Modal statements tell us something about what could be or must be thecase. Such claims can come in many forms. Consider:

1. No one can be both a bachelor and married. (‘Bachelor’means ‘unmarried man’.)
2. You could not have been born of different parents.(Someone born of different parents wouldn’t be you.)
3. Nothing can travel faster than light. (It’s a law ofnature.)
4. One cannot get from London to New York in less than one hour.(Planes that fast haven’t been built yet.)
5. You cannot leave the palace. (The doors are locked.)
6. You cannot promise to come and then stay at home. (It’swrong.)
7. You cannot start a job application cover letter with “heyguys”. (It’s just not done.)
8. You cannot castle if your king is in check. (It’s againstthe rules.)
9. You cannot deduct your holidays from your taxes. (It’sagainst the law.)
10. Fred cannot be the killer. (The evidence shows that he’sinnocent.)

Each of these claims appears to have a true reading. But it also seemsthat ‘cannot’ needs to be interpreted in different ways tomake the different sentences true. For one thing, we can, in the samebreath, accept a modal claim in one of the senses illustrated by(1)–(10) while rejecting it in another one of these senses, asin the following dialogue:

Caesar:

You’re lucky that I’m still here. The doors wereunlocked. I could have left the palace.

Cleopatra:

True. But then again, you couldn’t have left thepalace. That would have been wrong, given that you promised to meet mehere.

Moreover, the modal claims (1)–(10) appear to be true forcompletely different reasons. For example, it may be held that thetruth of (1) is due to the meanings of its constituent expressions;that (2) holds because it lies in your nature to be born of youractual parents; that (3) is true because the laws of nature precludesuperluminal motion; that (4) holds because of technologicallimitations; that (5) owes its truth to the presence of insurmountablepractical obstacles; that (6)–(9) are made true by the demandsof morality, etiquette, the rules of chess, and the law respectively;and that (10) holds because the known facts prove Fred’sinnocence.

It is one of the tasks of a philosophical theory of modality to give asystematic and unified account of this multiplicity of modal concepts.This article discusses a few of the main issues that need to beaddressed by anyone pursuing this goal. Sections 1 and 2 concern thequestion of what fundamental categories of modal notions there are.The focus will be on two contemporary debates: whether there areseparate forms of modality that are tied to the epistemic and themetaphysical domains (section 1), and whether there is a special kindof necessity associated with the laws of nature (section 2). Section 3discusses questions about the relations between different notions ofnecessity. Can some of them be reduced to other, more fundamentalones? If so, which concepts of necessity are the most fundamentalones? And if there are several fundamental kinds of necessity, what dothey have in common that makes them all kinds of necessity?

• 1. Epistemic and Metaphysical Modality
  • 1.1 The Data
  • 1.2 Dualism
  • 1.3 Monism
• 2. Metaphysical and Nomic Modality
• 3. The Structure of the Modal Realm
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Epistemic and Metaphysical Modality

There are many ways the world could have been. You could have gottenup later today. Your parents could have failed to meet, so that youwere never born. Life could never have developed on earth. The historyof the universe could even have been completely different from thebeginning. And many philosophers believe that the laws of nature couldhave been different as well (although that has been denied, asdiscussed insection 2). Maximally specific ways the world could have been are commonly called‘possible worlds.’ The apparatus of possible worlds allowsus to introduce a set of modal notions: a proposition is necessaryjust in case it is true in all possible worlds, a proposition ispossible just in case it is true in some possible worlds, and it iscontingent just in case it is true in some but not all possibleworlds. Asentence is necessary (possible, contingent) justin case it expresses a necessary (possible, contingent)proposition.

The modal notions considered in the last paragraph are not obviouslyepistemological. On the face it, we are not reporting a fact aboutwhat is or can be known or believed by anyone when we say that lifecould have failed to develop. But there is also a family of modalconcepts that are clearly epistemological. These are the notions weemploy when we say things like ‘Fredmust have stolenthe book (the evidence shows conclusively that he did it),’ or‘Marycannot be in London (she would have calledme).’ These modal utterances seem to make claims about what theavailable evidence shows, or about which scenarios can be ruled out onthe basis of the evidence. More formally, we can say that aproposition \(P\) isepistemically necessary for an agent Ajust in case the empirical evidence \(A\) possesses and idealreasoning (i.e., reasoning unrestricted by cognitive limitations) aresufficient to rule out \({\sim}P\). This notion of epistemic necessityis agent-relative: one and the same claim can be epistemicallynecessary for one agent, but not for another agent with less empiricalevidence. We obtain a notion of epistemic necessity of particularphilosophical interest by focusing on a limiting case, namely that ofa possible agent with no empirical evidence whatsoever.^[1] A proposition \(P\) is epistemically necessary for such an agent justin case ideal reasoning alone, unaided by empirical evidence, issufficient to rule out \({\sim}P\). A proposition that meets thiscondition can be calleda priori in at least one sense ofthis term, or we can call it simplyepistemically necessary(without relativization to an agent). Propositions that are notapriori are calleda posteriori.^[2]

It is an important and controversial question whether the necessarypropositions are all and only the epistemically necessary (apriori) ones, or whether the extensions of the two concepts cancome apart. One possible reason for thinking that the notions arecoextensive derives from a very natural picture of information andinquiry. On this picture, all information about the world isinformation about which of all possible worlds is realized (i.e.,about where in the space of all possible worlds the actual world islocated). My total information about the world can be identified withthe set of possible worlds that I cannot rule out on the basis of myempirical evidence and ideal reasoning. As I gather more and moreempirical evidence, I can progressively narrow down the range ofpossibilities. Suppose, for example, that I am ignorant of the currentweather conditions. The worlds compatible with my evidence includesome where the weather is good and others where it is bad. A look outof the window at the rain provides information about the matter. I cannow narrow down the set of possibilities by excluding all possibleworlds with fine weather. On this account, a proposition \(P\) isepistemically necessary for \(A\) just in case \(P\) is truein all possible worlds that cannot be ruled out on the basis of\(A\)’s empirical evidence and ideal reasoning. \(P\) isapriori just in case it is epistemically necessary for a possibleagent who has no empirical evidence. Since such an agent cannot ruleout any possible worlds, a proposition isa priori just incase it is true in all possible worlds. In other words, theapriori propositions are all and only the necessary propositions.^[3]

This approach is often combined with a certain account of semanticcontent. One of the main purposes of language is to transmitinformation about the world. Where \(P\) is any sentence used for thatpurpose (roughly speaking, a declarative sentence), it seems naturalto think of \(P\)’s content (the proposition expressed by it) asthe information that is semantically encoded in it. Combining thiswith the foregoing account of information, we can think of the contentof a sentence as a set of possible worlds (namely, the set containingjust those worlds of which the sentence is true) or, equivalently, asa function from worlds to truth-values.

