## 1. Introduction


### 1.1. Systematic introduction



In philosophy of science and metaphysics, laws of nature take center
stage: Many explications of philosophical key concepts in philosophy
of science essentially rely on laws of nature. For instance, a
majority of theories of causation, explanation, confirmation,
determinism, and counterfactuals presuppose laws of nature (see the
SEP-entry on laws of nature).



Until the second half of the 20th century, certain characteristics of
laws of nature were taken for granted by philosophers of science: Laws
of nature were taken to be true, logically contingent, universal
statements that support counterfactual claims. Laws were taken to play
a major role in explanation, induction, confirmation, causation, and
in counterfactual reasoning. The paradigm cases for such laws were
taken from (fundamental) physics. Newton's second law
\((F=ma)\) or the Schrodinger-equation fit the
bill of the traditional concept of a law of nature. Philosophers of
science were aware of the fact that there are hardly any universal
laws of this kind in biology, psychology, economics or other special
sciences. This posed no problem as long as there was a consensus that
this might be conceived as a shortcoming of the special sciences.
However, with the success especially of the biological sciences it
became clear that there is genuine scientific knowledge that does not
conform to the paradigm of physics. As a consequence, scientific
practice in the special sciences was no longer taken to be deficient
but was analyzed as a legitimate practice different from physics. The
generalizations of the special sciences including biology, psychology,
and economics seem to differ in an important way from the
generalizations of fundamental physics. In order to illustrate this
difference, here are some examples of generalizations that play an
explanatory role in various special sciences:


* *Snell's Law*: "At the interface between dielectric
media, there is (also) *a refracted ray* in the second medium
lying in the plane of incidence, making an angle
\(\theta\_t\), with the normal and obeying
Snell's law: 



\[
\frac{\sin \theta}{\sin \theta\_t} = \frac{n\_2}{n\_1}
\]


where \(v\_1\) and \(v\_2\) are the velocities of propagation in the two
media, and \(n\_1 =(c/v\_1), n\_2 =(c/v\_2)\) are the indices of
refraction." (Miles V. Klein's *Optics*, cited from
Cartwright 1983, 46)
* The lack of vitamin C causes scurvy.
* *Mendel's Law of Segregation*: "In a parent, the
alleles for each character separate in the production of gametes, so
that only one is transmitted to each individual in the next
generation." (Rosenberg & McShea 2008, 36)
* *The Area Law in Island-Biogeography*: "the equilibrium
number \(S\) of a species of a given taxonomic group on an island
(as far as creatures are concerned) increases [polynomially] with the
islands area [A]: \(S = c\times A^z\). The (positive-valued)
constants \(c\) and \(z\) are specific to the taxonomic group
and island group." (Lange 2000, 235f.; Lange 2002, 416f.)
* People's actions are goal-oriented, in the sense that if person
\(x\) wants \(A\) and believes \(B\) to be an optimal means
for achieving \(A\), then \(x\) will attempt to do \(B\)
(Fodor 1987; Dray 1957, 132ff).
* *The Law of Demand*: Under the condition of perfect
competition, an increase of demand of a commodity leads to an increase
of price, given that the quantity of the supply of the commodity
remains constant (Roberts 2004, 159; Kincaid 2004, 177).
* The lack of social integration results in a higher probability of
attempted suicides.



These generalizations may be explained in terms of lower level
sciences, but this is not our focus here. What is important for our
purposes is that all of these generalizations are non-universal. That
is, there are (actual and merely possible) situations in which the
above generalizations do *not* hold, although all the
conditions obtain that are explicitly stated in the antecedents of
these generalizations. For instance, there are situations in which the
following is the case: at the interface between dielectric media,
there is a refracted ray in the second medium lying in the plane of
incidence, making an angle \(\theta\_t\), with the normal and this ray
does not obey the equation \(\sin \theta/\sin \theta\_t = n\_2 /n\_1\),
the lack of vitamin C does not cause scurvy, the alleles separate
differently than Mendel's law of Segregation describes, the
equilibrium number of a species does not increase in accord with the
Area Law, people fail to act in a goal-oriented fashion, and an
increase of demand of a commodity does not lead to an increase of
price (even if the condition of perfect competition were
realized).



However, even though the generalizations are non-universal they do
play a role in explanations and predictions, they may be used for
purposes of manipulation and they support counterfactuals.



The question arises how to deal with this situation. Are these
generalizations to be classified as laws--laws that pertain only
to special conditions, i.e., *ceteris paribus* laws--or
are they something entirely different. Given the fact that laws play a
major role in most accounts of explanation, causation etc., there seem
to be basically three options:


1. These generalizations do not qualify as laws, because they are
non-universal. Therefore, there are no genuine explanations in those
areas of science that rely on these generalizations, most notably the
special sciences.
2. These generalizations do not qualify as laws, because they are
non-universal. However, explanation, causation etc. does not
presuppose the existence of universal laws.
3. These generalizations do qualify as laws, despite the fact that
they are non-universal, because they have important features in common
with universal laws, most importantly the support of counterfactuals.



The difference between options (b) and (c) may be to some extent
merely terminological. Woodward and Hitchcock, for example, replace
laws with 'invariant generalizations' (see section 6.2)
but admit that their account may be read as a
*reconceptualization* of lawhood (see Woodward and Hitchcock
2003, hereafter EG1, 3).



The motivation of those who take option (c) and stick to the concept
of law is that the generalizations cited above have enough in common
with Newton's first law or the Schrodinger-equation to
classify them as laws. For instance, they play the same kind of role
in explanation, prediction, and they support counterfactuals. There
are two beneficial consequences of this particular approach. First
(against option (a)), it need not be denied that there are genuine
explanations etc. in the special sciences. Second (against option
(b)), in analogy to the traditional view, the concept of a law (albeit
of a non-universal or *ceteris paribus* law) can be used in
explications of notions such as explanation, and causation.



The main problem for those who choose option (c) is that they have to
spell out what is meant by a non-universal law (statement) and how it
can perform similar explanatory and predictive tasks as a universal
law. A convenient way to reconstruct the debate about option (c) is to
understand non-universal laws as statements that are qualified by a
*ceteris paribus* (henceforth, "cp") clause. Yet,
how to understand the meaning of the cp-clause is an open question
that is answered differently by various philosophers.



Before we discuss various accounts of cp-laws, several issues
surrounding the notion of "*ceteris paribus*"
should be distinguished in order to structure the debate:


1. How can different kinds of cp-laws be distinguished and
classified? Are there criteria to distinguish genuine cp-laws from
strict laws? How and whether the laws of (fundamental) physics and the
laws of the special sciences differ is one of the crucial questions
driving the debate about *ceteris paribus* laws. (see section
3).
2. Are there any genuine cp-laws at all (see sections 4-5)?
Some authors deny that there are any genuine cp-laws (cf. Schiffer
1991, Earman and Roberts 1999, Earman, Roberts and Smith 2002,
Woodward 2002). While most of these authors agree that there are
sentences that may be reconstructed as containing (implicit)
cp-clauses, they claim that these sentences lack a clear meaning,
cannot be tested, and therefore do not qualify as laws.



However, if there are cp-laws further questions arise:


3. In what disciplines do we find cp-laws (see sections 6-7)?
Are the cp-laws confined to the special sciences or are there cp-laws
also in fundamental physics?
4. What different kinds of cp-laws are used in the sciences? It
seems that in the literature two very different kinds of situations
are envisaged. On the one hand, several authors (including Mill,
Marshall, Cartwright, Pietroski and Rey, Huttemann, and Lipton)
stress that cp-laws describe the behavior of systems under *ideal
or abstract conditions*. Such conditions are--if at all
--only rarely realized (see section 7). On the other hand, there
are authors who claim that, at least in certain disciplines,
cp-conditions are nothing but *normal conditions*, or cp-laws
are laws expressing *normality hypotheses* (see section 8).
5. What are the truth conditions of cp-law statements? Philosophers
have attempted to determine the meaning of cp-clauses in different
ways. The question of meaning is connected to the problem of empirical
content, i.e., the question whether cp-laws have non-trivial and
empirically testable content. Since many philosophers have argued that
cp-laws lack empirically testable content, this problem constitutes a
major challenge to a theory of cp-laws. Formally independent but
methodologically related is the question under what conditions it is
epistemically legitimate to accept cp-laws. (see sections 3-5).



Having distinguished these issues, it should be clear that
*the* debate about cp-clauses and cp-laws does not exist.
Instead, theories of "cp" engage in different and often
separable enterprises.


### 1.2. Overview



Section 2 gives, on the one hand, an account of the explicit use of
cp-clauses (mainly) in the literature on economic issues from
scholasticism to modern economics (section 2.1.). On the other hand,
we sketch two problems in philosophy of science and philosophy of mind
that explain why the issue of cp-laws became an intensely discussed
issue since the 1980s (section 2.2,).



Section 3 distinguishes different readings of the cp-clause and *a
fortiori* different kinds of cp-laws. In section 3.1, exclusive
cp-laws (factors not mentioned in the antecedent of the law are
assumed to be *absent*) are distinguished from comparative
cp-laws (factors not mentioned in the antecedent of the law are
assumed to be *constant*). In line with the bulk of the
relevant literature, we focus on exclusive cp-laws. In section 3.2, a
distinction between definite and indefinite cp-laws is presented.



Section 4 presents the main challenge to cp-laws. It appears that many
laws, e.g., of the special sciences, if taken as strict laws, turn out
to be false; however, if they are hedged by a cp-clause they appear to
lack a clear meaning and appear to be empirically untestable. All of
the major accounts of cp-laws discussed in sections 5 to 8 are
reactions to this challenge.



Section 5 deals with the most influential early accounts of cp-laws.
They take cp-laws to be incomplete laws that can be completed. Whereas
Fodor (1987) and others (see Section 5.1) focus on special science
cp-laws that can be completed by adding factors to the antecedent of
the law that are dealt with in more fundamental sciences, Pietroski
and Rey (1995) take cp-laws to contain (implicit) promises: that is,
when asserting a cp-law, one promises to be able to cite factors that
complete the law on a case by case basis if the consequent of the
cp-law does not obtain (although the antecedent does). In section 5.2,
we present several objections against such completer accounts.



Section 6 presents two versions of stability theories of laws in the
special sciences, namely Lange's (2000, 2002, 2005) stability
theory in section 6.1, and Woodward & Hitchcock's (see EG1 and Hitchcock & Woodward 2003, hereafter EG2)
invariance theory in section 6.2. The common general idea of
invariance or stability theories of laws is that the laws differ from
merely accidental non-laws in virtue of their invariance or stability
under counterfactual suppositions. According to this view, *ceteris
paribus* or non-universal laws differ from universal laws in
degree, not in kind: universal laws hold under all counterfactual
suppositions while *ceteris paribus* or non-universal laws
merely hold for a limited range of suppositions.



Section 7 deals with dispositional accounts of cp-laws. The unifying
idea is that the laws of nature are grounded in dispositional
properties. We distinguish two different dispositional accounts of
cp-laws: On Huttemann's (2014) version, that is based on pioneering
work of Mill and Cartwright, law-statements are statements about
dispositions, tendencies, or capacities rather than about overt
behavior, and thus cp-laws are turned into strict laws. Bird's
(2007) version of dispositionalism, by contrast, does not turn cp-laws
into strict laws. Instead, it identifies cp-laws with generalizations
hedged by a cp-clause that are grounded in non-fundamental
dispositional properties that admit of interferers.



Section 8 deals with normality theories of laws. The guiding idea of
normality theories is that "cp, all \(A\)s are
\(B\)s" means that *normally* \(A\)s are \(B\)s.
The various normality accounts differ in their explication of the
notion of normality. While Schurz (2001b, 2002) interprets normality
as a high conditional, objective probability of the consequent of the
law statement given the antecedent (see section 8.1), Spohn (2012)
suggests to explicate the normality conditions in terms of degrees of
belief and ranking functions over possible worlds (see section
8.2).



Section 9 sketches the relevance of cp-laws in other areas of
philosophy (such as epistemology and ethics).



Section 10 is a survey to recent work on cp-laws in metaphysics,
philosophy of language and linguistics as well as in other
sciences.



Section 11 presents a brief conclusion and an outlook for future
research.



Finally, section 12 provides a concise overview of the relevant
literature. 


## 2. History and Background


### 2.1. A brief historical survey: from scholasticism to modern economics



The Latin phrase "*ceteris paribus*" or
"*caeteris paribus*"--literally meaning
"other things being equal"-- was used in a
non-technical sense by
 Cicero.[1]
 However, most of the early uses of the *ceteris
paribus*-clause are found in economics. In economic contexts the
use of *ceteris paribus* clauses can be traced back to Petrus
 Olivi[2]
 in 1295. In the 16th century, Juan de
 Medina[3]
 and Luis de
 Molina[4]
 used "*ceteris paribus*" while discussing economic
issues.



In 1662, William Petty was probably the first to use the term in an
English language
 publication.[5]
 In his *Treatise of Taxes and Contributions*, Petty qualifies
his labor theory of value by a "*caeteris paribus*"
clause:



> 
> If a man can bring to London an ounce of Silver out of the Earth in
> Peru, in the same time that he can produce a bushel of Corn, then one
> is the natural price of the other; now if by reason of new and more
> easier Mines a man can get two ounces of Silver as easily as formerly
> he did one, then Corn will be as cheap at ten shillings the bushel, as
> it was before at five shillings *caeteris paribus*. (Petty
> 1662, 50, our emphasis)
> 



John Stuart Mill used the explicit phrase "*ceteris
paribus*" only
 occasionally[6]
 but it had an important impact because he characterized economy by
its way of coping with disturbing factors:



> 
> Political economy considers mankind as solely occupied in acquiring
> and consuming wealth [...] not that any political economist was
> ever so absurd as to suppose that mankind is really thus constituted
> [...] when a concurrence of causes produces an effect, these
> causes have to be studied one at a time, and their laws separately
> investigated [...] since the law of the effect is compounded of
> the laws of all the causes which determine it. (Mill 1843,
>  VI.9.3)[7]
> 



Since economy is thus concerned with one cause only, its laws describe
what would happen provided there are no other causal factors.



