## 1. The Basic Question: What is it to be a Law?



Here are four reasons philosophers examine what it is to be a law of
nature: First, as indicated above, laws at least appear to have a
central role in scientific practice. Second, laws are important to
many other philosophical issues. For example, sparked by the account
of counterfactuals defended by Chisholm (1946, 1955) and Goodman
(1947), and also prompted by Hempel and Oppenheim's (1948)
deductive-nomological model of explanation, philosophers have wondered
what makes counterfactual and explanatory claims true, have thought
that laws play some part, and so also have wondered what distinguishes
laws from nonlaws. Third, Goodman famously suggested that there is a
connection between lawhood and confirmability by an inductive
inference. So, some sympathetic to Goodman's idea come to the
problem of laws as a result of their interest in the problem of
induction. Fourth, philosophers love a good puzzle. Suppose that
everyone here is seated (cf., Langford 1941, 67). Then, trivially,
that everyone here is seated is true. Though true, this generalization
does not seem to be a law. It is just too accidental. Einstein's
principle that no signals travel faster than light is also a true
generalization but, in contrast, it is thought to be a law; it is not
nearly so accidental. What makes the difference?



This may not seem like much of a puzzle. That everyone here is seated
is spatially restricted in that it is about a specific place; the
principle of relativity is not similarly restricted. So, it is easy to
think that, unlike laws, accidentally true generalizations are about
specific places. But that is not what makes the difference. There are
true nonlaws that are not spatially restricted. Consider the
unrestricted generalization that all gold spheres are less than one
mile in diameter. There are no gold spheres that size and in all
likelihood there never will be, but this is still not a law. There
also appear to be generalizations that could express laws that are
restricted. Galileo's law of free fall is the generalization
that, *on Earth*, free-falling bodies accelerate at a rate of
9.8 meters per second squared. The perplexing nature of the puzzle is
clearly revealed when the gold-sphere generalization is paired with a
remarkably similar generalization about uranium spheres:



> 
> All gold spheres are less than a mile in diameter.
> 
> 
> All uranium spheres are less than a mile in diameter.
> 
> 
> 



Though the former is not a law, the latter arguably is. The latter is
not nearly so accidental as the first, since uranium's critical
mass is such as to guarantee that such a large sphere will never exist
(van Fraassen 1989, 27). What makes the difference? What makes the
former an accidental generalization and the latter a law?


## 2. Systems



One popular answer ties being a law to deductive systems. The idea
dates back to Mill (1843, 384), but has been defended in one form or
another by Ramsey (1978 [f.p. 1928]), Lewis (1973, 1983, 1986,
1994), Earman (1984) and Loewer (1996). Deductive systems
are individuated by their axioms. The logical consequences of the
axioms are the theorems. Some true deductive systems will be stronger
than others; some will be simpler than others. These two virtues,
strength and simplicity, compete. (It is easy to make a system
stronger by sacrificing simplicity: include all the truths as
axioms. It is easy to make a system simple by sacrificing strength:
have just the axiom that 2 + 2 = 4.) According to Lewis (1973, 73),
the laws of nature belong to all the true deductive systems with a
*best* combination of simplicity and strength. So, for example,
the thought is that it is a law that all uranium spheres are less than
a mile in diameter because it is, arguably, part of the best deductive
systems; quantum theory is an excellent theory of our universe and
might be part of the best systems, and it is plausible to think that
quantum theory plus truths describing the nature of uranium would
logically entail that there are no uranium spheres of that size
(Loewer 1996, 112). It is doubtful that the generalization that all
gold spheres are less than a mile in diameter would be part of the
best systems. It could be added as an axiom to any system, but it
would bring little or nothing of interest in terms of strength and
adding it would sacrifice something in terms of simplicity. (Lewis
later made significant revisions to his account in order to address
problems involving physical probability (Lewis 1986, 1994).



Many features of the systems approach are appealing. For one thing,
it deals with a challenge posed by vacuous laws. Some laws are
vacuously true: Newton's first law of motion -- that all
inertial bodies have no acceleration -- is a law, even though
there are no inertial bodies. But there are also lots of vacuously
true nonlaws: all plaid pandas weigh 5 lbs., all unicorns are
unmarried, etc. With the systems approach, there is no exclusion of
vacuous generalizations from the realm of laws, and yet only those
vacuous generalizations that belong to the best systems qualify (cf.,
Lewis 1986, 123). Furthermore, one goal of scientific theorizing
is the formulation of true theories that are well balanced in terms of
their simplicity and strength. So, the systems approach seems to
underwrite the truism that an aim of science is the discovery of laws
(Earman 1978, 180; Loewer 1996, 112). One last aspect of the systems
view that is appealing to many (though not all) is that it is in
keeping with broadly Humean constraints on a sensible metaphysics.
There is no overt appeal to closely related modal concepts (e.g., the
counterfactual conditional, causation, dispositions) and no overt
appeal to modality-supplying entities (e.g., universals or God; for
the supposed need to appeal to God, see Foster 2004). Indeed, the
systems approach is the centerpiece of Lewis's defense of
*Humean supervenience*, "the doctrine that all there is
in the world is a vast mosaic of local matters of particular fact,
just one little thing and then another" (1986, ix).



