisvery idea is reflected in the S5-denying formal treat-ment of modality by restriction of the accessibility rela-tionship between possible worlds. Since the S5-denyingapproach understands metaphysical modality in such away, the corresponding modal logic will recognise anyworld which represents a situation in which the thresh-old condition is not satisfied as representing an impos-sibility for Woody. Hence, the S5-denying approachaccommodates the case. In fact, it is designed havingWoody’s case in mind. What about 2D-interpretation?It seems to understand things slightly different. Thequestion of whether or not Woody could have beenmade from m

′′ does not have a straight answer. It ap-pears that the question involves some ambiguity thatneeds to be sorted out before the theory can answer.The response is going to depend on which subspace isin question or, in other words, which possible world isbeing held as indicatively actual. If we take the possi-ble world w1 as indicatively actual (take a look at theFigure 3 again), then the answer is ‘no’, Woody couldnot have been made from m

′′. However, if we take ei-ther the w2 or w3 as indicatively actual relativizing the


Fernando Furtado 34

answer to either subspace S2 or S3, then the answer isgoing to be ‘yes’, Woody could have been made fromm′′.My main concern with the account is that it looks

like there is no limiting case. There is no threshold torestrict how different the world that is being taken asindicatively actual can be. In reality, there is nothingin the theory preventing us from taking as indicativelyactual a world substantively different from the actualworld. Let me give an example. Suppose a world (wn)so dramatically different from the actual world thatis a possibility for Woody to be made of iron insteadof wood. As far as I understand it, 2D-interpretationdoes not offer any tool that could rule out that pos-sibility. In fact, 2D-interpretation cannot prevent usfrom taking wn as indicatively actual. That seems toshow that 2D-interpretation does not accommodate es-sentialism since the essentialist is going to want to saythat wn does not represent a genuine possibility forWoody: wn is not a metaphysically possible world.Since 2D-interpretation does not rule out cases suchas just described, it does not seem to be well suitedfor to be considered a theory for essentialist relativizedmetaphysical modality. And that is for me an entirelystrange conclusion because it looked like it had beendesigned to handle Woody’s data recognising it as acrucial limiting case of metaphysical modality. Fur-thermore, we seem to be able to conclude, followingthe principles of the theory itself, that there shouldnot be an essentialist restriction on what is possiblefor Woody. I take that as tension within the proposalthat at least requires further clarification.

5 Is 2D-interpretation a Conciliatory Theory?

The last objection that I want to present is addressedto Murray and Wilson’s intention to provide a concil-



iatory theory for relativized modality. Their theoryintends to be conciliatory in a few different ways. Inthis section, I will focus on one that I suppose to bethe most relevant for the purposes of this paper: theattempt to make 2D-interpretation an S5-friendly the-ory. The idea is that the 2D-approach can give a de-cent treatment of Woody’s data within an S5 modallogic framework. I argue that even if they providedan S5-friendly theory, I do not believe that any firmS5 supporter such as Plantinga or Williamson wouldconsider adopting it.

My point here is straightforward: among the fewarguments available in the literature supporting S5modal logic (for metaphysical modality), none of themseems to be supporting 2D-interpretation. Let us con-sider, for example, the case of Plantinga and thinkabout the quotation of The Nature of Necessity givenat the beginning of this paper. Among other things,he says, and I quote:

“[i] Are there propositions that in fact arepossible, but would have been impossiblehad things been different in some way? [ii][C]ould [they] have been impossible? Theanswer, I think, is clear; [they] could nothave been impossible.” (Plantinga, 1974,p. 52)

Suppose that the proposition p (Woody is made ofm′′) is considered impossible. Assuming we accept

Plantinga’s argument, we will be compelled to acceptthat p could not be possible. It seems to be clear thatthis is a problem for those who do not want to admitS5 modal logic (for metaphysical modality). However,one may ask, is the situation any better for those whothink that p would be possible if another world hadbeen taken as indicatively actual? All they are doingis saying that p is impossible relatively to some worldstaken as indicatively actual but not to others. Would


Fernando Furtado 36

that be enough to persuade Plantinga to accept 2D-interpretation? The only way to know is asking forhim. Until then, I bet he would not be willing to ac-cept 2D-interpretation. As far as it goes, I understandthat if Plantinga’s argument works against S5-deniers(which by the reasons given in the first part of thispaper, I do not think it does), then it will work forsimilar reasons against the 2D-interpretation.