The view of cp-laws promoted by Mill is the
absence-of-disturbing-factors view (see section 7). Another view is
the normal-tendency view of cp-laws (see section 8). The historical
roots of the view are, for example, found in John Elliot
Cairnes's description of the methodology of economics in his
*Character and Logical Method of Political Economy*:



> 
> The doctrines of political economy are to be understood as asserting,
> not what will take place, but what would or what tends to take place,
> in this sense only they are true. (Cairnes 1888, 69)
> 



Cairnes (1888, 103) uses the expression "*ceteris
paribus*" in order to refer to "what would or what
tends to take place" if normal conditions obtained.



The use of *ceteris paribus*-clauses was advocated and
popularized by Alfred Marshall in the late 19th century. It was
Marshall's genuine contribution to economics to advocate partial
equilibrium analysis. Marshall claimed that an analysis of this kind
holds merely '*ceteris paribus*'. In his
influential *Principles of Economics*, Marshall defines the
task of economists in terms of the phrase '*ceteris
paribus*':



> 
> [Economists answer] a complex question, studying one bit at a time,
> and at last combining his partial solutions into a more or less
> complete solution of the whole riddle. In breaking it up, he
> segregates those disturbing causes, whose wanderings happen to be
> inconvenient, for the time in a pound called *Caeteris
> Paribus*. (Marshall 1890, 366)
> 



In the introduction to the same work, Marshall explains why economics
is interested in the isolation of causes by assuming that *other
things are equal*:



> 
> The forces to be dealt with (in economics) are, however so numerous,
> that it is best to take a few at a time; and to work out a number of
> partial solutions as auxiliaries to our main study. Thus we begin by
> isolating the primary relations of supply, demand and price in regard
> to a particular commodity. We reduce to inaction all other forces by
> the phrase '*other things being equal*': We do not
> suppose that they are inert, but for the time we ignore their
> activity. This scientific device is a great deal older than science:
> it is the method by which, consciously or unconsciously, sensible men
> have dealt immemorial with every difficult problem of ordinary life.
> (Marshall 1890, xiii, our emphasis)
> 



In a similar vein, Lionel Robbins claims in his classic paper *An
Essay on the Nature and Significance of Economic Science* that
economic laws and antecedent facts deductively imply economic
predictions, but only if other things remain
 unchanged.[8]



The use of "*ceteris paribus*" in economics and in
philosophy of economics is not of merely historiographical interest.
In current philosophy of economics and economics, the use of and the
debate on "*ceteris paribus*" is a vital issue:


1. *Philosophy of Economics*. In the debate on philosophy of
economics it is widely recognized that generalizations in economics
are qualified by a *ceteris paribus*-clause--yet, its
interpretation is controversial. (Cf. Hutchison 1938, 40-46;
Blaug 1997, 335, 696; Blaug 1992, 59-62; Cartwright 1989,
161-164; Cartwright 1999, 137-139, 147f.; Hausman 1992,
chapter 8; Rosenberg 1992, 113f.; Kincaid 1996, 63-83; Kincaid
2004; Kincaid & Ross 2009, 5-8; Roberts 2004; Schlicht 1985;
cf. also the contributions by Rosenberg and Hausman in Kincaid &
Ross 2009; cf. also Weber 1906, 128f.; Marx 1867, 12, 1894, 839).
2. *Economics*. Economists themselves use the *ceteris
paribus*-clause. Expression such as "*ceteris
paribus*" and "other things being equal" are
commonly used in textbooks--often they are explicated in a
special section (Cf. Friedman 1953/2008, 154-159; Kaufer 1997;
Keynes 1891, 218, 233, a case study is to be found on pp. 235f.;
Krugman and Wells 2009, 21, 271f.; Mankiw 1998, 66; Mas-Colell,
Whinston and Green 1995; Persky 1990; Samuelson 1955, 9f.; Samuelson
1958, 8; Samuelson and Nordhaus 1958, 7f., 67f.; Schumpeter 1954,
34f.; Varian 1992; Whitaker 2008; Woolridge 2009, 12f.).


### 2.2. Background of the contemporary debate



Outside the economics literature the use of the *ceteris
paribus*-phrase became a central and also controversial issue in
some areas of philosophy of science and in philosophy of mind in the
1980s.



The fact that test procedures for scientific theories or hypotheses
are reliable only if disturbing factors are excluded or at least
controlled for had already been noted by Carnap (1956, 69). Canfield
and Lehrer (1961) have pointed out that in order to figure as premises
of deductive-nomological explanations, the laws of physics have to be
furnished with cp-clauses. They provide the following example to
support their claim:



> 
> 
> Letting '\(Tx\)' mean '\(x\) is a
> thread'; '\(Wx\)' mean the complex
> statement '\(x\) is loaded with a weight exceeding that which
> characterizes its tensile strength'; and
> '\(Bx\)' mean '\(x\) breaks' we
> write the inference schema:
> 
> 
> 
> > 
> > 
> > 
> > |  |  |  |
> > | --- | --- | --- |
> > | (I) | a) | \(L(Tx \amp Wx, Bx)\) |
> > |  | b) | \(Tx \amp Wx\) |
> > |  | c) | \(Bx\) |
> > 
> > 
> > 
> 
> 
> 
> where the function \(L(Tx \amp Wx,
> Bx)\) indicates that there is a lawlike connection between
> the conjunction '\(Tx \amp Wx\)'
> and '\(Bx\)' but leaves the nature of the
> connection unspecified. (Canfield & Lehrer 1961, 205)
> 
> 
> 



The envisaged prediction ('\(Bx\)') no longer
holds if a magnet neutralizes the effect of the weight
('\(Mx\)'). This can be accounted for if the
absence of the magnet is explicitly mentioned in the law. The problem
is that the law must be complete with respect to all such factors that
might prevent the breaking of the thread. The completeness condition
requires that there are no further disturbing factors--i.e., it
requires an exclusive cp-clause (see section 3) Canfield and Lehrer
also present an argument for why these cp-clauses cannot be defined
away, namely that any such attempt leads to an infinite regress of
further cp-clauses.



The issue of disturbing factors was taken up in 1970 by Lakatos who
raised it as a problem for the falsificationist methodology:



> 
> Some scientific theories forbid an event occurring [...] only on
> the condition that no other factor [...] has any influence on it.
> [...] Another way of putting this is to say that some scientific
> theories are normally interpreted as containing a *ceteris
> paribus* clause (Lakatos 1970, 101).
> 



Popper only briefly commented on this claim in a footnote (see Popper
1974, 1186f.), while other authors discussed it explicitly as a
problem for falsificationism (see, for instance, Johansen 1980). 



Hempel argues in his 1988 paper "On Provisos" that every
inferential construal of scientific theory testing is confronted with
the problem that a proviso always has to be added:



> 
> A proviso has to be conceived as a clause which pertains to some
> particular application of a given theory and which asserts that in the
> case at hand, no effective factors are present other than those
> explicitly taken into account. (Hempel 1988, 154)
> 



In fact, Hempel dismisses the phrase "*ceteris
paribus*" as not particularly helpful.



Carnap, Lakatos and Hempel all argue that in theory testing we usually
implicitly assume an additional premise that states that there are no
disturbing factors. Even though in the later literature Hempel's
paper has been taken to be evidence for the existence of
cp-*laws* even in fundamental physics, the issue of whether a
cp-*clause* has to be added as an extra premise in theory
testing has to be distinguished from the claim that the *laws*
themselves should be read as containing an implicit cp-clause (for a
criticism of this conflation cf. Earman and Roberts 1999, 442ff.;
Schrenk 2007a, 25-36; Eliot 2011).



Two further developments lead to the explicit discussion of
cp-*laws*.



*First*, in her paper "The Truth doesn't explain
much" (reprinted in Cartwright 1983), Nancy Cartwright
criticizes the DN-model of explanation on the grounds that the alleged
laws on which the explanation relies (according to the DN-model) are
not true, but rather generalizations that hold under special,
typically ideal, conditions only. These cp-laws, she claims, will not
do the work required:



> 
> *Ceteris paribus* generalizations, read literally without the
> '*ceteris paribus*' modifier, are false. They are
> not only false, but held by us to be false; and there is no ground in
> the covering law picture for false laws to explain anything. On the
> other hand, with the modifier the *ceteris paribus*
> generalizations may be true, but they cover only those few cases where
> the conditions are right. (Cartwright 1983, 45)
> 



Cartwright (1983, 46f.) illustrates her claim using Snell's law
(for details, see our first example in section 1.1 above).
Snell's law is false if read as a universal generalization
because, among other things, it is true only with respect to isotropic
media.



Cartwright explicitly calls Snell's law a cp-*law*. Many
laws we take to be strict universal laws are in fact cp-laws that hold
in special circumstances only. According to Cartwright, the problem
arises that these laws cover only special or ideal conditions (most
media are anisotropic), whereas in scientific practice they are used
to explain phenomena in non-ideal situations as well. Cp-laws seem to
play an essential role in explanations but there is no account
available of how cp-laws can do this work.



*Second*, the issue of cp-*laws* became important in the
philosophy of mind as well. Davidson argued for his anomalous monism
on the basis of a stark contrast between homonomic and heteronomic
generalizations. Homonomic generalizations are those that can be
improved in the same vocabulary, in which they are stated. This
requires, as Davidson puts it, a comprehensive closed system that is
only provided by physics. Outside of physics, we have only heteronomic
generalizations, i.e., generalizations that can be made precise only
by drawing on the vocabulary of another science. As a consequence,
Davidson assumes that only physics has laws, whereas there cannot be
psycho-physical or psychological laws. More generally,
Davidson's argument implies that there are no special science
laws (see Davidson 1963,
 219).[9]



This claim provoked discussions about whether there are genuine
psychological laws (cf. Fodor 1974, 1987, 1991, 1997; Kim 1985, LePore
and Loewer 1987, 1989; Carrier 1998). Some authors, for example
Schiffer (1991) and Earman, Roberts and Smith (2002), have even argued
that no special science (from biology upwards) has genuine laws of its
own. By contrast, most authors have supported only the weaker claim
that there are no *strict* special science laws, but have
argued that special science laws should be construed as laws that
contain implicit cp-clauses.



Fodor (1987) observes that we do use psychological generalizations (in
later papers, e.g., Fodor (1991) he uses 'law' instead of
'generalization') for predicting and explaining human
behavior. These generalizations or laws, says Fodor, are
non-strict--they are hedged by a cp-clause. Nevertheless, these
claims, he insisted are neither false nor uninformative.



> 
> It is, I expect, a long story, how the generalizations of the special
> sciences manage to be both hedged and informative [...]. Telling
> that story is part of making clear why we have the special sciences at
> all; why we don't just have basic physics. (Fodor 1987, 5)
> 



So, by the end of the 1980s cp-laws had become focal in two important
debates: one about the structure of scientific explanations in
general, the other about the status of the special sciences as
autonomous scientific disciplines.


## 3. A Framework for the Discussion: Distinguishing exclusive/ comparative and definite/indefinite *ceteris paribus* laws



It has been argued that "*ceteris paribus*" is an
*ambiguous* notion. Two distinctions will help to disambiguate
the notion: the distinction between comparative and exclusive cp-laws
(section 3.1), and the distinction between definite and indefinite
cp-laws (section 3.2).


### 3.1. Comparative vs. exclusive cp-laws



Schurz (2002) suggests distinguishing between two conceptions of
cp-law: *comparative* versus *exclusive*. Comparative
cp-laws require that factors not mentioned in the antecedent or the
consequent of the law remain unchanged. By contrast, exclusive cp-laws
assert the connection between antecedent and consequent only under the
condition that certain factors are excluded.



The comparative sense of cp-clauses derives from the literal meaning
of "*ceteris paribus*" as "the others being
equal". A comparative cp-law asserts that the increase (or
decrease) of the value of a 'variable' \(X\) leads to
an increase (or decrease) of the value of another variable, say
\(Y\), *provided* that all other (possibly unknown)
\(X\)-independent variables
\(Z\_1 ,\ldots ,Z\_n\) that describe the
states of the considered system (or at least those
\(X\)-independent variables that are potentially interfering)
remain at same values. Thereby, a variable \((Z\_i)\) is
called *\(X\)-independent* iff it is *not* causally
(or nomologically) influenced by \(X\). So, to repeat, that a
variable \(Z\) is \(X\)-independent means (by the above
definition) only that it is not caused by \(X\); though it may well
be a cause of \(X\) (see Schurz 2014 for a novel distinction
between *ceteris paribus* and *ceteris rectis*
laws).



The reason why cp-laws of this type are called
"comparative" in Schurz (2002) is that the requirement
that the other \(X\)-independent variables
\(Z\_1 ,\ldots ,Z\_n\) 'are
equal', or remain at the same values, only makes sense if the
law *compares* *two* states of a described (kind of)
system. These two systems differ in the value of the antecedent
variable \(X\), but agree in their values of the
\(X\)-independent variables
\(Z\_1 ,\ldots ,Z\_n\). If the latter
condition holds for all possible values of the
\(Z\_1 ,\ldots ,Z\_n\), the comparative
cp-law is called unrestricted; otherwise it is called restricted (see
below). Another plausible name for this kind of cp-law would be
'equality cp-law' (this was suggested by the referee),
because it merely requires that the remainder factors are
'equal' in the two compared states of the described
system, rather than that some of them have to be excluded (as in
exclusive cp-laws). 