This picture connects the modal, epistemic and semantic realms in asimple and elegant way, and various versions of it have informed thework of numerous contemporary philosophers (including David Lewis,Robert Stalnaker, David Chalmers, and Frank Jackson). However, theapproach has come under pressure from data to be considered in thenext section.

1.1 The Data

The idea that all and only thea priori truths are necessarywas thrown into serious doubt by the work of philosophers includingHilary Putnam (1972) and Saul Kripke (1980). Kripke distinguishesbetween two different kinds of singular terms,rigid andnon-rigid ones. A so-calledrigid designator is anexpression that singles out the same thing in all possible worlds.Kripke argues that ordinary proper names like ‘Al Gore’are rigid. We can use this name to describe how things actually are,e.g., by saying ‘Al Gore became vice president in 1993.’In such cases, the name picks out Al Gore. But we can equally use thename to describe how things stand in other possible worlds, e.g., bysaying, ‘If Bill Clinton had chosen a different running mate, AlGore would not have become vice president.’ In this case, we aretalking about a non-actualized possibility, and we use the name‘Al Gore’ to describe this possibility. Moreover, we usethe name to say something about how things stand with Al Gore in thatpossibility. In general, when we use the name to describe any possibleworld, we use it to talk about the same person, Al Gore. Otherexamples of rigid designators include indexical expressions like thefirst-person pronoun ‘I,’ or the expression‘now.’ When you use the term ‘I’ to describeany possible world, you are always picking out the same thing:yourself. Natural kind terms like ‘water’ and‘gold’ can also be regarded as rigid terms, as they singleout the same kinds in every possible world. Non-rigid singular term,by contrast, pick out different entities in different possiblescenarios. The paradigmatic examples of non-rigid terms aredescriptions that are satisfied by different objects in differentpossible worlds. For example, ‘the most annoying person in thehistory of the world’ may pick out Fred in the actual world,while picking out Cleopatra in some other possible worlds.

Singular terms can be introduced into the language with the help ofdescriptions. There are two ways in which that can be done. On the onehand, we can stipulate that the singular term is to besynonymous with the description, for example by laying downthat ‘the morning star’ is to mean the same as ‘thelast celestial body to be seen in the morning.’ When we use theexpression to describe another possible world, the new expression willsingle out whatever celestial body is the last one that can be seen inthe morning in that world. Since different things meet this conditionin different worlds, the expression is non-rigid. On the other hand,we may introduce a term with the stipulation that it is to be a rigiddesignator referring to whatever objectactually satisfiesthe description. For instance, we may lay it down that‘Phosphorus’ is to refer rigidly to the object that isactually the last celestial body visible in the morning. Since thatobject is Venus, the name will pick out Venus, not only when we use itto describe the actual world, but also when we (in the actual world)use it to describe other possible worlds, including worlds where Venusis not the last planet visible in the morning.When a description isused to introduce a singular term in the second way, it merely servesto fix the reference of the term, but is not synonymous with it.

Now consider a true identity statement that involves two rigiddesignators, such as

(1)

Mark Twain (if heexists) is Samuel Clemens.

Since ‘Mark Twain’ and ‘Samuel Clemens’ pickout the same entity in every possible world where they pick outanything, this identity statement is a necessary truth. (Note that thestatement is conditionalized on Mark Twain’s existence, whichmakes it possible to avoid the question whether (1) is true in worldswhere the two names pick out nothing.) But it is far from immediatelyobvious that (1) expresses something that can be knownapriori. At least on the face of it, we may think that someone whoknows her neighbor by the name of ‘Samuel Clemens,’ whohas read several stories by an author named ‘Mark Twain’and who fails to realize that her neighbor and the author areidentical may not know that which is expressed by (1). Moreover, itmay seem that her ignorance is irremediable by reasoning alone, thatshe requires empirical evidence to come to know that which is statedby (1).

Another type of apparent counterexample to the thesis that all andonly thea priori truths are necessary concerns sentenceslike

(2)

If gold exists, then it has atomic number 79.

It seems plausible that it is an essential property of gold to haveatomic number 79: gold could not have (existed but) failed to havethat property. (A substance in another possible world that fails tohave atomic number 79 simply isn’t gold, no matter how similarit may otherwise be to the gold of the actual world.) And yet it seemsclear that it can only be known empirically that gold has that atomicnumber. So, while (2) is a necessary truth, what it says cannot beknowna priori. For another illustration of this phenomenon,suppose that I point to the wooden desk in my office and say:

(3)

If this desk exists, it is made of wood.

It is arguably essential to this desk to be made of wood. A desk inanother possible world that isn’t wooden simply can’t bethis desk, no matter how similar it may otherwise be to mydesk. But it seems that we need empirical evidence to know that thedesk is made of wood. So, (3) is another apparent example of anecessarya posteriori truth.

Just as Kripke claims that some truths are necessary without beinga priori, he argues that a truth can bea prioriwithout being necessary. To use an example of Gareth Evans’s(1982), suppose that I introduce the term ‘Julius’ bystipulating that it is to refer rigidly to the person who is in factthe inventor of the zip (if such a person exists). Then it may appearthat I don’t need further empirical evidence to know that

(4)

If Julius exists, then Julius is the inventor of the zip.

But (4) does not seem to be a necessary truth. After all, Julius couldhave become a salesperson rather than an inventor.

