## 1. Epistemic and Metaphysical Modality



There are many ways the world could have been. You could have gotten
up later today. Your parents could have failed to meet, so that you
were never born. Life could never have developed on earth. The history
of the universe could even have been completely different from the
beginning. And many philosophers believe that the laws of nature could
have been different as well (although that has been denied, as
discussed in
 section 2).
 Maximally specific ways the world could have been are commonly called
'possible worlds.' The apparatus of possible worlds allows
us to introduce a set of modal notions: a proposition is necessary
just in case it is true in all possible worlds, a proposition is
possible just in case it is true in some possible worlds, and it is
contingent just in case it is true in some but not all possible
worlds. A *sentence* is necessary (possible, contingent) just
in case it expresses a necessary (possible, contingent)
proposition.



The modal notions considered in the last paragraph are not obviously
epistemological. On the face it, we are not reporting a fact about
what is or can be known or believed by anyone when we say that life
could have failed to develop. But there is also a family of modal
concepts that are clearly epistemological. These are the notions we
employ when we say things like 'Fred *must* have stolen
the book (the evidence shows conclusively that he did it),' or
'Mary *cannot* be in London (she would have called
me).' These modal utterances seem to make claims about what the
available evidence shows, or about which scenarios can be ruled out on
the basis of the evidence. More formally, we can say that a
proposition \(P\) is *epistemically necessary for an agent A*
just in case the empirical evidence \(A\) possesses and ideal
reasoning (i.e., reasoning unrestricted by cognitive limitations) are
sufficient to rule out \({\sim}P\). This notion of epistemic necessity
is agent-relative: one and the same claim can be epistemically
necessary for one agent, but not for another agent with less empirical
evidence. We obtain a notion of epistemic necessity of particular
philosophical interest by focusing on a limiting case, namely that of
a possible agent with no empirical evidence
 whatsoever.[1]
 A proposition \(P\) is epistemically necessary for such an agent just
in case ideal reasoning alone, unaided by empirical evidence, is
sufficient to rule out \({\sim}P\). A proposition that meets this
condition can be called *a priori* in at least one sense of
this term, or we can call it simply *epistemically necessary*
(without relativization to an agent). Propositions that are not *a
priori* are called *a
 posteriori*.[2]



It is an important and controversial question whether the necessary
propositions are all and only the epistemically necessary (*a
priori*) ones, or whether the extensions of the two concepts can
come apart. One possible reason for thinking that the notions are
coextensive derives from a very natural picture of information and
inquiry. On this picture, all information about the world is
information about which of all possible worlds is realized (i.e.,
about where in the space of all possible worlds the actual world is
located). My total information about the world can be identified with
the set of possible worlds that I cannot rule out on the basis of my
empirical evidence and ideal reasoning. As I gather more and more
empirical evidence, I can progressively narrow down the range of
possibilities. Suppose, for example, that I am ignorant of the current
weather conditions. The worlds compatible with my evidence include
some where the weather is good and others where it is bad. A look out
of the window at the rain provides information about the matter. I can
now narrow down the set of possibilities by excluding all possible
worlds with fine weather. On this account, a proposition \(P\) is
*epistemically necessary for* \(A\) just in case \(P\) is true
in all possible worlds that cannot be ruled out on the basis of
\(A\)'s empirical evidence and ideal reasoning. \(P\) is *a
priori* just in case it is epistemically necessary for a possible
agent who has no empirical evidence. Since such an agent cannot rule
out any possible worlds, a proposition is *a priori* just in
case it is true in all possible worlds. In other words, the *a
priori* propositions are all and only the necessary
 propositions.[3]



This approach is often combined with a certain account of semantic
content. One of the main purposes of language is to transmit
information about the world. Where \(P\) is any sentence used for that
purpose (roughly speaking, a declarative sentence), it seems natural
to think of \(P\)'s content (the proposition expressed by it) as
the information that is semantically encoded in it. Combining this
with the foregoing account of information, we can think of the content
of a sentence as a set of possible worlds (namely, the set containing
just those worlds of which the sentence is true) or, equivalently, as
a function from worlds to truth-values.



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


### 1.1 The Data



The idea that all and only the *a priori* truths are necessary
was thrown into serious doubt by the work of philosophers including
Hilary Putnam (1972) and Saul Kripke (1980). Kripke distinguishes
between two different kinds of singular terms, *rigid* and
*non-rigid* ones. A so-called *rigid designator* is an
expression 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 the
name to describe how things stand in other possible worlds, e.g., by
saying, 'If Bill Clinton had chosen a different running mate, Al
Gore would not have become vice president.' In this case, we are
talking about a non-actualized possibility, and we use the name
'Al Gore' to describe this possibility. Moreover, we use
the name to say something about how things stand with Al Gore in that
possibility. In general, when we use the name to describe any possible
world, we use it to talk about the same person, Al Gore. Other
examples of rigid designators include indexical expressions like the
first-person pronoun 'I,' or the expression
'now.' When you use the term 'I' to describe
any 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 single
out the same kinds in every possible world. Non-rigid singular term,
by contrast, pick out different entities in different possible
scenarios. The paradigmatic examples of non-rigid terms are
descriptions that are satisfied by different objects in different
possible worlds. For example, 'the most annoying person in the
history 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 of
descriptions. There are two ways in which that can be done. On the one
hand, we can stipulate that the singular term is to be
*synonymous* with the description, for example by laying down
that 'the morning star' is to mean the same as 'the
last celestial body to be seen in the morning.' When we use the
expression to describe another possible world, the new expression will
single out whatever celestial body is the last one that can be seen in
the morning in that world. Since different things meet this condition
in 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 rigid
designator referring to whatever object *actually* satisfies
the description. For instance, we may lay it down that
'Phosphorus' is to refer rigidly to the object that is
actually the last celestial body visible in the morning. Since that
object is Venus, the name will pick out Venus, not only when we use it
to describe the actual world, but also when we (in the actual world)
use it to describe other possible worlds, including worlds where Venus
is not the last planet visible in the morning.When a description is
used to introduce a singular term in the second way, it merely serves
to fix the reference of the term, but is not synonymous with it.