*A terminological note:* With a "variable" \(X\)
(in the mathematical sense) we mean a functional property of
individuals \(d\in D\) of a domain \(D\), i.e., a
function \(X: D\rightarrow \ran(X)\) from \(D\) into
\(\ran(X)\). The set \(\ran(X)\) (for "range of
\(X\)") is the set of possible values
\(x\in\ran(X)\) of the variable \(X\); if \(X\) is
quantitative, \(\ran(X) = \Re\), i.e., the set of real numbers.
(If \(ran(X) = \Re\) and the values of \(X\) are distributed
according to a given probability distribution, then \(X\) is what
is called a 'random variable' in probability theory; cf.
Hays and Winkler 1975). In what follows,
"\(d\_{(i)}\)" denotes proper individual
variables (in the logical sense) and
"\(x\_{(i)}\)" denotes possible values of
functional variables \(X\_{(i)}\). An important
subcase of comparative cp-laws are *probabilistic* comparative
cp-laws, in which the quantitative variables \(X\) express the
*probabilities* \(P(F)\) of some qualitative
properties expressed by predicates \(F\) (for instance, \(F\)
might be the increased probability of a car accident in example (2)
below).



Here are two examples of comparative cp-laws:



> 
> (1) *Ceteris paribus*, an increase of gas temperature leads to
> a (proportional) increase of gas volume (Gay-Lussac's gas law).
> 



> 
> (2) *Ceteris paribus*, an increase of the blood alcohol level
> of a driver leads to an increased probability of a car accident.
> 



While (1) states a *quantitative* relation between the
increases, only an *ordinal* relation between the increases is
predicted in (2).



Comparative cp-laws are connected to the invariance approach of Woodward and Hitchcock (see SS6.2 below). According to this
account, a cp-generalization expresses what would happen if an
intervention on \(X\) would
 occur.[10]
 In causal graph theory, an intervention is defined as an operation
which changes the value of \(X\) and decouples \(X\) from all
causal parents of \(X\) (cf. Pearl 2000, 23f, Woodward 2003, 98).
This notion of an intervention has the effect that a change of
\(X\)'s value (as the result of an intervention) will not
change the value of any other \(X\)-independent
variable--which is exactly what is required by a comparative
cp-law.



In the philosophical debate, cp-laws have often been understood in the
other exclusive sense. An *exclusive cp-law* asserts that a
certain state or event-type \(A\) leads to another state or
event-type \(B\), *provided* disturbing factors or
influences are *absent*.



*Terminological note:* '\(A\)' is called the
*antecedent* and '\(B\)' the *consequent
'*predicate'. In terms of quantitative variables
\(X\), the predicate formula \(A(d)\) may express, for
example, that \(d\) has a certain \(X\)-value \(x\), or has
changed its \(X\)-value from \(x\_1\) to
\(x\_2\).



Thus, an exclusive cp-clause does not merely require holding all other
\(X\)-independent and potentially disturbing factors constant; it
rather *excludes* the presence of *disturbing* factors.
More generally, it restricts the possible values of the remainder
variables \(Z\_1 ,\ldots ,Z\_n\) to those
value ranges where they cannot disturb the cp-law. Therefore, the
exclusion-clause of exclusive cp-laws may always be equivalently
reformulated as a clause which requires that certain truth conditions
for the exclusive cp-law hold, namely those conditions which exclude
disturbing factors. In this sense, Cartwright has remarked that
"the literal translation is 'other things being
equal', yet it would be more apt to read '*ceteris
paribus*' as 'other things being
*right*'" (1983, 45). Joseph (1980, 777) talked
about "*ceteris absentibus*" clauses, and Hempel
(1988, 29) calls exclusive cp-clauses "Provisos"
("... *provided* disturbing factors are
absent").



Here are two examples of exclusive cp-laws--(3) comes from
physics and (4) from psychology:



> 
> (3) *Ceteris paribus*, planets have elliptical orbits (see
> Lakatos 1970).
> 



> 
> (4) *Ceteris paribus*, people's actions are
> goal-oriented, in the sense that if person \(x\) wants \(A\) and
> believes \(B\) to be an optimal means for achieving \(A\), then
> \(x\) will attempt to do \(B\) (see Fodor 1987; Dray 1957,
> 132ff).
> 



In (3), the cp-clause requires that other (non-negligible) forces on
the planet except that of the sun are--not merely constant
but--*absent*. Likewise, the cp-clause of (4) requires
that any factors causing irrational behavior be absent.



The distinction between comparative and exclusive cp-laws is not
disjoint: some cp-laws are both comparative and exclusive, as for
example in the following example from theoretical economy:



> 
> (5) *Ceteris paribus*, an increase of demand leads to an
> increase of prices.
> 



Not only must the compared economies agree in remainder factors such
as the supply of the good (this is the comparative aspect); various
interferers, such as political regulations which prevent an increase
of prices, must be excluded (that is the exclusive aspect).



Comparative cp-laws which are not restricted by an exclusive cp-clause
are also called *unrestricted* comparative cp-laws. They assert
an invariant connection between an \(X\)-increase and an
\(Y\)-increase for *all* possible values of \(X\) and of
the \(X\)-independent remainder variables
\(Z\_1 ,\ldots ,Z\_n\). Unrestricted
*probabilistic* comparative cp-laws have been suggested as an
explication of generic *causal* relations (Cartwright 1989,
145f; Eells 1991, 85f): they assert that a variable \(X\) is a
probabilistic cause of some other variable \(Y\) in all possible
circumstances, where these circumstances are expressed in terms of the
values of \(X\) *and* the \(X\)-independent remainder
variables \(Z\_i (1\le i\le n)\).



Unfortunately, unrestricted invariance claims are rarely true. One
example mentioned by Cartwright (1989, SS5.2) in which they do
indeed hold is the force law of classical physics, \(f :=
S\_{i \in I}f\_i = m \cdot a\), which equates the total force
\(f\) with the *sum* of all component forces. In that case, an
increase of a component force \(f\_i\) will produce an increase of
acceleration for *all* possible values of \(m\) and the
remainder forces \(f\_i (i\in I)\). In Schurz (2002, SS2, theorem
1) it is proved that \(X\_i\) is connected with \(Y\) by an
unrestricted comparative cp-law, where \(Y\) is a function of
independent variables \(X\_1 ,\ldots ,X\_n\), iff the \(X\_1 ,\ldots
,X\_n\) are
*non-interacting causes* of \(Y\) in a certain (technically
defined) sense.



In moderately complex systems, the composition of causes is usually
*interactive*. Therefore, these systems will only obey
*restricted* comparative cp-laws (Cartwright 1983, 64ff.;
Dupre 1984). Restricted comparative cp-laws are nothing but
comparative cp-laws in the scope of an exclusive cp-clause, which
assert the comparative cp-relation only for a restricted class of
circumstances, expressed in terms of allowed values of the independent
variable \(X\) and the \(X\)-independent variables
\(Z\_i\). In this sense, examples (1) and (2) above
are exclusive-comparative cp-laws because in (1), the cp-connection
between temperature and volume holds only for approximately ideal
gases, and in (2) the cp-connection holds only under psychologically
normal conditions.



Although the *definiens* of a comparative cp-law relating
\(X\) with \(Y\) refers to "all \(X\)-independent
variables", comparative cp-laws are nevertheless empirically
testable by the method of randomized experiments (cf. Fisher 1951). In
this method, one randomly splits a sample into two subgroups, an
"experimental group" and a "control group",
then enforces an increase of the value of \(X\) on the experimental
but not on the control group, and finally compares the two samples
concerning the value of \(Y\). Since the split of the original
sample was random, experimental and control group will agree in their
distribution of *all* \(X\)-independent remainder variables,
apart from random errors. Thus, one can apply this method without
knowing which \(X\)-independent variables are causally relevant for
\(Y\) and which are not. If there is a significant change of the
value of \(Y\) in the experimental group compared with the control
group, the cp-law "\(X\)-increase leads to an
\(Y\)-change" is confirmed (strictly speaking only the
restriction of this law to the population from which the sample has
been taken is strongly confirmed, while confirmation of the
unrestricted cp-law requires tests for a variety of different
"populations", i.e., distributions of \(X\)-independent
remainder variables). On the other hand, if there is no significant
change of \(Y\)'s value, the cp-law is strongly disconfirmed,
even in its unrestricted form. This does not mean that the method of
randomized controlled experiment is free from error possibilities. In
particular, the experimentally induced change of \(X\)'s
value may have smuggled in unrecognized \(X\)-dependent variables
(i.e., unrecognized side-effects) which are partly responsible for the
resulting change of \(Y\). For example, in a teaching experiment
comparing two teaching methods (e.g., the new method and the old
method), the new teaching method may come out as more successful than
the old one--yet not because of intrinsic properties of the new
method, but just because the fact that the teaching method was
*new* has increased the motivation of its teachers.


### 3.2 Definite versus indefinite cp-laws



An important distinction concerning only exclusive cp-laws is that
between *definite* and *indefinite* exclusive cp-laws. A
definite exclusive cp-law *specifies* the disturbing factors
which are excluded (or the validity conditions which are required) in
the antecedent of the law. In other words, a definite exclusive cp-law
"exclusively cp, if \(A(d)\), then
\(B(d)\)" has a *strict completion* of the form
"For all \(d\): if \(A(d)\) and \(C(d)\),
then \(B(d)\)", where the completing condition
"\(C(d)\)" excludes the presence of the
specified disturbing factors in application \(d\). Earman, Roberts
and Smith (2002, 283f.) call definite exclusive cp-laws *lazy
cp-laws*.



However, in most cases such a strict completion is impossible. This is
especially clear for our non-physical example (2), as well as for the
following example (6) from biology:



> 
> (6) exclusively cp, birds can fly.
> 



The number of possible factors which may disturb a bird's flying
capability is potentially infinite. In other words, for every
condition \(C\) which excludes a finite list of such factors the
completed law "All birds satisfying \(C\) can fly" will
still have to face further exceptions--hence, a strict completion
of this law is impossible. Exclusive cp-laws of this sort are called
indefinite exclusive cp-laws, or in the terminology of Earman, Roberts
and Smith (2002), *non-lazy cp-laws*.



It is a widely agreed in philosophy that the real significance of
exclusive cp-laws lies in situations where strict completion is
impossible (for example, cf. Rescher 1994, 14; Pietroski & Rey
1995, 84, 102; Horgan & Tienson 1996, 119f.). In that case, the
exclusive cp-law is *indefinite*, which means that its
exclusive cp-clause consists in a universal second order condition
which excludes all kinds of disturbing factors to the law, whatever
they are. Formally, also an indefinite exclusive cp-law may be written
as a strictly completed law of the form "if disturbing factors
are excluded, then \(A\)s will *always* be
\(C\)s"--only that the meaning of "all disturbing
factors" is now unclear which leads to various sorts of
deficiencies that are discussed in the next section.


## 4. The Challenge: Exclusive *Ceteris Paribus* Laws Between Falsity and Triviality



Exclusive cp-laws of the form "exclusively cp, \(A\)s are
\(B\)s" admit exceptions, i.e., instantiations of \(A\)s
which are not-\(B\)s (Pietroski and Rey speak of
"abnormal" instances; 1995, 88). A philosophical
reconstruction of exclusive cp-laws--which includes restricted
comparative cp-laws and, thus, the majority of all cp-laws--faces
a severe problem. This problem can be articulated in the form of a
dilemma. This dilemma has been formulated by Lange (1993, 235), who
attributes it to Hempel (1988).



*First horn*: If exclusive cp-laws are reconstructed as some
sort of strict law, then they will tend to be false. Typically, it
will not be the case that all \(A\)s satisfying a completer
condition \(C\) are \(B\)s, since the range of potentially
disturbing factors is typically indefinable. For instance, the
relationship between supply and price is not always as the law of
supply says (or, as it seems to say *prima facie*), because an
interfering factor might occur. In other words, special science laws
that instantiate perfect regularities are--mildly
put--"scarce" (Cartwright 1983, 45). Yet, if one
supposes that the law is to be formalized as a universally quantified
conditional sentence, then one counter-instance (due to a disturbing
factor) to the universally quantified sentence means that it is
false.



*Second horn*: If we instead suppose that an indefinite
exclusive *ceteris paribus* clause is attached to the law so
that it means "All \(A\)s are \(B\)s, *if nothing
interferes*", then the cp-law in question is in danger of
lacking empirical content. It lacks empirical content because it seems
to say nothing more than "All \(A\)s are \(B\)s or
not-(All \(A\)s are \(B\)s)". If this is true, then
exclusive cp-laws are analytically true sentences and, therefore,
trivially true. This, however, is an unwelcome consequence because
laws of the special sciences should be reconstructed as empirical
statements--not as sentences being true in virtue of meaning.



Both horns are quite unpleasant results for any philosopher looking
for a theory of non-strict laws. To deal with this dilemma is a
central challenge for every theory of *ceteris paribus* laws
(Lange 1993; Earman and Roberts 1999; Earman, Roberts and Smith 2002).
In the following section, several attempts to cope with this dilemma
are presented and tenacious problems of them are discussed.