According to Kripke, our initial surprise at the divergent extensionsof a prioricity and necessity should be mitigated on reflection. Aprioricity (epistemic necessity) is anepistemologicalnotion: it has to do with what can be known. That is not true of theconcept of necessity. (2) is necessary because the atomic number ofgold is an essential feature of it, and on the face of it, that hasnothing to do with what is known or believed by anyone. This kind ofnecessity is ametaphysical notion, and we may use the term‘metaphysical necessity’ to distinguish it more clearlyfrom epistemic necessity.

Kripke’s examples are not the only ones that could be appealedto in order to shed doubt on the coextensiveness of necessity and aprioricity. Some other problematic cases are listed below (Chalmers2002a; cp. Chalmers 2012, ch. 6).

1. Mathematical truths. It is common to hold that allmathematical truths are necessary. But on the face of it, there is noguarantee that all mathematical truths are knowablea priori(or knowable in any way at all). For example, either the continuumhypothesis or its negation is true, and whichever of these claims istrue is also necessary. But for all we know, there is no way for us toknow that that proposition is true.
2. Laws of nature. Some necessitarians about the naturallaws (seesection 2) believe that the laws hold in all metaphysically possible worlds. Butthey are nota priori truths.
3. Metaphysical principles. It is often believed that manymetaphysical theses are necessary if true, e.g., theses about thenature of properties (e.g., about whether they are universals, sets ortropes) or ontological principles like the principle of unrestrictedmereological composition (which says that for any things there issomething that is their sum). But it is not obvious that all truths ofthis kind area priori. (For discussion, see Chalmers 2012,§§6.4–6.5; Schaffer 2017.)
4. Principles linking the physical and the mental. Somephilosophers hold that all truths about the mental are metaphysicallynecessitated by the physical truths, but deny that it is possible toderive the mental truths from the physical ones bya priorireasoning (see Hill & McLaughlin 1999; Yablo 1999; Loar 1999; andChalmers 1999 for discussion). On that account, some of theconditionals that link physical and mental claims are metaphysicallynecessary but nota priori.

These examples are controversial. For any given mathematical claimwhose truth-value is unknown, one could hold that it is only ourcognitive limitations that have prevented us from establishing orrefuting the statement, and that the question could be decided byideal reasoning (so that the truth of the matter isapriori). Alternatively, it may be held that the truth-value ofthe mathematical statement is indeterminate. (Perhaps our practices donot completely determine the references of all the terms used in themathematical claim). The same two options are available in the case ofmetaphysical principles. Alternatively, one may argue that therelevant metaphysical theses are merely contingent (see, e.g., Cameron2007). Necessitarianism about the natural laws is highly controversialand may simply be denied. And in response to (iv), one may deny thatthe physical truths metaphysically necessitate the mental truths(Chalmers 1996), or one may hold that the mental truths can be derivedfrom the physical ones bya priori reasoning (Jackson1998).

Philosophers have paid more attention to the examples given by Kripkethan to other possible cases of the necessarya posteriori,and for that reason the discussion in the rest of this section willmostly focus on Kripke’s cases. Two strategies for explainingthese examples can be distinguished. Dualists about metaphysical andepistemic modality (dualists, for short) hold that the phenomenareflect a deep and fundamental distinction between two kinds ofmodality. Monists, by contrast, believe that all the data canultimately be explained by appeal to a single kind of modality. Theymay agree that there are cases in which a single sentence is, in somesense, both necessary anda posteriori, or both contingentanda priori. But they insist that there is no similardistinction at the level of worlds or propositions. Rather, thephenomenon arises because a single sentence can be associated with twodifferent propositions, one that is necessary and another that iscontingent.

1.2 Dualism

Dualists distinguish between two concepts of propositional necessity,metaphysically necessity andepistemic necessity.The two notions are not coextensive. At least some of the sentences inKripke’s examples express propositions that possess the one kindof necessity but not the other.^[4]

Once the existence of a distinctively metaphysical form ofpropositional necessity is accepted, it is natural to wonder whetherit is possible to say more about its nature. Kit Fine (1994) offers anaccount of it that appeals to the traditional distinction betweenthose properties of a thing that it possesses by its very nature andthose that it has merely accidentally. For example, it lies in thenature of water to be composed of hydrogen and oxygen—beingcomposed in this way is part of what it is to be water—but it ismerely accidental to water that we use it to brush our teeth. Aproposition is metaphysically necessary just in case it is true invirtue of the natures of things. (Also see Kment 2014, chs. 6–7,and Ditter 2020.) Other philosophers (Rayo 2013, §2.2.1, ch.5; Dorr 2016) have discussed the idea that metaphysical necessity canbe explained in terms of the idiom “To be \(F\) is to be\(G\)” (as in “To be water is to beH₂O”). Yet another account ties the metaphysicalnotion of necessity constitutively to causation and explanation (Kment2006a,b, 2014, 2015a; also see the exchange between Lange 2015 andKment 2015b).

Dualism requires us to dismantle the picture of inquiry, informationand content sketched in the introduction tosection 1. Note that it is natural for a dualist to distinguish the space ofmetaphysically possible worlds from the space ofepistemically possible worlds, i.e., from the space of(maximally specific) ways the world might be that cannot be ruled outon the basis of ideal reasoning alone, without empirical evidence(Soames 2005, 2011). The range of epistemically possible worldsoutstrips the range of metaphysically possible worlds: there are someways the world couldn’t have been, but which cannot be ruled byideal reasoning alone. For example, there is no metaphysicallypossible world where gold has atomic number 78. But prior to carryingout the right chemical investigations, we don’t have enoughevidence to exclude all scenarios where gold has that atomic number,so some worlds where gold has atomic number 78 are epistemicallypossible. Empirical evidence is not used only to rule out(metaphysical) possibilities, but is sometimes needed to rule outmetaphysical impossibilities that are epistemically possible.Consequently, we cannot in general identify information with sets ofmetaphysically possible worlds, since we need to distinguish betweenstates of information in which the available evidence rules out thesame metaphysically possible worlds but different metaphysicallyimpossible worlds. By the same token, the information encoded in asentence cannot in general be identified with a set of metaphysicallypossible worlds, since two sentences may be true in all the samemetaphysically possible worlds, but not in all the same epistemicallypossible worlds. If we wanted to identify information and sententialcontents with sets of worlds, it would seem more promising to use setsof epistemically possible worlds. But the dualist may instead rejectthe possible-worlds account of information and propositions altogether(see, e.g., Soames 1987, 2003, 395f.).

