Laws of nature take center stage in philosophy of science andmetaphysics. Laws are usually believed to stand in tight relations tomany philosophical key concepts such as causation, explanation,confirmation, determinism, and counterfactuals. Traditionally,philosophers have focused on physical laws, which were taken to be atleast true, universal statements that support counterfactual claims.But, although this claim about laws might be true with respect tophysics, laws in the special sciences (such as biology, psychology,and economics) appear to have—maybe notsurprisingly—different features than the laws of physics.Special science laws—for instance, the economic law “Underthe condition of perfect competition, an increase of demand of acommodity leads to an increase of price, given that the quantity ofthe supplied commodity remains constant” and, in biology,Mendel’s Laws—are usually taken to “haveexceptions”, to be “non-universal” or to be“ceteris paribus laws”. How and whether the lawsof physics and the laws of the special sciences differ is one of thecrucial questions motivating the debate onceteris paribuslaws. Another major, controversial question concerns the determinationof the precise meaning of “ceteris paribus”.Philosophers have attempted to explicate the meaning ofceterisparibus clauses in different ways. The question of meaning isconnected to the problem of empirical content, i.e., the questionwhetherceteris paribus laws have non-trivial and empiricallytestable content. Since many philosophers have argued thatceterisparibus laws lack empirically testable content, this problemconstitutes a major challenge to a theory ofceteris paribuslaws.

• 1. Introduction
  • 1.1. Systematic introduction
  • 1.2. Overview
• 2. History and Background
  • 2.1. A brief historical survey: from scholasticism to modern economics
  • 2.2. Background of the contemporary debate
• 3. A Framework for the Discussion: Distinguishing exclusive/ comparative and definite/indefiniteceteris paribus laws
  • 3.1. Comparative vs. exclusive cp-laws
  • 3.2 Definite versus indefinite cp-laws
• 4. The Challenge: ExclusiveCeteris Paribus Laws Between Falsity and Triviality
• 5. Exclusive CP-laws: The Method of Completers
  • 5.1. Semantic and epistemic completers
  • 5.2. Criticisms: triviality and accidentality
• 6. Invariance & Stability Theories
  • 6.1. Counterfactually stable laws and pragmatic knowledge of disturbing factors
  • 6.2. Invariance under interventions of explanatory generalizations
• 7. Dispositional Accounts
• 8. Normality Theories
  • 8.1. Normic laws and evolution
  • 8.2. Normal conditions approach
  • 8.3.Ceteris paribus laws and non-monotonic reasoning
• 9.Prima Facie Reasons andCeteris Paribus Conditions in Ethics and Epistemology
• 10.Ceteris Paribus Laws: Metaphysics, Language, and Science
• 11. Conclusion
• 12. Suggested Reading
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

1.1. Systematic introduction

In philosophy of science and metaphysics, laws of nature take centerstage: Many explications of philosophical key concepts in philosophyof science essentially rely on laws of nature. For instance, amajority of theories of causation, explanation, confirmation,determinism, and counterfactuals presuppose laws of nature (see theSEP-entry on laws of nature).

Until the second half of the 20th century, certain characteristics oflaws of nature were taken for granted by philosophers of science: Lawsof nature were taken to be true, logically contingent, universalstatements that support counterfactual claims. Laws were taken to playa major role in explanation, induction, confirmation, causation, andin counterfactual reasoning. The paradigm cases for such laws weretaken from (fundamental) physics. Newton’s second law \((F=ma)\)or the Schrödinger-equation fit the bill of the traditionalconcept of a law of nature. Philosophers of science were aware of thefact that there are hardly any universal laws of this kind in biology,psychology, economics or other special sciences. This posed no problemas long as there was a consensus that this might be conceived as ashortcoming of the special sciences. However, with the successespecially of the biological sciences it became clear that there isgenuine scientific knowledge that does not conform to the paradigm ofphysics. As a consequence, scientific practice in the special scienceswas no longer taken to be deficient but was analyzed as a legitimatepractice different from physics. The generalizations of the specialsciences including biology, psychology, and economics seem to differin an important way from the generalizations of fundamental physics.In order to illustrate this difference, here are some examples ofgeneralizations that play an explanatory role in various specialsciences:

• Snell’s Law: “At the interface between dielectricmedia, there is (also)a refracted ray in the second mediumlying in the plane of incidence, making an angle \(\theta_t\), withthe 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 twomedia, and \(n_1 =(c/v_1), n_2 =(c/v_2)\) are the indices ofrefraction.” (Miles V. Klein’sOptics, cited fromCartwright 1983, 46)

• The lack of vitamin C causes scurvy.

• Mendel’s Law of Segregation: “In a parent, thealleles for each character separate in the production of gametes, sothat only one is transmitted to each individual in the nextgeneration.” (Rosenberg & McShea 2008, 36)

• The Area Law in Island-Biogeography: “the equilibriumnumber \(S\) of a species of a given taxonomic group on an island (asfar as creatures are concerned) increases [polynomially] with theislands area [A]: \(S = c\times A^z\). The (positive-valued) constants\(c\) and \(z\) are specific to the taxonomic group and islandgroup.” (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 forachieving \(A\), then \(x\) will attempt to do \(B\) (Fodor 1987; Dray1957, 132ff).

• The Law of Demand: Under the condition of perfectcompetition, an increase of demand of a commodity leads to an increaseof price, given that the quantity of the supply of the commodityremains constant (Roberts 2004, 159; Kincaid 2004, 177).

• The lack of social integration results in a higher probability ofattempted suicides.

These generalizations may be explained in terms of lower levelsciences, but this is not our focus here. What is important for ourpurposes is that all of these generalizations are non-universal. Thatis, there are (actual and merely possible) situations in which theabove generalizations donot hold, although all theconditions obtain that are explicitly stated in the antecedents ofthese generalizations. For instance, there are situations in which thefollowing is the case: at the interface between dielectric media,there is a refracted ray in the second medium lying in the plane ofincidence, making an angle \(\theta_t\), with the normal and this raydoes not obey the equation \(\sin \theta/\sin \theta_t = n_2 /n_1\),the lack of vitamin C does not cause scurvy, the alleles separatedifferently than Mendel’s law of Segregation describes, theequilibrium number of a species does not increase in accord with theArea Law, people fail to act in a goal-oriented fashion, and anincrease of demand of a commodity does not lead to an increase ofprice (even if the condition of perfect competition wererealized).

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

The question arises how to deal with this situation. Are thesegeneralizations to be classified as laws—laws that pertain onlyto special conditions, i.e.,ceteris paribus laws—orare they something entirely different. Given the fact that laws play amajor role in most accounts of explanation, causation etc., there seemto be basically three options:

1. These generalizations do not qualify as laws, because they arenon-universal. Therefore, there are no genuine explanations in thoseareas of science that rely on these generalizations, most notably thespecial sciences.
2. These generalizations do not qualify as laws, because they arenon-universal. However, explanation, causation etc. does notpresuppose the existence of universal laws.
3. These generalizations do qualify as laws, despite the fact thatthey are non-universal, because they have important features in commonwith universal laws, most importantly the support ofcounterfactuals.

The difference between options (b) and (c) may be to some extentmerely terminological. Woodward and Hitchcock, for example, replacelaws with ‘invariant generalizations’ (see section 6.2)but admit that their account may be read as areconceptualization of lawhood (see Woodward and Hitchcock2003, hereafter EG1, 3).

The motivation of those who take option (c) and stick to the conceptof law is that the generalizations cited above have enough in commonwith Newton’s first law or the Schrödinger-equation toclassify them as laws. For instance, they play the same kind of rolein explanation, prediction, and they support counterfactuals. Thereare two beneficial consequences of this particular approach. First(against option (a)), it need not be denied that there are genuineexplanations etc. in the special sciences. Second (against option(b)), in analogy to the traditional view, the concept of a law (albeitof a non-universal orceteris paribus law) can be used inexplications of notions such as explanation, and causation.

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

Before we discuss various accounts of cp-laws, several issuessurrounding the notion of “ceteris paribus”should be distinguished in order to structure the debate:

1. How can different kinds of cp-laws be distinguished andclassified? Are there criteria to distinguish genuine cp-laws fromstrict laws? How and whether the laws of (fundamental) physics and thelaws of the special sciences differ is one of the crucial questionsdriving the debate aboutceteris paribus laws. (see section3).
2. Are there any genuine cp-laws at all (see sections 4–5)?Some authors deny that there are any genuine cp-laws (cf. Schiffer1991, Earman and Roberts 1999, Earman, Roberts and Smith 2002,Woodward 2002). While most of these authors agree that there aresentences 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-lawsalso in fundamental physics?
4. What different kinds of cp-laws are used in the sciences? It seemsthat in the literature two very different kinds of situations areenvisaged. On the one hand, several authors (including Mill, Marshall,Cartwright, Pietroski and Rey, Hüttemann, and Lipton) stress thatcp-laws describe the behavior of systems underideal or abstractconditions. Such conditions are—if at all —onlyrarely realized (see section 7). On the other hand, there are authorswho claim that, at least in certain disciplines, cp-conditions arenothing butnormal conditions, or cp-laws are laws expressingnormality hypotheses (see section 8).
5. What are the truth conditions of cp-law statements? Philosophershave attempted to determine the meaning of cp-clauses in differentways. The question of meaning is connected to the problem of empiricalcontent, i.e., the question whether cp-laws have non-trivial andempirically testable content. Since many philosophers have argued thatcp-laws lack empirically testable content, this problem constitutes amajor challenge to a theory of cp-laws. Formally independent butmethodologically related is the question under what conditions it isepistemically legitimate to accept cp-laws. (see sections3–5).

Having distinguished these issues, it should be clear thatthe debate about cp-clauses and cp-laws does not exist.Instead, theories of “cp” engage in different and oftenseparable enterprises.

1.2. Overview

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

Section 3 distinguishes different readings of the cp-clause andafortiori different kinds of cp-laws. In section 3.1, exclusivecp-laws (factors not mentioned in the antecedent of the law areassumed to beabsent) are distinguished from comparativecp-laws (factors not mentioned in the antecedent of the law areassumed to beconstant). In line with the bulk of therelevant literature, we focus on exclusive cp-laws. In section 3.2, adistinction between definite and indefinite cp-laws is presented.

Section 4 presents the main challenge to cp-laws. It appears that manylaws, e.g., of the special sciences, if taken as strict laws, turn outto be false; however, if they are hedged by a cp-clause they appear tolack a clear meaning and appear to be empirically untestable. All ofthe major accounts of cp-laws discussed in sections 5 to 8 arereactions 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. WhereasFodor (1987) and others (see Section 5.1) focus on special sciencecp-laws that can be completed by adding factors to the antecedent ofthe law that are dealt with in more fundamental sciences, Pietroskiand Rey (1995) take cp-laws to contain (implicit) promises: that is,when asserting a cp-law, one promises to be able to cite factors thatcomplete the law on a case by case basis if the consequent of thecp-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 thespecial sciences, namely Lange’s (2000, 2002, 2005) stabilitytheory in section 6.1, and Woodward & Hitchcock’s (see EG1and Hitchcock & Woodward 2003, hereafter EG2) invariance theory insection 6.2. The common general idea of invariance or stabilitytheories of laws is that the laws differ from merely accidentalnon-laws in virtue of their invariance or stability undercounterfactual suppositions. According to this view,ceterisparibus or non-universal laws differ from universal laws indegree, not in kind: universal laws hold under all counterfactualsuppositions whileceteris paribus or non-universal lawsmerely hold for a limited range of suppositions.

Section 7 deals with dispositional accounts of cp-laws. The unifyingidea is that the laws of nature are grounded in dispositionalproperties. We distinguish two different dispositional accounts ofcp-laws: On Hüttemann’s (2014) version, that is based onpioneering work of Mill and Cartwright, law-statements are statementsabout dispositions, tendencies, or capacities rather than about overtbehavior, and thus cp-laws are turned into strict laws. Bird’s(2007) version of dispositionalism, by contrast, does not turn cp-lawsinto strict laws. Instead, it identifies cp-laws with generalizationshedged by a cp-clause that are grounded in non-fundamentaldispositional properties that admit of interferers.

Section 8 deals with normality theories of laws. The guiding idea ofnormality theories is that “cp, all \(A\)s are \(B\)s”means thatnormally \(A\)s are \(B\)s. The various normalityaccounts differ in their explication of the notion of normality. WhileSchurz (2001b, 2002) interprets normality as a high conditional,objective probability of the consequent of the law statement given theantecedent (see section 8.1), Spohn (2012) suggests to explicate thenormality conditions in terms of degrees of belief and rankingfunctions over possible worlds (see section 8.2).

Section 9 sketches the relevance of cp-laws in other areas ofphilosophy (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 othersciences.

Section 11 presents a brief conclusion and an outlook for futureresearch.

Finally, section 12 provides a concise overview of the relevantliterature.

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 anon-technical sense by Cicero.^[1] However, most of the early uses of theceterisparibus-clause are found in economics. In economic contexts theuse ofceteris 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 economicissues.

In 1662, William Petty was probably the first to use the term in anEnglish language publication.^[5] In hisTreatise of Taxes and Contributions, Petty qualifieshis labor theory of value by a “caeteris paribus”clause:

If a man can bring to London an ounce of Silver out of the Earth inPeru, in the same time that he can produce a bushel of Corn, then oneis the natural price of the other; now if by reason of new and moreeasier Mines a man can get two ounces of Silver as easily as formerlyhe did one, then Corn will be as cheap at ten shillings the bushel, asit was before at five shillingscaeteris paribus. (Petty1662, 50, our emphasis)

John Stuart Mill used the explicit phrase “ceterisparibus” only occasionally^[6] but it had an important impact because he characterized economy byits way of coping with disturbing factors:

Political economy considers mankind as solely occupied in acquiringand consuming wealth […] not that any political economist wasever so absurd as to suppose that mankind is really thus constituted[…] when a concurrence of causes produces an effect, thesecauses have to be studied one at a time, and their laws separatelyinvestigated […] since the law of the effect is compounded ofthe 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 describewhat would happen provided there are no other causal factors.