Other aspects of the systems approach make philosophers wary. (See,
especially, Armstrong 1983, 66-73; van Fraassen 1989,
40-64; Carroll 1990, 197-206.) Some argue that this
approach will have the untoward consequence that laws are
inappropriately mind-dependent in virtue of the account's appeal
to the concepts of simplicity, strength and best balance, concepts
whose instantiation seems to depend on cognitive abilities, interests,
and purposes. The appeal to simplicity raises further questions
stemming from the apparent need for a regimented language to permit
reasonable comparisons of the systems (Lewis 1983, 367.) More
recently, Roberts questions the systems approach at a point sometimes
thought to be a strength of the view: "We have no practice of
weighing competing virtues of simplicity and information content for
the purpose of choosing one deductive system over others, where all
are presumed to be true" (2008, 10). There is the practice of
curve-fitting, which involves weighing the competing virtues of
simplicity and closeness of fit, but this is a practice that is part
of the process of discovering *what is true*. Also, the systems
approach is ill-suited to rule out widespread and striking
regularities as laws, even those that are clearly determined by the
initial conditions. That the universe is closed, that entropy
generally increases, that the planets of our solar system are
co-planar, and others (if true) could be added to any true deductive
system, greatly increasing the strength of the system, with only a
small cost in terms of simplicity (Maudlin 2007, 16; Roberts 2008,
23). Interestingly, sometimes the systems view is abandoned
*because* it satisfies the broadly Humean constraints
on laws of nature; some argue that what generalizations are laws
is not determined by local matters of particular fact. (See Section 4
below.) Though Humeans like Lewis generally favor realism to any form
of anti-realism (Section 5 below), Berenstain and Ladyman (2012)
have argued that scientific realism is incompatible with Humeanism
because realism requires a notion of natural necessity not susceptible
to Humean analysis.


## 3. Universals



In the late 1970s, there emerged a competitor for the systems approach
and all other Humean attempts to say what it is to be a law. Led by
Armstrong (1978, 1983, 1991, 1993), Dretske (1977), and Tooley (1977,
1987), the rival approach appeals to universals (i.e., certain kinds
of properties and relations) to distinguish laws from nonlaws.



Focusing on Armstrong's development of the view, here is a
concise statement of the framework characteristic of the
universals approach:



> 
> Suppose it to be a law that *F*s are *G*s.
> *F*-ness and *G*-ness are taken to be universals. A
> certain relation, a relation of non-logical or contingent
> necessitation, holds between *F*-ness and *G*-ness. This
> state of affairs may be symbolized as
> '*N*(*F*,*G*)' (1983, 85).
> 



This framework promises to address familiar puzzles and problems:
Maybe the difference between the uranium-spheres generalization and
the gold-spheres generalization is that being uranium does necessitate
being less than one mile in diameter, but being gold does not. Worries
about the subjective nature of simplicity, strength and best balance
do not emerge; there is no threat of lawhood being mind-dependent so
long as necessitation is not mind-dependent. Some think that the
framework supports the idea that laws play a special explanatory role
in inductive inferences, since a law is not just a universal
generalization, but is an entirely different creature -- a
relation holding between two other universals (Armstrong 1991, Dretske
1977). The framework is also consistent with lawhood not supervening
on local matters of particular fact; the denial of Humean
supervenience often accompanies acceptance of the universals
approach.



For there truly to be this payoff, however, more has to be said about
what *N* is. This is a problem van Fraassen calls *the
identification problem,* which he couples this with a second
problem, what he calls *the inference problem* (1989, 96). The
essence of this pair of problems was captured early on by Lewis with
his usual flair:



> 
> Whatever *N* may be, I cannot see how it could be absolutely
> impossible to have *N*(*F*,*G*) and *Fa*
> without *Ga*. (Unless *N* just is constant conjunction,
> or constant conjunction plus something else, in which case
> Armstrong's theory turns into a form of the regularity theory he
> rejects.) The mystery is somewhat hidden by Armstrong's
> terminology. He uses 'necessitates' as a name for the
> lawmaking universal *N*; and who would be surprised to hear
> that if *F* 'necessitates' *G* and
> *a* has *F*, then *a* must have *G*? But I
> say that *N* deserves the name of 'necessitation'
> only if, somehow, it really can enter into the requisite necessary
> connections. It can't enter into them just by bearing a name,
> any more than one can have mighty biceps just by being called
> 'Armstrong' (1983, 366).
> 



Basically, there needs to be a specification of what the lawmaking
relation is (the identification problem). Then, there needs to be a
determination of whether it is suited to the task (the inference
problem): Does *N*'s holding between *F* and
*G* entail that *F*s are *G*s? Does its holding
support corresponding counterfactuals? Do laws really turn out not to
supervene, to be mind-independent, to be explanatory? Armstrong does
say more about what his lawmaking relation is. He states in reply to
van Fraassen:



> 
> It is at this point that, I claim, the Identification problem has been
> solved. The required relation is the causal relation, ... now
> hypothesized to relate types not tokens (1993, 422).
> 



Questions remain about the nature of this causal relation understood
as a relation that relates both token events and universals. (See van
Fraassen 1993, 435-437, and Carroll 1994, 170-174.)


## 4. Humean Supervenience



Rather than detailing all the critical issues that divide the
systems approach and the universals approach, attention has been on
the divisive issue of supervenience (i.e., determination). It concerns
whether Humean considerations really determine what the laws are.
There are some important examples that appear to show that they do
not.