## 5. Exclusive CP-laws: The Method of Completers


### 5.1. Semantic and epistemic completers



The general idea behind completer approaches is that the best way to
explicate exclusive cp-laws is to add the missing conditions for a
strict implication into the antecedent of the law statement. There are
two quite different possibilities to do this. The first possibility is
to add these conditions explicitly, by appropriate descriptions of the
first order individual variables involved in the law. If this were the
correct account, all correct cp-laws would turn out to be definite
exclusive cp-laws (or lazy cp-laws in the terminology of Earman,
Roberts and Smith 2002). The second possibility is to add these
conditions by way of a second order quantification over first order
predicate variables, which apply to the first order individual
variables of the law. In this account, cp-laws are turned into
indefinite exclusive cp-laws.



Fodor's starting point for his account of cp-laws is
Davidson's distinction between homonomic and heteronomic
generalizations (see section 2.2):



> 
> Exceptions to the generalizations of a special science are typically
> *inexplicable* from the point of view of (that is, in the
> vocabulary of) that science. That's one of the things that makes
> it a *special* science. But, of course, it may nevertheless be
> perfectly possible to explain the exceptions *in the vocabulary of
> some other science*. [...]. On the one hand the [special
> sciences'] ceteris paribus clauses are ineliminable from the
> point of view of its propriety conceptual resources. But, on the other
> hand, we have--so far at least--no reason to doubt that they
> can be discharged in the vocabulary of some lower-level science
> (neurology, say, of biochemistry; at worst physics). (Fodor 1987, 6)
> 



So, Fodor's idea is that the additional factors whose existence
is required by the exclusive cp-clause cannot be completely specified
within the conceptual resources of the special sciences, although this
could be done (at least in principle) within the vocabulary of some
more fundamental science such as neurophysiology or, ultimately,
fundamental physics. Fodor calls the missing factors
'completers'. A physical microdescription of the
antecedent condition \(A\) is called a realizer of \(A\) (the
same \(A\) may have several different realizers).



> 
> 
> (7) A factor \(C\) is a completer relative to a realizer \(R\)
> of \(A\) and a consequent predicate \(B\) iff:
> 
> 
> 1. \(R\) and \(C\) is strictly sufficient for \(B\)
> 2. \(R\) on its own is not strictly sufficient for \(B\)
> 3. \(C\) on its own is not strictly sufficient for \(B\).
> 
> 
> 
> (Fodor 1991, 23)
> 
> 
> 



A first option (which Fodor conceives of as insufficient) would be to
define the truth conditions of a cp-law as follows:



> 
> (8) cp\((A\rightarrow B)\) is true iff for every realizer \(R\)
> of \(A\) there is a completer \(C\) such that
> \(A \amp C\rightarrow B\) (cf. Hausman 1992, 133-139
> for a similar account).
> 



Schiffer has objected to this account on the grounds that if \(A\)
is a (mental) functional state, which can be realized by very
different states \(R\_i\), it is highly unlikely
that there is a completer for every realization (cf. Schiffer 1991, 5)
Fodor accepts this objection and provides the following account:



> 
> (9) cp\((A\rightarrow B)\) is true iff either (i) for every
> realizer of \(A\) there is a completer \(C\) such that
> \(A \amp C\rightarrow B\) or (ii) if there is no such
> completer for a realization \(R\_i\) of \(A\)
> there must be many *other* laws in the network for \(A\) for
> which \(R\_i\) has completers (see Fodor 1991, 27).
> 



While condition (9)(i) repeats the simpler definition in (8),
condition (9)(ii) expresses the idea that if \(A\) is, for
instance, an intentional state and goes into the antecedent of many
laws, then even if \(R\_i\) does not have
completers in the cp-law in question, cp\((A\rightarrow B)\) is
still true, if \(R\_i\) does have completers in
many other cp-laws.



Mott (1992) provided the following counterexample to condition (ii) of
this account of truth-conditions: cp, if a person is thirsty, then she
will eat salt. This comes out as a true cp-law, because on the one
hand, very likely many, maybe all, realizers of "being
thirsty" lack completers (because nobody tends to eat salt when
thirsty). On the other hand, 'being thirsty' might very
well go into the antecedent of many other cp-laws for which there are
such completers. Mott's counter-example can be avoided if one
adds to (ii) the requirement that for sufficiently many realizers
\(R\_j\) of \(A\) there exist completers with
respect to \(B\) (see Silverberg 1996 and Earman and Roberts 1999,
section 9 for discussion).



Although Fodor starts with the hope that the missing conditions of
exclusive cp-laws could eventually be specified in the fundamental
science, his definition of exclusive cp-laws involves a second order
quantification over variable (unspecified) completers of independent
exception-explainers. Hence, his account of exclusive cp-laws belongs
to the family of indefinite exclusive cp-laws. Because of this fact
Fodor's and related accounts face severe problems, as will be
discussed in section
 5.2.[11]




Schiffer, Fodor and Mott try to explicate cp-laws by ordinary
truth-conditions. In contrast, Pietroski and Rey provide stronger
conditions for what they call "non-vacuous truth", in
terms of *epistemic* conditions such as explanation and
independent confirmability. Pietroski and Rey compare cp-clauses to
cheques:



> 
> These cheques represent a 'promise' to the effect that all
> [counterinstances] of the putative law can be explained by citing
> factors that are [...] independent of that law. If the promise
> cannot be kept the cheque was no good to begin with. (Pietroski and
> Rey 1995, 89)
> 



Completion is conceived here as explanatory completion and it is
required only *post factum*. If there is a counter-instance of
the law in question, we are committed to explain why the law was not
instantiated:



> 
> a cp law holds in a 'closed system', i.e. a system
> considered in abstraction from other, independently existing factors.
> Such a systematization is non-vacuous only to the extent that
> deviations from the regularities that are constitutive of it can be
> explained by those factors. (Pietroski and Rey 1995, 89)
> 



For the cp-law to be acceptable in spite of the counter-instance we
need independent evidence for the existence of the disturbing factor.
It would not be acceptable if the only evidence for the existence of
the disturbing factor were the counter-instance we started with.



Pietroski and Rey (1995, 92) take a cp-law to be non-vacuously true
and hence epistemically acceptable iff (in simplified words) the
following conditions are met:



> 
> 
> (10) cp\((A\rightarrow B)\) is non-vacuously true iff
> 
> 
> 1. '\(A\)' and '\(B\)' are otherwise
> nomological and
> 2. For all \(x\), if \(Ax\), then (either
> \(Bx\) or there exists an independently confirmable factor
> that explains why \(\neg Bx)\), and
> 3. cp\((A\rightarrow B)\) explains at least something as assumed
> in condition (ii).
> 
> 
> 



Pietroski & Rey's (1995) account involves a second order
quantification over unspecified exception-explainers. Thus, as in the
case of semantic completer accounts, their account explicates
indefinite exclusive cp-laws and faces severe problems to be explained
in the next section.


### 5.2. Criticisms: triviality and accidentality



Several authors have independently shown that completer approaches are
unsatisfactory. Earman and Roberts (1999, 454f) provide arguments
which show that an exclusive cp-law in the sense of Pietroski and Rey
cannot escape the problem of *vacuity.* Their example is the
alleged cp-law "cp, all spherical bodies conduct
electricity". Every failure of this law can be explained in
terms of a factor, for which we have independent evidence,
*viz.* the molecular structure of the body in question. Schurz
(2001a) proves that Pietroski & Rey's exclusive cp-laws are
not vacuous but *almost* vacuous. More precisely, the content
of "exclusively cp, \(A\)s are \(B\)s" in the sense
of Pietroski & Rey's definition (10) is equivalent to the
claim that for every event of the form "\(d\) is an \(A\)
and \(d\) is (or is not) a \(B\)" there exists a true
strict completer \(C(d)\) such that either "All
\((A\wedge C)\)s are \(B\)s" or "All
\((A\wedge C)\)s are \(\neg B\)s" is a strictly true
law. This means that conditional on \(A\), every \(B\)-event or
not\(-B\)-event is assumed a deterministic and independently
identifiable cause, whatever this cause may be. This presupposition of
determinism is both too strong and too weak: It is too weak because it
does not establish a relevant nomological connection between \(A\)
and \(B\). It is too strong because in all areas involving random
processes, determinism does not hold. Woodward (2002, SS2) has
demonstrated that a similar criticism affects Fodor's
account.



There is an even more serious *accidentality*-problem that is
connected to the almost-vacuity of indefinite exclusive cp-laws. An
exclusive cp-law may be true although its antecedent is not at all
nomologically or causally relevant for its consequent. All sorts of
events are connected by indefinite exclusive cp-laws, provided only
that they have deterministic causes, which for Pietroski and Rey
(1995) have additionally to be independently testable, and for Fodor
(1991) have to be nomologically relevant (cf. Woodward 2002, 309, (ii,
iii)). For example, Woodward (2002, 310) demonstrates that according
to the explications of Pietroski and Rey (1995), Fodor (1991) and
Hausman (1992), the exclusive cp-law "all charged particles
accelerate at a rate of \(n\) meters/sec\(^2\)" is a
true exclusive cp-law for arbitrary values of \(n\). Schurz (2002,
364) demonstrates that according to these accounts "if a person
looks to the right, she will see a kangaroo" is a true exclusive
cp-law. Also, Earman & Robert's example can be read as
demonstrating the *accidentality* of "cp, all spherical
bodies conduct electricity", although the example was originally
used to show that this statement is vacuously true.


## 6. Invariance & Stability Theories



The common guiding idea of invariance or stability theories is that
the laws differ from non-laws in virtue of their invariance or
stability under counterfactual suppositions.



"Stability" and "invariance" are often taken
to be synonyms. For the sake of clarity, we will use
"stability" to refer to Lange's stability theory of
laws (section 6.1) and we will use "invariance" to refer
to Woodward & Hitchcock's invariance theory of laws (section
6.2).



In stability/invariance accounts, laws qualified by a cp-clause are
considered to be non-strict in the sense that they hold true only
under a limited range of counterfactual suppositions. Different
versions of invariance or stability theories differ in how they
*determine* this limited range of counterfactual suppositions.
In this sense, invariance or stability theories can be understood as
an alternative to exclusive interpretations of non-strict laws and as
an attempt to deal with the dilemma of triviality and falsity.


### 6.1. Counterfactually stable laws and pragmatic knowledge of disturbing factors



According to Lange (2000, 2002, 2005), the universal fundamental laws
in physics and the cp-laws in the inexact special sciences differ only
in degree (for a related account that does not focus on stability
alone see Mitchell (2000)). The lawhood of universal laws and cp-laws
is due to the same property of these statements: their stability.
Lange's (2000, 8f.) approach consists in two steps. First, the
role of laws in scientific reasoning is determined: they figure in
explanations and predictions, they support counterfactuals, and they
are inductively confirmed. And second, it is claimed that the laws can
play this role because of their characteristic stability. So, in order
to understand Lange's account of cp-laws, we must first clarify the
crucial notion of stability.



Lange's basic idea is that laws are lawlike because they are
true in a wide range of possible conditions. Most importantly, the
laws do not only hold true in any (non-nomic) *actual*
conditions but also remain true under all (non-nomic)
*counterfactual* suppositions. Along these lines, Lange (2000,
48f.; cf. 2009a, 20) proposes a preliminary definition of laws in
terms of nomic preservation:



> 
> (11) Some proposition \(l\) is a law iff its truth is preserved
> under all those counterfactual suppositions that are consistent with
> every physical necessity, i.e., under all physically possible
> counterfactual suppositions.
> 



This definition, however, raises an obvious question: what does it
mean to be physically possible? Roughly, a proposition \(p\) is a
physical possibility iff there is some possible world \(w\) where
the same laws of nature hold as in the actual world and \(p\)
obtains at \(w\). But if this is true, Lange's definition of
lawhood seems to be circular, since being consistent with the
*laws* (of the actual world) is ultimately part of the
*definiens* of being a law (of the actual world). Lange (2005,
2009a, 25-28) recognizes the problem of circularity and offers
an alternative account of laws in terms of the notion of stability. It
says, roughly put: 



> 
> (12) A set of statements \(G\) is stable iff every member of
> \(G\) is true, \(G\) is logically closed, and \(G\) remains
> true under every (non-nomic) counterfactual supposition \(p\) that
> is consistent with every member of \(G\) (cf. Lange 2000, 100, 103;
> 2005, 420; Lange 2009a, 29).
> 



This notion of stability in turn leads to a theory of laws:



> 
> (13) A proposition \(l\) is a law iff it is a member of a
> non-maximal stable set \(G\).
> 



According to Lange (cf. 2005, chapter 4; 2009a, chapter 2), there
exist several different stable sets: for example, the set of logical
truths, the set of all physical laws, and the set of all truths (which
is trivially stable). The requirement of \(G\) to be non-maximal,
i.e., to be strictly smaller than the set of all truths, is necessary
to avoid the identification of the set of laws with the set of all
truths.



Lange argues that definition (13) is not affected by the circularity
problem, since stability does not presuppose the notion of a law, and
that it provides a sharp distinction between laws and merely
accidentally true non-laws. Consider some accidentally true universal
statement as for example:



> 
> \(A\): All fruits in my basket are red.
> 



According to Lange, this generalization is an accidental truth because
it is not a member of a (non-maximal) stable set. For assume \(A\)
is a member of some non-maximal set \(\Delta\) (which may contain all
physical laws). By non-maximality, there exists some further
accidental truth \(B\), for example,



> 
> \(B\): I want to put a green fruit into my basket.
> 



which is not in \(\Delta\). If \(\Delta\) is stable, every member of \(\Delta\)
must remain true under the counterfactual supposition that
\(\neg\)A\(\vee \neg B\), since this supposition is compatible with
every member of \(\Delta\). However, Lange (2005, 421) argues that there
are at least some conversational contexts in which \(A\) does not
have priority over \(B\). Hence,
\(\neg A \vee \neg B \rightarrow A\) is not the
case, and therefore \(\Delta\) is not stable under the counterfactual
supposition \(\neg A\vee \neg B\). By contrast, according to
Lange, a law of nature has priority over any accidental truth in every
context, and thus the laws of nature form a stable set. 