The view of cp-laws promoted by Mill is theabsence-of-disturbing-factors view (see section 7). Another view isthe normal-tendency view of cp-laws (see section 8). The historicalroots of the view are, for example, found in John ElliotCairnes’s description of the methodology of economics in hisCharacter 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 “ceterisparibus” in order to refer to “what would or whattends to take place” if normal conditions obtained.

The use ofceteris paribus-clauses was advocated andpopularized by Alfred Marshall in the late 19th century. It wasMarshall’s genuine contribution to economics to advocate partialequilibrium analysis. Marshall claimed that an analysis of this kindholds merely ‘ceteris paribus’. In hisinfluentialPrinciples of Economics, Marshall defines thetask of economists in terms of the phrase ‘ceterisparibus’:

[Economists answer] a complex question, studying one bit at a time,and at last combining his partial solutions into a more or lesscomplete solution of the whole riddle. In breaking it up, hesegregates those disturbing causes, whose wanderings happen to beinconvenient, for the time in a pound calledCaeterisParibus. (Marshall 1890, 366)

In the introduction to the same work, Marshall explains why economicsis interested in the isolation of causes by assuming thatotherthings 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 ofpartial solutions as auxiliaries to our main study. Thus we begin byisolating the primary relations of supply, demand and price in regardto a particular commodity. We reduce to inaction all other forces bythe phrase ‘other things being equal’: We do notsuppose that they are inert, but for the time we ignore theiractivity. This scientific device is a great deal older than science:it is the method by which, consciously or unconsciously, sensible menhave dealt immemorial with every difficult problem of ordinary life.(Marshall 1890, xiii, our emphasis)

In a similar vein, Lionel Robbins claims in his classic paperAnEssay on the Nature and Significance of Economic Science thateconomic laws and antecedent facts deductively imply economicpredictions, but only if other things remain unchanged.^[8]

The use of “ceteris paribus” in economics and inphilosophy of economics is not of merely historiographical interest.In current philosophy of economics and economics, the use of and thedebate on “ceteris paribus” is a vital issue:

1. Philosophy of Economics. In the debate on philosophy ofeconomics it is widely recognized that generalizations in economicsare qualified by aceteris paribus-clause—yet, itsinterpretation 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; Kincaid2004; 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 theceterisparibus-clause. Expression such as “ceterisparibus” and “other things being equal” arecommonly used in textbooks—often they are explicated in aspecial 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.; Samuelson1958, 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 theceterisparibus-phrase became a central and also controversial issue insome areas of philosophy of science and in philosophy of mind in the1980s.

The fact that test procedures for scientific theories or hypothesesare reliable only if disturbing factors are excluded or at leastcontrolled for had already been noted by Carnap (1956, 69). Canfieldand Lehrer (1961) have pointed out that in order to figure as premisesof deductive-nomological explanations, the laws of physics have to befurnished with cp-clauses. They provide the following example tosupport their claim:

Letting ‘\(Tx\)’ mean ‘\(x\) is a thread’;‘\(Wx\)’ mean the complex statement ‘\(x\) is loadedwith a weight exceeding that which characterizes its tensilestrength’; 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 alawlike connection between the conjunction ‘\(Tx \ampWx\)’ and ‘\(Bx\)’ but leaves the nature of theconnection unspecified. (Canfield & Lehrer 1961, 205)

The envisaged prediction (‘\(Bx\)’) no longer holds if amagnet neutralizes the effect of the weight (‘\(Mx\)’).This can be accounted for if the absence of the magnet is explicitlymentioned in the law. The problem is that the law must be completewith respect to all such factors that might prevent the breaking ofthe thread. The completeness condition requires that there are nofurther disturbing factors—i.e., it requires an exclusivecp-clause (see section 3) Canfield and Lehrer also present an argumentfor why these cp-clauses cannot be defined away, namely that any suchattempt leads to an infinite regress of further cp-clauses.

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

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

Popper only briefly commented on this claim in a footnote (see Popper1974, 1186f.), while other authors discussed it explicitly as aproblem for falsificationism (see, for instance, Johansen 1980).

Hempel argues in his 1988 paper “On Provisos” that everyinferential construal of scientific theory testing is confronted withthe problem that a proviso always has to be added:

A proviso has to be conceived as a clause which pertains to someparticular application of a given theory and which asserts that in thecase at hand, no effective factors are present other than thoseexplicitly taken into account. (Hempel 1988, 154)

In fact, Hempel dismisses the phrase “ceterisparibus” as not particularly helpful.

Carnap, Lakatos and Hempel all argue that in theory testing we usuallyimplicitly assume an additional premise that states that there are nodisturbing factors. Even though in the later literature Hempel’spaper has been taken to be evidence for the existence ofcp-laws even in fundamental physics, the issue of whether acp-clause has to be added as an extra premise in theorytesting has to be distinguished from the claim that thelawsthemselves should be read as containing an implicit cp-clause (for acriticism of this conflation cf. Earman and Roberts 1999, 442ff.;Schrenk 2007a, 25–36; Eliot 2011).

Two further developments lead to the explicit discussion ofcp-laws.

First, in her paper “The Truth doesn’t explainmuch” (reprinted in Cartwright 1983), Nancy Cartwrightcriticizes the DN-model of explanation on the grounds that the allegedlaws on which the explanation relies (according to the DN-model) arenot true, but rather generalizations that hold under special,typically ideal, conditions only. These cp-laws, she claims, will notdo the work required:

Ceteris paribus generalizations, read literally without the‘ceteris paribus’ modifier, are false. They arenot only false, but held by us to be false; and there is no ground inthe covering law picture for false laws to explain anything. On theother hand, with the modifier theceteris paribusgeneralizations may be true, but they cover only those few cases wherethe 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 generalizationbecause, among other things, it is true only with respect to isotropicmedia.

Cartwright explicitly calls Snell’s law a cp-law. Manylaws we take to be strict universal laws are in fact cp-laws that holdin special circumstances only. According to Cartwright, the problemarises that these laws cover only special or ideal conditions (mostmedia are anisotropic), whereas in scientific practice they are usedto explain phenomena in non-ideal situations as well. Cp-laws seem toplay an essential role in explanations but there is no accountavailable of how cp-laws can do this work.

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

This claim provoked discussions about whether there are genuinepsychological laws (cf. Fodor 1974, 1987, 1991, 1997; Kim 1985, LePoreand Loewer 1987, 1989; Carrier 1998). Some authors, for exampleSchiffer (1991) and Earman, Roberts and Smith (2002), have even arguedthat no special science (from biology upwards) has genuine laws of itsown. By contrast, most authors have supported only the weaker claimthat there are nostrict special science laws, but haveargued that special science laws should be construed as laws thatcontain implicit cp-clauses.

Fodor (1987) observes that we do use psychological generalizations (inlater papers, e.g., Fodor (1991) he uses ‘law’ instead of‘generalization’) for predicting and explaining humanbehavior. These generalizations or laws, says Fodor, arenon-strict—they are hedged by a cp-clause. Nevertheless, theseclaims, he insisted are neither false nor uninformative.

It is, I expect, a long story, how the generalizations of the specialsciences manage to be both hedged and informative […]. Tellingthat story is part of making clear why we have the special sciences atall; 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 importantdebates: one about the structure of scientific explanations ingeneral, the other about the status of the special sciences asautonomous scientific disciplines.

3. A Framework for the Discussion: Distinguishing exclusive/ comparative and definite/indefiniteceteris paribus laws

It has been argued that “ceteris paribus” is anambiguous notion. Two distinctions will help to disambiguatethe notion: the distinction between comparative and exclusive cp-laws(section 3.1), and the distinction between definite and indefinitecp-laws (section 3.2).

3.1. Comparative vs. exclusive cp-laws

Schurz (2002) suggests distinguishing between two conceptions ofcp-law:comparative versusexclusive. Comparativecp-laws require that factors not mentioned in the antecedent or theconsequent of the law remain unchanged. By contrast, exclusive cp-lawsassert the connection between antecedent and consequent only under thecondition that certain factors are excluded.

The comparative sense of cp-clauses derives from the literal meaningof “ceteris paribus” as “the others beingequal”. A comparative cp-law asserts that the increase (ordecrease) of the value of a ‘variable’ \(X\) leads to anincrease (or decrease) of the value of another variable, say \(Y\),provided that all other (possibly unknown) \(X\)-independentvariables \(Z_1 ,\ldots ,Z_n\) that describe the states of theconsidered system (or at least those \(X\)-independent variables thatare potentially interfering) remain at same values. Thereby, avariable \((Z_i)\) is called\(X\)-independent iff it isnot causally (or nomologically) influenced by \(X\). So, torepeat, that a variable \(Z\) is \(X\)-independent means (by the abovedefinition) only that it is not caused by \(X\); though it may well bea cause of \(X\) (see Schurz 2014 for a novel distinction betweenceteris paribus andceteris rectis laws).

The reason why cp-laws of this type are called“comparative” in Schurz (2002) is that the requirementthat the other \(X\)-independent variables \(Z_1 ,\ldots ,Z_n\)‘are equal’, or remain at the same values, only makessense if the lawcomparestwo states of a described(kind of) system. These two systems differ in the value of theantecedent variable \(X\), but agree in their values of the\(X\)-independent variables \(Z_1 ,\ldots ,Z_n\). If the lattercondition holds for all possible values of the \(Z_1 ,\ldots ,Z_n\),the comparative cp-law is called unrestricted; otherwise it is calledrestricted (see below). Another plausible name for this kind of cp-lawwould be ‘equality cp-law’ (this was suggested by thereferee), because it merely requires that the remainder factors are‘equal’ in the two compared states of the describedsystem, rather than that some of them have to be excluded (as inexclusive cp-laws).

A terminological note: With a “variable” \(X\)(in the mathematical sense) we mean a functional property ofindividuals \(d\in D\) 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 possiblevalues \(x\in\ran(X)\) of the variable \(X\); if \(X\) isquantitative, \(\ran(X) = \Re\), i.e., the set of real numbers. (If\(ran(X) = \Re\) and the values of \(X\) are distributed according toa given probability distribution, then \(X\) is what is called a‘random variable’ in probability theory; cf. Hays andWinkler 1975). In what follows, “\(d_{(i)}\)” denotesproper individual variables (in the logical sense) and“\(x_{(i)}\)” denotes possible values of functionalvariables \(X_{(i)}\). An important subcase of comparative cp-laws areprobabilistic comparative cp-laws, in which the quantitativevariables \(X\) express theprobabilities \(P(F)\) of somequalitative 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 toa (proportional) increase of gas volume (Gay-Lussac’s gas law).

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

While (1) states aquantitative relation between theincreases, only anordinal relation between the increases ispredicted in (2).

Comparative cp-laws are connected to the invariance approach ofWoodward and Hitchcock (see §6.2 below). According to thisaccount, a cp-generalization expresses what would happen if anintervention on \(X\) would occur.^[10] In causal graph theory, an intervention is defined as an operationwhich changes the value of \(X\) and decouples \(X\) from all causalparents of \(X\) (cf. Pearl 2000, 23f, Woodward 2003, 98). This notionof an intervention has the effect that a change of \(X\)’s value(as the result of an intervention) will not change the value of anyother \(X\)-independent variable—which is exactly what isrequired by a comparative cp-law.

In the philosophical debate, cp-laws have often been understood in theother exclusive sense. Anexclusive cp-law asserts that acertain state or event-type \(A\) leads to another state or event-type\(B\),provided disturbing factors or influences areabsent.

Terminological note: ‘\(A\)’ is called theantecedent and ‘\(B\)’ theconsequent‘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 \(X\)-valuefrom \(x_1\) to \(x_2\).

Thus, an exclusive cp-clause does not merely require holding all other\(X\)-independent and potentially disturbing factors constant; itratherexcludes the presence ofdisturbing factors.More generally, it restricts the possible values of the remaindervariables \(Z_1 ,\ldots ,Z_n\) to those value ranges where they cannotdisturb the cp-law. Therefore, the exclusion-clause of exclusivecp-laws may always be equivalently reformulated as a clause whichrequires that certain truth conditions for the exclusive cp-law hold,namely those conditions which exclude disturbing factors. In thissense, Cartwright has remarked that “the literal translation is‘other things being equal’, yet it would be more apt toread ‘ceteris paribus’ as ‘other thingsbeingright’” (1983, 45). Joseph (1980, 777)talked about “ceteris absentibus” clauses, andHempel (1988, 29) calls exclusive cp-clauses “Provisos”(“…provided disturbing factors areabsent”).

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

(3)Ceteris paribus, planets have elliptical orbits (seeLakatos 1970).

(4)Ceteris paribus, people’s actions aregoal-oriented, in the sense that if person \(x\) wants \(A\) andbelieves \(B\) 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 onthe planet except that of the sun are—not merely constantbut—absent. Likewise, the cp-clause of (4) requiresthat any factors causing irrational behavior be absent.

The distinction between comparative and exclusive cp-laws is notdisjoint: some cp-laws are both comparative and exclusive, as forexample in the following example from theoretical economy:

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

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

Comparative cp-laws which are not restricted by an exclusive cp-clauseare also calledunrestricted comparative cp-laws. They assertan invariant connection between an \(X\)-increase and an\(Y\)-increase forall possible values of \(X\) and of the\(X\)-independent remainder variables \(Z_1 ,\ldots ,Z_n\).Unrestrictedprobabilistic comparative cp-laws have beensuggested as an explication of genericcausal relations(Cartwright 1989, 145f; Eells 1991, 85f): they assert that a variable\(X\) is a probabilistic cause of some other variable \(Y\) in allpossible circumstances, where these circumstances are expressed interms of the values of \(X\)and the \(X\)-independentremainder variables \(Z_i (1\le i\le n)\).