> 
> 
> Suppose that there are ten different kinds of fundamental particles.
> So, there are fifty-five possible kinds of two-particle interactions.
> Suppose that fifty-four of these kinds have been studied and
> fifty-four laws have been discovered. The interaction of *X*
> and *Y* particles have not been studied because conditions are
> such that they never will interact. Nevertheless, it seems that it
> might be a law that, when *X* particles and *Y*
> particles interact, *P* occurs. Similarly it might be a law
> that when *X* and *Y* particles interact, *Q*
> occurs. There seems to be nothing about the local matters of
> particular fact in this world that fixes which of these
> generalizations is a law (Tooley 1977, 669).
> 
> 
> 



The failure of supervenience arises in other cases. Consider the
possibility that there is a lone particle traveling through otherwise
empty space at a constant velocity of, say, one meter per second. It
seems that this might just be a nearly empty Newtonian universe in
which it is accidentally true that all bodies have a velocity of one
meter per second; it just so happens that there is nothing to alter
the particle's motion. But, it might also be the case that this
world is not Newtonian and that it is a law that all bodies have
velocity at one meter per second; it could be that this generalization
is not accidental and would have held true even if there were other
bodies slamming into the lone particle. (Earman 1986, 100; Lange 2000,
85-90.)



Maudlin presses the case against the Humeans by focusing on the common
practice among physicists of considering models of a theory's
laws.



> 
> Minkowski space-time, the space-time of Special Relativity, is a model
> of the field equations of General Relativity (in particular, it is a
> vacuum solution). So an empty Minkowski space-time is one way the
> world could be if it is governed by the laws of General Relativity.
> But is Minkowski space-time a model *only* of the General
> Relativistic laws? Of course not! One could, for example, postulate
> that Special Relativity is the complete and accurate account of
> space-time structure, and produce another theory of gravitation, which
> would still have the vacuum Minkowski space-time as a model. So under
> the assumption that no possible world can be governed by the laws of
> General Relativity and by a rival theory of gravity, the total
> physical state of the world cannot always determine the laws (2007,
> 67).
> 



The suggestion here is that there is the possibility of a matter-less
universe with the laws of General Relativity and another with laws of
a conflicting theory of gravitation. (For additional examples, see
Carroll 1994, 60-80). What Maudlin sees as a consequence of
standard scientific reasoning, Humeans will see as an example exposing
the absurdity of nonsupervenience.



Humeans contend that the various pairs of so-called possible worlds
are not really possible. Sometimes this contention turns on the issue
of whether laws govern, sometimes on epistemological or ontological
concerns, and sometimes on concerns about about how our language
works. One objection to the nonsupervenience arguments from the Humean
camp is that, if one comes to the debate with the governing conception
in mind, one is likely to find the antisupervenience examples
convincing, but using this conception to reject Humean analyses of
lawhood is somehow to beg the question or to otherwise be unconvincing
because it is a conception Humeans reject (Beebee, 2000). (Also see
Loewer 1996 and Roberts 1998.) In contrast, some are sympathetic to
Humeanism and aspects of the governing conception (Schneider 2007,
Ward 2007, Roberts 2008). In particular, when we consider laws
governing the nation, the laws don't do anything to the
governed. What governs is the government that creates and
enforces the laws. "The proposition that we call the law is not
the agent of the governing, but the content of the governing"
(Roberts 2008, 46).



Some argue based on skeptical considerations that their brand of
Humean supervenience is true (Earman and Roberts 2005ab). Others
reject skeptical concerns (Schaffer 2008, 94-99, Carroll
2008, 75-79). Schaffer presses an ontological concern
to the effect that nonsupervening laws are ungrounded entities
(Schaffer 2008, 84-85).



An original manner of responding to apparent counterexamples to
supervenience takes a semantic turn. In the lone-particle example
reported above, there is a world with the lone particle traveling at
one meter per second, though it is not a law that all particles travel
at that speed. There is also a world with the lone particle traveling
at one meter per second, though it is a law that all particles are
traveling at that speed. This reasoning does not contradict
supervenience because of the context sensitivity of the predicate,
'is a law'. Though the sentence, 'It is a law that
all particles travel at one meter per second' is (i) true
relative to one context/world pair and (ii) false relative to a
different context/world pair. This difference in truth-value could
merely be the result of a difference between two contexts (Roberts
2008, 357-61).



For Roberts, a possible world *w* in which there exists only a
single particle traveling at constant velocity throughout all of
history and relative to a context in which the salient theory is, say,
Newtonian Mechanics, 'It is a law that all particles have a
constant velocity of one meter per second' is true just in case
the reference of the 'that' clause plays the law role in
the salient theory, which it doesn't in this case. It might
play the law role relative to some other theory, but this would
be a different context. A single generalization cannot
both play the law role and also not play the law role relative to a
single theory, and so a different salient theory and so a different
context is required for 'It is a law that all bodies travel at
one meter per second' to be true (Roberts 2008, 357-61). 
What is enticing about this reply is that it does not reject any
intuitive claim about the laws in the various possible worlds. The
antisupervenience judgments about what are the laws are reasonable
claims given the contexts. It is just that there is a failure to
recognize the influence of context. So, for example, Maudlin's
so-called two possibilities would be seen by Roberts as descriptions
of a single possibility that are made relative to two contexts with
different salient theories: General Relativity and some rival theory
of gravity. (Parallel points could be made about Tooley's
examples involving the 10 different kinds of fundamental particles.)
The key is the context sensitivity that is built into the truth
conditions of lawhood sentences. Other views that take lawhood
sentences to be context sensitive might also be able to avail
themselves of Roberts' challenge to the antisupervenience
examples. What is not compelling about Roberts' position,
though, is his view on the context dependence of lawhood ascriptions.
His view is devised for one particular phrase of English:
'law of nature', but it would be better if the
contextual treatment of 'law of nature' melded neatly
with the context dependence of other natural language words and
phrases. We should try to understand the context dependence of our
'law of nature' talk by appeal to linguistic principles,
and the investigation should be driven by considerations of
conversational practice (Carroll 2018, 131-32). 'Law of
nature' should not be an isolated freak of our language
(cf., Unger 1971, 202) on the verb 'to know'.