1.3 Monism

As mentioned above, monists explain the data described by Kripke byholding that the sentences that figure in Kripke’s examples areassociated with two different propositions, one that is necessary andanother that is contingent. This view comes in two main versions.According to the first version, both propositions are semanticallyexpressed by the sentence. Proponents of this account need toformulate a semantic theory that explains how that is possible.According to the second version, only one of these propositions issemantically expressed by the sentence, while the other is theproposition that is communicated by a typical assertoric use of thesentence. A philosopher holding this view needs to explain thepragmatic mechanism by which an utterance of the sentence comes tocommunicate the second proposition.

The first version of monism has been developed by David Chalmers andFrank Jackson (Chalmers 1996, 1999, 2002a,b, 2004, 2006a,b; Chalmersand Jackson 2001; Jackson 1998, 2004, 2011), who build on earlier workby David Kaplan (1989a,b), Gareth Evans (1979) and Martin Davies andLloyd Humberstone (1980), and others. On Chalmers’s andJackson’s view, what explains the phenomena uncovered by Kripkeis not a difference between two spaces of possible worlds. There isonly a single space of possible worlds: the metaphysically possibleworlds—the ways the world could have been—just are theepistemically possible worlds: the ways the world might be for all wecan know independently of empirical evidence. What explains the datais a difference between two different ways in which sentences can beused todescribe the worlds in that space, i.e., between twodifferent notions of a sentence’s being true in a world. Thedistinction can be illustrated by appeal to our example of the propername ‘Phosphorus.’ Suppose that we have just introducedthis name by using the description ‘the last celestial bodyvisible in the morning’ to fix its reference. Consider apossible world \(w\) where the description singles out, not Venus (asin our world), but Saturn. Assume further that in \(w\) (as in theactual world), Venus is the second planet from the sun, but Saturn isnot. Consider:

(5)

Phosphorus is the second planet from the sun.

Is (5) true in \(w\)? There are two different ways of understandingthis question. On the one hand, it could mean something roughly likethis: if \(w\) actually obtains (contrary to what astronomers tellus), is Phosphorus the second planet from the sun? The answer to thatquestion is surely ‘no.’ ‘Phosphorus’ refersto whatever is actually the last celestial body visible in themorning, and on the assumption that \(w\) actually obtains, thatobject is Saturn, and is therefore not the second planet. As Chalmerswould put it, (5) is nottrue at w considered as actual.^[5] But we can also interpret the question differently: if \(w\)had obtained, thenwould Phosphorus have been thesecond planet? In considering that question, we are not hypotheticallyassuming that the object that actually satisfies the reference-fixingdescription is Saturn. Instead, we can draw freely on our belief thatthe object actually fitting the description is Venus, so that the namepicks out Venus in all possible worlds. Since Venus is the secondplanet in \(w\), it is true to say: if \(w\) had obtained, thenPhosphorus would have been the second planet. In Chalmers’sterminology, (5) istrue at w considered ascounterfactual.

The distinction between the two concepts of truth in a world can beexplained within a theoretical framework known astwo-dimensionalsemantics, which assigns to a sentence like (5) an intension thatis a function, not from worlds to truth-values, but frompairs of worlds to truth-values. The intension of (5) is thefunction that assigns the true to a pair of worlds \(\langle u; w\rangle\) just in case the object that is the last celestial bodyvisible in the morning in \(u\) is the second planet in \(w\).^[6] This account makes it easy to define the two notions of truth in aworld. A sentence \(P\) is true in \(w\) considered as actual just incase the two-dimensional function assigns the true to \(\langle w;w\rangle\). \(P\) is true in \(w\) considered as counterfactual justin case, where \(u\) is the actual world, the two-dimensional functionassigns the true to \(\langle u; w\rangle\). Note that thetwo-dimensional intension of (5) determines whether (5) is true at aworld \(w\) considered as actual. But it does not in general determinewhether (5) is true at \(w\) considered as counterfactual. That alsodepends on which world is actual. Knowledge of a sentence’stwo-dimensional intension is therefore not in general sufficient toknow whether the sentence is true at \(w\) considered ascounterfactual. Further empirical evidence may be required.

When combined with the conception of a sentence’s content as theset of worlds where it is true, the distinction between the twoconcepts of truth in a world yields a distinction between twodifferent propositions expressed by a sentence. The first of thesepropositions is the function that assigns the true to a world \(w\)just in case the sentence is true in \(w\) considered as actual, whilethe second proposition is the function that assigns the true to aworld \(w\) just in case the sentence is true in \(w\) considered ascounterfactual. Jackson calls the former proposition thesentence’s ‘A-intension’ (for ‘actual’)and the latter its ‘C-intension’ (for‘counterfactual’), while Chalmers calls the former the‘primary intension’ and the latter the ‘secondaryintension.’ The distinction between the two propositionsexpressed by a sentence yields a distinction between two notions ofsentential necessity: primary necessity, which applies to sentenceswith necessary primary intensions, and secondary necessity, whichapplies to sentences with necessary secondary intensions. If asentence has primary necessity, then that fact, andafortiori the fact that the sentence is true, can be read off itstwo-dimensional intension. Therefore, if we know the two-dimensionalintension, then that is enough to know that the sentence is true. Nofurther empirical evidence is required. That motivates the thoughtthat the notion of primary necessity captures the idea of a prioricityor epistemic necessity. The notion of secondary necessity, on theother hand, may be taken to capture the Kripkean idea of metaphysicalnecessity.