Unfortunately, unrestricted invariance claims are rarely true. Oneexample mentioned by Cartwright (1989, §5.2) in which they doindeed hold is the force law of classical physics, \(f := S_{i \inI}f_i = m \cdot a\), which equates the total force \(f\) with thesum of all component forces. In that case, an increase of acomponent force \(f_i\) will produce an increase of acceleration forall possible values of \(m\) and the remainder forces \(f_i(i\in I)\). In Schurz (2002, §2, 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 \(X_1 ,\ldots ,X_n\) arenon-interactingcauses of \(Y\) in a certain (technically defined) sense.

In moderately complex systems, the composition of causes is usuallyinteractive. Therefore, these systems will only obeyrestricted comparative cp-laws (Cartwright 1983, 64ff.;Dupré 1984). Restricted comparative cp-laws are nothing butcomparative cp-laws in the scope of an exclusive cp-clause, whichassert the comparative cp-relation only for a restricted class ofcircumstances, expressed in terms of allowed values of the independentvariable \(X\) and the \(X\)-independent variables \(Z_i\). In thissense, examples (1) and (2) above are exclusive-comparative cp-lawsbecause in (1), the cp-connection between temperature and volume holdsonly for approximately ideal gases, and in (2) the cp-connection holdsonly under psychologically normal conditions.

Although thedefiniens of a comparative cp-law relating \(X\)with \(Y\) refers to “all \(X\)-independent variables”,comparative cp-laws are nevertheless empirically testable by themethod 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 experimentalbut not on the control group, and finally compares the two samplesconcerning the value of \(Y\). Since the split of the original samplewas random, experimental and control group will agree in theirdistribution ofall \(X\)-independent remainder variables,apart from random errors. Thus, one can apply this method withoutknowing which \(X\)-independent variables are causally relevant for\(Y\) and which are not. If there is a significant change of the valueof \(Y\) in the experimental group compared with the control group,the cp-law “\(X\)-increase leads to an \(Y\)-change” isconfirmed (strictly speaking only the restriction of this law to thepopulation from which the sample has been taken is strongly confirmed,while confirmation of the unrestricted cp-law requires tests for avariety of different “populations”, i.e., distributions of\(X\)-independent remainder variables). On the other hand, if there isno significant change of \(Y\)’s value, the cp-law is stronglydisconfirmed, even in its unrestricted form. This does not mean thatthe method of randomized controlled experiment is free from errorpossibilities. In particular, the experimentally induced change of\(X\)’s value may have smuggled in unrecognized \(X\)-dependentvariables (i.e., unrecognized side-effects) which are partlyresponsible for the resulting change of \(Y\). For example, in ateaching experiment comparing two teaching methods (e.g., the newmethod and the old method), the new teaching method may come out asmore successful than the old one—yet not because of intrinsicproperties of the new method, but just because the fact that theteaching method wasnew has increased the motivation of itsteachers.

3.2 Definite versus indefinite cp-laws

An important distinction concerning only exclusive cp-laws is thatbetweendefinite andindefinite exclusive cp-laws. Adefinite exclusive cp-lawspecifies the disturbing factorswhich are excluded (or the validity conditions which are required) inthe antecedent of the law. In other words, a definite exclusive cp-law“exclusively cp, if \(A(d)\), then \(B(d)\)” has astrict completion of the form “For all \(d\): if\(A(d)\) and \(C(d)\), then \(B(d)\)”, where the completingcondition “\(C(d)\)” excludes the presence of thespecified disturbing factors in application \(d\). Earman, Roberts andSmith (2002, 283f.) call definite exclusive cp-lawslazycp-laws.

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

(6) exclusively cp, birds can fly.

The number of possible factors which may disturb a bird’s flyingcapability is potentially infinite. In other words, for everycondition \(C\) which excludes a finite list of such factors thecompleted law “All birds satisfying \(C\) can fly” willstill have to face further exceptions—hence, a strict completionof this law is impossible. Exclusive cp-laws of this sort are calledindefinite exclusive cp-laws, or in the terminology of Earman, Robertsand Smith (2002),non-lazy cp-laws.

It is a widely agreed in philosophy that the real significance ofexclusive cp-laws lies in situations where strict completion isimpossible (for example, cf. Rescher 1994, 14; Pietroski & Rey1995, 84, 102; Horgan & Tienson 1996, 119f.). In that case, theexclusive cp-law isindefinite, which means that itsexclusive cp-clause consists in a universal second order conditionwhich excludes all kinds of disturbing factors to the law, whateverthey are. Formally, also an indefinite exclusive cp-law may be writtenas a strictly completed law of the form “if disturbing factorsare excluded, then \(A\)s willalways be\(C\)s”—only that the meaning of “all disturbingfactors” is now unclear which leads to various sorts ofdeficiencies that are discussed in the next section.

4. The Challenge: ExclusiveCeteris 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 whichare not-\(B\)s (Pietroski and Rey speak of “abnormal”instances; 1995, 88). A philosophical reconstruction of exclusivecp-laws—which includes restricted comparative cp-laws and, thus,the majority of all cp-laws—faces a severe problem. This problemcan be articulated in the form of a dilemma. This dilemma has beenformulated by Lange (1993, 235), who attributes it to Hempel(1988).

First horn: If exclusive cp-laws are reconstructed as somesort of strict law, then they will tend to be false. Typically, itwill not be the case that all \(A\)s satisfying a completer condition\(C\) are \(B\)s, since the range of potentially disturbing factors istypically indefinable. For instance, the relationship between supplyand price is not always as the law of supply says (or, as it seems tosayprima facie), because an interfering factor might occur.In other words, special science laws that instantiate perfectregularities are—mildly put—“scarce”(Cartwright 1983, 45). Yet, if one supposes that the law is to beformalized as a universally quantified conditional sentence, then onecounter-instance (due to a disturbing factor) to the universallyquantified sentence means that it is false.

Second horn: If we instead suppose that an indefiniteexclusiveceteris paribus clause is attached to the law sothat it means “All \(A\)s are \(B\)s,if nothinginterferes”, then the cp-law in question is in danger oflacking empirical content. It lacks empirical content because it seemsto 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 areanalytically true sentences and, therefore, trivially true. This,however, is an unwelcome consequence because laws of the specialsciences should be reconstructed as empirical statements—not assentences being true in virtue of meaning.

Both horns are quite unpleasant results for any philosopher lookingfor a theory of non-strict laws. To deal with this dilemma is acentral challenge for every theory ofceteris paribus laws(Lange 1993; Earman and Roberts 1999; Earman, Roberts and Smith 2002).In the following section, several attempts to cope with this dilemmaare 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 toexplicate exclusive cp-laws is to add the missing conditions for astrict implication into the antecedent of the law statement. There aretwo quite different possibilities to do this. The first possibility isto add these conditions explicitly, by appropriate descriptions of thefirst order individual variables involved in the law. If this were thecorrect account, all correct cp-laws would turn out to be definiteexclusive cp-laws (or lazy cp-laws in the terminology of Earman,Roberts and Smith 2002). The second possibility is to add theseconditions by way of a second order quantification over first orderpredicate variables, which apply to the first order individualvariables of the law. In this account, cp-laws are turned intoindefinite exclusive cp-laws.

Fodor’s starting point for his account of cp-laws isDavidson’s distinction between homonomic and heteronomicgeneralizations (see section 2.2):

Exceptions to the generalizations of a special science are typicallyinexplicable from the point of view of (that is, in thevocabulary of) that science. That’s one of the things that makesit aspecial science. But, of course, it may nevertheless beperfectly possible to explain the exceptionsin the vocabulary ofsome other science. […]. On the one hand the [specialsciences’] ceteris paribus clauses are ineliminable from thepoint of view of its propriety conceptual resources. But, on the otherhand, we have—so far at least—no reason to doubt that theycan 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 existenceis required by the exclusive cp-clause cannot be completely specifiedwithin the conceptual resources of the special sciences, although thiscould be done (at least in principle) within the vocabulary of somemore fundamental science such as neurophysiology or, ultimately,fundamental physics. Fodor calls the missing factors‘completers’. A physical microdescription of theantecedent 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 todefine 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 \(A\) isa (mental) functional state, which can be realized by very differentstates \(R_i\), it is highly unlikely that there is a completer forevery realization (cf. Schiffer 1991, 5) Fodor accepts this objectionand provides the following account:

(9) cp\((A\rightarrow B)\) is true iff either (i) for every realizerof \(A\) there is a completer \(C\) such that \(A \amp C\rightarrowB\) or (ii) if there is no such completer for a realization \(R_i\) of\(A\) there must be manyother 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, theneven 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 completersin many other cp-laws.

Mott (1992) provided the following counterexample to condition (ii) ofthis account of truth-conditions: cp, if a person is thirsty, then shewill eat salt. This comes out as a true cp-law, because on the onehand, very likely many, maybe all, realizers of “beingthirsty” lack completers (because nobody tends to eat salt whenthirsty). On the other hand, ‘being thirsty’ might verywell go into the antecedent of many other cp-laws for which there aresuch completers. Mott’s counter-example can be avoided if oneadds to (ii) the requirement that for sufficiently many realizers\(R_j\) of \(A\) there exist completers with respect to \(B\) (seeSilverberg 1996 and Earman and Roberts 1999, section 9 fordiscussion).

Although Fodor starts with the hope that the missing conditions ofexclusive cp-laws could eventually be specified in the fundamentalscience, his definition of exclusive cp-laws involves a second orderquantification over variable (unspecified) completers of independentexception-explainers. Hence, his account of exclusive cp-laws belongsto the family of indefinite exclusive cp-laws. Because of this factFodor’s and related accounts face severe problems, as will bediscussed in section 5.2.^[11]

Schiffer, Fodor and Mott try to explicate cp-laws by ordinarytruth-conditions. In contrast, Pietroski and Rey provide strongerconditions for what they call “non-vacuous truth”, interms ofepistemic conditions such as explanation andindependent confirmability. Pietroski and Rey compare cp-clauses tocheques:

These cheques represent a ‘promise’ to the effect that all[counterinstances] of the putative law can be explained by citingfactors that are […] independent of that law. If the promisecannot be kept the cheque was no good to begin with. (Pietroski andRey 1995, 89)

Completion is conceived here as explanatory completion and it isrequired onlypost factum. If there is a counter-instance ofthe law in question, we are committed to explain why the law was notinstantiated:

a cp law holds in a ‘closed system’, i.e. a systemconsidered in abstraction from other, independently existing factors.Such a systematization is non-vacuous only to the extent thatdeviations from the regularities that are constitutive of it can beexplained by those factors. (Pietroski and Rey 1995, 89)

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

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

(10) cp\((A\rightarrow B)\) is non-vacuously true iff

1. ‘\(A\)’ and ‘\(B\)’ are otherwisenomological and
2. For all \(x\), if \(Ax\), then (either \(Bx\) or there exists anindependently confirmable factor that explains why \(\neg Bx)\),and
3. cp\((A\rightarrow B)\) explains at least something as assumed incondition (ii).

Pietroski & Rey’s (1995) account involves a second orderquantification over unspecified exception-explainers. Thus, as in thecase of semantic completer accounts, their account explicatesindefinite exclusive cp-laws and faces severe problems to be explainedin the next section.

5.2. Criticisms: triviality and accidentality

Several authors have independently shown that completer approaches areunsatisfactory. Earman and Roberts (1999, 454f) provide argumentswhich show that an exclusive cp-law in the sense of Pietroski and Reycannot escape the problem ofvacuity. Their example is thealleged cp-law “cp, all spherical bodies conductelectricity”. Every failure of this law can be explained interms 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 arenot vacuous butalmost vacuous. More precisely, the contentof “exclusively cp, \(A\)s are \(B\)s” in the sense ofPietroski & Rey’s definition (10) is equivalent to the claimthat 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” isa strictly true law. This means that conditional on \(A\), every\(B\)-event or not\(-B\)-event is assumed a deterministic andindependently identifiable cause, whatever this cause may be. Thispresupposition of determinism is both too strong and too weak: It istoo weak because it does not establish a relevant nomologicalconnection between \(A\) and \(B\). It is too strong because in allareas involving random processes, determinism does not hold. Woodward(2002, §2) has demonstrated that a similar criticism affectsFodor’s account.

There is an even more seriousaccidentality-problem that isconnected to the almost-vacuity of indefinite exclusive cp-laws. Anexclusive cp-law may be true although its antecedent is not at allnomologically or causally relevant for its consequent. All sorts ofevents are connected by indefinite exclusive cp-laws, provided onlythat 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 accordingto the explications of Pietroski and Rey (1995), Fodor (1991) andHausman (1992), the exclusive cp-law “all charged particlesaccelerate at a rate of \(n\) meters/sec\(^2\)” is a trueexclusive cp-law for arbitrary values of \(n\). Schurz (2002, 364)demonstrates that according to these accounts “if a person looksto the right, she will see a kangaroo” is a true exclusivecp-law. Also, Earman & Robert’s example can be read asdemonstrating theaccidentality of “cp, all sphericalbodies conduct electricity”, although the example was originallyused to show that this statement is vacuously true.

6. Invariance & Stability Theories

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

“Stability” and “invariance” are often takento be synonyms. For the sake of clarity, we will use“stability” to refer to Lange’s stability theory oflaws (section 6.1) and we will use “invariance” to referto Woodward & Hitchcock’s invariance theory of laws (section6.2).

In stability/invariance accounts, laws qualified by a cp-clause areconsidered to be non-strict in the sense that they hold true onlyunder a limited range of counterfactual suppositions. Differentversions of invariance or stability theories differ in how theydetermine this limited range of counterfactual suppositions.In this sense, invariance or stability theories can be understood asan alternative to exclusive interpretations of non-strict laws and asan 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 lawsin physics and the cp-laws in the inexact special sciences differ onlyin degree (for a related account that does not focus on stabilityalone see Mitchell (2000)). The lawhood of universal laws and cp-lawsis due to the same property of these statements: their stability.Lange’s (2000, 8f.) approach consists in two steps. First, therole of laws in scientific reasoning is determined: they figure inexplanations and predictions, they support counterfactuals, and theyare inductively confirmed. And second, it is claimed that the laws canplay this role because of their characteristic stability. So, in orderto understand Lange’s account of cp-laws, we must first clarifythe crucial notion of stability.