## 5. Antirealism



The majority of contemporary philosophers are *realists* about
laws; they believe that some reports of what the laws are succeed in
describing reality. There are, however, some *antirealists* who
disagree.



For example, van Fraassen, Giere, and also Mumford believe that
there are no laws. Van Fraassen finds support for his view in the
problems facing accounts like Lewis's and Armstrong's, and
the perceived failure of Armstrong and others to describe an adequate
epistemology that permits rational belief in laws (1989,
130, 180-181). Giere appeals to the origins of the use of
the concept of law in the history of science (1999 [f.p. 1995],
86-90) and contends that the generalizations often described as
laws are not in fact true (90-91). Mumford's reasons are
more metaphysical; he maintains that, in order to govern, laws must be
external to the properties they govern, but, to be external in this
way, the governed properties must lack proper identity conditions
(2004, 144-145). Others adopt a subtly different sort of
antirealism. Though they will utter sentences like 'It is a law
that no signals travel faster than light', they are antirealists
in virtue of thinking that such sentences are not (purely)
fact-stating. Whether this Einsteinian generalization is a law is not
a fact about the universe; it is not something out there waiting to be
discovered. Reports of what are laws only project a certain attitude
(in addition to belief) about the contained generalizations (Blackburn
1984, 1986, Ward 2002, 197). Ward takes the attitude to be one
regarding the suitability of the generalization for prediction and
explanation. 



The challenge for antirealism is to minimize the havoc lawless reality
would play with our folk and scientific practices. Regarding science,
the examples and uses of laws described at the start of this entry
attest to 'law' having a visible role in science that
scientists seem prepared to take as factive. Regarding our folk
practices, though 'law' is not often part of
run-of-the-mill conversations, an antirealism about lawhood would
still have wide-ranging consequences. This is due to lawhood's
ties to other concepts, especially the *nomic* ones, concepts
like the counterfactual conditional, dispositions, and causation. For
example, it seems that, for there to be any interesting counterfactual
truths, there must be at least one law of nature. Would an ordinary
match in ordinary conditions light if struck? It seems it would, but
only because we presume nature to be regular in certain ways. We think
this counterfactual is true because we believe there are laws. Were
there no laws, it would not be the case that, if the match were
struck, it would light. As a result, it would also not be the case
that the match was *disposed* to ignite, nor the case that
striking the match would *cause* it to light.



Could an antirealist deflect this challenge by denying the connections
between lawhood and other concepts? Would this allow one to be an
antirealist about laws and still be a realist about, say,
counterfactuals? The danger lurking here is that the resulting
position seems bound to be *ad hoc*. Concepts like the
counterfactual conditional, dispositions, and causation exhibit many
of the same puzzling features that lawhood does; there are parallel
philosophical questions and puzzles about these concepts. It is hard
to see what would warrant antirealism about lawhood, but not the other
nomic concepts.


## 6. Antireductionism



Some advocate antireductionist, antisupervenience views (Carroll 1994,
2008, Ismael 2015, Lange 2000, 2009, Maudlin 2007, Woodward
1992). Regarding the question of what it is to be a law, they
reject the answers given by Humeans; they often deny Humean
supervenience, and they see no advantage in an appeal to universals.
They reject all attempts to say what it is to be a law that do not
appeal to nomic concepts. Yet they still believe that there really are
laws of nature; they are not antirealists. Maudlin takes
lawhood to be a primitive status and laws to be ontological primitives
-- fundamental entities in our ontology. His project is to show
what work laws can do, defining physical possibility in terms of laws
and sketching law-based accounts of the counterfactual conditional and
of explanation. Carroll's analysis of lawhood is in
terms of causal/explanatory concepts. The starting point is the
intuition that laws are not accidental, that they are not
coincidences. Not being a coincidence, however, is not all there is to
being a law. For example, it might be true that there are no gold
spheres greater than 1000 miles in diameter because there is so little
gold in the universe. In that case, strictly speaking, that
generalization would be true, suitably general, and not a coincidence.
Nevertheless, that would not be a law. Arguably, what blocks this
generalization from being a law is that something *in* nature
-- really, an initial condition of the universe, the limited
amount of gold -- accounts for the generalization. Contrast this
with the law that inertial bodies have no acceleration. With this and
other laws, it seems that it holds because of nature (itself). 
Lange's (2000, 2009) treatment includes an account of what it is
to be a law in terms of a counterfactual notion of stability. The
overall account is intricate, but the basic idea is this: Call a
logically closed set of true propositions stable if and only if the
members of the set would remain true given any antecedent that is
consistent with the set itself. So, for example, the set of logical
truths is trivially stable, because logical truths would be true no
matter what. A set that included the accidental generalization that
all the people in the room are sitting, but is consistent with the
proposition that someone in the room shouts 'Fire!' would
not be a stable set; if someone were to shout 'Fire', then
someone in the room would not be sitting. Lange argues that no
stable set of sub-nomic facts -- except maybe the set of all
truths -- contains an accidental truth. "By identifying the
laws as the members of at least one non-maximal stable set, we
discover how a sub-nomic fact's lawhood is fixed by the
sub-nomic facts and the subjunctive facts about them" (2009,
43).



Attempts to undermine antireductionism often include challenges to
antisupervenience like those mentioned at the end of Section 4. 
Hildebrand challenges Carroll's and Maudlin's
antireductionisms based on the failure of primitive laws to explain
the uniformity of nature (Hildebrand, 2013). A symposium on
Lange's (2009)  *Laws and Lawmakers* includes, along with
Lange's replies, a variety of criticisms from Carroll, Loewer,
and Woodward. (See Lange *et al.*, 2011.) Demerest (2012)
raises three challenges to Lange's antireductionism all centered
on whether subjunctives are suited to play the role of lawmakers.