Even assuming that Lange's argumentation is valid (see, however,
Demarest (2012) and Hall (2011) for criticism based on the
context-sensitivity of counterfactuals), there is the worry that the
circularity problem has not entirely been dissolved: It reappears at
the level of the truth conditions for counterfactuals, since usually
they are formulated in terms of the laws (see, e.g., Goodman 1947;
Lewis 1973; Maudlin 2007: 21). Lange (2009a: chapter 4) answers this
objection by arguing that subjunctive facts are ontologically
primitive and the truth conditions for counterfactuals do not refer to
laws. He concludes that, therefore, his explications (12) and (13) are
not circular. 



We can now turn to the question how cp-laws differ from universal
laws. According to Lange, cp-laws are stable (sets of) propositions
whose application is *pragmatically* restricted to the purposes
of a scientific discipline. He elaborates this idea with respect to
the laws in the special sciences, or inexact sciences as he calls
them: "A set is stable *for the purpose of an inexact
science* if and only if it is invariant under every counterfactual
supposition of interest to the science and consistent with the
set" (Lange 2002, 416). Lange tries to avoid the horns of the
Falsity-or-Triviality-Dilemma by treating '*ceteris
paribus*' as a name for a set \(I\) of interfering
factors. Notice that \(I\) does not list *all* the possible
interferences which prevent the occurrence of \(B\) in 'cp,
all \(A\)s are \(B\)s', but only those factors that are
*relevant* (for a discipline). One can understand Lange in a
way that he provides two
 strategies[12]
 to determine the members of \(I\): (A) the strategy of
non-negligibility and (B) the strategy of intended interest of a
science. Both strategies try to explicate a methodology that is
implicitly used by scientists in a particular discipline (cf. Lange
2000, 170-174).



(A) *The strategy of non-negligibility*: Instead of providing a
complete list of all interfering factors, scientists merely refer to
those interfering factors "that arise sufficiently often, and
can cause sufficiently great deviations from \(G\)-hood, that a
policy of inferring \(F\)s to be \(G\) [...] would not be
good enough for the relevant purposes" (Lange 2002, 411; Lange
2000, 170f). For instance, consider the economic law "cp, if the
supply of a commodity increases then the price decreases".
According to Lange, it may happen that the increase in supply is so
small that no decrease in price results. But it might as well happen
that the price does not decrease although the supply increases
significantly, because a gigantic comet hitting the planet Earth and
destroying all life on its surface disturbs the instantiation of this
law. The comet causes sufficiently great deviation from a decrease in
the price of a good. Nevertheless, comets are negligible for the
purposes of economists because their occurrence does not arise
sufficiently often to count as interfering factor that is to be
explicitly listed in the cp-conditions.



(B) *The strategy of intended interest of a science*: A law may
still count as stable if it fails to hold under those counterfactual
suppositions that do not fall into the range of the laws intended
purpose and application. This point is best illustrated by an example
from island biogeography--the area law--provided by
Lange:



> 
> It has been suggested that *ceteris paribus*, the equilibrium
> number \(S\) of a species of a given taxonomic group on an island
> (as far as creatures are concerned) increases
>  [polynomially][13]
>  with the islands area [A]: \(S = c\times A^z\). The (positive-valued)
> constants \(c\) and \(z\) are specific to the taxonomic group
> and island group. (Lange 2002, 416f.; cf. Lange 2000, 235f.)
> 



There are counterfactual suppositions for which the area law is not
true. For example, imagine an island where the animals of the species
"chicken" exclusively live on chicken farms. Suppose
further that on these farms chicken are bred and held under extremely
crowded conditions. So, the counterfactual supposition stemming from
this example is "chicken on the island in question are bred
under extremely crowded, artificial conditions set up by
farmers". Obviously, the area law will drastically fail to hold
for this case. Moreover, cases of this kind are not far-fetched
philosophical thought experiments: they do *not* occur
*rarely* in times of cultivated breeding of animals (as
required by *the strategy of non-negligibility*). Nevertheless,
scientists exclude this kind of exceptions because it conflicts with
the intended purpose and application of their discipline (in this
case: island biogeography) (cf. Lange 2002, 417; Lange 2000, 232f. for
further
 examples[14]).



It is illuminating to contrast Lange's strategies of determining
a set of disturbing influences \(I\) (in the light of intended
applications of a law) with the use of cp-clauses that does indeed
render a statement trivial. Lange illustrates this point by an example
of a mere "excuse clause":



> 
> Suppose someone says 'I can run a four-minute mile' but
> with each failure reveals a proviso that she had not stated earlier:
> 'except on this track', 'except on sunny Tuesdays in
> march' and so on. It quickly becomes apparent that this person
> will not acknowledge having committed herself to any claim by
> asserting 'I can run a four-minute mile.' (Lange 2000,
> 172; cf. also Lange 2002, 410)
> 



According to Lange, excuse clauses of this kind differ from cp-clauses
as used in the sciences, because the latter refer to strategies of
determining disturbing factors. Although the relevant set of
disturbing factors is not listed explicitly, it is implicit in the
scientific practice (and the education and studies) in a particular
field of inquiry.



Based on his stability theory of laws, Lange also proposes an argument
for the autonomy and irreducibility of special science laws. Recall
that arguing for the autonomy and irreducibility of the special
sciences with respect to physics was one of the crucial motivations to
start the debate on cp-laws (see section 2.2). Suppose that the laws
of a certain special science \(D\) are stable with respect to the
purposes of \(D\). This pragmatic restriction is unique to
\(D\). In other words, \(D\)'s laws are stable for
counterfactual suppositions that violate other disciplines'
laws, for example the laws of fundamental physics. For instance, Lange
(2002, 420) claims that 'the area law would still have held had
there been birds equipped with organs weakening gravity's pull
somewhat'. Having organs that weaken gravity's pull is
understood as a violation of the law of gravitation. Thus,
\(D\)'s laws are stable under some counterfactual supposition
for which fundamental physics is unstable. This unique range of
stability makes \(D\) epistemically autonomous as a scientific
discipline because \(D\)'s laws can figure, for instance, in
explanations answering some why-questions that no other discipline can
answer as adequately as \(D\) itself (cf. Lange 2000, chapter 8;
Lange 2002, 420f.; also Reutlinger & Koch 2008).


### 6.2. Invariance under interventions of explanatory generalizations



Similar to Lange's stability theory, the invariance approach
(most prominently advocated in EG1, EG2;
Woodward 2000, 2002, 2003) also identifies the stability or invariance
of a generalization as its key feature for performing the role of a
law and figuring in explanations and predictions. As in Lange's
theory, to be invariant means for Woodward and Hitchcock to be true
under several (not necessarily all) counterfactual suppositions.
Despite this basic agreement there are important respects in which the
invariance theory differs from Lange's stability theory. In what
follows we focus on the distinguishing features of the invariance
approach:


1. *The Quantitative Language of Law Statements*: Although
the idea might be implicit in Lange's theory, invariance
theories conceive candidates for law statements explicitly and
straightforwardly as *quantitative* statements. Thus, the
predicates occurring in the statement are variables \(X\) which may
attain certain values \(x\) in ran\((X)\), as explained in
section 3.1. One might object that there may be laws in the historical
and social sciences that are not quantitative but merely qualitative.
But this qualitative character of statements can nevertheless be
translated into quantitative statements with binary variables, that
may only take two possible values, '\(X=1\)' meaning
that some event type \(X\) occurs and '\(X=0\)'
meaning that some event type \(X\) does not occur (see EG1, 10f.).
2. *Counterfactual Suppositions Viewed
Differently--Interventions*: Invariance theories characterize
the counterfactually supposed antecedent conditions under which a
generalization remains stable differently than Lange's stability
theory. Lange simply assumes these conditions, e.g., the temperature
of the gas is 32oC, to be the result of counterfactual
suppositions. We suppose that \(p\) is true and see whether the law
\(l\) remains true supposing \(p\). So, stability theory is
indifferent concerning how the supposed state of affairs comes about.
Invariance theory is more restrictive in characterizing the
counterfactual situations: A counterfactual situation refers to a
*change* in the value of a variable. For an invariance
theorist, it is important to explain *how* the change of a
variable is brought about: the change of the value of a variable must
be the causal outcome of an *intervention*. An intervention
consists in a local change that *sets* a variable \(X\) to a
certain value and at the same time decouples \(X\) from all
\(X\)-independent variables, as explained in section 3. So, an
intervention on \(X\) is a direct, exogenous causal influence on
some variable \(X\) which (1) does only act on \(X\) itself
while (2) other variables \((Z)\) are only changed in virtue of
being directly or indirectly caused by the change in \(X\). Notice
that interventions need not be carried out by human beings. Instead
they are conceived as hypothetical (not necessarily human) causes.
(cf. Woodward 2003, 98; and Woodward 2003, 103f., 123-127 for
arguments against an anthropomorphic interpretation of interventions.)
3. *Different kinds of Variables*: Invariance theories distinguish
two kinds of variables: (1) Explicit variables, that figure explicitly
in a generalization and (2) background variables, that describe
background conditions which are not mentioned in the generalization in
question. Consequently, interventions can be carried out with respect
to both kinds of
 variables.[15]




Similar to Lange's theory, Woodward and Hitchcock use the
intended purposes of inquiry of a certain discipline to distinguish
between interventions into explicit variables and background variables
(cf., e.g., Woodward 2003, 262f). According to their invariance
theory, only the former are of importance to laws in a specific
scientific discipline. Woodward uses an example from economics to
illustrate the point:



> 
> In microeconomics, individual economic agents are often assumed to
> conform to the behavioral generalizations constituting rational choice
> theory (RCT). [...] Even if we assume, for the sake of the
> argument, that these generalizations are roughly accurate descriptions
> of the behavior of many participants in markets, it is clear that
> there are many changes and interventions over which the
> generalizations will fail to be invariant. For example, there are many
> pharmaceutical interventions and surgically produced changes in brain
> structure that will lead previously selfish agents to act in
> non-self-interested ways [...]. *However, economists have not
> generally regarded these sorts of failures of invariance as
> interesting or important, at least if* [...] *they occur
> relatively rarely in the population*. (Woodward 2003, 263)
> 



Obviously, the reason to ignore rare but possible changes in the
background variables is very similar to Lange's *strategy of
non-negligibility*. Woodward contrasts these negligible changes of
background variables with important changes of explicit variables:



> 
> For example, microeconomists often require that fundamental
> explanatory generalizations such as the principles of RCT [rational
> choice theory] be invariant under changes in information available to
> economic agents or under changes in their beliefs and under changes in
> the incentives or relative prices they face. (Woodward 2003, 263; cf.
> Woodward 2003, 264 for a case study from macro-economics)
>
4. *Conditions of Invariance*: Invariance theories differ
drastically from Lange's approach in defining their key notion,
i.e., stability or invariance. According to Woodward and Hitchcock
(EG1, 17) and Woodward (2003, 250):



> 
> (14) A statement of the form \(Y=f(X)\) is invariant
> iff the following so- called *testing intervention condition*
> holds: there are at least two different possible values of an explicit
> variable \(X, x\_1\) and \(x\_2\), for
> which \(Y\) realizes a value in the way that the function \(f\)
> describes, and the fact that \(X\) takes \(x\_1\) or,
> alternatively, \(x\_2\) is the result of an intervention.
> 



The most intuitive case of a testing intervention might be the
following one: \(X=x\_1\) describes an actual state
of affairs while \(X=x\_2\) describes a possible
counterfactual state of affairs. For instance, suppose that
Boyle's gas law --\(p \cdot V = N \cdot k cdot T\)--is true for the actual
temperature of a gas \(g\) of 30degC. According to the testing
intervention condition, Boyle's gas law is stable if it also
holds for (counterfactual) temperature of, say, 40degC. One might
call this kind of invariance "minimal" invariance.
Contrary to Lange's theory, minimal invariance is necessary and
sufficient for the statement to count as invariant. Notice, however,
that invariance under variations of background variables is neither
necessary nor sufficient for being minimally invariant (cf. Woodward
2003, 248; EG1, 7f.).
5. *Degrees of Invariance*: One can still distinguish more or
less stable generalizations. So, invariance is not an all or nothing
matter--instead it admits of degree. Roughly put, the degree of
invariance of a generalization \(G\) is measured by the range of
possible values of variables contained in \(G\) for which \(G\)
remains invariant. This view implies two extremes of invariance: (1) A
generalization that holds for all possible values of its variables is
maximally invariant, (2) a generalization \(G\) that holds for
merely two of the possible values of the dependent variables in
\(G\) is minimally invariant. Between these extremes of invariance
lie generalizations which hold only for a certain range of values.
Woodward and Hitchcock (EG1) illustrate the latter kind of
generalization by the example of an equation that describes the growth
of plant depending on the amount of water and
 fertilizer.[16]



According to Woodward & Hitchcock, laws or generalizations in the
special sciences do not hold under all interventions. They merely hold
for a certain range of possible values (of those variable figuring in
the law statement). By being non-strict the generalizations in the
special sciences do not satisfy a condition that is traditionally
associated with laws of nature, namely the condition of universality.
Nonetheless, being invariant for a limited range of values is enough
for a proposition to play a lawlike role in the sciences, as is argued in EG2. Woodward and Hitchcock (EG2,
184-189) develop further criteria and examples of determining
the degree of invariance of a generalization \(G\) (e.g., comparing
two generalizations \(G\) and \(G\*, G\) might be more
exact than \(G^\*\) with respect to the same range of intended
application; or \(G\) might be more or less sensitive to variations
in the background conditions than \(G^\*\) etc.).