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



(1)
 Mark Twain (if he
exists) is Samuel Clemens.


Since 'Mark Twain' and 'Samuel Clemens' pick
out the same entity in every possible world where they pick out
anything, this identity statement is a necessary truth. (Note that the
statement is conditionalized on Mark Twain's existence, which
makes it possible to avoid the question whether (1) is true in worlds
where the two names pick out nothing.) But it is far from immediately
obvious that (1) expresses something that can be known *a
priori*. At least on the face of it, we may think that someone who
knows her neighbor by the name of 'Samuel Clemens,' who
has read several stories by an author named 'Mark Twain'
and who fails to realize that her neighbor and the author are
identical may not know that which is expressed by (1). Moreover, it
may seem that her ignorance is irremediable by reasoning alone, that
she requires empirical evidence to come to know that which is stated
by (1).



Another type of apparent counterexample to the thesis that all and
only the *a priori* truths are necessary concerns sentences
like



(2)
 If gold exists, then it has atomic number 79.


It seems plausible that it is an essential property of gold to have
atomic number 79: gold could not have (existed but) failed to have
that property. (A substance in another possible world that fails to
have atomic number 79 simply isn't gold, no matter how similar
it may otherwise be to the gold of the actual world.) And yet it seems
clear that it can only be known empirically that gold has that atomic
number. So, while (2) is a necessary truth, what it says cannot be
known *a 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 in
another possible world that isn't wooden simply can't be
*this* desk, no matter how similar it may otherwise be to my
desk. But it seems that we need empirical evidence to know that the
desk is made of wood. So, (3) is another apparent example of a
necessary *a posteriori* truth.



Just as Kripke claims that some truths are necessary without being
*a priori*, he argues that a truth can be *a priori*
without being necessary. To use an example of Gareth Evans's
(1982), suppose that I introduce the term 'Julius' by
stipulating that it is to refer rigidly to the person who is in fact
the inventor of the zip (if such a person exists). Then it may appear
that 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 could
have become a salesperson rather than an inventor.



According to Kripke, our initial surprise at the divergent extensions
of a prioricity and necessity should be mitigated on reflection. A
prioricity (epistemic necessity) is an *epistemological*
notion: it has to do with what can be known. That is not true of the
concept of necessity. (2) is necessary because the atomic number of
gold is an essential feature of it, and on the face of it, that has
nothing to do with what is known or believed by anyone. This kind of
necessity is a *metaphysical* notion, and we may use the term
'metaphysical necessity' to distinguish it more clearly
from epistemic necessity.



Kripke's examples are not the only ones that could be appealed
to in order to shed doubt on the coextensiveness of necessity and a
prioricity. Some other problematic cases are listed below (Chalmers
2002a; cp. Chalmers 2012, ch. 6).


1. *Mathematical truths*. It is common to hold that all
mathematical truths are necessary. But on the face of it, there is no
guarantee that all mathematical truths are knowable *a priori*
(or knowable in any way at all). For example, either the continuum
hypothesis or its negation is true, and whichever of these claims is
true is also necessary. But for all we know, there is no way for us to
know that that proposition is true.
2. *Laws of nature*. Some necessitarians about the natural
laws (see
 section 2)
 believe that the laws hold in all metaphysically possible worlds. But
they are not *a priori* truths.
3. *Metaphysical principles*. It is often believed that many
metaphysical theses are necessary if true, e.g., theses about the
nature of properties (e.g., about whether they are universals, sets or
tropes) or ontological principles like the principle of unrestricted
mereological composition (which says that for any things there is
something that is their sum). But it is not obvious that all truths of
this kind are *a priori*. (For discussion, see Chalmers 2012,
SSSS6.4-6.5; Schaffer 2017.)
4. *Principles linking the physical and the mental*. Some
philosophers hold that all truths about the mental are metaphysically
necessitated by the physical truths, but deny that it is possible to
derive the mental truths from the physical ones by *a priori*
reasoning (see Hill & McLaughlin 1999; Yablo 1999; Loar 1999; and
Chalmers 1999 for discussion). On that account, some of the
conditionals that link physical and mental claims are metaphysically
necessary but not *a priori*.