Lange’s basic idea is that laws are lawlike because they aretrue in a wide range of possible conditions. Most importantly, thelaws do not only hold true in any (non-nomic)actualconditions 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 interms of nomic preservation:

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

This definition, however, raises an obvious question: what does itmean to be physically possible? Roughly, a proposition \(p\) is aphysical possibility iff there is some possible world \(w\) where thesame laws of nature hold as in the actual world and \(p\) obtains at\(w\). But if this is true, Lange’s definition of lawhood seemsto be circular, since being consistent with thelaws (of theactual world) is ultimately part of thedefiniens of being alaw (of the actual world). Lange (2005, 2009a, 25–28) recognizesthe problem of circularity and offers an alternative account of lawsin terms of the notion of stability. It says, roughly put:

(12) A set of statements \(G\) is stable iff every member of \(G\) istrue, \(G\) is logically closed, and \(G\) remains true under every(non-nomic) counterfactual supposition \(p\) that is consistent withevery member of \(G\) (cf. Lange 2000, 100, 103; 2005, 420; Lange2009a, 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-maximalstable set \(G\).

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

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

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

According to Lange, this generalization is an accidental truth becauseit is not a member of a (non-maximal) stable set. For assume \(A\) isa member of some non-maximal set \(\Delta\) (which may contain allphysical laws). By non-maximality, there exists some furtheraccidental 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 withevery member of \(\Delta\). However, Lange (2005, 421) argues thatthere are at least some conversational contexts in which \(A\) doesnot have priority over \(B\). Hence, \(\neg A \vee \neg B \rightarrowA\) is not the case, and therefore \(\Delta\) is not stable under thecounterfactual supposition \(\neg A\vee \neg B\). By contrast,according to Lange, a law of nature has priority over any accidentaltruth in every context, and thus the laws of nature form a stableset.

Even assuming that Lange’s argumentation is valid (see, however,Demarest (2012) and Hall (2011) for criticism based on thecontext-sensitivity of counterfactuals), there is the worry that thecircularity problem has not entirely been dissolved: It reappears atthe level of the truth conditions for counterfactuals, since usuallythey are formulated in terms of the laws (see, e.g., Goodman 1947;Lewis 1973; Maudlin 2007: 21). Lange (2009a: chapter 4) answers thisobjection by arguing that subjunctive facts are ontologicallyprimitive and the truth conditions for counterfactuals do not refer tolaws. He concludes that, therefore, his explications (12) and (13) arenot circular.

We can now turn to the question how cp-laws differ from universallaws. According to Lange, cp-laws are stable (sets of) propositionswhose application ispragmatically restricted to the purposesof a scientific discipline. He elaborates this idea with respect tothe laws in the special sciences, or inexact sciences as he callsthem: “A set is stablefor the purpose of an inexactscience if and only if it is invariant under every counterfactualsupposition of interest to the science and consistent with theset” (Lange 2002, 416). Lange tries to avoid the horns of theFalsity-or-Triviality-Dilemma by treating ‘ceterisparibus’ as a name for a set \(I\) of interfering factors.Notice that \(I\) does not listall the possibleinterferences which prevent the occurrence of \(B\) in ‘cp, all\(A\)s are \(B\)s’, but only those factors that arerelevant (for a discipline). One can understand Lange in away that he provides two strategies^[12] to determine the members of \(I\): (A) the strategy ofnon-negligibility and (B) the strategy of intended interest of ascience. Both strategies try to explicate a methodology that isimplicitly used by scientists in a particular discipline (cf. Lange2000, 170–174).

(A)The strategy of non-negligibility: Instead of providing acomplete list of all interfering factors, scientists merely refer tothose interfering factors “that arise sufficiently often, andcan cause sufficiently great deviations from \(G\)-hood, that a policyof inferring \(F\)s to be \(G\) […] would not be good enoughfor the relevant purposes” (Lange 2002, 411; Lange 2000, 170f).For instance, consider the economic law “cp, if the supply of acommodity increases then the price decreases”. According toLange, it may happen that the increase in supply is so small that nodecrease in price results. But it might as well happen that the pricedoes not decrease although the supply increases significantly, becausea gigantic comet hitting the planet Earth and destroying all life onits surface disturbs the instantiation of this law. The comet causessufficiently great deviation from a decrease in the price of a good.Nevertheless, comets are negligible for the purposes of economistsbecause their occurrence does not arise sufficiently often to count asinterfering factor that is to be explicitly listed in thecp-conditions.

(B)The strategy of intended interest of a science: A law maystill count as stable if it fails to hold under those counterfactualsuppositions that do not fall into the range of the laws intendedpurpose and application. This point is best illustrated by an examplefrom island biogeography—the area law—provided byLange:

It has been suggested thatceteris paribus, the equilibriumnumber \(S\) of a species of a given taxonomic group on an island (asfar 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 andisland group. (Lange 2002, 416f.; cf. Lange 2000, 235f.)

There are counterfactual suppositions for which the area law is nottrue. For example, imagine an island where the animals of the species“chicken” exclusively live on chicken farms. Supposefurther that on these farms chicken are bred and held under extremelycrowded conditions. So, the counterfactual supposition stemming fromthis example is “chicken on the island in question are bredunder extremely crowded, artificial conditions set up byfarmers”. Obviously, the area law will drastically fail to holdfor this case. Moreover, cases of this kind are not far-fetchedphilosophical thought experiments: they donot occurrarely in times of cultivated breeding of animals (asrequired bythe strategy of non-negligibility). Nevertheless,scientists exclude this kind of exceptions because it conflicts withthe intended purpose and application of their discipline (in thiscase: island biogeography) (cf. Lange 2002, 417; Lange 2000, 232f. forfurther examples^[14]).

It is illuminating to contrast Lange’s strategies of determininga set of disturbing influences \(I\) (in the light of intendedapplications of a law) with the use of cp-clauses that does indeedrender a statement trivial. Lange illustrates this point by an exampleof a mere “excuse clause”:

Suppose someone says ‘I can run a four-minute mile’ butwith each failure reveals a proviso that she had not stated earlier:‘except on this track’, ‘except on sunny Tuesdays inmarch’ and so on. It quickly becomes apparent that this personwill not acknowledge having committed herself to any claim byasserting ‘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-clausesas used in the sciences, because the latter refer to strategies ofdetermining disturbing factors. Although the relevant set ofdisturbing factors is not listed explicitly, it is implicit in thescientific practice (and the education and studies) in a particularfield of inquiry.

Based on his stability theory of laws, Lange also proposes an argumentfor the autonomy and irreducibility of special science laws. Recallthat arguing for the autonomy and irreducibility of the specialsciences with respect to physics was one of the crucial motivations tostart the debate on cp-laws (see section 2.2). Suppose that the lawsof a certain special science \(D\) are stable with respect to thepurposes of \(D\). This pragmatic restriction is unique to \(D\). Inother words, \(D\)’s laws are stable for counterfactualsuppositions that violate other disciplines’ laws, for examplethe laws of fundamental physics. For instance, Lange (2002, 420)claims that ‘the area law would still have held had there beenbirds equipped with organs weakening gravity’s pullsomewhat’. Having organs that weaken gravity’s pull isunderstood as a violation of the law of gravitation. Thus,\(D\)’s laws are stable under some counterfactual suppositionfor which fundamental physics is unstable. This unique range ofstability makes \(D\) epistemically autonomous as a scientificdiscipline because \(D\)’s laws can figure, for instance, inexplanations answering some why-questions that no other discipline cananswer as adequately as \(D\) itself (cf. Lange 2000, chapter 8; Lange2002, 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 itskey feature for performing the role of a law and figuring inexplanations and predictions. As in Lange’s theory, to beinvariant means for Woodward and Hitchcock to be true under several(not necessarily all) counterfactual suppositions. Despite this basicagreement there are important respects in which the invariance theorydiffers from Lange’s stability theory. In what follows we focuson the distinguishing features of the invariance approach:

1. The Quantitative Language of Law Statements: Although theidea might be implicit in Lange’s theory, invariance theoriesconceive candidates for law statements explicitly andstraightforwardly asquantitative statements. Thus, thepredicates occurring in the statement are variables \(X\) which mayattain certain values \(x\) in ran\((X)\), as explained in section3.1. One might object that there may be laws in the historical andsocial sciences that are not quantitative but merely qualitative. Butthis qualitative character of statements can nevertheless betranslated into quantitative statements with binary variables, thatmay only take two possible values, ‘\(X=1\)’ meaning thatsome event type \(X\) occurs and ‘\(X=0\)’ meaning thatsome event type \(X\) does not occur (see EG1, 10f.).

2. Counterfactual Suppositions ViewedDifferently—Interventions: Invariance theories characterizethe counterfactually supposed antecedent conditions under which ageneralization remains stable differently than Lange’s stabilitytheory. Lange simply assumes these conditions, e.g., the temperatureof the gas is 32ºC, to be the result of counterfactualsuppositions. We suppose that \(p\) is true and see whether the law\(l\) remains true supposing \(p\). So, stability theory isindifferent concerning how the supposed state of affairs comes about.Invariance theory is more restrictive in characterizing thecounterfactual situations: A counterfactual situation refers to achange in the value of a variable. For an invariancetheorist, it is important to explainhow the change of avariable is brought about: the change of the value of a variable mustbe the causal outcome of anintervention. An interventionconsists in a local change thatsets a variable \(X\) to acertain value and at the same time decouples \(X\) from all\(X\)-independent variables, as explained in section 3. So, anintervention on \(X\) is a direct, exogenous causal influence on somevariable \(X\) which (1) does only act on \(X\) itself while (2) othervariables \((Z)\) are only changed in virtue of being directly orindirectly caused by the change in \(X\). Notice that interventionsneed not be carried out by human beings. Instead they are conceived ashypothetical (not necessarily human) causes. (cf. Woodward 2003, 98;and Woodward 2003, 103f., 123–127 for arguments against ananthropomorphic interpretation of interventions.)

3. Different kinds of Variables: Invariance theories distinguishtwo kinds of variables: (1) Explicit variables, that figure explicitlyin a generalization and (2) background variables, that describebackground conditions which are not mentioned in the generalization inquestion. Consequently, interventions can be carried out with respectto both kinds of variables.^[15]

   Similar to Lange’s theory, Woodward and Hitchcock use theintended purposes of inquiry of a certain discipline to distinguishbetween interventions into explicit variables and background variables(cf., e.g., Woodward 2003, 262f). According to their invariancetheory, only the former are of importance to laws in a specificscientific discipline. Woodward uses an example from economics toillustrate the point:

   In microeconomics, individual economic agents are often assumed toconform to the behavioral generalizations constituting rational choicetheory (RCT). […] Even if we assume, for the sake of theargument, that these generalizations are roughly accurate descriptionsof the behavior of many participants in markets, it is clear thatthere are many changes and interventions over which thegeneralizations will fail to be invariant. For example, there are manypharmaceutical interventions and surgically produced changes in brainstructure that will lead previously selfish agents to act innon-self-interested ways […].However, economists have notgenerally regarded these sorts of failures of invariance asinteresting or important, at least if […]they occurrelatively rarely in the population. (Woodward 2003, 263)

   Obviously, the reason to ignore rare but possible changes in thebackground variables is very similar to Lange’sstrategy ofnon-negligibility. Woodward contrasts these negligible changes ofbackground variables with important changes of explicit variables:

   For example, microeconomists often require that fundamentalexplanatory generalizations such as the principles of RCT [rationalchoice theory] be invariant under changes in information available toeconomic agents or under changes in their beliefs and under changes inthe 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 differdrastically 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 followingso- calledtesting intervention condition holds: there are atleast two different possible values of an explicit variable \(X, x_1\)and \(x_2\), for which \(Y\) realizes a value in the way that thefunction \(f\) 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 thefollowing one: \(X=x_1\) describes an actual state of affairs while\(X=x_2\) describes a possible counterfactual state of affairs. Forinstance, 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 30°C. According to the testing intervention condition,Boyle’s gas law is stable if it also holds for (counterfactual)temperature of, say, 40°C. One might call this kind of invariance“minimal” invariance. Contrary to Lange’s theory,minimal invariance is necessary and sufficient for the statement tocount as invariant. Notice, however, that invariance under variationsof background variables is neither necessary nor sufficient for beingminimally invariant (cf. Woodward 2003, 248; EG1, 7f.).

5. Degrees of Invariance: One can still distinguish more orless stable generalizations. So, invariance is not an all or nothingmatter—instead it admits of degree. Roughly put, the degree ofinvariance of a generalization \(G\) is measured by the range ofpossible values of variables contained in \(G\) for which \(G\)remains invariant. This view implies two extremes of invariance: (1) Ageneralization that holds for all possible values of its variables ismaximally invariant, (2) a generalization \(G\) that holds for merelytwo of the possible values of the dependent variables in \(G\) isminimally invariant. Between these extremes of invariance liegeneralizations which hold only for a certain range of values.Woodward and Hitchcock (EG1) illustrate the latter kind ofgeneralization by the example of an equation that describes the growthof plant depending on the amount of water and fertilizer.^[16]

According to Woodward & Hitchcock, laws or generalizations in thespecial sciences do not hold under all interventions. They merely holdfor a certain range of possible values (of those variable figuring inthe law statement). By being non-strict the generalizations in thespecial sciences do not satisfy a condition that is traditionallyassociated with laws of nature, namely the condition of universality.Nonetheless, being invariant for a limited range of values is enoughfor a proposition to play a lawlike role in the sciences, as is arguedin EG2. Woodward and Hitchcock (EG2, 184–189) develop furthercriteria and examples of determining the degree of invariance of ageneralization \(G\) (e.g., comparing two generalizations \(G\) and\(G*, G\) might be more exact than \(G^*\) with respect to the samerange of intended application; or \(G\) might be more or lesssensitive 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 ofcounterfactual suppositions or, alternatively, interventions to whichthe law is applicable. One can say that Lange’s stability theoryand Woodward & Hitchcock’s invariance theory start fromopposed extremes of the spectrum of stability: (1) Lange starts at themaximum of stability, i.e., he starts with the claim that alaw-proposition \(l\) remains true forall counterfactualsuppositions that are consistent with \(l\), where by definition acounterfactual supposition is admissible if it is consistent with thelaws of nature. In a second step, Lange reduces the set of allcounterfactual suppositions consistent with \(l\) because of certainpragmatic goals of the discipline which uses the proposition \(l\).(2) Woodward and Hitchcock start with minimal stability, i.e., thesatisfaction of the testing intervention condition for someproposition \(l\). In a second step, Woodward and Hitchcock add tominimal stability that \(l\)’s degree of invariance increaseswith the number of possible interventions for which \(l\) holds (amongother criteria).