Let us compare Lange's stability theory and Woodward &
Hitchcock's invariance theory with respect to the range of
counterfactual suppositions or, alternatively, interventions to which
the law is applicable. One can say that Lange's stability theory
and Woodward & Hitchcock's invariance theory start from
opposed extremes of the spectrum of stability: (1) Lange starts at the
maximum of stability, i.e., he starts with the claim that a
law-proposition \(l\) remains true for *all* counterfactual
suppositions that are consistent with \(l\), where by definition a
counterfactual supposition is admissible if it is consistent with the
laws of nature. In a second step, Lange reduces the set of all
counterfactual suppositions consistent with \(l\) because of
certain pragmatic goals of the discipline which uses the proposition
\(l\). (2) Woodward and Hitchcock start with minimal stability,
i.e., the satisfaction of the testing intervention condition for some
proposition \(l\). In a second step, Woodward and Hitchcock add to
minimal stability that \(l\)'s degree of invariance increases
with the number of possible interventions for which \(l\) holds
(among other criteria).


6. *Different strategy to distinguish laws from accidentally true
generalizations*: Traditionally, philosophers of science believe
that genuine laws differ from accidentally true generalization *in
kind*. Invariance theorists, such as Woodward and Hitchcock, agree
that such a distinction can be drawn. Accidentally true
generalizations are not minimally invariant in the sense of satisfying
the testing intervention condition. To take an example, Woodward and Hitchcock think that an accidentally true generalization such as
"All golden spheres have a diameter of less than a mile"
is not invariant under *any* interventions, i.e., the
generalization does *not* remain minimally invariant under
testing interventions on their explicit variables (see definition
(14)). Therefore, laws and accidental generalizations differ *in
kind* (cf. Woodward 2003, 239f.). However, Woodward and Hitchcock
acknowledge the fact that minimally invariant generalizations may fail
to be stable because these generalizations may be true only if very
specific *background conditions* obtain (see the preceding
paragraphs about degrees of invariance).



Analogously, Lange argues for a principled distinction between laws
and accidents within the framework of his stability theory (section
6.1). However, Lange's strategy to distinguish laws from
accidents seems to be partly motivated by the intuition that
*minimal* invariance in Woodward & Hitchcock's sense
is too weak to account for *real life* stability displayed by
scientific generalizations. Correspondingly, Lange (2009b:
297-302) argues that Woodward & Hitchcock's account
fails to distinguish laws from mere accidents and therefore also fails
to account for the laws' characteristic explanatory power. 


## 7. Dispositional Accounts



John Stuart Mill objected to the claim that laws might have
exceptions. The problem of exceptions disappears if laws are taken to
refer to tendencies.



> 
> With regard to *exceptions*; in any tolerably advanced science
> there is properly no such thing as an exception. What is thought to be
> an exception to a principle is always some other and distinct
> principle cutting into the former: some other force which impinges
> against the first force, and deflects it from its direction. There are
> not a *law* and an *exception* to that law--the law
> acting in ninety-nine cases, and the exception in one. There are two
> laws, each possibly acting in the whole hundred cases, and bringing
> about a common effect by their conjunct operation. [...] Thus if
> it were stated to be a law of nature, that all heavy bodies fall to
> the ground, it would probably be said that the resistance of the
> atmosphere, which prevents a balloon from falling, constitutes the
> balloon as an exception to that pretended law of nature. But the real
> law is that all heavy bodies *tend* to fall [...]. (Mill
> 1836 quoted after Mill 2000, 56, original emphasis)
> 



This idea was taken up by Cartwright and others. Cartwright's
discussion of cp-laws focused on the question why cp-laws are of
interest in non-ideal situations. If what cp-laws say is confined to
special, namely ideal, circumstances it seems *prima facie*
that these laws are irrelevant for all other, non-ideal, situations.
That, however, is not the case in scientific practice. Cp-laws are
used to explain phenomena outside the ideal circumstances. Cartwright
argues that this practice requires the postulation of capacities or
tendencies:



> 
> The logic that uses what happens in ideal circumstances to explain
> what happens in real ones is the logic of tendencies or capacities.
> What *is* an ideal situation for studying a particular factor?
> It is a situation in which all other 'disturbing' factors
> are missing. And what is special about that? *When all other
> factors are absent, the factor manifests its power explicitly in its
> behaviour*. [...] This tells you something about what will
> happen in very different, mixed circumstances--but only if you
> assume that the factor has a fixed capacity that it carries with it
> from situation to situation. (Cartwright 1989, 190f.)
> 



Subsequently, her suggestion has been taken up by several authors that
attempt to provide a semantics for cp-laws or even of laws in general
in terms of dispositions (cf. Huttemann 1998, 2007; Lipton 1999;
Drewery 2001; and Bird 2005, 2007, chapter 3).



The *prima facie* advantage of a dispositional account of laws
vis-a-vis the cp-law-problem is the following: The
dispositional account provides a *semantics* for cp-laws.
According to one version of the dispositionalist account of cp-laws
(see Huttemann 2014), a law statement is true provided the type
of system in question has the disposition that the law statement
attributes to the system. Reconstructing law-statements as statements
about dispositions, tendencies, or capacities rather than about overt
behavior turns cp-laws into strict laws. The claim is--as in
Mill--that certain kinds of systems have certain kinds of
tendencies or dispositions. In stating the law, it is no longer
necessary to appeal to cp-clauses. One advantage is that the so-called
problem of instantiation, "the problem that many cp-laws appear
not to have any instances" (Lipton 1999, 164) can be solved. The
law is no longer considered to be a description of the systems'
occurrent behavior that is only manifest under very special
conditions--if at all. The law concerns the underlying stable
tendencies or dispositions. Furthermore, this dispositionalist account
is able to provide a rationale for why scientists are interested in
cp-laws. Cp-laws describe how systems behave in the absence of
disturbing factors, i.e., if the disposition, tendency etc. can
manifest itself completely. This knowledge can be used to account for
more complex situations, in which various systems and their
dispositions are intertwined--provided laws of superposition are
available.



There is a different version of dispositionalism about cp-laws
according to which they are not turned into strict laws (see Bird
2007: section 3.3). In a nutshell, Bird's general idea is that the
"laws are those regularities whose truth is guaranteed by the
essentially dispositional nature of one or more of the constituent
properties" (Bird 2007: 46-47). According to Bird, fundamental
essentially dispositional properties or potencies give rise to strict
laws, since there are no (or only very few) *finks* or
*antidotes* that would prevent the manifestation of the potency
from occurring if the stimulus had occurred (see Bird 2007 chapters
3.3.2 and 3.3.3, and the SEP-entry on dispositions for details on
interferes such *finks* and *antidotes*). However, the
manifestation process of non-fundamental dispositional properties can
be interfered with and so the manifestation only occurs *ceteris
paribus*, i.e., in conditions where the trigger occurs and
*finks* and *antidotes* are absent (see Bird 2007:
chapter 3.3.1). Therefore, these non-fundamental properties give rise
to *ceteris paribus* laws only. 



We want to focus on two kinds of concerns that have been raised with
respect to dispositionalist accounts:


1. Dispositions, powers, potencies or tendencies may be present without
being manifest (cf. Earman and Roberts 1999, 451f.):



> 
> Thus if what one wants explained is the *actual* pattern, how
> does citing a tendency--which for all we know may or may not be
> dominant and, thus, by itself may or may not produce something like
> the actually observed pattern--serve to explain this pattern?
> 



As a rejoinder, dispositionalists can point to laws of composition. As
a consequence, it is not the disposition on its own that can explain
the actual pattern in such cases but rather the dispositions plus the
law of composition. For example, Newton's second law with (i)
the gravitational force and (ii) the Coulomb-force describe two
dispositions that explain the actual behavior of a particle relying on
the law for the superposition of forces. The law for the superposition
of forces describes how the two forces contribute to the actual
behavior. (Note that the dispositions the laws of nature attribute to
systems need not in general be taken to be macroscopic dispositions
like fragility or solubility.) To the extent that such laws are
available (as for example for physical forces) the contribution of the
various tendencies or dispositions can be determined even if they fail
to be (completely) manifest. It can, however, be disputed whether such
laws of superposition/composition are available in general, especially
if one holds like Bird (2007) that every law needs to be grounded in
potencies. But even if composition laws are taken to be underpinned by
dispositions, a further law seems to be needed that tells us what
happens if the disposition for the law of composition fails to be
manifest. We need a meta-law that tells what the law of composition
contributes to the actual behavior. So, the assumption that laws of
composition are grounded in dispositions attributed to physical
systems threatens to lead to an infinite regress.
2. A second problem concerns the question whether a dispositional account
can indeed avoid the dilemma for exclusive cp-laws (see section 4).
Even though it is true, at least on one version, that the laws no
longer appeal explicitly to cp-clauses, the
triviality-or-falsity-dilemma turns up at a different place (see
Lipton 1999, section 5: Hume's revenge). The dispositionalist
still has to specify the stimulus or triggering conditions for the
disposition in order to give the disposition's ascription a
determinate content. The conditions under which the disposition will
manifest itself are exactly those that are required to explicate the
cp-clause (see Lipton 1999, section 5; for an extensive discussion of
this problem cf. also Schrenk 2007b). So, explicating the triggering
conditions of a disposition requires that something informative is
said about the cp-clauses, and more generally, leads us back to the
problems discussed in the previous sections (see Huttemann 2014
for a recent defense of the dispositionalist approach to cp-laws; Ward
2009 provides a critical discussion).


## 8. Normality Theories



The reconstructions of exclusive cp-laws in terms of strict
completions do not imply anything about the *probability* with
which undisturbed antecedent-events will produce the consequent (cf.
Pietroski and Rey 1995, p. 84; Schiffer 1991, p. 8). According to
proponents of normality theories this seems to be counterintuitive,
because cp-laws should be asserted only if the situation without
(non-negligible) disturbing factors is a *normal* or at least a
rather probable situation. Examples of correct exclusive cp-laws which
violate this normality condition are, for example: "exclusively
cp, no tire blows out", "exclusively cp, there are no
clouds in the sky", etc. Several authors have therefore
developed so-called *normality accounts* of cp-laws, including
Silverberg (1996), Earman and Roberts (1999, 463), Spohn (2002), Spohn
(2012, chapter 13), and Schurz (2001b, 2002), who argues that the
normality interpretation is the preferred interpretation of exclusive
cp-laws in the *life sciences*, from biology upwards to the
social sciences.



Normality theories can be understood as an alternative to the
(definite and/or indefinite) exclusive interpretation of non-strict
laws. The general idea of normality theories is that "cp, all
\(A\)s are \(B\)s" means that *normally* \(A\)s
are \(B\)s. They can also be combined with a comparative
interpretation into normic-comparative law-statements such as
"normally an increase of \(X\) leads to an increase of
\(Y\) when other \(X\)-independent variables are held
constant". However, the normality accounts differ in their
explication of the notion of "normally". One possibility
(suggested by Schurz 2001b, 2002, see section 8.1) is to explicate the
normality condition in terms of a high probability of the consequent
predicate, given the antecedent predicate, where the underlying
conditional probabilities are objective statistical probabilities
based on the dispositions of evolutionary systems. Another possibility
(suggested by Spohn 2012, see section 8.2) is to explicate the
normality conditions in terms of degrees of belief and ranking
functions over possible worlds.



Largely independently of Schurz's and Spohn's specific
approaches but in agreement with the general idea of the normality
approach, Huttemann and Reutlinger (2013), Kowalenko (2014),
Reutlinger (2014), Roberts (2014) and Strevens (2014) have recently
explored a statistical approach to cp-laws, according to which cp-law
statements are statistical claims.


### 8.1. Normic laws and evolution



Schurz (2001b; 2002, SS5) analyses cp-laws in non-physical
sciences as *normic* laws of the form "\(A\)s are
normally \(B\)s". Here are some examples (4\* and 6\* are the
normic reconstructions of 4 and 6 above):



> 
> (4\*) Birds normally can fly.
> 



> 
> (6\*) People's actions are normally goal-oriented
> 



> 
> (15) Governments normally try to keep the economy of their country
> intact.
> 



> 
> (16) Turning the ignition key normally turns on the engine of my car.
> 



According to the *statistical consequence thesis*, normic laws
imply numerically unspecified statistical generalizations of the form
"Most \(A\)s are \(B\)s", by which they can be
empirically tested. The statistical consequence thesis has been
challenged by cognitive scientists (e.g., McCarthy 1986) and by
philosophers of biology (e.g., Millikan 1984). Schurz (2001b) defends
the statistical consequence thesis by the following argument which is
based on the generalized theory of evolution that does not only apply
to biological evolution but also to cultural evolution (cf. Mesoudi
*et al.* 2006 for an excellent overview on generalized
evolution theory; see Strossner 2015 for a recent
discussion).



The common domain of the life sciences (which, according to Schurz,
include biology, psychology as well as the social sciences and the
humanities) are evolutionary systems or their products. Evolutionary
systems are systems whose self-regulatory properties have been
gradually selected according to their contribution to reproductive
success. The temporal persistence of self-regulatory systems is
governed by a certain range of prototypical norm *states*, in
which these systems constantly have to be in order to stay alive. They
manage this with the help of *regulatory mechanisms* which
compensate for *disturbing influences* of the environment.
Although the self-regulatory capacities of evolutionary systems are
the product of a long adaptation history, they are not perfect.
Dysfunctions may occur, whence their normic behavior may have various
*exceptions*. Yet, it must be the case that these systems are
in their prototypical norm states in the high *statistical
majority* of cases and times, since otherwise, they would not have
*survived* in evolution.