These examples are controversial. For any given mathematical claim
whose truth-value is unknown, one could hold that it is only our
cognitive limitations that have prevented us from establishing or
refuting the statement, and that the question could be decided by
ideal reasoning (so that the truth of the matter is *a
priori*). Alternatively, it may be held that the truth-value of
the mathematical statement is indeterminate. (Perhaps our practices do
not completely determine the references of all the terms used in the
mathematical claim). The same two options are available in the case of
metaphysical principles. Alternatively, one may argue that the
relevant metaphysical theses are merely contingent (see, e.g., Cameron
2007). Necessitarianism about the natural laws is highly controversial
and may simply be denied. And in response to (iv), one may deny that
the physical truths metaphysically necessitate the mental truths
(Chalmers 1996), or one may hold that the mental truths can be derived
from the physical ones by *a priori* reasoning (Jackson
1998).



Philosophers have paid more attention to the examples given by Kripke
than to other possible cases of the necessary *a posteriori*,
and for that reason the discussion in the rest of this section will
mostly focus on Kripke's cases. Two strategies for explaining
these examples can be distinguished. Dualists about metaphysical and
epistemic modality (dualists, for short) hold that the phenomena
reflect a deep and fundamental distinction between two kinds of
modality. Monists, by contrast, believe that all the data can
ultimately be explained by appeal to a single kind of modality. They
may agree that there are cases in which a single sentence is, in some
sense, both necessary and *a posteriori*, or both contingent
and *a priori*. But they insist that there is no similar
distinction at the level of worlds or propositions. Rather, the
phenomenon arises because a single sentence can be associated with two
different propositions, one that is necessary and another that is
contingent.


### 1.2 Dualism



Dualists distinguish between two concepts of propositional necessity,
*metaphysically necessity* and *epistemic necessity*.
The two notions are not coextensive. At least some of the sentences in
Kripke's examples express propositions that possess the one kind
of necessity but not the
 other.[4]



Once the existence of a distinctively metaphysical form of
propositional necessity is accepted, it is natural to wonder whether
it is possible to say more about its nature. Kit Fine (1994) offers an
account of it that appeals to the traditional distinction between
those properties of a thing that it possesses by its very nature and
those that it has merely accidentally. For example, it lies in the
nature of water to be composed of hydrogen and oxygen--being
composed in this way is part of what it is to be water--but it is
merely accidental to water that we use it to brush our teeth. A
proposition is metaphysically necessary just in case it is true in
virtue of the natures of things. (Also see Kment 2014, chs. 6-7,
and Ditter 2020.) Other philosophers (Rayo 2013, SS2.2.1, ch.
5; Dorr 2016) have discussed the idea that metaphysical necessity can
be explained in terms of the idiom "To be \(F\) is to be
\(G\)" (as in "To be water is to be
H2O"). Yet another account ties the metaphysical
notion of necessity constitutively to causation and explanation (Kment
2006a,b, 2014, 2015a; also see the exchange between Lange 2015 and
Kment 2015b).



Dualism requires us to dismantle the picture of inquiry, information
and content sketched in the introduction to
 section 1.
 Note that it is natural for a dualist to distinguish the space of
*metaphysically possible worlds* from the space of
*epistemically possible worlds*, i.e., from the space of
(maximally specific) ways the world might be that cannot be ruled out
on the basis of ideal reasoning alone, without empirical evidence
(Soames 2005, 2011). The range of epistemically possible worlds
outstrips the range of metaphysically possible worlds: there are some
ways the world couldn't have been, but which cannot be ruled by
ideal reasoning alone. For example, there is no metaphysically
possible world where gold has atomic number 78. But prior to carrying
out the right chemical investigations, we don't have enough
evidence to exclude all scenarios where gold has that atomic number,
so some worlds where gold has atomic number 78 are epistemically
possible. Empirical evidence is not used only to rule out
(metaphysical) possibilities, but is sometimes needed to rule out
metaphysical impossibilities that are epistemically possible.
Consequently, we cannot in general identify information with sets of
metaphysically possible worlds, since we need to distinguish between
states of information in which the available evidence rules out the
same metaphysically possible worlds but different metaphysically
impossible worlds. By the same token, the information encoded in a
sentence cannot in general be identified with a set of metaphysically
possible worlds, since two sentences may be true in all the same
metaphysically possible worlds, but not in all the same epistemically
possible worlds. If we wanted to identify information and sentential
contents with sets of worlds, it would seem more promising to use sets
of epistemically possible worlds. But the dualist may instead reject
the 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 by
holding that the sentences that figure in Kripke's examples are
associated with two different propositions, one that is necessary and
another that is contingent. This view comes in two main versions.
According to the first version, both propositions are semantically
expressed by the sentence. Proponents of this account need to
formulate a semantic theory that explains how that is possible.
According to the second version, only one of these propositions is
semantically expressed by the sentence, while the other is the
proposition that is communicated by a typical assertoric use of the
sentence. A philosopher holding this view needs to explain the
pragmatic mechanism by which an utterance of the sentence comes to
communicate the second proposition.



The first version of monism has been developed by David Chalmers and
Frank Jackson (Chalmers 1996, 1999, 2002a,b, 2004, 2006a,b; Chalmers
and Jackson 2001; Jackson 1998, 2004, 2011), who build on earlier work
by David Kaplan (1989a,b), Gareth Evans (1979) and Martin Davies and
Lloyd Humberstone (1980), and others. On Chalmers's and
Jackson's view, what explains the phenomena uncovered by Kripke
is not a difference between two spaces of possible worlds. There is
only a single space of possible worlds: the metaphysically possible
worlds--the ways the world could have been--just are the
epistemically possible worlds: the ways the world might be for all we
can know independently of empirical evidence. What explains the data
is a difference between two different ways in which sentences can be
used to *describe* the worlds in that space, i.e., between two
different notions of a sentence's being true in a world. The
distinction can be illustrated by appeal to our example of the proper
name 'Phosphorus.' Suppose that we have just introduced
this name by using the description 'the last celestial body
visible in the morning' to fix its reference. Consider a
possible world \(w\) where the description singles out, not Venus (as
in our world), but Saturn. Assume further that in \(w\) (as in the
actual world), Venus is the second planet from the sun, but Saturn is
not. Consider:



(5)
 Phosphorus is the second planet from the sun.