This account makes it straightforward to explain cases ofaposteriori necessity: they are simply cases of sentences whosesecondary intensions are necessary, but whose primary intensions arecontingent. Suppose that ‘Hesperus’ and‘Phosphorus’ were introduced, respectively, by thereference-fixing descriptions ‘the first celestial body visiblein the evening (if it exists)’ and ‘the last celestialbody visible in the morning (if it exists).’ Since the twodescriptions single out the same object in the actual world, thesentence ‘If Hesperus exists, then Hesperus is Phosphorus’is true in all worlds considered as counterfactual, and therefore hasa necessary secondary intension. However, in some non-actual worlds,the two descriptions single out different objects. The sentence isfalse in such a world considered as actual. The primary intension ofthe sentence is therefore contingent.

An analogous account can be given of Kripke’s examples of thecontingenta priori: these concern sentences whose primaryintensions are necessary and whose secondary intensions arecontingent. Assume again that the reference of ‘Julius’ isfixed by the description ‘the inventor of the zip (if such aperson exists).’ Then in every world considered as actual, thename singles out the person who is the inventor of the zip in thatworld (if there is such a person) or nothing (if no such person existsin the world). The primary intension of (4) is necessary. However,when we evaluate (4) in a world \(w\) considered as counterfactual,‘Julius’ picks out the individual who is theactual inventor of the zip (provided that there actually issuch an individual and that he or she exists in \(w)\). And sincethere are possible worlds where that individual exists but is not theinventor of the zip, the secondary intension of (4) is contingent.

Chalmers (2002a, 2010) and Jackson (1998) have tried to support theirmodal monism by arguing that it is gratuitous to postulate two formsof modality, given that all the phenomena pointed out by Kripke can beaccommodated by appeal to a single kind of modality. Dualists mayreply that the greater simplicity in the view of modality has beenachieved only by adding complexity to the semantic theory. Thatresponse could be answered by arguing that two-dimensional semanticscan be motivated by independent considerations. That, of course, iscontroversial, as is the general viability of two-dimensionalsemantics (see the entrytwo-dimensional semantics for detailed discussion).

In addition, it is not obvious that the view of Chalmers and Jacksoncan satisfactorily explain all the phenomena discussed insection 1.1. Some commentators have denied that it can give a viable generalaccount of Kripkean examples (see, e.g., Soames 2005; Vaidya 2008;Roca-Royes 2011). In any case, it is clear that the view can onlyexplain how necessity and epistemic necessity can come apart forsentences whose primary and secondary intensions differ. That may betrue of the cases considered by Kripke, but it seems doubtful for theother examples considered insection 1.1 (mathematical and metaphysical truths, laws, and principlesconnecting the physical to the mental). In response, Chalmers hasargued that none of the latter cases are genuine examples of thenecessarya posteriori (1999, 2002a).

Thesecond version of monism allows us to accommodate thephenomena considered insection 1.1 while staying much closer to the picture sketched in the introductiontosection 1. On this view, the data can be explained by appeal to a single spaceof possible worlds and a single notion of truth in a world. Theproposition semantically expressed by a sentence containing a propername or natural-kind term is a function from individual worlds totruth-values. The proposition expressed by ‘Phosphorusexists,’ e.g., is a function that assigns the true to thoseworlds where Venus exists and the false to the other worlds. (If thereference of ‘Phosphorus’ was determined by areference-fixing description together with the facts about whichentity meets the description, then that fact itself is not asemantic fact, but ametasemantic one, i.e., it doesnot concern the question of what the meaning of the word is, but thequestion of how the meaning of the word is determined.) What explainsthe impression that a sentence like (1) expresses anaposteriori claim is the fact that the proposition asserted by atypical utterance of the sentence is not the one that is semanticallyexpressed by it, but a different proposition that is contingent andcan only be known empirically.

Robert Stalnaker (1978, 2001) has given a detailed account of thepragmatic mechanism by which a contingent proposition comes to beasserted by the utterance of a sentence that semantically expresses anecessary proposition. On his account, linguistic communicationevolves in a context characterized by background assumptions that areshared between the participants. These assumptions can be representedby the set of worlds at which they are jointly true, which Stalnakercalls the ‘context set.’ The point of assertion is to addthe proposition asserted to the set of background assumptions andthereby eliminate worlds where it is not true from the context set. Toachieve this, every assertion needs to conform to the rule that theproposition asserted is false in some of the worlds that were in thecontext set before the utterance (otherwise there are no worlds toeliminate) and true in others (since the audience cannot eliminate allworlds from the context set). Now consider a context where the sharedbackground assumptions include the proposition that the references of‘\(A\)’ and ‘\(B\)’ were fixed by certaindescriptions but leave open whether the two descriptions single outthe same object. Suppose that someone says ‘\(A\) is\(B\).’ In every world in the context set, the sentencesemantically expresses either a necessary truth (if the twodescriptions single out the same object in the world) or a necessaryfalsehood (if they don’t). If the proposition that the speakerintends to assert were the one that is semantically expressed by thesentence, the aforementioned rule would be violated.^[7] To avoid attributing this rule violation to the speaker, the audiencewill construe the utterance as expressing a different proposition, andthe most natural candidate is the proposition that the sentenceuttered semantically expresses a true proposition. (Stalnaker callsthis the ‘diagonal proposition.’) By exploiting thismechanism of reinterpretation, a speaker can use the sentence toexpress the diagonal proposition. This proposition is true in justthose worlds in the context set where the two descriptions single outthe same object. It is clearly a contingent proposition, and empiricalevidence is required to know it. Stalnaker suggests an analogousexplanation of Kripke’s proposed cases of contingentapriori truth (1978, 83f.).

Stalnaker’s account of the necessarya posteriorirequires that the proposition semantically expressed by the sentenceand the proposition that the sentence semantically expresses a truthhold in different worlds in the context set. And that seems to requirethat the assumptions shared between the participants of theconversation don’t determine what proposition is semanticallyexpressed by the sentence. It has been argued that that assumption isimplausible in some cases of Kripkeana posteriorinecessities (Soames 2005, 96–105). Suppose that I point to thedesk in my office in broad daylight and say ‘That desk (if itexists) is made of wood.’ Unless the context is highly unusual,the shared assumptions, so the argument goes, uniquely determine whatproposition is expressed by the sentence.