6. Different strategy to distinguish laws from accidentally truegeneralizations: Traditionally, philosophers of science believethat genuine laws differ from accidentally true generalizationinkind. Invariance theorists, such as Woodward and Hitchcock, agreethat such a distinction can be drawn. Accidentally truegeneralizations are not minimally invariant in the sense of satisfyingthe testing intervention condition. To take an example, Woodward andHitchcock think that an accidentally true generalization such as“All golden spheres have a diameter of less than a mile”is not invariant underany interventions, i.e., thegeneralization doesnot remain minimally invariant undertesting interventions on their explicit variables (see definition(14)). Therefore, laws and accidental generalizations differinkind (cf. Woodward 2003, 239f.). However, Woodward and Hitchcockacknowledge the fact that minimally invariant generalizations may failto be stable because these generalizations may be true only if veryspecificbackground conditions obtain (see the precedingparagraphs about degrees of invariance).

Analogously, Lange argues for a principled distinction between lawsand accidents within the framework of his stability theory (section6.1). However, Lange’s strategy to distinguish laws fromaccidents seems to be partly motivated by the intuition thatminimal invariance in Woodward & Hitchcock’s senseis too weak to account forreal life stability displayed byscientific generalizations. Correspondingly, Lange (2009b:297–302) argues that Woodward & Hitchcock’s accountfails to distinguish laws from mere accidents and therefore also failsto account for the laws’ characteristic explanatory power.

7. Dispositional Accounts

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

With regard toexceptions; in any tolerably advanced sciencethere is properly no such thing as an exception. What is thought to bean exception to a principle is always some other and distinctprinciple cutting into the former: some other force which impingesagainst the first force, and deflects it from its direction. There arenot alaw and anexception to that law—the lawacting in ninety-nine cases, and the exception in one. There are twolaws, each possibly acting in the whole hundred cases, and bringingabout a common effect by their conjunct operation. […] Thus ifit were stated to be a law of nature, that all heavy bodies fall tothe ground, it would probably be said that the resistance of theatmosphere, which prevents a balloon from falling, constitutes theballoon as an exception to that pretended law of nature. But the reallaw is that all heavy bodiestend to fall […]. (Mill1836 quoted after Mill 2000, 56, original emphasis)

This idea was taken up by Cartwright and others. Cartwright’sdiscussion of cp-laws focused on the question why cp-laws are ofinterest in non-ideal situations. If what cp-laws say is confined tospecial, namely ideal, circumstances it seemsprima faciethat these laws are irrelevant for all other, non-ideal, situations.That, however, is not the case in scientific practice. Cp-laws areused to explain phenomena outside the ideal circumstances. Cartwrightargues that this practice requires the postulation of capacities ortendencies:

The logic that uses what happens in ideal circumstances to explainwhat happens in real ones is the logic of tendencies or capacities.Whatis an ideal situation for studying a particular factor?It is a situation in which all other ‘disturbing’ factorsare missing. And what is special about that?When all otherfactors are absent, the factor manifests its power explicitly in itsbehaviour. […] This tells you something about what willhappen in very different, mixed circumstances—but only if youassume that the factor has a fixed capacity that it carries with itfrom situation to situation. (Cartwright 1989, 190f.)

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

Theprima facie advantage of a dispositional account of lawsvis-á-vis the cp-law-problem is the following: Thedispositional account provides asemantics for cp-laws.According to one version of the dispositionalist account of cp-laws(see Hüttemann 2014), a law statement is true provided the typeof system in question has the disposition that the law statementattributes to the system. Reconstructing law-statements as statementsabout dispositions, tendencies, or capacities rather than about overtbehavior turns cp-laws into strict laws. The claim is—as inMill—that certain kinds of systems have certain kinds oftendencies or dispositions. In stating the law, it is no longernecessary to appeal to cp-clauses. One advantage is that the so-calledproblem of instantiation, “the problem that many cp-laws appearnot to have any instances” (Lipton 1999, 164) can be solved. Thelaw is no longer considered to be a description of the systems’occurrent behavior that is only manifest under very specialconditions—if at all. The law concerns the underlying stabletendencies or dispositions. Furthermore, this dispositionalist accountis able to provide a rationale for why scientists are interested incp-laws. Cp-laws describe how systems behave in the absence ofdisturbing factors, i.e., if the disposition, tendency etc. canmanifest itself completely. This knowledge can be used to account formore complex situations, in which various systems and theirdispositions are intertwined—provided laws of superposition areavailable.

There is a different version of dispositionalism about cp-lawsaccording to which they are not turned into strict laws (see Bird2007: section 3.3). In a nutshell, Bird’s general idea is thatthe “laws are those regularities whose truth is guaranteed bythe essentially dispositional nature of one or more of the constituentproperties” (Bird 2007: 46–47). According to Bird,fundamental essentially dispositional properties or potencies giverise to strict laws, since there are no (or only very few)finks orantidotes that would prevent themanifestation of the potency from occurring if the stimulus hadoccurred (see Bird 2007 chapters 3.3.2 and 3.3.3, and the SEP-entry ondispositions for details on interferes suchfinks andantidotes). However, the manifestation process ofnon-fundamental dispositional properties can be interfered with and sothe manifestation only occursceteris paribus, i.e., inconditions where the trigger occurs andfinks andantidotes are absent (see Bird 2007: chapter 3.3.1).Therefore, these non-fundamental properties give rise toceterisparibus laws only.

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

1. Dispositions, powers, potencies or tendencies may be present withoutbeing manifest (cf. Earman and Roberts 1999, 451f.):

   Thus if what one wants explained is theactual pattern, howdoes citing a tendency—which for all we know may or may not bedominant and, thus, by itself may or may not produce something likethe actually observed pattern—serve to explain this pattern?

   As a rejoinder, dispositionalists can point to laws of composition. Asa consequence, it is not the disposition on its own that can explainthe actual pattern in such cases but rather the dispositions plus thelaw of composition. For example, Newton’s second law with (i)the gravitational force and (ii) the Coulomb-force describe twodispositions that explain the actual behavior of a particle relying onthe law for the superposition of forces. The law for the superpositionof forces describes how the two forces contribute to the actualbehavior. (Note that the dispositions the laws of nature attribute tosystems need not in general be taken to be macroscopic dispositionslike fragility or solubility.) To the extent that such laws areavailable (as for example for physical forces) the contribution of thevarious tendencies or dispositions can be determined even if they failto be (completely) manifest. It can, however, be disputed whether suchlaws of superposition/composition are available in general, especiallyif one holds like Bird (2007) that every law needs to be grounded inpotencies. But even if composition laws are taken to be underpinned bydispositions, a further law seems to be needed that tells us whathappens if the disposition for the law of composition fails to bemanifest. We need a meta-law that tells what the law of compositioncontributes to the actual behavior. So, the assumption that laws ofcomposition are grounded in dispositions attributed to physicalsystems threatens to lead to an infinite regress.

2. A second problem concerns the question whether a dispositional accountcan indeed avoid the dilemma for exclusive cp-laws (see section 4).Even though it is true, at least on one version, that the laws nolonger appeal explicitly to cp-clauses, thetriviality-or-falsity-dilemma turns up at a different place (seeLipton 1999, section 5: Hume’s revenge). The dispositionaliststill has to specify the stimulus or triggering conditions for thedisposition in order to give the disposition’s ascription adeterminate content. The conditions under which the disposition willmanifest itself are exactly those that are required to explicate thecp-clause (see Lipton 1999, section 5; for an extensive discussion ofthis problem cf. also Schrenk 2007b). So, explicating the triggeringconditions of a disposition requires that something informative issaid about the cp-clauses, and more generally, leads us back to theproblems discussed in the previous sections (see Hüttemann 2014for a recent defense of the dispositionalist approach to cp-laws; Ward2009 provides a critical discussion).

8. Normality Theories

The reconstructions of exclusive cp-laws in terms of strictcompletions do not imply anything about theprobability withwhich undisturbed antecedent-events will produce the consequent (cf.Pietroski and Rey 1995, p. 84; Schiffer 1991, p. 8). According toproponents of normality theories this seems to be counterintuitive,because cp-laws should be asserted only if the situation without(non-negligible) disturbing factors is anormal or at least arather probable situation. Examples of correct exclusive cp-laws whichviolate this normality condition are, for example: “exclusivelycp, no tire blows out”, “exclusively cp, there are noclouds in the sky”, etc. Several authors have thereforedeveloped so-callednormality accounts of cp-laws, includingSilverberg (1996), Earman and Roberts (1999, 463), Spohn (2002), Spohn(2012, chapter 13), and Schurz (2001b, 2002), who argues that thenormality interpretation is the preferred interpretation of exclusivecp-laws in thelife sciences, from biology upwards to thesocial sciences.

Normality theories can be understood as an alternative to the(definite and/or indefinite) exclusive interpretation of non-strictlaws. The general idea of normality theories is that “cp, all\(A\)s are \(B\)s” means thatnormally \(A\)s are\(B\)s. They can also be combined with a comparative interpretationinto normic-comparative law-statements such as “normally anincrease of \(X\) leads to an increase of \(Y\) when other\(X\)-independent variables are held constant”. However, thenormality 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 interms of a high probability of the consequent predicate, given theantecedent predicate, where the underlying conditional probabilitiesare objective statistical probabilities based on the dispositions ofevolutionary systems. Another possibility (suggested by Spohn 2012,see section 8.2) is to explicate the normality conditions in terms ofdegrees of belief and ranking functions over possible worlds.

Largely independently of Schurz’s and Spohn’s specificapproaches but in agreement with the general idea of the normalityapproach, Hüttemann and Reutlinger (2013), Kowalenko (2014),Reutlinger (2014), Roberts (2014) and Strevens (2014) have recentlyexplored a statistical approach to cp-laws, according to which cp-lawstatements are statistical claims.

8.1. Normic laws and evolution

Schurz (2001b; 2002, §5) analyses cp-laws in non-physicalsciences asnormic laws of the form “\(A\)s arenormally \(B\)s”. Here are some examples (4* and 6* are thenormic 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 countryintact.

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

According to thestatistical consequence thesis, normic lawsimply numerically unspecified statistical generalizations of the form“Most \(A\)s are \(B\)s”, by which they can be empiricallytested. The statistical consequence thesis has been challenged bycognitive scientists (e.g., McCarthy 1986) and by philosophers ofbiology (e.g., Millikan 1984). Schurz (2001b) defends the statisticalconsequence thesis by the following argument which is based on thegeneralized theory of evolution that does not only apply to biologicalevolution but also to cultural evolution (cf. Mesoudiet al.2006 for an excellent overview on generalized evolution theory; seeStrößner 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 thehumanities) are evolutionary systems or their products. Evolutionarysystems are systems whose self-regulatory properties have beengradually selected according to their contribution to reproductivesuccess. The temporal persistence of self-regulatory systems isgoverned by a certain range of prototypical normstates, inwhich these systems constantly have to be in order to stay alive. Theymanage this with the help ofregulatory mechanisms whichcompensate fordisturbing influences of the environment.Although the self-regulatory capacities of evolutionary systems arethe product of a long adaptation history, they are not perfect.Dysfunctions may occur, whence their normic behavior may have variousexceptions. Yet, it must be the case that these systems arein their prototypical norm states in the highstatisticalmajority of cases and times, since otherwise, they would not havesurvived in evolution.

In this way, the evolution-theoretic foundation of normic laws doesnot only explain why normic laws are the typical form of the laws oflife sciences. It also explains why normic laws are not strict butnevertheless are associated with high conditional statisticalprobabilities, at least for most cases and times in evolution.

According to Schurz (2002), the difference between idealizationconditions in physics and normality conditions in evolutionarysciences can be explained as follows: In physics, one traditionallythinks ofcomplexity as a source ofdisorder—regularities are obtained byabstractingaway from complexities. In evolutionary systems, by contrast,complexity is usually asource oforder—complexity which has been selected byevolution to stabilize normic behavior. As Wachbroit (1994, 587f.)puts it, ideal planets are theoretical abstractions: mass points underthe influence of a centripetal force and ‘nothing else’.They do not literally exist. In contrast, normal birds reallydo exist because they are what has been selected throughevolution. When speaking of a normal bird, we do not abstract from itsadmirable complexity, but werely on it as the cause of itsnormal behavior. This does not mean that abstract laws of physics(e.g., the laws of aerodynamics) play no role in the explanation offunctional behaviors of evolutionary systems (e.g., that birds canfly). However, even when we explain a bird’s flying ability interms of the laws of aerodynamics, we still have to assume a‘normal’ bird which possesses the typical capabilities ofbirds that have been selected through evolution. The idealizationprocedures needed for planets would not make good sense for birds:there are no disturbing parameters which, when going to zero, turn areal bird into an ideal bird which can necessarily fly and which isapproximated by the real bird.