Is (5) true in \(w\)? There are two different ways of understanding
this question. On the one hand, it could mean something roughly like
this: if \(w\) actually obtains (contrary to what astronomers tell
us), is Phosphorus the second planet from the sun? The answer to that
question is surely 'no.' 'Phosphorus' refers
to whatever is actually the last celestial body visible in the
morning, and on the assumption that \(w\) actually obtains, that
object is Saturn, and is therefore not the second planet. As Chalmers
would put it, (5) is not *true at w considered as
 actual*.[5]
 But we can also interpret the question differently: if \(w\)
*had* obtained, then *would* Phosphorus have been the
second planet? In considering that question, we are not hypothetically
assuming that the object that actually satisfies the reference-fixing
description is Saturn. Instead, we can draw freely on our belief that
the object actually fitting the description is Venus, so that the name
picks out Venus in all possible worlds. Since Venus is the second
planet in \(w\), it is true to say: if \(w\) had obtained, then
Phosphorus would have been the second planet. In Chalmers's
terminology, (5) is *true at w considered as
counterfactual*.



The distinction between the two concepts of truth in a world can be
explained within a theoretical framework known as *two-dimensional
semantics*, which assigns to a sentence like (5) an intension that
is a function, not from worlds to truth-values, but from
*pairs* of worlds to truth-values. The intension of (5) is the
function that assigns the true to a pair of worlds \(\langle u; w
\rangle\) just in case the object that is the last celestial body
visible 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 a
world. A sentence \(P\) is true in \(w\) considered as actual just in
case the two-dimensional function assigns the true to \(\langle w;
w\rangle\). \(P\) is true in \(w\) considered as counterfactual just
in case, where \(u\) is the actual world, the two-dimensional function
assigns the true to \(\langle u; w\rangle\). Note that the
two-dimensional intension of (5) determines whether (5) is true at a
world \(w\) considered as actual. But it does not in general determine
whether (5) is true at \(w\) considered as counterfactual. That also
depends on which world is actual. Knowledge of a sentence's
two-dimensional intension is therefore not in general sufficient to
know whether the sentence is true at \(w\) considered as
counterfactual. Further empirical evidence may be required.



When combined with the conception of a sentence's content as the
set of worlds where it is true, the distinction between the two
concepts of truth in a world yields a distinction between two
different propositions expressed by a sentence. The first of these
propositions is the function that assigns the true to a world \(w\)
just in case the sentence is true in \(w\) considered as actual, while
the second proposition is the function that assigns the true to a
world \(w\) just in case the sentence is true in \(w\) considered as
counterfactual. Jackson calls the former proposition the
sentence'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 'secondary
intension.' The distinction between the two propositions
expressed by a sentence yields a distinction between two notions of
sentential necessity: primary necessity, which applies to sentences
with necessary primary intensions, and secondary necessity, which
applies to sentences with necessary secondary intensions. If a
sentence has primary necessity, then that fact, and *a
fortiori* the fact that the sentence is true, can be read off its
two-dimensional intension. Therefore, if we know the two-dimensional
intension, then that is enough to know that the sentence is true. No
further empirical evidence is required. That motivates the thought
that the notion of primary necessity captures the idea of a prioricity
or epistemic necessity. The notion of secondary necessity, on the
other hand, may be taken to capture the Kripkean idea of metaphysical
necessity.



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



An analogous account can be given of Kripke's examples of the
contingent *a priori*: these concern sentences whose primary
intensions are necessary and whose secondary intensions are
contingent. Assume again that the reference of 'Julius' is
fixed by the description 'the inventor of the zip (if such a
person exists).' Then in every world considered as actual, the
name singles out the person who is the inventor of the zip in that
world (if there is such a person) or nothing (if no such person exists
in 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 the
*actual* inventor of the zip (provided that there actually is
such an individual and that he or she exists in \(w)\). And since
there are possible worlds where that individual exists but is not the
inventor of the zip, the secondary intension of (4) is contingent.



Chalmers (2002a, 2010) and Jackson (1998) have tried to support their
modal monism by arguing that it is gratuitous to postulate two forms
of modality, given that all the phenomena pointed out by Kripke can be
accommodated by appeal to a single kind of modality. Dualists may
reply that the greater simplicity in the view of modality has been
achieved only by adding complexity to the semantic theory. That
response could be answered by arguing that two-dimensional semantics
can be motivated by independent considerations. That, of course, is
controversial, as is the general viability of two-dimensional
semantics (see the entry
 two-dimensional semantics
 for detailed discussion).