2. Metaphysical and Nomic Modality

It often seems very natural to use modal terminology when talkingabout the laws of nature. We are inclined to say that nothingcan move faster than light to express the fact that the lawsrule out superluminal motion, and to state Newton’s First Law bysaying that an objectcannot depart from uniform rectilinearmotion unless acted on by an external force. This motivates thethought that there is a form of necessity associated with the natural laws.^[8] It is controversial, however, whether that form of necessity issimply metaphysical necessity, or another kind of necessity. Theformer view is taken bynecessitarians (Swoyer 1982;Shoemaker 1980, 1998; Tweedale 1984; Fales 1993; Ellis 2001; Bird2005), who believe that the laws (or the laws conditionalized on theexistence of the properties mentioned in them) are metaphysicallynecessary.Contingentists deny that, but many contingentistshold that there is a kind of necessity distinct from metaphysicalnecessity that is characteristic of the laws (e.g., Fine 2002), andwhich may be callednatural ornomic necessity. Itis often assumed that nomic necessity is a weaker form of necessitythan metaphysical necessity: it attaches to the laws and to all truthsthat are metaphysically necessitated by them, so that anything that ismetaphysically necessary is also nomically necessary, but not viceversa.

Necessitarians have given several arguments for their position. Hereare two.

The argument from causal essentialism (e.g., Shoemaker 1980,1998). Some philosophers believe that the causal powers that aproperty confers on its instances are essential to it. Assuming thatcausal laws describe the causal powers associated with properties, itfollows that these laws (or versions of them that are conditionalizedon the existence of the relevant properties) are necessary truths.This is, in the first instance, only an argument for the necessity ofcausal laws, but perhaps it can be argued that all laws ofnature are of this kind. Of course, even if this assumption isgranted, the argument is only as strong as the premise that propertieshave their associated causal powers essentially. To support this view,Sydney Shoemaker (1980) has given a battery of epistemologicalarguments. He points out that our knowledge of the properties that anobject possesses can only rest on their effects on us, and musttherefore be grounded in the causal powers associated with theseproperties. But, he goes on to argue that, without a necessaryconnection between the properties and the associated causal powers, anobject’s effects on us could not serve as a source of all theknowledge about an object’s properties that we take ourselves topossess.

The argument from counterfactual robustness (Swoyer 1982;Fales 1990, 1993; also see Lange 2004 for discussion). Natural lawsare often believed to differ from accidental generalizations by theircounterfactual robustness (counterfactual-supporting power).Setting aside some minor complications (see Lewis 1979a), we candescribe this feature as follows: if it is a law that all \(F\)sare \(G\), then this generalization would still have been true ifthere had been more \(F\)s than there actually are, or if some \(F\)shad found themselves in conditions different from the ones thatactually obtain. For example, it would still have been true thatnothing moves faster than light if there had been more objects thanthere actually are, or if some bodies had been moving in a differentdirection. Contrast this withNo emerald has ever decorated aroyal crown. That may be true, but it is not very robust. Itwould have been false if some rulers of the past had made differentfashion choices. Some necessitarians have argued that contingentismabout the laws cannot provide a plausible explanation of the specialcounterfactual robustness of the laws. Note that a counterfactual“if it had been the case that \(P\), then it would have been thecase that \(Q\)” is usually taken to be true if \(Q\) is true inthose metaphysically possible \(P\)-worlds that are closest toactuality. On this view, the special counterfactual robustness of thelawAll \(F\)s are \(G\) amounts, roughly speaking, to this:of all the metaphysically possible worlds that contain some additional\(F\)s, or where some actual \(F\)s are in somewhat differentcircumstances, the ones where the actual law holds are closer than therest. If the laws hold in some metaphysically possible worlds but notin others, then the reason why the former are closer than the lattermust be that the rules we are using for deciding which worlds count asthe closest say so. But which such rules we use is a matter ofconvention. The counterfactual-supporting power of the laws does notseem to be a purely conventional matter, however. Necessitarianism,the argument continues, offers a better explanation: the laws hold inthe closest possible worlds simply because they hold inallmetaphysically possible worlds. Conventions don’t come into it.The contingentist may reply that, even though the counterfactualrobustness of the laws is grounded in a convention, that convention isnotarbitrary. Instead, it may have its rationale incertain features of the laws that make them different from accidentalgeneralizations in an objectively important way (Sidelle 2002),e.g., the fact that laws relate to pervasive and conspicuous patternsin the history of the world, or the fact that the truth of a law(unlike the truth of an accidental generalization) is explanatorilyindependent of matters of particular fact (Kment 2015b,11–12).

Contingentism has often been defended by pointing out that the laws ofnature can be known onlya posteriori, and that theirnegations are conceivable (see Sidelle 2002). Necessitarians may replyto the first point that Kripke’s work has given us reasons forthinking thata posteriori truths can be metaphysicallynecessary (seesection 1.1). In response to the second point, they may grant that the negation ofa law is conceivable, but deny that conceivability is a good guide topossibility (see the entryepistemology of modality). Alternatively, they may deny that we can really conceive of asituation in which, say, bodies violate the law of gravitation. Whatwe can conceive of is a situation in which objects move in ways thatappear to violate the law. But that situation cannot becorrectly be described as involving objects withmass.Rather, the objects in the imagined situation have a differentproperty that is very similar to mass (call it ‘schmass’)but which is governed by slightly different laws. Contingentists mayreply that the non-existence of schmass (or the non-existence ofobjects that move in the way imagined) is itself a law, so that wehave, after all, conceived of a situation where one of the actual lawsfails (see Fine 2002).

3. The Structure of the Modal Realm

The concepts of metaphysical, epistemic, and nomic necessity are onlya few of the modal notions that figure in our thought and discourse(as should be clear from the long list of uses of modal terms given inthe introduction to this entry). We also speak of

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Practical necessity
Logical necessity
Biological necessity
Medical necessity
Moral necessity
Legal necessity

and of a whole lot more. One would expect that some of these modalconcepts can be defined in terms of others. But how can that be done?And is it possible to single out a small number of fundamental notionsof necessity in terms of which all the others can be defined?

It may be helpful in approaching these questions to distinguishbetween two salient ways in which one modal property can be defined interms of another (Fine 2002, 254f.).