In this way, the evolution-theoretic foundation of normic laws does
not only explain why normic laws are the typical form of the laws of
life sciences. It also explains why normic laws are not strict but
nevertheless are associated with high conditional statistical
probabilities, at least for most cases and times in evolution.



According to Schurz (2002), the difference between idealization
conditions in physics and normality conditions in evolutionary
sciences can be explained as follows: In physics, one traditionally
thinks of *complexity* as a source of
*disorder*--regularities are obtained by *abstracting
away* from complexities. In evolutionary systems, by contrast,
complexity is usually a *source of
order*--*complexity* which has been selected by
evolution to stabilize normic behavior. As Wachbroit (1994, 587f.)
puts it, ideal planets are theoretical abstractions: mass points under
the influence of a centripetal force and 'nothing else'.
They do not literally exist. In contrast, normal birds really
*do* exist because they are what has been selected through
evolution. When speaking of a normal bird, we do not abstract from its
admirable complexity, but we *rely* on it as the cause of its
normal behavior. This does not mean that abstract laws of physics
(e.g., the laws of aerodynamics) play no role in the explanation of
functional behaviors of evolutionary systems (e.g., that birds can
fly). However, even when we explain a bird's flying ability in
terms of the laws of aerodynamics, we still have to assume a
'normal' bird which possesses the typical capabilities of
birds that have been selected through evolution. The idealization
procedures needed for planets would not make good sense for birds:
there are no disturbing parameters which, when going to zero, turn a
real bird into an ideal bird which can necessarily fly and which is
approximated by the real bird.


### 8.2. Normal conditions approach



Another way to understand "*ceteris paribus*" as
referring to normal conditions is introduced by Wolfgang Spohn (1997,
section 5; 2002, section 4; 2012, chapter 13.2; 2014). Spohn takes the
connection between *ceteris paribus* qualifications and
normality to be straightforward: "*ceteris
paribus*" means *other things being normal*. Let us
call this the *Normal Condition Approach*. The basic idea of
the Normal Conditions Approach is that a cp-law \(L\) holds when
normal conditions obtain. Spohn characterizes normal conditions as
"exactly those conditions that normally, usually, mostly obtain
[...] in the small space-time region inhabited by us"
(Spohn 1997, 278). Translated in probabilistic terms, the expression
"conditions that normally, usually, mostly obtain" means
that the occurrence of some conditions is *highly
probable*.



For Spohn, a condition is "normal" if it is (i) expected
or (ii) at least not ruled out to obtain by a rational, epistemic
agent (for formal definitions of normal and exceptional conditions cf.
Spohn 2002, section 4, and Spohn 2012, chapter 13.4). He illustrates
his Normal Conditions Approach with Hooke's Law:



> 
> Hooke's law, for instance, about the proportionality of the
> force applied to a spring and its extension. It needs qualifications
> in many ways, as was clear from the outset, even though the
> qualifications could be neither fully nor precisely specified. One
> must not overstretch the spring, the material the spring is made of
> must be elastic and homogeneous, its shape regular, the thermal
> distribution uniform, etc. Physicists have cleared up most of the
> conditions--all of them?--under which Hooke's law
> holds and provided deeper explanation in terms of the intermolecular
> forces within a molecular lattice. After all, our sophisticated
> technology of spring balances that is (or was) used in each lab and
> each deli-shop depends on it. (Spohn 2012, 305)
> 



Spohn concludes that Hooke's law holds under normal conditions
and draws an analogy to other law statements in the special
sciences:



> 
> Hooke's law holds under normal conditions, i.e., for diligently
> manufactured springs handled with the usual care. The rules of thumb
> of folk psychology normally apply; this is why they are such
> tremendously useful common knowledge. Likewise, the laws of scientific
> psychology and of economics at best hold under the conditions
> explicitly specified, along with further unspecified normal
> conditions. (Spohn 2012, 307)
> 



Spohn's Normal Conditions approach differs from Schurz's
Normic Laws approach in at least two respects.


1. According to Spohn (and contrary to Schurz's theory which refers
to objective probabilities), the fact that normal conditions obtain is
conceived epistemically and, more specifically, doxastically. An
epistemic agent forms beliefs about the normality of conditions, i.e.,
someone strongly believes that some condition obtains (cf. Spohn 2002,
385).



Spohn describes "the epistemic functioning" of beliefs
(doxastic states) about normality by modeling degrees of belief in
terms of ranking functions (originally developed in Spohn 1988 under
the label "ordinal conditional functions"; cf. also Spohn
2012 and the SEP-entry 'Formal Representations of Belief',
section 3.3)
2. Normality in Spohn's theory is explicated in terms of background
conditions, not in terms of probabilistic relations between the
antecedent predicate and the consequent predicate. If someone believes
that the law \(f(X)=Y\) holds, then she believes that
this functional relation holds under normal conditions \(N\), which
means in terms of Spohn's ranking functions that the law
\(f(X)=Y\) is believed with rank zero, i.e., it is
true in all normal worlds of a ranked world model. Such a ranked world
model consists of a set of possible worlds together with a ranking
function which attaches to each world a natural number \(0, 1, \ldots ,n\) specifying its rank. Worlds with rank 0 are the most normal
worlds, worlds with rank 1 contain exceptions from normal conditions
(1st degree exceptions), worlds with rank 2 exceptions from those
exceptions (2nd degree exceptions), etc.



Spohn (2014) is the most recent and in-depth elaboration of his normal
conditions based "epistemic account" of *ceteris
paribus* laws and conditions.



Since the normal conditions "\(N\)" express the
proposition that the given situation or world is normal, it would seem
that the ranking-of-worlds account entails the possibility of a strict
completion by adding the normality condition \(N\) to the
antecedent: "if \(N\), then \(Y = f(X)\)". Indeed, it follows from the truth condition
for the assertion "normally (or cp) \(L\)" in a ranked
world model (where \(L\) is e.g., \(Y = f(X))\)
that there exists a proposition \(N\)--namely the set of all
worlds with rank 0--such that "normally \(L\)" is
true in the given ranked model iff the material implication
\(N\rightarrow L\) is true in all worlds of this model. This does
neither imply, however, that (a) the completing proposition \(N\)
can be linguistically expressed, nor (b) that it can be expressed in a
non-trivial way, in the sense that \(N\) does not already imply
\(L\) on logical reasons alone. Since one can only speak of a
strict completion of "normally \(L\)" if (a) and (b)
hold, it follows that the ranking-of-worlds account does in fact not
entail the possibility of a strict completion. This result is
adequate, because as we have argued above, in most cases (especially
in all non-deterministic situations) a strict completion is
impossible. Thus, it has to be kept in mind that speaking of
"normality conditions" does not imply that these
"normality conditions" can be expressed by any
(non-trivial) proposition. For example, there is not any non-trivial
proposition \(N\) which could turn the conditional "normally
a Cs\(^{137}\) atom decays after 60 years" into a strict
implication of the form "if \(N\), then a Cs\(^{137}\)
atom decays after 60 years". Also note that the ranked-worlds
accounts and the conditional probability accounts are equivalent as a
semantics for logical axiomatizations of conditional reasoning (see
section 8.3).


### 8.3. *Ceteris paribus* laws and non-monotonic reasoning



Reasoning from exclusive or normic cp-laws has an important logical
feature: the inferences are in contrast to deductively valid arguments
*not monotonic*. An inference is monotonic iff adding arbitrary
new premises to a valid argument preserves its validity. But the
inference from an exclusive cp-law of the form "exclusively cp,
\(A\)s are \(B\)s" and a singular statement
\(A(d)\) (for "\(d\) is an \(A\)") to a
conclusion \(B(d)\) is no longer correct if \(d\) is
known to instantiate a disturbing factor \(D\) which blocks the
nomic connection between \(A\) and \(B\). For instance, the
exclusive cp-law may state that if something is a bird (\(A\)), then it
normally can fly (\(B\)), and \(D\) may assert that the given
bird-instance has broken wings.



Assume the exclusive cp-law is formulated with the help of a
non-strict conditional \(A \rightarrow B\). Drawing correct
inferences from non-strict conditionals requires non-classical rules
of valid inferences: The non-monotonicity of Default *Modus
Ponens* means formally that although the inference from
\(A\rightarrow B\) and \(A(d)\) to \(B(d)\) is
correct (i.e., \(A\rightarrow B, A(d) \dproves B(d)\) holds, where '\(\dproves\)' stands for
'non-monotonically correct inference'), the inference from
the extended premise set \(A\rightarrow B,
H\rightarrow \neg B, A(d)\), and
\(H(d)\) to \(B(d)\) is incorrect (i.e.,
\(A\rightarrow B\), A\((d), H(d) \dproves B(d))\). These non-monotonic effects are reflected in
non-classical rules for inferences from conditionals to conditionals.
For example, take the inference from "If \(A\), then
\(B\)" to "If \(A\wedge D\), then
\(B\)" (monotonicity \(M)\) or from "If \(A\),
then \(B\)" and "If \(B\), then \(D\)" to
"If \(A\), then \(D\)" (Cut \(C)\). Both
inferences are valid for strict (logical or material) conditionals,
but they are not generally correct for non-strict conditionals. Only
the more cautious inferences from \(A\rightarrow B\) and
\(A\rightarrow D\) to \(A\wedge D\rightarrow B\)
(cautious monotonicity CM) or from \(A\rightarrow B\) and
\(A\wedge B\rightarrow D\) to \(A\rightarrow D\)
(Cautious Cut CC) are correct for non-strict conditionals.



Two prominent *semantic criteria* for the truth of a non-strict
conditional and the correctness of inference from non-strict
conditionals have been suggested in the literature.


1. The *high-probability-semantics* understands normic laws in
the sense of high conditional probability assertions (as in section
8.1). This semantics considers a non-strict conditional
\(A\rightarrow B\) as true in a probability model if the
uncertainty of the conditional, which is defined as 1 minus the
conditional probability, is sufficiently low. An inference of a
conclusion conditional from a set of premise conditionals is regarded
as valid in this semantics iff the uncertainty of the conclusion
conditional is not greater than the sum of the uncertainties of the
premises. High probabilistic semantics goes back to Adams (1975) and
has been extended in Schurz (1998, 2005).
2. The second semantics, the *normality-semantics*,
corresponds to the normal-condition account (see section 8.2). This
semantics considers a conditional \(A\rightarrow B\) as true in a
ranked-world-model iff all lowest-rank \(A\)-worlds are
\(B\)-worlds. An inference is considered as valid in this semantics
iff all ranked-worlds-models which verify all premise conditionals
verify the conclusion conditional.



Remarkably, both semantics lead to the same conditional logics. Both
semantics possess the system \(P\) ("preferential
entailment") as a correct and complete axiomatization of
inferences among pure conditionals, and the system \(R\)
("rational entailment") for inferences among
truth-function combinations of conditionals (for more details cf.
Kraus *et al.* 1990, Hawthorne 1996, Adams 1975, Schurz 1998,
2004, 2005, Leitgeb 2004, and the SEP-entry on non-monotonic
logic).


## 9. *Prima Facie* Reasons and *Ceteris Paribus* Conditions in Ethics and Epistemology



The use of "cp" is not restricted to philosophy of
science. It is used in other areas of theoretical and practical
philosophy as well.



For instance, *prima facie* reasons are reasons under the
constraint of exclusive cp-conditions. They play an important role in
ethics and epistemology.



In ethics, a *prima facie* norm is an obligation (e.g.,
"*prima facie*, you should not hurt anybody") which
holds as long as exceptional circumstances (e.g., defending your own
life) are absent (cf. Ross 1930 and the SEP-entry on moral reasoning).
Dancy (2004, 17, 35f) uses the expression "*pro
tanto*" instead of "*prima facie*", which
is standardly used in the current debate. Pietroski (1993) has
promoted the *ceteris paribus* understanding of moral
obligations as a solution to paradoxes of moral obligations.
Especially for the solution of moral dilemmas, the cp-reading of moral
obligations has proven to be helpful, because although two conflicting
cp-obligations may block each other, they do not produce a strict
consistency in the sense of deductive logic (see Horty 1994).



In epistemology, a *prima facie* reason for a belief is a
reason which justifies this belief in the absence of defeaters, i.e.,
of exceptional information to the contrary. *Prima facie*
reasons have been suggested in epistemology as a means of providing
weak and defeasible *a priori* justifications for
epistemologically fundamental beliefs such as the belief in the
existence of an external reality, for which a strict and non-circular
*a priori*-justification seems to be impossible. Nevertheless,
the inference from visual appearances to reality counts as defeasibly
justified as long as there do not exist serious defeaters, i.e.,
reasons for serious doubt (cf. Williams 1996, 2001, and the SEP-entry
on *a priori*-justification and knowledge). An example is the
defeasible inference from visual appearances to beliefs concerning
external reality. My awareness of my visual appearance of a tree in
front of me is a *prima facie* reason for my belief that there
really is a tree in front of me, provided that information about
exceptional circumstances is absent, for example the information that
I am under the influence of drugs which cause hallucinations (cf.
Pollock 1986, Moser 2002). One may object that a similar kind of
possible exception to the inference from visual appearances to reality
is constituted by the possibility of a Cartesian demon who infuses
humans with illusionary perceptions, which is one of the major
skeptical counterarguments to the inference from appearances to
reality. In response to this objection, several epistemologists have
argued that these kinds of exceptions are not serious exceptions, and
therefore, the inference from appearances to reality is defeasibly
valid.