In addition, it is not obvious that the view of Chalmers and Jackson
can satisfactorily explain all the phenomena discussed in
 section 1.1.
 Some commentators have denied that it can give a viable general
account of Kripkean examples (see, e.g., Soames 2005; Vaidya 2008;
Roca-Royes 2011). In any case, it is clear that the view can only
explain how necessity and epistemic necessity can come apart for
sentences whose primary and secondary intensions differ. That may be
true of the cases considered by Kripke, but it seems doubtful for the
other examples considered in
 section 1.1
 (mathematical and metaphysical truths, laws, and principles
connecting the physical to the mental). In response, Chalmers has
argued that none of the latter cases are genuine examples of the
necessary *a posteriori* (1999, 2002a).



The *second* version of monism allows us to accommodate the
phenomena considered in
 section 1.1
 while staying much closer to the picture sketched in the introduction
to
 section 1.
 On this view, the data can be explained by appeal to a single space
of possible worlds and a single notion of truth in a world. The
proposition semantically expressed by a sentence containing a proper
name or natural-kind term is a function from individual worlds to
truth-values. The proposition expressed by 'Phosphorus
exists,' e.g., is a function that assigns the true to those
worlds where Venus exists and the false to the other worlds. (If the
reference of 'Phosphorus' was determined by a
reference-fixing description together with the facts about which
entity meets the description, then that fact itself is not a
*semantic* fact, but a *metasemantic* one, i.e., it does
not concern the question of what the meaning of the word is, but the
question of how the meaning of the word is determined.) What explains
the impression that a sentence like (1) expresses an *a
posteriori* claim is the fact that the proposition asserted by a
typical utterance of the sentence is not the one that is semantically
expressed by it, but a different proposition that is contingent and
can only be known empirically.



Robert Stalnaker (1978, 2001) has given a detailed account of the
pragmatic mechanism by which a contingent proposition comes to be
asserted by the utterance of a sentence that semantically expresses a
necessary proposition. On his account, linguistic communication
evolves in a context characterized by background assumptions that are
shared between the participants. These assumptions can be represented
by the set of worlds at which they are jointly true, which Stalnaker
calls the 'context set.' The point of assertion is to add
the proposition asserted to the set of background assumptions and
thereby eliminate worlds where it is not true from the context set. To
achieve this, every assertion needs to conform to the rule that the
proposition asserted is false in some of the worlds that were in the
context set before the utterance (otherwise there are no worlds to
eliminate) and true in others (since the audience cannot eliminate all
worlds from the context set). Now consider a context where the shared
background assumptions include the proposition that the references of
'\(A\)' and '\(B\)' were fixed by certain
descriptions but leave open whether the two descriptions single out
the same object. Suppose that someone says '\(A\) is
\(B\).' In every world in the context set, the sentence
semantically expresses either a necessary truth (if the two
descriptions single out the same object in the world) or a necessary
falsehood (if they don't). If the proposition that the speaker
intends to assert were the one that is semantically expressed by the
sentence, the aforementioned rule would be
 violated.[7]
 To avoid attributing this rule violation to the speaker, the audience
will construe the utterance as expressing a different proposition, and
the most natural candidate is the proposition that the sentence
uttered semantically expresses a true proposition. (Stalnaker calls
this the 'diagonal proposition.') By exploiting this
mechanism of reinterpretation, a speaker can use the sentence to
express the diagonal proposition. This proposition is true in just
those worlds in the context set where the two descriptions single out
the same object. It is clearly a contingent proposition, and empirical
evidence is required to know it. Stalnaker suggests an analogous
explanation of Kripke's proposed cases of contingent *a
priori* truth (1978, 83f.).



Stalnaker's account of the necessary *a posteriori*
requires that the proposition semantically expressed by the sentence
and the proposition that the sentence semantically expresses a truth
hold in different worlds in the context set. And that seems to require
that the assumptions shared between the participants of the
conversation don't determine what proposition is semantically
expressed by the sentence. It has been argued that that assumption is
implausible in some cases of Kripkean *a posteriori*
necessities (Soames 2005, 96-105). Suppose that I point to the
desk in my office in broad daylight and say 'That desk (if it
exists) is made of wood.' Unless the context is highly unusual,
the shared assumptions, so the argument goes, uniquely determine what
proposition is expressed by the sentence.


## 2. Metaphysical and Nomic Modality



It often seems very natural to use modal terminology when talking
about the laws of nature. We are inclined to say that nothing
*can* move faster than light to express the fact that the laws
rule out superluminal motion, and to state Newton's First Law by
saying that an object *cannot* depart from uniform rectilinear
motion unless acted on by an external force. This motivates the
thought that there is a form of necessity associated with the natural
 laws.[8]
 It is controversial, however, whether that form of necessity is
simply metaphysical necessity, or another kind of necessity. The
former view is taken by *necessitarians* (Swoyer 1982;
Shoemaker 1980, 1998; Tweedale 1984; Fales 1993; Ellis 2001; Bird
2005), who believe that the laws (or the laws conditionalized on the
existence of the properties mentioned in them) are metaphysically
necessary. *Contingentists* deny that, but many contingentists
hold that there is a kind of necessity distinct from metaphysical
necessity that is characteristic of the laws (e.g., Fine 2002), and
which may be called *natural* or *nomic* necessity. It
is often assumed that nomic necessity is a weaker form of necessity
than metaphysical necessity: it attaches to the laws and to all truths
that are metaphysically necessitated by them, so that anything that is
metaphysically necessary is also nomically necessary, but not vice
versa.



Necessitarians have given several arguments for their position. Here
are two.