1. Restriction. To say that property \(N\) can be definedfrom kind of necessity \(N^*\) byrestriction is to say thata proposition’s having \(N\) can be defined as the combinationof two things: (i) the proposition’s having \(N^*\), and (ii)its meeting certain additional conditions.
2. Relativization / quantifier restriction. To say that aproperty \(N\) can be defined from a kind of necessity \(N^*\) byrelativization to a class of propositions \(S\) is to saythat a proposition’s having \(N\) can be defined as its being\(N^*\)-necessitated by \(S\). A closely related way in which a modalproperty can be defined in terms of another is byquantifierrestriction. Suppose that \(P^*\) is a kind of possibility thatis the dual of \(N^*\) (in the sense that it is \(P^*\)-possible that\(p\) just in case it’s not \(N^*\)-necessary that not-\(p)\),and that we have at our disposal the notion of a \(P^*\)-possibleworld (a world that could \(P^*\)-possibly have been actualized). Tosay that the property \(N\) can be defined from \(N^*\) by quantifierrestriction is to say that that a proposition’s having \(N\) canbe defined as its being true in all \(P^*\)-possible worlds that meeta certain condition \(C\). (This is only the simplest way of defininga modal property from a kind of necessity by quantifier restriction.Much more sophisticated methods have been proposed. See, e.g., Kratzer1977, 1991.) Given reasonable assumptions, every definition byrelativization corresponds to a definition by quantifier restriction,and vice versa.^[9]

Restriction allows us to define narrower modal properties from broaderones. For example, it seems natural to define a proposition’sbeing mathematically necessary as its being both metaphysicallynecessary and a mathematical truth (Fine 2002, 255), or as its beingmetaphysically necessarybecause it is a mathematicaltruth. Maybe a proposition’s beinglogically necessary can similarly be defined as its beingmetaphysically necessary because it is true in virtue of the naturesof the logical constants (Fine 1994, 9–10; 2005, 237).(For a very different account that does not define logical necessityfrom metaphysical necessity by restriction, see Bacon 2018.)

Relativization and quantifier restriction allow us to define broadermodal properties in terms of narrower ones. For example, it may beheld that biological necessity can be defined as the property of beingmetaphysically (or perhaps nomically) necessitated by the basicprinciples of biology.

A modal property \(N\) is calledalethic just in case theclaim that a proposition has \(N\) entails that the proposition istrue. Metaphysical, epistemic and nomic necessity are all alethic. Bycontrast, moral and legal necessity are not. It is both morally andlegally necessary (i.e., it is required both by morality and by thelaw) that no murders are committed, even though murders are in factbeing committed. A modal property defined by restriction from analethic kind of necessity must itself be alethic. By contrast,relativization allows us to define non-alethic modal properties fromalethic ones, by relativizing to a class of propositions that containssome falsehoods. Similarly, we can define a non-alethic modal propertyfrom an alethic one by restricting the quantifier over possible worldsto some class that does not include the actual world. For example,legal necessity can perhaps be defined from metaphysical necessity byrestricting the quantifier to worlds where everybody conforms to theactual laws.

The properties listed in (6) can very naturally be called ‘kindsof necessity,’ and in some contexts they are the propertiesexpressed by necessity operators like ‘must’ and‘could not have been otherwise.’ But that is not true ofevery property that can be defined from some kind of necessity byrelativization or restriction. For example, we can define a propertyby relativizing metaphysical necessity to the class of truths statedin a certain book, but it would not be natural at all to call thisproperty a kind of necessity. It is not plausible that there is aspecial form of necessity that attaches to all and only thepropositions necessitated by the truths in the book. Similarly, theproperty defined by restricting metaphysical necessity to the truthsabout cheddar cheese cannot naturally be called a kind of necessity.There is no form of necessity that applies to just those necessarypropositions that deal with cheddar and to none of the others. It is agood question what distinguishes those properties defined byrelativization and restriction that we are willing to count as formsof necessity from the rest. Perhaps the most natural answer is thatthe distinction is dictated by our interests and concerns, and doesnot reflect a deep metaphysical difference.

A more pressing question is whether some of the forms of necessitydiscussed in sections1 and2 can be defined in terms of the others by relativization orrestriction. Consider epistemic and metaphysical necessity first, andsuppose for the sake of the argument that dualism is true and the twoproperties are indeed different forms of necessity. Can one of them bedefined in terms of the other by one of the aforementioned methods?Not if there are both necessarya posteriori and contingenta priori propositions, since relativization and restrictiononly allow us to define one property in terms of another if theextension of one is a subclass of that of the other. However, theexistence of contingenta priori truths is more controversialthan that of necessarya posteriori propositions, and someonetrying to define epistemic necessity in terms of metaphysicalnecessity or vice versa may repudiate the contingenta prioriand hold that the extension of epistemic necessity is included in thatof metaphysical necessity. Then such a philosopher could try (a) todefine metaphysical necessity from epistemic necessity byrelativization to some suitable class, or (b) to define epistemicnecessity from metaphysical necessity by restriction.

Such a definition may get the extension of the definiendum right. Buta definition may be intended to do much more than that: it may bemeant to tell us what it \(is\) for something to fall under theconcept to be defined. Suppose that someone tried to define theproperty of being an equiangular triangle as that of being a trianglewhose sides are of equal length. While this is extensionally correct,it does not give us the right account of what it is for something tobe an equiangular triangle (what it is for something to have thatproperty has something to do with the sizes of its angles, not withthe lengths of its sides). It could be argued that definitions of type(a) and (b) face similar difficulties. For example, a definition ofkind (a) entails that a proposition’s being metaphysicallynecessary consists in its being epistemically necessitated by acertain class of propositions. But that would make metaphysicalnecessity an epistemic property, and dualists typically want to resistthat idea. Similarly for definitions of type (b). Whether something isepistemically necessary (in the sense of beinga priori)seems to be a purely epistemic matter.A priori propositionsmay also be metaphysically necessary, but their metaphysical necessityisn’t part of whatmakes thema priori, andtherefore shouldn’t be mentioned in a definition of aprioricity.