8.2. Normal conditions approach

Another way to understand “ceteris paribus” asreferring to normal conditions is introduced by Wolfgang Spohn (1997,section 5; 2002, section 4; 2012, chapter 13.2; 2014). Spohn takes theconnection betweenceteris paribus qualifications andnormality to be straightforward: “ceterisparibus” meansother things being normal. Let uscall this theNormal Condition Approach. The basic idea ofthe Normal Conditions Approach is that a cp-law \(L\) holds whennormal 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” meansthat the occurrence of some conditions ishighlyprobable.

For Spohn, a condition is “normal” if it is (i) expectedor (ii) at least not ruled out to obtain by a rational, epistemicagent (for formal definitions of normal and exceptional conditions cf.Spohn 2002, section 4, and Spohn 2012, chapter 13.4). He illustrateshis Normal Conditions Approach with Hooke’s Law:

Hooke’s law, for instance, about the proportionality of theforce applied to a spring and its extension. It needs qualificationsin many ways, as was clear from the outset, even though thequalifications could be neither fully nor precisely specified. Onemust not overstretch the spring, the material the spring is made ofmust be elastic and homogeneous, its shape regular, the thermaldistribution uniform, etc. Physicists have cleared up most of theconditions—all of them?—under which Hooke’s lawholds and provided deeper explanation in terms of the intermolecularforces within a molecular lattice. After all, our sophisticatedtechnology of spring balances that is (or was) used in each lab andeach deli-shop depends on it. (Spohn 2012, 305)

Spohn concludes that Hooke’s law holds under normal conditionsand draws an analogy to other law statements in the specialsciences:

Hooke’s law holds under normal conditions, i.e., for diligentlymanufactured springs handled with the usual care. The rules of thumbof folk psychology normally apply; this is why they are suchtremendously useful common knowledge. Likewise, the laws of scientificpsychology and of economics at best hold under the conditionsexplicitly specified, along with further unspecified normalconditions. (Spohn 2012, 307)

Spohn’s Normal Conditions approach differs from Schurz’sNormic Laws approach in at least two respects.

1. According to Spohn (and contrary to Schurz’s theory which refersto objective probabilities), the fact that normal conditions obtain isconceived epistemically and, more specifically, doxastically. Anepistemic 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 interms of ranking functions (originally developed in Spohn 1988 underthe label “ordinal conditional functions”; cf. also Spohn2012 and the SEP-entry ‘Formal Representations of Belief’,section 3.3)

2. Normality in Spohn’s theory is explicated in terms of backgroundconditions, not in terms of probabilistic relations between theantecedent predicate and the consequent predicate. If someone believesthat the law \(f(X)=Y\) holds, then she believes that this functionalrelation holds under normal conditions \(N\), which means in terms ofSpohn’s ranking functions that the law \(f(X)=Y\) is believedwith rank zero, i.e., it is true in all normal worlds of a rankedworld model. Such a ranked world model consists of a set of possibleworlds together with a ranking function which attaches to each world anatural number \(0, 1, \ldots ,n\) specifying its rank. Worlds withrank 0 are the most normal worlds, worlds with rank 1 containexceptions from normal conditions (1st degree exceptions), worlds withrank 2 exceptions from those exceptions (2nd degree exceptions),etc.

   Spohn (2014) is the most recent and in-depth elaboration of his normalconditions based “epistemic account” ofceterisparibus laws and conditions.

Since the normal conditions “\(N\)” express theproposition that the given situation or world is normal, it would seemthat the ranking-of-worlds account entails the possibility of a strictcompletion by adding the normality condition \(N\) to the antecedent:“if \(N\), then \(Y = f(X)\)”. Indeed, it follows from thetruth condition for the assertion “normally (or cp) \(L\)”in a ranked world model (where \(L\) is e.g., \(Y = f(X))\) that thereexists a proposition \(N\)—namely the set of all worlds withrank 0—such that “normally \(L\)” is true in thegiven ranked model iff the material implication \(N\rightarrow L\) istrue in all worlds of this model. This does neither imply, however,that (a) the completing proposition \(N\) can be linguisticallyexpressed, nor (b) that it can be expressed in a non-trivial way, inthe sense that \(N\) does not already imply \(L\) on logical reasonsalone. Since one can only speak of a strict completion of“normally \(L\)” if (a) and (b) hold, it follows that theranking-of-worlds account does in fact not entail the possibility of astrict completion. This result is adequate, because as we have arguedabove, in most cases (especially in all non-deterministic situations)a strict completion is impossible. Thus, it has to be kept in mindthat speaking of “normality conditions” does not implythat these “normality conditions” can be expressed by any(non-trivial) proposition. For example, there is not any non-trivialproposition \(N\) which could turn the conditional “normally aCs\(^{137}\) atom decays after 60 years” into a strictimplication of the form “if \(N\), then a Cs\(^{137}\) atomdecays after 60 years”. Also note that the ranked-worldsaccounts and the conditional probability accounts are equivalent as asemantics for logical axiomatizations of conditional reasoning (seesection 8.3).

8.3.Ceteris paribus laws and non-monotonic reasoning

Reasoning from exclusive or normic cp-laws has an important logicalfeature: the inferences are in contrast to deductively valid argumentsnot monotonic. An inference is monotonic iff adding arbitrarynew premises to a valid argument preserves its validity. But theinference 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 longercorrect if \(d\) is known to instantiate a disturbing factor \(D\)which blocks the nomic connection between \(A\) and \(B\). Forinstance, the exclusive cp-law may state that if something is a bird(\(A\)), then it normally can fly (\(B\)), and \(D\) may assert thatthe given bird-instance has broken wings.

Assume the exclusive cp-law is formulated with the help of anon-strict conditional \(A \rightarrow B\). Drawing correct inferencesfrom non-strict conditionals requires non-classical rules of validinferences: The non-monotonicity of DefaultModus Ponensmeans formally that although the inference from \(A\rightarrow B\) and\(A(d)\) to \(B(d)\) is correct (i.e., \(A\rightarrow B, A(d) \dprovesB(d)\) holds, where ‘\(\dproves\)’ stands for‘non-monotonically correct inference’), the inference fromthe extended premise set \(A\rightarrow B, H\rightarrow \neg B,A(d)\), and \(H(d)\) to \(B(d)\) is incorrect (i.e., \(A\rightarrowB\), A\((d), H(d) \dproves B(d))\). These non-monotonic effects arereflected in non-classical rules for inferences from conditionals toconditionals. 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 correctfor non-strict conditionals. Only the more cautious inferences from\(A\rightarrow B\) and \(A\rightarrow D\) to \(A\wedge D\rightarrowB\) (cautious monotonicity CM) or from \(A\rightarrow B\) and\(A\wedge B\rightarrow D\) to \(A\rightarrow D\) (Cautious Cut CC) arecorrect for non-strict conditionals.

Two prominentsemantic criteria for the truth of a non-strictconditional and the correctness of inference from non-strictconditionals have been suggested in the literature.

1. Thehigh-probability-semantics understands normic laws inthe sense of high conditional probability assertions (as in section8.1). This semantics considers a non-strict conditional \(A\rightarrowB\) as true in a probability model if the uncertainty of theconditional, which is defined as 1 minus the conditional probability,is sufficiently low. An inference of a conclusion conditional from aset of premise conditionals is regarded as valid in this semantics iffthe uncertainty of the conclusion conditional is not greater than thesum of the uncertainties of the premises. High probabilistic semanticsgoes back to Adams (1975) and has been extended in Schurz (1998,2005).
2. The second semantics, thenormality-semantics,corresponds to the normal-condition account (see section 8.2). Thissemantics considers a conditional \(A\rightarrow B\) as true in aranked-world-model iff all lowest-rank \(A\)-worlds are \(B\)-worlds.An inference is considered as valid in this semantics iff allranked-worlds-models which verify all premise conditionals verify theconclusion conditional.

Remarkably, both semantics lead to the same conditional logics. Bothsemantics possess the system \(P\) (“preferentialentailment”) as a correct and complete axiomatization ofinferences among pure conditionals, and the system \(R\)(“rational entailment”) for inferences amongtruth-function combinations of conditionals (for more details cf.Krauset al. 1990, Hawthorne 1996, Adams 1975, Schurz 1998,2004, 2005, Leitgeb 2004, and the SEP-entry on non-monotoniclogic).

9.Prima Facie Reasons andCeteris Paribus Conditions in Ethics and Epistemology

The use of “cp” is not restricted to philosophy ofscience. It is used in other areas of theoretical and practicalphilosophy as well.

For instance,prima facie reasons are reasons under theconstraint of exclusive cp-conditions. They play an important role inethics and epistemology.

In ethics, aprima facie norm is an obligation (e.g.,“prima facie, you should not hurt anybody”) whichholds as long as exceptional circumstances (e.g., defending your ownlife) are absent (cf. Ross 1930 and the SEP-entry on moral reasoning).Dancy (2004, 17, 35f) uses the expression “protanto” instead of “prima facie”, whichis standardly used in the current debate. Pietroski (1993) haspromoted theceteris paribus understanding of moralobligations as a solution to paradoxes of moral obligations.Especially for the solution of moral dilemmas, the cp-reading of moralobligations has proven to be helpful, because although two conflictingcp-obligations may block each other, they do not produce a strictconsistency in the sense of deductive logic (see Horty 1994).

In epistemology, aprima facie reason for a belief is areason which justifies this belief in the absence of defeaters, i.e.,of exceptional information to the contrary.Prima faciereasons have been suggested in epistemology as a means of providingweak and defeasiblea priori justifications forepistemologically fundamental beliefs such as the belief in theexistence of an external reality, for which a strict and non-circulara priori-justification seems to be impossible. Nevertheless,the inference from visual appearances to reality counts as defeasiblyjustified as long as there do not exist serious defeaters, i.e.,reasons for serious doubt (cf. Williams 1996, 2001, and the SEP-entryona priori-justification and knowledge). An example is thedefeasible inference from visual appearances to beliefs concerningexternal reality. My awareness of my visual appearance of a tree infront of me is aprima facie reason for my belief that therereally is a tree in front of me, provided that information aboutexceptional circumstances is absent, for example the information thatI am under the influence of drugs which cause hallucinations (cf.Pollock 1986, Moser 2002). One may object that a similar kind ofpossible exception to the inference from visual appearances to realityis constituted by the possibility of a Cartesian demon who infuseshumans with illusionary perceptions, which is one of the majorskeptical counterarguments to the inference from appearances toreality. In response to this objection, several epistemologists haveargued that these kinds of exceptions are not serious exceptions, andtherefore, the inference from appearances to reality is defeasiblyvalid.

Finally, applications of cp-clauses can also be found in the form ofpresumptions in philosophy of language. Here it is assumed thatunderstanding linguistic expressions requires defeasible assumptionsor “presumptions”, for example presumptions concerning therationality of the speaker (cf. Quine 1960; Davidson 1973, Scholz2016, 148–159).

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

In the more recent literature, a number of trends and innovativeapproaches to cp-laws have emerged that cannot easily be subsumedunder the accounts presented above. For this reason, they deservespecial attention. There are at least four clearly discernibletendencies in the recent work on cp-laws:

1. A reconnection to metaphysics: One of the most salient trendsin the current literature is that the discussion of cp-laws(re)connects philosophy of science to analytic metaphysics in at leastthree respects: (i) current accounts of cp-laws that build on DavidLewis’s (1973, 1983) best systems account of laws; moreover,(ii) the interpretation of objective probabilities and (iii) themetaphysics of dispositions are brought into play for making progressin understanding cp-law statements.

   The most vivid sign of this reconnection consists in the fact thatseveral authors have recently defended friendly amendments of DavidLewis’s best systems account of laws of nature. WhileLewis’s (1973, 73) original version primarily focuses onfundamental laws, the amended versions are intended to capture alsonon-fundamental laws of the special sciences. Schrenk (2007a, 2014),Callender and Cohen (2009, 2010), and Frisch (2014a) defend aso-called “better best system account” of special sciencelaws. The central deviation from Lewis’s original account isthat the language a best system is framed in needs not be confined toone whose predicates refer to perfectly natural, fundamental physicalproperties only (see Lewis 1983).Instead “relativized” best systems can be formed forthe set of predicates used in chemistry, the set of predicates used inbiology, and so on. Jaag and Loew (2020) argue for a pragmaticversion of the best systems account that also lifts Lewis’snaturalness-requirement and captures central insights of the betterbest system account. On their view, the invariance of laws is not anend in itself but only in so far that it maximizes the usefulness forlimited creatures like us. If a generalization’s invariance is,e.g., limited to our cosmic epoch only, this limited invariancearguably would be enough to make it useful for us and thus a lawaccording to this account. Wilhelm (2022) argues that addingcalculational tractability to the criteria determining a best systemhelps to account for special science laws. Unterhuber (2014) developsa version of the “better best system” account utilizinggenerics. Reutlinger (2009) and Backmann and Reutlinger (2014) providea critical discussion of better best system accounts. Braddon-Mitchell(2001) and Wheeler (2018) present versions of a best systems accountdeploying the idea of algorithmic compression from data science.Albert (2000, 2015) and Loewer (2009) advocate a “statisticalmechanical” version of the best systems approach to specialscience laws which is, unlike the better best system account,committed to a particular Boltzmann-inspired interpretation ofstatistical mechanics (including assumptions about the early historyof the universe, in particular the so-called “pasthypothesis”). Weslake (2014) and Frisch (2014b) provide criticalresponses, while Fenton-Glynn (2016) constructively uses examples byAlbert and Loewer for an analysis of cp-laws. Finally, Albert andLoewer’s account also connects the debate on cp-laws with theon-going discussion on whether causation has a place in fundamentalphysics or whether causation is an ‘emergent’ phenomenoncaptured by the cp-law statements of the special sciences (Price andCorry 2007 and Frisch 2014b are representative of the opposing viewsin the debate on causation in physics).