*The argument from causal essentialism* (e.g., Shoemaker 1980,
1998). Some philosophers believe that the causal powers that a
property confers on its instances are essential to it. Assuming that
causal laws describe the causal powers associated with properties, it
follows that these laws (or versions of them that are conditionalized
on the existence of the relevant properties) are necessary truths.
This is, in the first instance, only an argument for the necessity of
*causal* laws, but perhaps it can be argued that all laws of
nature are of this kind. Of course, even if this assumption is
granted, the argument is only as strong as the premise that properties
have their associated causal powers essentially. To support this view,
Sydney Shoemaker (1980) has given a battery of epistemological
arguments. He points out that our knowledge of the properties that an
object possesses can only rest on their effects on us, and must
therefore be grounded in the causal powers associated with these
properties. But, he goes on to argue that, without a necessary
connection between the properties and the associated causal powers, an
object's effects on us could not serve as a source of all the
knowledge about an object's properties that we take ourselves to
possess.



*The argument from counterfactual robustness* (Swoyer 1982;
Fales 1990, 1993; also see Lange 2004 for discussion). Natural laws
are often believed to differ from accidental generalizations by their
*counterfactual robustness* (counterfactual-supporting power).
Setting aside some minor complications (see Lewis 1979a), we can
describe this feature as follows: if it is a law that all \(F\)s
are \(G\), then this generalization would still have been true if
there had been more \(F\)s than there actually are, or if some \(F\)s
had found themselves in conditions different from the ones that
actually obtain. For example, it would still have been true that
nothing moves faster than light if there had been more objects than
there actually are, or if some bodies had been moving in a different
direction. Contrast this with *No emerald has ever decorated a
royal crown*. That may be true, but it is not very robust. It
would have been false if some rulers of the past had made different
fashion choices. Some necessitarians have argued that contingentism
about the laws cannot provide a plausible explanation of the special
counterfactual robustness of the laws. Note that a counterfactual
"if it had been the case that \(P\), then it would have been the
case that \(Q\)" is usually taken to be true if \(Q\) is true in
those metaphysically possible \(P\)-worlds that are closest to
actuality. On this view, the special counterfactual robustness of the
law *All \(F\)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 different
circumstances, the ones where the actual law holds are closer than the
rest. If the laws hold in some metaphysically possible worlds but not
in others, then the reason why the former are closer than the latter
must be that the rules we are using for deciding which worlds count as
the closest say so. But which such rules we use is a matter of
convention. The counterfactual-supporting power of the laws does not
seem to be a purely conventional matter, however. Necessitarianism,
the argument continues, offers a better explanation: the laws hold in
the closest possible worlds simply because they hold in *all*
metaphysically possible worlds. Conventions don't come into it.
The contingentist may reply that, even though the counterfactual
robustness of the laws is grounded in a convention, that convention is
not *arbitrary*. Instead, it may have its rationale in
certain features of the laws that make them different from accidental
generalizations in an objectively important way (Sidelle 2002),
e.g., the fact that laws relate to pervasive and conspicuous patterns
in the history of the world, or the fact that the truth of a law
(unlike the truth of an accidental generalization) is explanatorily
independent of matters of particular fact (Kment 2015b,
11-12).



Contingentism has often been defended by pointing out that the laws of
nature can be known only *a posteriori*, and that their
negations are conceivable (see Sidelle 2002). Necessitarians may reply
to the first point that Kripke's work has given us reasons for
thinking that *a posteriori* truths can be metaphysically
necessary (see
 section 1.1).
 In response to the second point, they may grant that the negation of
a law is conceivable, but deny that conceivability is a good guide to
possibility (see the entry
 epistemology of modality).
 Alternatively, they may deny that we can really conceive of a
situation in which, say, bodies violate the law of gravitation. What
we can conceive of is a situation in which objects move in ways that
*appear* to violate the law. But that situation cannot be
correctly be described as involving objects with *mass*.
Rather, the objects in the imagined situation have a different
property that is very similar to mass (call it 'schmass')
but which is governed by slightly different laws. Contingentists may
reply that the non-existence of schmass (or the non-existence of
objects that move in the way imagined) is itself a law, so that we
have, after all, conceived of a situation where one of the actual laws
fails (see Fine 2002).


## 3. The Structure of the Modal Realm



The concepts of metaphysical, epistemic, and nomic necessity are only
a 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 in
the introduction to this entry). We also speak of



(6)
 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 modal
concepts 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 notions
of necessity in terms of which all the others can be defined?



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


1. *Restriction*. To say that property \(N\) can be defined
from kind of necessity \(N^\*\) by *restriction* is to say that
a proposition's having \(N\) can be defined as the combination
of two things: (i) the proposition's having \(N^\*\), and (ii)
its meeting certain additional conditions.
2. *Relativization / quantifier restriction*. To say that a
property \(N\) can be defined from a kind of necessity \(N^\*\) by
*relativization* to a class of propositions \(S\) is to say
that a proposition's having \(N\) can be defined as its being
\(N^\*\)-necessitated by \(S\). A closely related way in which a modal
property can be defined in terms of another is by *quantifier
restriction*. Suppose that \(P^\*\) is a kind of possibility that
is 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 \(P^\*\)-possible
world (a world that could \(P^\*\)-possibly have been actualized). To
say that the property \(N\) can be defined from \(N^\*\) by quantifier
restriction is to say that that a proposition's having \(N\) can
be defined as its being true in all \(P^\*\)-possible worlds that meet
a certain condition \(C\). (This is only the simplest way of defining
a modal property from a kind of necessity by quantifier restriction.
Much more sophisticated methods have been proposed. See, e.g., Kratzer
1977, 1991.) Given reasonable assumptions, every definition by
relativization corresponds to a definition by quantifier restriction,
and vice
 versa.[9]