## 7. Induction



Goodman thought that the difference between laws of nature and
accidental truths was linked inextricably with the problem of
induction. In his "The New Riddle of Induction" (1983,
[f.p. 1954], 73), Goodman says,



> 
> Only a statement that is *lawlike* -- regardless of its
> truth or falsity or its scientific importance -- is capable of
> receiving confirmation from an instance of it; accidental statements
> are not.
> 



(Terminology: *P* is lawlike only if *P* is a law if
true.) Goodman claims that, if a generalization is accidental (and so
not lawlike), then it is not capable of receiving confirmation from
one of its instances.



This has prompted much discussion, including some challenges. For
example, suppose there are ten flips of a fair coin, and that the
first nine land heads (Dretske 1977, 256-257). The first nine
instances -- at least in a sense -- confirm the
generalization that all the flips will land heads; the probability of
that generalization is raised from (.5)10
up to .5. But this generalization is not
lawlike; if true, it is not a law. It is standard to respond to such
an example by arguing that this is not the pertinent notion of
confirmation (that it is mere "content-cutting") and by
suggesting that what does require lawlikeness is confirmation of the
generalization's unexamined instances. Notice that, in the coin
case, the probability that the tenth flip will land heads does not
change after the first nine flips land heads. There are, however,
examples that generate problems for this idea too.



> 
> Suppose the room contains one hundred men and suppose you ask fifty of
> them whether they are third sons and they reply that they are; surely
> it would be reasonable to at least increase somewhat your expectation
> that the next one you ask will also be a third son (Jackson and
> Pargetter 1980, 423)
> 



It does no good to revise the claim to say that no generalization
*believed* to be accidental is capable of confirmation. About
the third-son case, one would know that the generalization, even if
true, would not be a law. The discussion continues. Frank Jackson and
Robert Pargetter have proposed an alternative connection between
confirmation and laws on which certain counterfactual truths must
hold: observation of *A*s that are *F*-and-*B*
confirms that all non-*F* *A*s are *B*s only if
the *A*s would still have been both *A* and *B*
if they had not been *F*. (This suggestion is criticized by
Elliott Sober 1988, 97-98.) Lange (2000, 111-142) uses a
different strategy. He tries to refine further the relevant notion of
confirmation, characterizing what he takes to be an intuitive notion
of inductive confirmation, and then contends that only generalizations
that are not believed not to be lawlike can be (in his sense)
inductively confirmed.



Sometimes the idea that laws have a special role to play in induction
serves as the starting point for a criticism of Humean analyses.
Dretske (1977, 261-262) and Armstrong (1983, 52-59, and
1991) adopt a model of inductive inference that involves an inference
to the best explanation. (Also see Foster 1983 and 2004.) On its
simplest construal, the model describes a pattern that begins with an
observation of instances of a generalization, includes an inference to
the corresponding law (this is the inference to the best explanation),
and concludes with an inference to the generalization itself or to its
unobserved instances. The complaint lodged against Humeans is that, on
their view of what laws are, laws are not suited to explain their
instances and so cannot sustain the required inference to the best
explanation.



This is an area where work on laws needs to be done. Armstrong and
Dretske make substantive claims on what can and can't be
instance confirmed: roughly, Humean laws can't,
laws-as-universals can. But, at the very least, these claims cannot be
quite right. Humean laws can't? As the discussion above
illustrates, Sober, Lange and others have argued that even
generalizations known to be accidental can be confirmed by their
instances. Dretske and Armstrong need some plausible and suitably
strong premise connecting lawhood to confirmability and it is not
clear that there is one to be had. Here is the basic problem: As many
authors have noticed (e.g., Sober 1988, 98; van Fraassen 1987, 255),
the confirmation of a hypothesis or its unexamined instances will
always be sensitive to what background beliefs are in place. So much
so that, with background beliefs of the right sort, just about
anything can be confirmed irrespective of its status as a law or
whether it is lawlike. Thus, stating a plausible principle describing
the connection between laws and the problem of induction will be
difficult.


## 8. Necessity



Philosophers have generally held that some contingent truths are (or
could be) laws of nature. Furthermore, they have thought that, if it
is a law that all *F*s are *G*s, then there need not be
any (metaphysically) necessary connection between *F*-ness and
*G*-ness, that it is (metaphysically) possible that something
be *F* without being *G*. For example, any possible
world that, as a matter of law, obeys the general principles of
Newtonian physics is a world in which Newton's first law is
true, and a world containing accelerating inertial bodies is a world
in which Newton's first law is false. The latter world is also a
world where inertia is instantiated but does not necessitate zero
acceleration. Some *necessitarians*, however, hold that all
laws are necessary truths. (See Shoemaker 1980 and 1998, Swoyer 1982,
Fales 1990, Bird 2005. See Vetter 2012 for criticism of Bird 2005 from
within the dispositional essentialist camp.) Others have held
something that is only slightly different. Maintaining that some laws
are singular statements about universals, they allow that some laws
are contingently true. So, on this view, an
*F*-ness/*G*-ness law could be false if *F*-ness
does not exist. Still, this difference is minor. These authors think
that, for there to be an *F*-ness/*G*-ness law, it must
be necessarily true that all *F*s are *G*s. (See
Tweedale 1984, Bigelow, Ellis, and Lierse 1992, Ellis and Lierse 1994,
and Ellis 2001, 203-228; 2009, 51-72.)