   Also other metaphysical issues besides Lewis’s account oflawhood influence the recent literature on cp-laws. The metaphysics ofprobabilities and disposition stand out in this respect. Taking upresults from Earman and Roberts (1999) and Schurz (2002), severalauthors including Hüttemann and Reutlinger (2013), Kowalenko(2014), Roberts (2014), Reutlinger (2014), and Strevens (2014) discussthe 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 thenormality account presented in section 8 above is but a special caseof the broader statistical approach to cp-laws).

   A third metaphysical trend concerns the metaphysics of dispositions:As we have explained in section 7 above, Hüttemann (2014, 2021)and Pemberton and Cartwright (2014) argue in favor of adispositionalist account of cp-laws and provide responses to standardobjections to the dispositionalist view (for instance, to objectionsvoiced by Earman and Roberts 1999).

2. Drawing on linguistics and philosophy of language: Severalrecent approaches to cp-laws explicitly draw on research inlinguistics and philosophy of language. Taking into account linguisticwork on generics, Nickel (2010, 2014) argues that our understanding ofcp-law statements is improved if one takes them to be generics anduses the formal semantics for generics to shed light on the meaning ofcp-law statements (for approaches to cp-laws deploying generics, seealso Unterhuber and Schurz 2013; Unterhuber 2014; and Claveau undGirard 2019). Roberts (2014) makes contact with philosophy of languagein analyzing cp-laws as vague and indexical statements aboutfrequencies. Ward (2007), Karbasizadeh (2008) and Nickel (2014)explore the link between cp-laws and different accounts of naturalkind terms.

3. Sensitivity to the sciences: Several philosophers have madean effort to draw attention to the fact that an adequate discussion oflaws has to be sensitive to the differences between distinctscientific disciplines. Their motivation is to provide a theory oflaws that fits generalizations (and models) in the life sciences, thesocial sciences and, more generally, the sciences of complex systems(for instance, see Mitchell 1997, 2002a, 2002b; 2008, 2009; Sober1997; Steel 2007; and Reiss 2008). In particular, Mitchell (2000),Strevens (2003, 2008), Craver (2007), Hüttemann (2007), Wimsatt(2007), Tobin (2005), Woodward (2010) and Reutlinger (2011, 2013)distinguish various dimensions of characterizing how generalizationsin these scientific disciplines may be non-universal or allow forexceptions. These dimensions include stability, robustness,specificity, and historical contingency (see Reutlinger and Unterhuber2014b for an overview). In a similar vein, Schurz (2014) introduces anovel distinction betweenceteris paribus andceterisrectis laws. Fenton-Glynn (2016) holds that some higher-levellaws about the macro-behavior of complex systems (his central exampleis the second law of thermodynamics) are “minutisrectis” laws and should be distinguished from cp-laws,because their non-universal character stems from a probabilitydistribution over initial micro-conditions (see Strevens 2003 forfurther examples). Finally, motivated by examples from populationecology, earth science, and statistical mechanics, Strevens (2008) andWeisberg (2013) argue for a new interpretation of laws and modelsinvolving idealized assumptions, the minimalist account ofidealizations. Strevens (2012, 2014) proposes treating cp-laws andidealized laws separately. What seems to emerge from this literatureis that theceteris paribus character of generalizations ismulti-faceted. There is a number of ways in which a generalization canbe non-universal: due to idealizations, by expressing statisticalregularities and probability distributions over initial conditions, bydrawing on the notion of normality, by being sensitive to changes ininitial and background conditions, and so on (see Reutlinger andUnterhuber 2014b for a more detailed analysis).

4. Using scientific concepts and methods: recent philosophicalattempts of capturing cp-laws increasingly involve importing keyconcepts from the sciences and using these concepts for developing ageneral theory of cp-laws. Inspired by classical mechanics, Maudlin(2007) and Hüttemann (2014) analyze cp-conditions in terms ofdefault or inertial behavior. Both authors capture a violation ofcp-conditions as a deviation from inertial behavior, where deviatingfrom inertial behavior is itself a process governed by laws ofdeviation (Maudlin) or laws of interaction and composition(Hüttemann). Other philosophers are influenced by the lifesciences such as Nickel (2014), Pemberton and Cartwright (2014), andStrevens (2012, 2014) who rely on the concept of a mechanism in orderto explicate the content of cp-conditions more generally. Kowalenko(2014) argues that cp-laws are best understood by drawing on methodsfrom statistics, such as multivariate regression analysis (for adiscussion of similar statistical approaches see Hüttemann andReutlinger 2013, Reutlinger 2014, Roberts 2014, and Strevens 2014).Schurz (2014) uses causal Bayesian networks as a formal framework fora causal reconstruction ofceteris paribus andceterisrectis laws. Finally, Jhun (2018) argues that focusing on theequilibrium methodology used in thermodynamics as well as in economicreasoning facilitates a new understanding of the explanatory role ofcp-laws in terms of efficient performance.

11. Conclusion

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

1. The debate on cp-laws since the 1980s has not particularly focusedon detailed case studies of laws in different scientific disciplines.But this just seems to be what a satisfying theory of cp-lawsrequires. A possible result of detailed case studies might well bethat generalizations reveal quite different features in differentdisciplines. In other words, the explication of cp-laws in, forinstance, physics, biology and economics might differ significantly.If so, this result also raises interesting methodological questionsfor philosophers of science, such as “What degree of unification(with respect to laws in different scientific disciplines) do werequire 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? Whenapplied to the special sciences, many theories of (causal) explanation(cf. Woodward 2003, EG1, EG2), causation (cf. Hausman 1998; Pearl2000; Hitchcock 2001; Woodward 2003) and mechanisms (Machamer, Dardenand Craver 2000; Glennan 2002; Craver 2007) seem to presupposegeneralizations. Interestingly, these generalizations tend to beclassified as non-strict. Consequently, these generalizations are inthe domain of the debate on cp-laws. These theories of explanation,causation and mechanisms are important test cases for the adequacy oftheories of cp-laws.
3. What metaphysical claims do the various theories of cp-laws commitus to? Do exclusive cp-laws commit us to assume that the fundamentallaws of nature are deterministic? Or are they compatible with aprobabilistic metaphysics? Furthermore, do cp-laws commit us to adispositionalist metaphysics? Or are they compatible with a Humeanontology? Is structural realism an adequate metaphysics for cp-laws ingeneral (see Wright 2017 for discussion) and in particular for thelaws of the special sciences?
4. What role do cp-laws play in formal philosophy? For example, it isimportant to answer the following questions: How are the logicalprinciples of non-monotonic reasoning connected to cp-laws? How docp-laws relate to Bayesian probabilities, and to the rules of Bayesianupdating of these probabilities?

These areas of research are, of course, not a complete list of futureresearch topics. Nevertheless, it seems to us that these questions areinteresting enough to stimulate productive further research onceteris paribus laws.

12. Suggested Reading

The critical article by Earman and Roberts (1999) provides anexcellent introduction to the contemporary debate, because itreconstructs the most important theories of cp-laws and also pointsout the problems of these approaches. The volumes by Earmanetal. (2002) and Reutlinger and Unterhuber (2014) contain arepresentative collection of contemporary essays on the problem ofcp-laws.

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

The important distinctions between (a) comparative and exclusive and(b) indefinite and definite cp-laws are established and argued for inSchurz (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 arecriticized 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 inbook-length (see his 2000 and 2009a, although the focus of the latteris not on cp-laws). Lange (2002) is a very accessible paper on hisview of cp-laws. Invariance accounts are best introduced in two jointpapers by James Woodward and Christopher Hitchcock (see their EG1,EG2) and in Woodward’s bookMaking 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 amodern classic (cf. her recent 2002, and Elgin and Sober 2002 for acritique). Smith (2002), Mumford (2004), Bird (2005, 2007),Hüttemann (2007, 2014, 2021), and Kistler (2020) developdifferent more recent dispositional accounts of cp-laws.

Concerning normality approaches, Schurz (2001b, 2002) argues for anormic laws approach. Spohn (2002) and Spohn (2012, chapter 13) aregood 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 Cohenand Callender (2009, 2010), Schrenk (2007a, 2014), and Unterhuber(2014). Wheeler (2018: chapter 3.2) gives an accessible briefexposition of various versions; and Reutlinger (2009) and Backmann andReutlinger (2014) provide a critical discussion. Hicks, Jaag, and Loew(2023) is a collection presenting recent developments of pragmaticversions of the best systems account.

There are also various approaches in the current debate that cannoteasily be classified: Mitchell (1997, 2002a, b; 2008, 2009), Sober1997, Steel (2007, chapter 6) and Reiss (2008) attempt to explicate atheory of laws that fits scientific practice in biology, the socialsciences 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 inorder 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 ametaphysics of generalizations in the social sciences (and for thespecial sciences in general). Finally, Maudlin (2007, chapter 5)advocates an explication of laws in the special sciences that is basedon the notion of quasi-Newtonian laws.