If this argument is correct, then it is impossible to define epistemicmodal properties in terms of non-epistemic ones, or vice versa. Butwhat about metaphysical and nomic necessity? Suppose for the sake ofthe argument that there is such a thing as nomic necessity (a form ofnecessity associated with the laws of nature) but that contingentismabout the natural laws is true, so that nomic necessity is indeeddistinct from metaphysical necessity. Can we define one of theseproperties in terms of the other? The most natural way of doing thiswould be to say that

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Nomic necessity can be defined as the property of beingmetaphysically necessitated by the laws of nature.

Such a definition may be extensionally accurate, and many philosopherswould not hesitate to endorse it. But others have doubted that itcaptures what it \(is\) for a proposition to be nomically necessary(Fine 2002). Nomic necessity is a special modal status enjoyed by alland only the propositions that are metaphysically necessitated by thenatural laws. Now, if \(P\) is metaphysically necessitated by the lawswithout itself being a law, then it may seem plausible to say, in somesense, that \(P\) has that special modal statusbecause \(P\)is metaphysically necessitated by the laws. But the reason why beingmetaphysically necessitated by the laws confers that special modalstatus on \(P\) is presumably that the laws themselves have that modalstatus and that this modal status gets transmitted across metaphysicalnecessitation. But if we now ask what makes it so that the lawsthemselves have that special modal status, (7) does not seem to giveus the correct answer: the special necessity of the laws doesn’tconsist in the fact that they are metaphysically necessitated by thelaws. Hence, (7) cannot be a correct general account of whatconstitutes that special modal status.

It is open to debate which kinds of necessity are fundamental, in thesense that all others can be defined in terms of them, while they arenot themselves definable in terms of others. The monist viewconsidered insection 1.3, when combined with (7), may inspire the hope that we can make do witha single fundamental kind of necessity. Others have argued that thereare several kinds of necessity that are not mutually reducible. Forexample, Fine (2002) suggests (in a discussion that sets asideepistemic modality) that there are three fundamental kinds ofnecessity, which he calls ‘metaphysical,’‘nomic’ and ‘normative’ necessity. (Forfurther discussion of normative necessity, see Rosen 2020.)

The reduction of the various kinds of necessity to a small number offundamental ones would be an important step towards the goal of aunified account of modality.^[10] But those who believe that there are several different fundamentalkinds of necessity need to address another question: What is thecommon feature of these fundamental kinds of necessity that makes themall kinds of necessity? Why do they count as kinds of necessity, whileother properties don’t?

One strategy for answering this question, which centers onnon-epistemic forms of necessity, starts from a certain conception ofwhat (non-epistemic) necessity consists in: for a proposition to benecessary is for its truth to be, in a certain sense, particularlysecure, unshakable, and unconditional in a wholly objective way.A necessary truth could not easily have been false (it could lesseasily have been false than a contingent truth). We may call thisfeature of a proposition ‘modal force.’ It is natural toapply this conception to metaphysical and nomic necessity. Each ofthese properties may be held to consist in having a certain grade ofmodal force, though if contingentism is true, the degree of modalforce required for nomic necessity is lower than that required formetaphysical necessity. We could then say that a property is one ofthe fundamental forms of necessity just in case a proposition\(P\)’s possessing that property consists entirely in\(P\)’s having a specific grade of modal force. Other kinds ofnecessity, like those listed in (6) can be defined from thefundamental ones by relativization or restriction. Having theseproperties doesnot consist simply in having a specific gradeof modal force (and these properties therefore aren’t among thefundamental kinds of necessity). For example, if a property is definedby relativizing metaphysical necessity to a class of propositions\(S\), then the fact that a proposition \(P\) has that propertyconsists in the fact thatthe connection between \(S\) and\(P\) has a certain grade of modal force. But that is not the samething asP itself having a certain grade of modal force.Similarly, if a property is defined from, say, metaphysical necessityby restriction, then having that property does not consistmerely in possessing such-and-such a grade of modal force,but in the conjunction of that feature with some other property.

This approach evidently leaves the question of how to understand theidea of modal force (of a proposition’s truth being veryunshakable). Some authors have attempted to explain this notion incounterfactual terms (see Lewis 1973a, §2.5; Lewis 1973b,§2.1; McFetridge 1990, 150ff.; Lange 1999, 2004, 2005; Williamson2005, 2008; Hill 2006; Kment 2006a; cp. Jackson 1998, Chalmers 2002a):the necessary truths are distinguished from the contingent ones by thefact that they are not only true as things actually are, but that theywould still have been true if things had been different in variousways. To capture this idea more precisely, Lange (2005) introduces theconcept of ‘stability’: a deductively closed set \(S\) oftruths is stable just in case, for any claim \(P\) in \(S\) and anyclaim \(Q\) consistent with \(S\), it is true in any context to saythat it would still have been the case that \(P\) if it had been thecase that \(Q\). The different forms of necessity have in common thattheir extensions are stable sets.

Kment (2006a, 2014, chs. 1–2) argues that modal force, and hencenecessity and possibility, come in many degrees (cp. Williamson 2016).We often talk about such degrees of possibility when we say thingslike ‘Team \(A\) could more easily have won than Team\(B\),’ ‘Team \(A\) could easily have won’ or‘Team \(A\) almost won.’ The first utterance states that\(A\)’s winning had a greater degree of possibility than\(B\)’s winning, while the second and third simply ascribe ahigh degree of possibility to \(A\)’s winning. Aproposition’s degree of possibility is the higher the less of adeparture from actuality is required for it to be true. Suppose, e.g.,that Team \(A\) would have won if one of their players had stood justan inch further to the left at a crucial moment during the game. Thenwe can truly say that the team could easily have won. More formally,\(P\)’s degree of possibility is the higher the closer theclosest \(P\)-worlds are to actuality (also see Lewis 1973a,§2.5; Lewis 1973b, §2.1; Kratzer 1991). Similarly, atruth’s degree of necessity is measured by the distance fromactuality to the closest worlds where it is false. What metaphysicalnecessity, nomic necessity and the other grades of necessity have incommon is that each of them is the property of having a degree ofpossibility that is above a certain threshold. What distinguishes themis a difference in their associated thresholds.

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