Two reasons can be given for believing that being a law does not
depend on any necessary connection between properties. The first
reason is the conceivability of it being a law in one possible world
that all *F*s are *G*s even though there is another
world with an *F* that is not *G*. The second is that
there are laws that can only be discovered in an *a posteriori*
manner. If necessity is always associated with laws of nature, then it
is not clear why scientists cannot always get by with *a
priori* methods. Naturally, these two reasons are often
challenged. The necessitarians argue that conceivability is not a
guide to possibility. They also appeal to Saul Kripke's (1972)
arguments meant to reveal certain *a posteriori* necessary
truths in order to argue that the *a posteriori* nature of some
laws does not prevent their lawhood from requiring a necessary
connection between properties. In further support of their own view,
the necessitarians argue that their position is a consequence of their
favored theory of dispositions, according to which dispositions have
their causal powers essentially. So, for example, on this theory,
charge has as part of its essence the power to repel like charges.
Laws, then, are entailed by the essences of dispositions (cf., Bird
2005, 356). As necessitarians see it, it is also a virtue of their
position that they can explain why laws are counterfactual-supporting;
they support counterfactuals in the same way that other necessary
truths do (Swoyer 1982, 209; Fales 1990, 85-87).



The primary worry for necessitarians concerns their ability to sustain
their dismissals of the traditional reasons for thinking that some
laws are contingent. The problem (cf., Sidelle 2002, 311) is that they
too make distinctions between necessary truths and contingent ones,
and even seem to rely on considerations of conceivability to do so.
Prima facie, there is nothing especially suspicious about the judgment
that it is possible that an object travel faster than light. How is it
any worse than the judgment that it is possible that it is raining in
Paris? Another issue for necessitarians is whether their essentialism
regarding dispositions can sustain all the counterfactuals that are
apparently supported by laws of nature (Lange 2004).


## 9. Laws, Circularity and Prospects for Explanation



Going back to Armstrong (1983, 40), there have been
challenges to those who hold a Humean account of laws, and about
whether Humean laws are explanatory. More recently, Maudlin has
put the challenge in a perspicuous way:



> 
> 
> If one is a Humean, then the Humean Mosaic itself appears to admit of
> no further explanation. Since it is the ontological bedrock in terms
> of which all other existent things are to be explicated, none of these
> further things can really account for the structure of the Mosaic
> itself. This complaint has been long voiced, commonly as an objection
> to any Humean account of laws. If the laws are nothing but generic
> features of the Humean Mosaic, then there is a sense in which one
> cannot appeal to those very laws to explain the particular features of
> the Mosaic itself: the laws are what they are in virtue of the Mosaic
> rather than vice versa (Maudlin 2007, 172).
> 
> 
> 



Loewer (2012, 131) offers a response to the issue
that Maudlin highlights. Loewer's response is that the
great Humean mosaic makes the laws of nature true. The move he makes
to avoid the circularity is that Humean laws do not
*metaphysically* explain elements of the mosaic, but they do
*scientifically* explain aspects of the mosaic, suggesting that
there are two notions of explanation and so no circularity. This
move has spawned a recent slew of excellent journal articles regarding
the viability of Loewer's move (see especially Lange 2013,
Miller 2015, Roski 2018, and Shumener 2017).




An increasingly popular way to look at the relation between laws and
their instances is taking instances as grounding laws. No individual
instance of a law can fully ground the law, but a conjunction of
instances does more fully ground the law. Another plausible way of
viewing the relation between laws and their instances is to see laws
as grounding their instances (Emery 2019). Because the grounding
relation is non-symmetric, both of these views cannot be true. The way
out of this dilemma is one that illuminates the debate about
explanation in an interesting way. Consider that while (*P*
& *Q*) is a full ground for *Q*, it seems wrong to
claim that (*P* & *Q*) explains why *Q*
(Roski 2018). This is because the content of the explanandum (what is
to be explained) is embedded in the content of the explanans (what is
intended to do the explaining), and something cannot explain itself
(or be an essential part of an explanation of itself). Notice that
this formulation exposes the problem: if the explanans includes the
explanandum as part of its content, it makes the explanation devoid of
understanding. Ones audience would have to already have had an
understanding of the explanandum. Successful explanations are not
circular, so anyone taking laws as grounds for their instances ought
not to think that the grounding relation is explanatory. The point
here is not to show that grounding is not an explanatory relation, but
rather to show that *laws of nature* are not suited to explain
their instances. Circularity also infects the DN model of
explanations. As the authors of the DN model pointed out:



> 
> ... the content of the explanandum is contained in the
> explanans. That is correct since the explanandum is a semantic
> consequence of the explanans (Hempel and Oppenheim, cf. 1948, 162;
> also see Shumener 2017, 793).



The issue here undermines the importance of the role for explanations
to provide understanding. The required validity brings semantic
circularity, because the content of the explanans would then be
sufficient for the truth of the explanandum. In line with the regular
presentations of the DN model, at least one law of nature is required
to be a premise in an "explanatory argument". Indeed, at
least one law needs to be essential to the validity of the argument,
and the laws being part of the explanans are clearly a factor
regarding the circularity. To add to these challenges, it is good
to remember what Dretske pointed out regarding laws and
explanation.