David Lewis gave one more argument in the samedirection in his On Plurality of Worlds. Let us lookinto it:

“In these questions of haecceitism and essence,by what right do we ignore worlds thatare deemed inaccessible? Accessible or not,they’re still worlds. We still believe in them.Why don’t they count?” (Lewis, 1986, p.246)

David Lewis is precisely discussing issues related toWoody’s case in that quotation. And he puts for-ward an argument quite similar to the one presentedin the first part of this paper to support the non-relativized approach. The general idea is the same asPlantinga’s. If metaphysical modality is the one be-ing taken into consideration, there will not be a worldthat is not accessible from the actual world. The pointhere is to show that metaphysical modality must vali-date S5 modal logic, even if one is willing to accept aweaker system to a restricted notion of modality, suchas nomological or physical.

Supposing that Lewis’ argument supporting the non-relativized approach is convincing, is the 2D-theoristin a better position in comparison to its S5-denying op-ponent? I think that if Lewis’ argument is convincingagainst the later, then it is also convincing against theformer. From the S5-denying perspective, the propo-sition p – Woody is originally made of m′′ – is notpossible regarding the actual world, which means that



a world representing such a possibility (w3) is consid-ered an impossible (inaccessible) world (from the per-spective of the actual world), which does not precludew3 of being possible from some other possible world.Here comes the rhetorical appeal of Lewis’ quotation –“Accessible or not, they’re still worlds. We still believein them. Why don’t they count?”. Lewis is pushingto the idea according to which if we recognise w3 asa world, we should recognise it as possible (accessible)from any possible world; against S5-denying approach.The question now is that supposing that the rhetori-cal appeal of Lewis quotation against S5-deniers hasconvinced you, would you still be in a position to ac-cept the 2D-interpretation workaround? In the end,what they are doing is saying that p is possible re-garding some isolated subspaces but not others. Whyshould we grant that there are such thing as isolatedsubspaces? One may ask adapting Lewis’ idea. Oncethey are there, why should we concede that they areisolated? I think no answer to this question can bemore plausible than a similar explanation given by anS5-denier.

In this respect, I think that the 2D-interpretation isin a position at least as complicated as the S5-denier.Worse for the 2D-theorist who motivated his approachpromising an S5-friendly theory. It might be a com-patible theory, but is it any attractive to a supporterof the standard interpretation? I do not think so.

In a more general perspective, it is a well-knownfact that S5 modal logic system is compatible withisolated subspaces of possible worlds; each of them be-ing modelled by a semantic structure which employs anequivalence relation on the accessibility relation amongpossible worlds. However, I think, the appeal of whatLewis and Plantinga say is not only that any possi-ble world must be accessible to all the other possibleworlds in a given subspace (compatible with S5), but


Fernando Furtado 38

also that that subspace is the total universe of possibleworlds (what S5 does not offer). In this case, the acces-sibility relation between possible worlds is called ‘to-tal’, which means that for any possible world one picksout, this possible world will be accessible to any otherpossible world that exists (no qualification needed).The requirement for accessibility relationship betweenpossible worlds to be total is stronger than what S5can deliver, and then a theory as strong as S5 modallogic cannot meet the technical demands. Therefore,even if the 2D-interpretation does manage to accom-modate Woody’s data within an S5 framework, it stillunable to deliver a notion of accessibility relation be-tween possible worlds characterised as a total. In thebest-case scenario, the accessibility relation betweenpossible worlds in the 2D-interpretation is an equiva-lence relation, but it cannot be a total relation. To behonest, I think that only because no one had thoughtof something to make sense of isolated subspace thatpeople have not realised yet what they want is a to-tal relation instead of an equivalence relation. Nowthat Murray and Wilson have come up with the 2D-interpretation, people will need to be clear about that.And that’s a good thing.

5 References

Chandler, H. ‘Plantinga and the contingently pos-sible’, Analysis, 36, 106–109, 1976.

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Documents

S5-denying Approach to Relativized Metaphysical Modality