Restriction allows us to define narrower modal properties from broader
ones. For example, it seems natural to define a proposition's
being mathematically necessary as its being both metaphysically
necessary and a mathematical truth (Fine 2002, 255), or as its being
metaphysically necessary *because* it is a mathematical
truth. Maybe a proposition's being
logically necessary can similarly be defined as its being
metaphysically necessary because it is true in virtue of the natures
of the logical constants (Fine 1994, 9-10; 2005, 237).
(For a very different account that does not define logical necessity
from metaphysical necessity by restriction, see Bacon 2018.)



Relativization and quantifier restriction allow us to define broader
modal properties in terms of narrower ones. For example, it may be
held that biological necessity can be defined as the property of being
metaphysically (or perhaps nomically) necessitated by the basic
principles of biology.



A modal property \(N\) is called *alethic* just in case the
claim that a proposition has \(N\) entails that the proposition is
true. Metaphysical, epistemic and nomic necessity are all alethic. By
contrast, moral and legal necessity are not. It is both morally and
legally necessary (i.e., it is required both by morality and by the
law) that no murders are committed, even though murders are in fact
being committed. A modal property defined by restriction from an
alethic kind of necessity must itself be alethic. By contrast,
relativization allows us to define non-alethic modal properties from
alethic ones, by relativizing to a class of propositions that contains
some falsehoods. Similarly, we can define a non-alethic modal property
from an alethic one by restricting the quantifier over possible worlds
to some class that does not include the actual world. For example,
legal necessity can perhaps be defined from metaphysical necessity by
restricting the quantifier to worlds where everybody conforms to the
actual laws.



The properties listed in (6) can very naturally be called 'kinds
of necessity,' and in some contexts they are the properties
expressed by necessity operators like 'must' and
'could not have been otherwise.' But that is not true of
every property that can be defined from some kind of necessity by
relativization or restriction. For example, we can define a property
by relativizing metaphysical necessity to the class of truths stated
in a certain book, but it would not be natural at all to call this
property a kind of necessity. It is not plausible that there is a
special form of necessity that attaches to all and only the
propositions necessitated by the truths in the book. Similarly, the
property defined by restricting metaphysical necessity to the truths
about cheddar cheese cannot naturally be called a kind of necessity.
There is no form of necessity that applies to just those necessary
propositions that deal with cheddar and to none of the others. It is a
good question what distinguishes those properties defined by
relativization and restriction that we are willing to count as forms
of necessity from the rest. Perhaps the most natural answer is that
the distinction is dictated by our interests and concerns, and does
not reflect a deep metaphysical difference.



A more pressing question is whether some of the forms of necessity
discussed in sections
 1
 and
 2
 can be defined in terms of the others by relativization or
restriction. Consider epistemic and metaphysical necessity first, and
suppose for the sake of the argument that dualism is true and the two
properties are indeed different forms of necessity. Can one of them be
defined in terms of the other by one of the aforementioned methods?
Not if there are both necessary *a posteriori* and contingent
*a priori* propositions, since relativization and restriction
only allow us to define one property in terms of another if the
extension of one is a subclass of that of the other. However, the
existence of contingent *a priori* truths is more controversial
than that of necessary *a posteriori* propositions, and someone
trying to define epistemic necessity in terms of metaphysical
necessity or vice versa may repudiate the contingent *a priori*
and hold that the extension of epistemic necessity is included in that
of metaphysical necessity. Then such a philosopher could try (a) to
define metaphysical necessity from epistemic necessity by
relativization to some suitable class, or (b) to define epistemic
necessity from metaphysical necessity by restriction.



Such a definition may get the extension of the definiendum right. But
a definition may be intended to do much more than that: it may be
meant to tell us what it \(is\) for something to fall under the
concept to be defined. Suppose that someone tried to define the
property of being an equiangular triangle as that of being a triangle
whose sides are of equal length. While this is extensionally correct,
it does not give us the right account of what it is for something to
be an equiangular triangle (what it is for something to have that
property has something to do with the sizes of its angles, not with
the lengths of its sides). It could be argued that definitions of type
(a) and (b) face similar difficulties. For example, a definition of
kind (a) entails that a proposition's being metaphysically
necessary consists in its being epistemically necessitated by a
certain class of propositions. But that would make metaphysical
necessity an epistemic property, and dualists typically want to resist
that idea. Similarly for definitions of type (b). Whether something is
epistemically necessary (in the sense of being *a priori*)
seems to be a purely epistemic matter. *A priori* propositions
may also be metaphysically necessary, but their metaphysical necessity
isn't part of what *makes* them *a priori*, and
therefore shouldn't be mentioned in a definition of a
prioricity.