> 
> 
> To say that a law is a universal truth having explanatory power is
> like saying a chair is a breath of air used to seat people. You cannot
> make a silk purse out of a sow's ear, not even a very good
> sow's ear; and you cannot make a generalization, not even a
> purely universal generalization, explain its instances. The fact that
> every F is G fails to explain why any F is G, and it fails to explain
> it, not because its explanatory efforts are too feeble to have
> attracted our attention, but because the explanatory attempt is never
> even made ... Subsuming an instance under a universal
> generalization has exactly as much explanatory power as deriving Q
> from P & Q. None (1977, 26).
> 
> 
> 



Dretske's reaction to this quotation was to conclude that
laws of nature are not universally quantified conditionals; that they
are not mere generalizations. Instead, it was thought that laws
had to be a different kind of thing: a relation between universals,
physically necessary generalizations, or a true axiom or theorem of an
ideal system, or even a metaphysically necessary generalization. 
Another approach needs to be considered, maybe, just maybe, laws of
nature are generalizations and just aren't explanatory in any
very significant way. This is an approach that identifies what sort of
entity a law of nature is.


## 10. Physics and the Special Sciences



Two separate (but related) questions have received much recent
attention in the philosophical literature surrounding laws. Neither
has much to do with what it is to be a law. Instead, they have to do
with the nature of the generalizations scientists try to discover.
First: Does any science try to discover exceptionless regularities in
its attempt to discover laws? Second: Even if one science --
fundamental physics -- does, do others?


### 10.1 Do Physicists try to discover Exceptionless Regularities?



Philosophers draw a distinction between *strict*
generalizations and *ceteris-paribus* generalizations. The
contrast is supposed to be between universal generalizations of the
sort discussed above (e.g., that all inertial bodies have no
acceleration) and seemingly less formal generalizations like that,
other things being equal, smoking causes cancer. The idea is that the
former would be contradicted by a single counterinstance, say, one
accelerating inertial body, though the latter is consistent with there
being one smoker who never gets cancer. Though in theory this
distinction is easy enough to understand, in practice it is often
difficult to distinguish strict from *ceteris-paribus*
generalizations. This is because many philosophers think that many
utterances which include no explicit *ceteris-paribus* clause
implicitly do include such a clause.



For the most part, philosophers have thought that if scientists have
discovered any exceptionless regularities that are laws, they have
done so at the level of fundamental physics. A few philosophers,
however, are doubtful that there are exceptionless regularities at
even this basic level. For example, Cartwright has argued that
the descriptive and the explanatory aspects of laws conflict.
"Rendered as descriptions of fact, they are false; amended to be
true, they lose their fundamental explanatory force" (1980, 75).
Consider Newton's gravitational principle, *F* =
G*mm*'/*r*2.
Properly understood, according to Cartwright, it says that for any two
bodies the force between them is
G*mm*'/*r*2. But
if that is what the law says then the law is not an exceptionless
regularity. This is because the force between two bodies is influenced
by other properties than just their mass and the distance between
them, by properties like the charge of the two bodies as described by
Coulomb's law. The statement of the gravitational principle can
be amended to make it true, but that, according to Cartwright, at
least on certain standard ways of doing so, would strip it of its
explanatory power. For example, if the principle is taken to hold only
that *F* = G*mm*'/*r*2
if there are no forces other than
gravitational forces at work, then though it would be true it would
not apply except in idealized circumstances. Lange (1993) uses a
different example to make a similar point. Consider a standard
expression of the law of thermal expansion: 'Whenever the
temperature of a metal bar of length *L*0
changes by *T*, the length of the bar
changes by *L* = *k**L*0*T*,'
where *k* is a
constant, the thermal expansion coefficient of the metal. If this
expression were used to express the strict generalization
straightforwardly suggested by its grammar, then such an utterance
would be false since the length of a bar does not change in the way
described in cases where someone is hammering on the ends of the bar.
It looks like the law will require provisos, but so many that the only
apparent way of taking into consideration all the required provisos
would be with something like a *ceteris-paribus* clause. Then
the concern becomes that the statement would be empty. Because of the
difficulty of stating plausible truth conditions for *ceteris-paribus*
sentences, it is feared that 'Other things being equal,
*L* = *kL*0*T*'
could only mean
'*L* = *kL*0*T*
provided that *L* =
*kL*0*T*.'



Even those who agree with the arguments of Cartwright and Lange
sometimes disagree about what ultimately the arguments say about laws.
Cartwright believes that the true laws are not exceptionless
regularities, but instead are statements that describe causal powers.
So construed, they turn out to be both true and explanatory. Lange
ends up holding that there are propositions properly adopted as laws,
though in doing so one need not also believe any exceptionless
regularity; there need not be one. Giere (1999) can usefully be
interpreted as agreeing with Cartwright's basic arguments but
insisting that law-statements don't have implicit provisos or
implicit *ceteris-paribus* clauses. So, he concludes that there
are no laws.



Earman and Roberts hold that there are exceptionless and lawful
regularities. More precisely, they argue that scientists doing
fundamental physics do attempt to state strict generalizations that
are such that they would be strict laws if they were true:



> 
> Our claim is only that ... typical theories from fundamental
> physics are such that *if* they were true, there would be
> precise proviso free laws. For example, Einstein's gravitational
> field law asserts -- without equivocation, qualification,
> proviso, *ceteris paribus* clause -- that the Ricci
> curvature tensor of spacetime is proportional to the total
> stress-energy tensor for matter-energy; the relativistic version of
> Maxwell's laws of electromagnetism for charge-free flat
> spacetime asserts -- without qualification or proviso --
> that the curl of the **E** field is proportional to the
> partial time derivative, etc. (1999, 446).
> 



About Cartwright's gravitational example, they think (473, fn.
14) that a plausible understanding of the gravitational principle is
as describing only the *gravitational* force between the two
massive bodies. (Cartwright argues that there is no such component
force and so thinks such an interpretation would be false. Earman and
Roberts disagree.) About Lange's example, they think the law
should be understood as having the single proviso that there be no
external stresses on the metal bar (461). In any case, much more would
need to be said to establish that *all* the apparently strict
and explanatory generalizations that have been or will be stated by
physicists have turned or will turn out to be false. (Earman, *et
al*., 2003 includes more recent papers by both Cartwright and
Lange, and also many other papers on *ceteris-paribus*
laws.)