Finally, applications of cp-clauses can also be found in the form of
presumptions in philosophy of language. Here it is assumed that
understanding linguistic expressions requires defeasible assumptions
or "presumptions", for example presumptions concerning the
rationality of the speaker (cf. Quine 1960; Davidson 1973, Scholz
2016, 148-159).


## 10. *Ceteris Paribus* Laws: Metaphysics, Language, and Science



In the current literature, a number of novel trends and innovative
approaches to cp-laws have emerged that cannot easily be subsumed
under the already existing accounts. For this reason they deserve
special mention. We believe there are at least four clearly
discernable novel tendencies in the current work on cp-laws:


1. *A reconnection to metaphysics:* One of the most salient trends
in the current literature is that the discussion of cp-laws
(re)connects philosophy of science to analytic metaphysics in at least
three respects: (i) current accounts of cp-laws make use of David
Lewis's (1973) best system account of laws; moreover, (ii) the
interpretation of objective probabilities and (iii) the metaphysics of
dispositions are brought into play for making progress in
understanding cp-law statements. 



The most vivid sign of this reconnection consists in the fact that
several authors have recently defended a friendly amendment of David
Lewis's best system account of laws of nature. While
Lewis's (1973, 73) original version primarily focuses on
fundamental laws, the amended versions are intended to capture also
non-fundamental laws of the special sciences. Schrenk (2007a, 2014),
Callender and Cohen (2009, 2010), and Frisch (2014a) defend a
so-called "better best system" account of special science
laws. Unterhuber (2014) develops a version of the "better best
system" account utilizing generics. Jaag and Loew (forthcoming)
argue for a pragmatic version of the best systems account that
captures also central insights of the "better best system"
theory. On their view, the universality of laws is not an end in
itself. If a generalization's invariance is, e.g., limited to
our cosmic epoch only, this limited invariance arguably would be
enough to make it useful for us and thus a law according to this
account. Reutlinger (2009) and Backmann and Reutlinger (2014) provide
a critical discussion of "better best system" accounts.
Braddon-Mitchell (2001) and Wheeler (2018) present versions of a best
systems account deploying the idea of algorithmic compression from
data science. Albert (2000, 2015) and Loewer (2009) advocate a
"statistical mechanical" version of the best systems
approach to special science laws which is, unlike the "better
best system" account, committed to a particular
Boltzmann-inspired interpretation of statistical mechanics (including
assumptions about the early history of the universe). Weslake (2014)
and Frisch (2014b) provide critical responses, while Fenton-Glynn
(2016) constructively uses examples by Albert and Loewer for an
analysis of cp-laws. Finally, Albert and Loewer's account also
connects the debate on cp-laws with the on-going discussion on whether
causation has a place in fundamental physics or whether causation is
an 'emergent' phenomenon captured by the cp-law statements
of the special sciences (Price and Corry 2007 and Frisch 2014b are
representative of the opposing views in the debate on causation in
physics).



Also other metaphysical issues besides Lewis's account of
lawhood influence the recent literature on cp-laws. The metaphysics of
probabilities and disposition stand out in this respect. Taking up
results from Earman and Roberts (1999) and Schurz (2002), several
authors including Huttemann and Reutlinger (2013), Kowalenko
(2014), Roberts (2014), Reutlinger (2014), and Strevens (2014) discuss
the prospects of a statistical approach to cp-laws (or a
"softness" approach, to use Strevens's term),
according to which cp-laws are statistical claims. (Notice that the
normality account presented in section 8 above is but a special case
of the broader statistical approach to cp-laws).



A third metaphysical trend concerns the metaphysics of dispositions:
As we have explained in section 7 above, Huttemann (2014) and
Pemberton and Cartwright (2014) argue in favor of a dispositionalist
account of cp-laws and provide responses to standard objections to the
dispositionalist view (for instance, to objections voiced by Earman
and Roberts 1999).
2. *Drawing on linguistics and philosophy of language:* Several
recent approaches to cp-laws explicitly draw on research in
linguistics and philosophy of language. Taking into account linguistic
work on generics, Nickel (2010, 2014) argues that our understanding of
cp-law statements is improved if one takes them to be generics and
uses the formal semantics for generics to shed light on the meaning of
cp-law statements (for approaches to cp-laws deploying generics, see
also Unterhuber and Schurz 2013; Unterhuber 2014; and Claveau und
Girard 2019). Roberts (2014) makes contact with philosophy of language
in analyzing cp-laws as vague and indexical statements about
frequencies. Ward (2007), Karbasizadeh (2008) and Nickel (2014)
explore the link between cp-laws and different accounts of natural
kind terms.
3. *Sensitivity to the sciences:* Several philosophers have made
an effort to draw attention to the fact that an adequate discussion of
laws has to be sensitive to the differences between distinct
scientific disciplines. Their motivation is to provide a theory of
laws that fits generalizations (and models) in the life sciences, the
social sciences and, more generally, the sciences of complex systems
(for instance, see Mitchell 1997, 2002a,b; 2008, 2009; Sober 1997;
Steel 2007; and Reiss 2008). In particular, Mitchell (2000), Strevens
(2003, 2008), Craver (2007), Huttemann (2007), Wimsatt (2007),
Tobin (2005), Woodward (2010) and Reutlinger (2011, 2013) distinguish
various dimensions of characterizing how generalizations in these
scientific disciplines may be non-universal or allow for exceptions.
These dimensions include stability, robustness, specificity, and
historical contingency (see Reutlinger and Unterhuber 2014b for an
overview). In a similar vein, Schurz (2014) introduces a novel
distinction between *ceteris paribus* and *ceteris
rectis* laws. Fenton-Glynn (2016) holds that some higher-level
laws about the macro-behavior of complex systems (his central example
is the second law of thermodynamics) are "*minutis
rectis*" laws and should be distinguished from cp-laws,
because their non-universal character stems from a probability
distribution over initial micro-conditions (see Strevens 2003 for
further examples). Finally, motivated by examples from population
ecology, earth science, and statistical mechanics, Strevens (2008) and
Weisberg (2013) argue for a new interpretation of laws and models
involving idealized assumptions, the minimalist account of
idealizations. Strevens (2012, 2014) proposes treating cp-laws and
idealized laws separately. What seems to emerge from this literature
is that the *ceteris paribus* character of generalizations is
multi-faceted. There is a number of ways in which a generalization can
be non-universal: due to idealizations, by expressing statistical
regularities and probability distributions over initial conditions, by
drawing on the notion of normality, by being sensitive to changes in
initial and background conditions, and so on (see Reutlinger and
Unterhuber 2014b for a more detailed analysis).
4. *Using scientific concepts and methods:* recent philosophical
attempts of capturing cp-laws increasingly involve importing key
concepts from the sciences and using these concepts for developing a
general theory of cp-laws. Inspired by classical mechanics, Maudlin
(2007) and Huttemann (2014) analyze cp-conditions in terms of
default or inertial behavior. Both authors capture a violation of
cp-conditions as a deviation from inertial behavior, where deviating
from inertial behavior is itself a process governed by laws of
deviation (Maudlin) or laws of interaction and composition
(Huttemann). Other philosophers are influenced by the life
sciences such as Nickel (2014), Pemberton and Cartwright (2014), and
Strevens (2012, 2014) who rely on the concept of a mechanism in order
to explicate the content of cp-conditions more generally. Kowalenko
(2014) argues that cp-laws are best understood by drawing on methods
from statistics, such as multivariate regression analysis (for a
discussion of similar statistical approaches see Huttemann and
Reutlinger 2013, Reutlinger 2014, Roberts 2014, and Strevens 2014).
Schurz (2014) uses causal Bayesian networks as a formal framework for
a causal reconstruction of *ceteris paribus* and *ceteris
rectis* laws. Finally, Jhun (2018) argues that focusing on the
equilibrium methodology used in thermodynamics as well as in economic
reasoning facilitates a new understanding of the explanatory role of
cp-laws in terms of efficient performance.


## 11. Conclusion



Where are the debates about cp-laws leading? What are the important
challenges to be met in future research? There are at least four main
areas of productive future research:


1. The debate on cp-laws since the 1980s has not particularly
focused on detailed case studies of laws in different scientific
disciplines. But this just seems to be what a satisfying theory of
cp-laws requires. A possible result of detailed case studies might
well be that generalizations reveal quite different features in
different disciplines. In other words, the explication of cp-laws in,
for instance, physics, biology and economics might differ
significantly. If so, this result also raises interesting
methodological questions for philosophers of science, such as
"What degree of unification (with respect to laws in different
scientific disciplines) do we require in order to approve of a theory
of cp-laws?".
2. What is the connection between cp-laws and theories of (causal)
explanation, various theories of causation and mechanisms? When
applied to the special sciences, many theories of (causal) explanation
(cf. Woodward 2003, EG1, EG2), causation (cf.
Hausman 1998; Pearl 2000; Hitchcock 2001; Woodward 2003) and
mechanisms (Machamer, Darden and Craver 2000; Glennan 2002; Craver
2007) seem to presuppose generalizations. Interestingly, these
generalizations tend to be classified as non-strict. Consequently,
these generalizations are in the domain of the debate on cp-laws.
These theories of explanation, causation and mechanisms are important
test cases for the adequacy of theories of cp-laws.
3. What metaphysical claims do the various theories of cp-laws
commit us to? Do exclusive cp-laws commit us to assume that the
fundamental laws of nature are deterministic? Or are they compatible
with a probabilistic metaphysics? Furthermore, do cp-laws commit us to
a dispositionalist metaphysics? Or are they compatible with a Humean
ontology? Is structural realism an adequate metaphysics for cp-laws in
general (see Wright 2017 for discussion) and in particular for the
laws of the special sciences?
4. What role do cp-laws play in formal philosophy? For example, it
is important to answer the following questions: How are the logical
principles of non-monotonic reasoning connected to cp-laws? How do
cp-laws relate to Bayesian probabilities, and to the rules of Bayesian
updating of these probabilities?



These areas of research are, of course, not a complete list of future
research topics. Nevertheless, it seems to us that these questions are
interesting enough to stimulate productive further research on
*ceteris paribus* laws.


## 12. Suggested Reading



The critical article by Earman and Roberts (1999) provides an
excellent introduction to the contemporary debate, because it
reconstructs the most important theories of cp-laws and also points
out the problems of these approaches. The volumes by Earman *et
al.* (2002) and Reutlinger and Unterhuber (2014) contain a
representative collection of contemporary essays on the problem of
cp-laws.



Persky (1990), Blaug (1996) and Kaufer (1997) are accessible surveys
of the history of cp-clauses and cp-laws. Notice that these articles
focus on economics and the social sciences. For a systematic
discussion of laws in the social sciences, the debate between Kincaid
(2004) and Roberts (2004) on whether there are laws in the social
sciences is very instructive.



The important distinctions between (a) comparative and exclusive and
(b) indefinite and definite cp-laws are established and argued for in
Schurz (2002).



The dilemma of triviality or falsity is discussed in Lange (1993),
which is also a helpful survey of the debate until 1993.



Concerning completer accounts, Fodor (1991), Hausman (1992: chapter 8)
and Pietroski and Rey (1995) are central papers. These accounts are
criticized by Earman and Roberts (1999), Schurz (2001a) and Woodward
(2002). A more recent defense of completer accounts is Strevens
(2012).



Concerning stability accounts, Marc Lange develops his version in
book-length (see his 2000 and 2009a, although the focus of the latter
is not on cp-laws). Lange (2002) is a very accessible paper on his
view of cp-laws. Invariance accounts are best introduced in two joint
papers by James Woodward and Christopher Hitchcock (see their EG1, EG2) and in Woodward's book *Making Things Happen* (2003:
chapter 6). Hall (2007) provides an insightful discussion of Woodward
& Hitchcock's invariance account. 



For dispositionalist approaches to cp-laws, Cartwright (1989) is a
modern classic (cf. her recent 2002, and Elgin and Sober 2002 for a
critique). Smith (2002), Mumford (2004), Bird (2005, 2007) and
Huttemann (2007, 2014) develop different more recent
dispositional accounts of cp-laws.



Concerning normality approaches, Schurz (2001b, 2002) argues for a
normic laws approach. Spohn (2002) and Spohn (2012, chapter 13) are
good introductions to the normal conditions approach. See also Glymour
(2002) for a formal approach to cp-laws.



Humean better best system accounts of cp-laws are presented in Cohen
and Callender (2009, 2010), Schrenk (2007a, 2014), and Unterhuber
(2014). Wheeler (2018: chapter 3.2) gives an accessible brief
exposition of various versions; and Reutlinger (2009) and Backmann and
Reutlinger (2014) provide a critical discussion. 



There are also various approaches in the current debate that cannot
easily be classified: Mitchell (1997, 2002a, b; 2008, 2009), Sober
1997, Steel (2007, chapter 6) and Reiss (2008) attempt to explicate a
theory of laws that fits scientific practice in biology, the social
sciences and generalization describing complex systems. Further,
Mitchell (2000), Craver (2007), Tobin (2005) and Reutlinger (2011)
distinguish various dimensions of the non-universality of (cp-)laws in
order to argue for an explication of laws in the special sciences.
Karbasizadeh (2008) links the debates on cp-laws and natural kinds.
Ladyman and Ross (2007, chapters 4 and 5), Kincaid (2008), Ross
(2008), and Ladyman (2008) argue for ontic structural realism as a
metaphysics of generalizations in the social sciences (and for the
special sciences in general). Finally, Maudlin (2007, chapter 5)
advocates an explication of laws in the special sciences that is based
on the notion of quasi-Newtonian laws.