If this argument is correct, then it is impossible to define epistemic
modal properties in terms of non-epistemic ones, or vice versa. But
what about metaphysical and nomic necessity? Suppose for the sake of
the argument that there is such a thing as nomic necessity (a form of
necessity associated with the laws of nature) but that contingentism
about the natural laws is true, so that nomic necessity is indeed
distinct from metaphysical necessity. Can we define one of these
properties in terms of the other? The most natural way of doing this
would be to say that



(7)
 Nomic necessity can be defined as the property of being
metaphysically necessitated by the laws of nature.


Such a definition may be extensionally accurate, and many philosophers
would not hesitate to endorse it. But others have doubted that it
captures what it \(is\) for a proposition to be nomically necessary
(Fine 2002). Nomic necessity is a special modal status enjoyed by all
and only the propositions that are metaphysically necessitated by the
natural laws. Now, if \(P\) is metaphysically necessitated by the laws
without itself being a law, then it may seem plausible to say, in some
sense, that \(P\) has that special modal status *because* \(P\)
is metaphysically necessitated by the laws. But the reason why being
metaphysically necessitated by the laws confers that special modal
status on \(P\) is presumably that the laws themselves have that modal
status and that this modal status gets transmitted across metaphysical
necessitation. But if we now ask what makes it so that the laws
themselves have that special modal status, (7) does not seem to give
us the correct answer: the special necessity of the laws doesn't
consist in the fact that they are metaphysically necessitated by the
laws. Hence, (7) cannot be a correct general account of what
constitutes that special modal status.



It is open to debate which kinds of necessity are fundamental, in the
sense that all others can be defined in terms of them, while they are
not themselves definable in terms of others. The monist view
considered in
 section 1.3,
 when combined with (7), may inspire the hope that we can make do with
a single fundamental kind of necessity. Others have argued that there
are several kinds of necessity that are not mutually reducible. For
example, Fine (2002) suggests (in a discussion that sets aside
epistemic modality) that there are three fundamental kinds of
necessity, which he calls 'metaphysical,'
'nomic' and 'normative' necessity. (For
further discussion of normative necessity, see Rosen 2020.)



The reduction of the various kinds of necessity to a small number of
fundamental ones would be an important step towards the goal of a
unified account of
 modality.[10]
 But those who believe that there are several different fundamental
kinds of necessity need to address another question: What is the
common feature of these fundamental kinds of necessity that makes them
all kinds of necessity? Why do they count as kinds of necessity, while
other properties don't?



One strategy for answering this question, which centers on
non-epistemic forms of necessity, starts from a certain conception of
what (non-epistemic) necessity consists in: for a proposition to be
necessary is for its truth to be, in a certain sense, particularly
secure, unshakable, and unconditional in a wholly objective way.
A necessary truth could not easily have been false (it could less
easily have been false than a contingent truth). We may call this
feature of a proposition 'modal force.' It is natural to
apply this conception to metaphysical and nomic necessity. Each of
these properties may be held to consist in having a certain grade of
modal force, though if contingentism is true, the degree of modal
force required for nomic necessity is lower than that required for
metaphysical necessity. We could then say that a property is one of
the 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 of
necessity, like those listed in (6) can be defined from the
fundamental ones by relativization or restriction. Having these
properties does *not* consist simply in having a specific grade
of modal force (and these properties therefore aren't among the
fundamental kinds of necessity). For example, if a property is defined
by relativizing metaphysical necessity to a class of propositions
\(S\), then the fact that a proposition \(P\) has that property
consists in the fact that *the connection* between \(S\) and
\(P\) has a certain grade of modal force. But that is not the same
thing as *P itself* having a certain grade of modal force.
Similarly, if a property is defined from, say, metaphysical necessity
by restriction, then having that property does not consist
*merely* 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 the
idea of modal force (of a proposition's truth being very
unshakable). Some authors have attempted to explain this notion in
counterfactual terms (see Lewis 1973a, SS2.5; Lewis 1973b,
SS2.1; McFetridge 1990, 150ff.; Lange 1999, 2004, 2005; Williamson
2005, 2008; Hill 2006; Kment 2006a; cp. Jackson 1998, Chalmers 2002a):
the necessary truths are distinguished from the contingent ones by the
fact that they are not only true as things actually are, but that they
would still have been true if things had been different in various
ways. To capture this idea more precisely, Lange (2005) introduces the
concept of 'stability': a deductively closed set \(S\) of
truths is stable just in case, for any claim \(P\) in \(S\) and any
claim \(Q\) consistent with \(S\), it is true in any context to say
that it would still have been the case that \(P\) if it had been the
case that \(Q\). The different forms of necessity have in common that
their extensions are stable sets.



Kment (2006a, 2014, chs. 1-2) argues that modal force, and hence
necessity and possibility, come in many degrees (cp. Williamson 2016).
We often talk about such degrees of possibility when we say things
like '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 a
high degree of possibility to \(A\)'s winning. A
proposition's degree of possibility is the higher the less of a
departure 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 just
an inch further to the left at a crucial moment during the game. Then
we can truly say that the team could easily have won. More formally,
\(P\)'s degree of possibility is the higher the closer the
closest \(P\)-worlds are to actuality (also see Lewis 1973a,
SS2.5; Lewis 1973b, SS2.1; Kratzer 1991). Similarly, a
truth's degree of necessity is measured by the distance from
actuality to the closest worlds where it is false. What metaphysical
necessity, nomic necessity and the other grades of necessity have in
common is that each of them is the property of having a degree of
possibility that is above a certain threshold. What distinguishes them
is a difference in their associated thresholds.