### 10.2 Could there be any Special-Science Laws?



Supposing that physicists do try to discover exceptionless
regularities, and even supposing that our physicists will sometimes be
successful, there is a further question of whether it is a goal of any
science other than fundamental physics -- any so-called special
science -- to discover exceptionless regularities and whether
these scientists have any hope of succeeding. Consider an economic law
of supply and demand that says that, when demand increases and supply
is held fixed, price increases. Notice that, in some places, the price
of gasoline has sometimes remained the same despite an increase in
demand and a fixed supply, because the price of gasoline was
government regulated. It appears that the law has to be understood as
having a *ceteris-paribus* clause in order for it to be true. This
problem is a very general one. As Jerry Fodor (1989, 78) has pointed
out, in virtue of being stated in a vocabulary of a special science,
it is very likely that there will be limiting conditions --
especially underlying physical conditions -- that will undermine
any interesting strict generalization of the special sciences,
conditions that themselves could not be described in the
special-science vocabulary. Donald Davidson prompted much of the
recent interest in special-science laws with his "Mental
Events" (1980 [f.p. 1970], 207-225). He gave an argument
specifically directed against the possibility of strict
psycho-physical laws. More importantly, he made the suggestion that
the absence of such laws may be relevant to whether mental events ever
cause physical events. This prompted a slew of papers dealing with the
problem of reconciling the absence of strict special-science laws with
the reality of mental causation (e.g., Loewer and Lepore 1987 and
1989, Fodor 1989, Schiffer 1991, Pietroski and Rey 1995).



Progress on the problem of provisos depends on three basic issues
being distinguished. First, there is the question of what it is to be
a law, which in essence is the search for a necessarily true
completion of: "*P* is a law if and only if
...". Obviously, to be a true completion, it must hold for
all *P*, whether *P* is a strict generalization or a
*ceteris-paribus* one. Second, there is also a need to
determine the truth conditions of the generalization sentences used by
scientists. Third, there is the *a posteriori* and scientific
question of which generalizations expressed by the sentences used by
the scientists are true. The second of these issues is the one where
the action needs to be.



On this score, it is striking how little attention is given to the
possible effects of context. Mightn't it be that, when the
economist utters a certain strict generalization sentence in an
"economic setting" (say, in an economics textbook or at an
economics conference), context-sensitive considerations affecting its
truth conditions will have it turn out that the utterance is true?
This might be the case despite the fact that the same sentence uttered
in a different context (say, in a discussion among fundamental
physicists or better yet in a philosophical discussion of laws) would
result in a clearly false utterance. These changing truth conditions
might be the result of something as plain as a contextual shift in the
domain of quantification or perhaps something less obvious. Whatever
it is, the important point is that this shift could be a function of
nothing more than the linguistic meaning of the sentence and familiar
rules of interpretation (e.g., the rule of accommodation).



Consider a situation where an engineering professor utters,
"When a metal bar is heated, the change in its length is
proportional to the change in its temperature" and suppose a
student offers, "Not when someone is hammering on both ends of
the bar." Has the student shown that the teacher's
utterance was false? Maybe not. Notice that the student comes off
sounding a bit insolent. In all likelihood, such an unusual situation
as someone hammering on both ends of a heated bar would not have been
in play when the professor said what he did. In fact, the reason the
student comes off sounding insolent is because it seems that he should
have known that his example was irrelevant. Notice that the
professor's sentence needn't include some implicit
*ceteris-paribus* clause in order for his utterance to be true;
as this example illustrates, in ordinary conversations, plain old
strict generalization sentences are not always used to cover the full
range of actual cases. Indeed, they are rarely used in this way. If
special scientists do make true utterances of generalization sentences
(sometimes *ceteris-paribus* generalization sentences,
sometimes not), then apparently nothing stands in the way of them
uttering true special-science lawhood sentences. The issue here has
been the truth of special-science generalizations, not any other
requirements of lawhood.


## 11. Concluding Remarks: What is Next?



How will matters progress? How can philosophy advance beyond the
current disputes about laws of nature? Three issues are
especially interesting and important ones. The first concerns
whether lawhood is a part of the content of scientific theories. This
is a question often asked about causation, but less frequently
addressed about lawhood. Roberts offers an analogy in support of the
thought that it is not: It is a postulate of Euclidean geometry that
two points determine a line. But it is not part of the content of
Euclidean geometry that this proposition is a postulate. Euclidean
geometry is not a theory about postulates; it is a theory about
points, lines, and planes ... (2008, 92). This may be a
plausible first step toward understanding the absence of some
nomic terms from formal statements of scientific theories. The
second issue is whether there are any contingent laws of nature.
Necessitarians continue to work on filling in their view, while
Humeans and others pay relatively little attention to what they are up
to; new work needs to explain the source of the underlying commitments
that divide these camps. Finally, more attention needs to be
paid to the language used to report what are the laws and the language
used to express the laws themselves and whether the laws
explain. It is clear that recent disputes about generalizations
in physics and the special sciences turn on precisely these matters,
but exploring them may also pay dividends on central matters regarding
ontology, realism vs. antirealism, and supervenience.