Bibliography

• Adams, E. W., 1975,The Logic of Conditionals, Dordrecht:Reidel.
• Albert, D., 2015,After Physics, Cambridge, MA: HarvardUniversity Press.
• Backmann, M. and A. Reutlinger (2014): “Better Best Systems.Too Good To Be True”,Dialectica, 68:375–390.
• Bird, A., 2005, “The Dispositionalist Conception ofLaws”,Foundations of Science, 10: 353–370.
• –––, 2007,Nature’s Metaphysics,Oxford: Oxford University Press.
• Blaug, M., 1992,The Methodology of Economics or HowEconomists Explain, New York: Cambridge University Press, 2ndEdition.
• –––, 1997,Economic Theory inRetrospect, Cambridge: Cambridge University Press, 2ndEdition.
• Braddon-Mitchell, D., 2001, “Lossy Laws”,Noûs, 35(2): 260–277.
• Cairnes, J., 1888,The Character and Logical Method ofPolitical Economy, London: Longman, Brown, Green, andRoberts.
• Callender, C., and J. Cohen, 2010, “Special Sciences,Conspiracy and the Better Best System Account of Laws”,Erkenntnis.
• Canfield, J., and K. Lehrer, 1961, “A Note on Prediction andDeduction”,Philosophy of Science, 28:204–208
• Carnap, R., 1956, “The Methodological Character ofTheoretical Concepts”, inThe Foundations of Science and theConcepts of Psychology and Psychoanalysis (Minnesota Studies inthe Philosophy of Science, Vol. I), H. Feigl, and M. Scriven (eds.),Minneapolis: Minnesota University Press, pp. 38–76.
• Carrier, M., 1998, “In Defense of Psychological Laws”,International Studies in the Philosophy of Science, 12:217–232.
• Cartwright, N., 1983,How the Laws of Physics Lie,Oxford: Oxford University Press.
• –––, 1989,Nature’s Capacities andtheir Measurement, Cambridge: Cambridge University Press.
• –––, 1999,The Dappled World. A Study of theBoundaries of Science, Cambridge: Cambridge UniversityPress.
• –––, 2002, “In favor of Laws that are notCeteris Paribus After All”, inCeteris Paribus laws, J.Earman et al. (eds),Erkenntnis, 52 (Special Issue):425–439.
• –––, 2007,Hunting Causes and using them.Approaches in Philosophy and Economics, Cambridge: CambridgeUniversity Press.
• Churchland, P., 1970, “The Logical Character of ActionExplanations”,Philosophical Review, 79:214–236.
• Cicero, M. T., 1838,De Officiis, J.F. Heusinger (ed.),Braunschweig: Vieweg.
• Claveau, F., and J. Girard, 2019, “Generic Generalizationsin Science”,Erkenntnis, 84(4): 839–859.
• Cohen, J., and C. Callender, 2009, “A Better Best SystemAccount of Lawhood”,Philosophical Studies, 145(1):1–34.
• Collins, J., N. Hall, and L.A. Paul (eds.), 2004,Causationand Counterfactuals, Cambridge, Massachusetts: MIT Press.
• Craver, C., 2007,Explaining the Brain. Mechanisms and theMosaic Unity of Neuroscience, Oxford: Clarendon Press.
• Dancy, J., 2004,Ethics Without Principles, Oxford:Oxford University Press.
• Davidson, D. 1980,Essays on Actions and Events, Oxford:Oxford University Press.
• –––, 1984,Inquiries into Truth andInterpretation, Oxford: Oxford University Press.
• Dray, W., 1957,Laws and Explanation in History, Oxford:Oxford University Press.
• Demarest, H., 2012, “Do Counterfactuals Ground the Laws? ACritique of Lange”,Philosophy of Science, 79(3):333–344.
• Drewery, A., 2000, “Laws, Regularities andExceptions”,Ratio, 13(1): 1–12.
• –––, 2001, “Dispositions andCeterisParibus Laws”,British Journal for the Philosophy ofScience, 52(4): 723–733.
• Dupré, J., 1984, “Probabilistic CausalityEmancipated”, Midwest Studies in Philosophy, 9:169–75.
• Earman, J, and J. Roberts, 1999, “Ceteris Paribus, There isno Problem of Provisos”,Synthese, 118:439–478.
• Earman, J., J. Roberts, and S. Smith (2002): “CeterisParibus Lost”, inCeteris Paribus laws, J. Earman,et al. (eds.),Erkenntnis, 52 (Special Issue):281–301.
• Earman, J., C. Glymour, and S. Mitchell (eds.), 2002,CeterisParibus Laws,Erkenntnis, 57 (Special Issue).
• Eells, E., 1991,Probabilistic Causality, Cambridge:Cambridge University Press.
• Elgin, M., and E. Sober, 2002, “Cartwright on Explanationand Idealization”, inCeteris Paribus laws, J. Earman,et al. (eds),Erkenntnis 52 (Special Issue):441–450.
• Eliot, C., 2011, “Hempel’s Provisos and CeterisParibus Clauses”,Journal for General Philosophy ofScience, 42 (2): 207–218.
• Feigl, H., and M. Scriven (eds.), 1956,The Foundations ofScience and the Concepts of Psychology and Psychoanalysis(Minnesota Studies in the Philosophy of Science, Vol. I), Minneapolis:Minnesota University Press.
• Fenton-Glynn, L., 2016, “Ceteris Paribus Laws and MinutisRectis Laws”,Philosophy and Phenomenological Research,93(2): 274–305.
• Fisher, R., 1951,The Design of Experiments, Edinburgh:Oliver and Boyd.
• Fodor, J., 1974, “Special Sciences, or the Disunity ofScience as a Working Hypothesis”,Synthese, 28:97–115.
• –––, 1987,Psychosemantics: The Problem ofMeaning in the Philosophy of Mind, Cambridge, MA: MIT Press.
• –––, 1991, “You Can Fool Some People Allof the Time, Everything Else Being Equal; Hedged Laws andPsychological Explanations”,Mind, 100:19–34.
• –––, 1997, “Special Sciences. StillAutonomous After all these Years”,PhilosophicalPerspectives, 11: 149–163.
• Friedman, M., 1953/2008, “The Methodology of PositiveEconomics”, in D. Hausman (ed.),The Philosophy ofEconomics. An Anthology, Third Edition, New York: CambridgeUniversity Press, pp. 145–178.
• Frisch, M., 2014a, “Why Physics Can’t ExplainEverything”, in A. Wilson (Ed.),Asymmetries of chance andtime, Oxford: Oxford University Press, 221–240.
• –––, 2014b,Causal Reasoning inPhysics, Cambridge: Cambridge University Press.
• Gabbay, D.M., et al. (eds.), 1994,Handbook of Logic inArtificial Intelligence and LogicProgramming, Vol. 3:Nonmonotonic Reasoning and Uncertain Reasoning, Oxford: ClarendonPress.
• Glennan, S., 2002, “Rethinking Mechanist Explanation”,Philosophy of Science (Supplement), 69: S342–353.
• Glymour, C., 2002, “A Semantics and Methodology for CeterisParibus Hypotheses”, inCeteris Paribus laws, J.Earman,et al. (eds.),Erkenntnis, 52 (SpecialIssue): 395–404.
• Goodman, N., 1947, “The Problem of CounterfactualConditionals”,Journal of Philosophy, 44(5):113–128.
• Hall, N., 2007, “Structural Equations and Causation”,Philosophical Studies, 132: 109–136.
• –––, 2011, “Review ofLaws andLawmakers: Science, Metaphysics, and the Laws of Nature”,Notre Dame Philosophical Reviews, 27 September 2011,available online.
• Hausman, D., 1992,The Separate and Inexact Science ofEconomics, Cambridge, Massachusetts: Cambridge UniversityPress.
• –––, 1998,Causal Asymmetries, NewYork: Cambridge University Press.
• Hawthorne, J., 1996, “On the Logic of Non-MonotonicConditionals and Conditional Probabilities”,Journal ofPhilosophical Logic, 25: 185–218.
• Hays, W., and Winkler, R., 1975,Statistics: Probability,Inference, and Decision, New York: Holt, 2nd edition.
• Hempel, C., 1942, “The Function of General Laws inHistory”, in C. Hempel, 1965, pp. 231–243.
• –––, 1952, “Typological Methods in theNatural and the Social Sciences”, in C. Hempel, 1965, pp.155–171.
• –––, 1961–62, “RationalAction”,Proceedings of the American PhilosophicalAssociation, 35: 5–23.
• –––, 1965,Aspects of Scientific Explanationand other Essays, New York: The Free Press.
• –––, 1988, “Provisoes: A Problemconcerning the Inferential Function of Scientific Theories”,Erkenntnis, 28: 147–164.
• Hempel, C., and P. Oppenheim, 1948, “Studies in the Logic ofExplanation”, in C. Hempel, 1965, pp. 245–290
• Hicks, M., S. Jaag, and C. Loew (eds.), 2023,Humean Laws forHuman Agents, Oxford: Oxford University Press.
• Hitchcock, C., 2001, “The Intransitivity of CausationRevealed in Equations and Graphs”,The Journal ofPhilosophy, 98: 273–299.
• Hitchcock, C., and J. Woodward, 2003 [EG2], “ExplanatoryGeneralizations, Part II: Plumbing Explanatory Depth”,Noûs, 37(2): 181–199. [For EG1, see Woodward& Hitchcock 2003.]
• Hollander, S., 1985,The Economics of John Stuart Mill. Vol.I: Theory and Method, Oxford: Blackwell.
• Horgan, T., and J. Tienson, 1996,Connectionism and thePhilosophy of Psychology, Cambridge, Massachusetts: MITPress.
• Horty, J., 1994, “Moral Dilemmas and Non-monotonicLogic”,Journal of Philosophical Logic, 23:35–65.
• Hutchison, T.W., 1938,The Significance and Basic Postulatesin Economic Theory, New York: Kelly, 2nd Edition.
• Hüttemann, A., 1998, “Laws and Dispositions”,Philosophy of Science, 65: 121– 135.
• –––, 2007, “Causation, Laws andDispositions”, in M. Kistler, and B. Gnassounou (eds.),Dispositions and Causal Powers, Aldershot: Ashgate, pp.207–219.
• –––, 2014, “Ceteris Paribus Laws inPhysics”,Erkenntnis, 79: 1715–1728.
• –––, 2021,A Minimal Metaphysics forScientific Practice, Cambridge: Cambridge University Press.
• Hüttemann, A. and A. Reutlinger, 2013, “Against theStatistical Account of Special Science Laws”, in V. Karakostasand D. Dieks (eds.),EPSA11: Perspectives and FoundationalProblems, Dordrecht: Springer, 181–192.
• Jaag, S. and C. Loew, 2020, “Making best systems best forus”,Synthese, 197: 2525–2550.
• Jeffrey, R.C., 1971, “Probability Measures andIntegrals”, in R. Carnap, and R. Jeffrey (eds.), 1971,Studies in Inductive Logic and Probability, Berkeley:University of California Press, pp. 167–224.
• Johanson, I., 1980, “Ceteris Paribus Clauses, ClosureClauses and Falsifiability”,Journal for the GeneralPhilosophy of Science, 10: 16–22.
• Joseph, G., 1980, “The Many Sciences and the OneWorld”,Journal of Philosophy, 77: 773–790.
• Jhun, J., S., 2018, “What’s the Point of CeterisParibus? or, How to Understand Supply and Demand Curves”,Philosophy of Science, 85(2): 271–292.
• Karbasizadeh, A. E., 2008, “Revising the Concept of Lawhood:Special Sciences and Natural Kinds”,Synthese, 162:115–130.
• Kaufer, E., 1997, “Reply to Persky”,Journal ofEconomic Perspectives, 11(2): 190–191.
• Keynes, J.M., 1891,The General Theory of Employment,Interest, and Money, London: Macmillan.
• Kim, J., 1985, “Psychological Laws”, inActionsand Events, E. LePore, and B. McLaughlin (eds.), Oxford: OxfordUniversity Press, pp. 369–386.
• Kincaid, H., 1996,Philosophical Foundations of the SocialSciences, Cambridge: Cambridge University Press.
• –––, 2004, “Are There Laws in the SocialSciences?: Yes”, inContemporary Debates in the Philosophyof Science, C. Hitchcock (ed.), Oxford: Blackwell, pp.168–187.
• –––, 2008, “Structural Realism and theSocial Sciences”,Philosophy of Science, 75:720–731
• Kincaid, H., and D. Ross, 2009,The Oxford Handbook ofEconomics, Oxford: Oxford University Press.
• Kistler, M., 2020, “Laws, Exceptions andDispositions”,The Journal for the Philosophy of Language,Mind, and Arts, 1(1): 53–74.
• Kowalenko, R., 2014, “Ceteris Paribus Laws: A NaturalisticAccount”,International Studies in the Philosophy ofScience, 28(2): 133–155.
• Kraus, S., D. Lehmann, and M. Magidor, 1990, “NonmonotonicReasoning, Preferential Models and Cumulative Logics”,Artificial Intelligence, 44: 167–207.
• Krugman, P., and R. Wells, 2009,Economics, London:Palgrave Macmillan, 2nd Edition.
• Ladyman, J., 2008, “Structural Realism and the Relationbetween the Special Sciences and Physics”,Philosophy ofScience, 75: 744–755.
• Lakatos, I., 1970,Criticism and the Growth of Knowledge,New York: Cambridge University Press.
• Lange, M., 1993, “Natural Laws and the Problem ofProvisos”,Erkenntnis, 38: 233–248.
• –––, 2000,Natural Laws in ScientificPractice, Oxford: Oxford University Press.
• –––, 2002, “Who’s Afraid of CeterisParibus Laws? Or: How I Learned to Stop Worrying and Love Them”,inCeteris Paribus laws, J. Earman,et al. (eds.),Erkenntnis, 52 (Special Issue): 407–423.
• –––, 2005, “Laws and theirStability”,Synthese, 144: 415–432.
• –––, 2009a,Laws and Lawmakers. Science,Metaphysics and the Laws of Nature, Oxford: Oxford UniversityPress.
• –––, 2009b, “Why Do the Laws ExplainWhy?”, inDispositions and Causes, T. Handfield (ed.),Oxford: Oxford University Press, pp. 286–321.
• Leitgeb, H., 2004,Inference at the Low Level, Dordrecht:Kluwer.
• Lepore, E., and B. Loewer, 1987, “Mind Matters”,Journal of Philosophy, 93: 630–642.
• –––, 1989, “More on Making MindMatter”,Philosophical Topics, 17: 175–191.
• Lewis, D., 1973,Counterfactuals. Oxford: Blackwell.
• –––, 1983, “New Work for a Theory ofUniversals”,Australasian Journal of Philosophy, 61:343–377.
• Lipton, P., 1999, “All Else Being Equal”,Philosophy, 74: 155–168.
• Loewer, B., 2009, “Why is there anything exceptphysics?”Synthese, 170: 217–233.
• Machamer, P., L. Darden, and C. Craver, 2000, “ThinkingAbout Mechanisms”,Philosophy of Science, 67:1–25.
• Mankiw, N. G., 1998,Principles of Microeconomics,Orlando: The Dryden Press.
• Marx, K., 1867,Kritik der Politischen Ökonomie. DasKapital Band I, in Marx- Engels-Werke, Vol. 23, Berlin:Dietz.
• –––, 1894,Kritik der PolitischenÖkonomie. Das Kapital Band III, inMarx-Engels-Werke, Vol. 25, Berlin: Dietz.
• Marshall, A., 1890,Principles of Economics, Macmillan:London, Eighth Edition.
• Mas-Colell, A., M. Whinston, and J. Green, 1995,MicroeconomicTheory, Oxford: Oxford University Press.
• Maudlin, T., 2007,The Metaphysics Within Physics,Oxford: Oxford University Press.
• McCarthy, J., 1986, “Application of Circumscription toFormalizing Common-Sense Knowledge”,ArtificialIntelligence, 13: 89–116.
• Medina, J. de, 1581,De Poenitentia Restitutione etContractibus, Ingolstadt, First Edition 1544.
• Mesoudi, A., A. Whiten, and K.N. Laland, 2006, “Towards aunified science of cultural evolution”,Behavioral and BrainScience, 29: 329–347.
• Mill, J. S., 1836/2008, “On the Definition and Method ofPolitical Economy”, inThe Philosophy of Economics. AnAnthology, D. Hausman (ed.), New York: Cambridge UniversityPress, Third Edition, pp. 41–58.
• –––, 1843,A System of Logic, London:J. W. Parker.
• –––, 1871,Principles of PoliticalEconomy, New York: A.M. Kelly.
• Millikan, R. G., 1984,Language, Thought, and Other BiologicalCategories. Cambridge, Massachusetts: MIT Press.
• Mitchell, S., 1997, “Pragmatic Laws”,Philosophyof Science, 64: 242–265.
• –––, 2000, “Dimensions of ScientificLaw”,Philosophy of Science, 67: 242–265.
• –––, 2002a, “Ceteris Paribus– An Inadequate representation of biological Contingency”,in inCeteris Paribus laws, J. Earman,et al.(eds.),Erkenntnis, 52 (Special Issue): 329–350.
• –––, 2002b, “Contingent Generalizations:Lessons from Biology”, inAkteure, Mechanismen, Modelle, ZurTheoriefähigkeit makro-sozialer Analysen, R. Mayntz (ed.),Frankfurt/New York: Campus, pp. 179–195.
• –––, 2008,Komplexitäten. Warum wirerst anfangen, die Welt zu verstehen, Frankfurt: Suhrkamp.(English translation forthcoming.)
• –––, 2009, “Complexity and Explanation inthe Social Sciences”, inPhilosophy of the Social Sciences.Philosophical Theory and Scientific Practice, C. Mantzavinos(ed.), Cambridge: Cambridge University Press, pp. 130–145.
• Morreau, M., 1999, “Other Things Being Equal”,Philosophical Studies, 96: 163–182.
• Molina, L. de, 1659,De Instita et Iure, Mainz, FirstEdition 1593.
• Moser, P. (ed.), 2002,The Oxford Handbook ofEpistemology, Oxford: Oxford University Press.
• Mott, P., 1992, “Fodor and Ceteris Paribus Laws”,Mind, 101: 335–346.
• Mumford, S., 2004,Laws in Nature, Oxford: OxfordUniversity Press.
• Nickel, B., 2009, “Generics and the ways ofnormality”,Linguistics and Philosophy, 31:629–648.
• Nickel, B., 2014, “The Role of Kinds in the Semantics ofCeteris Paribus Laws”,Erkenntnis, 79(10):1729–1744.
• Pearl, J., 2000,Causality: Models, Reasoning andInference, Cambridge: Cambridge University Press.
• Pemberton, J., and Cartwright, N., 2014, “Ceteris paribuslaws need machines to generate them”,Erkenntnis, 79:1745–1758.
• Persky, J., 1990, “Ceteris Paribus”,Journal ofEconomic Perspectives, 4: 187– 193.
• Pietroski, P., 1993, “Prima Facie Obligations, CeterisParibus Laws in Moral Theory,”Ethics, 103:489–515.
• Pietroski, P., and R. Rey, 1995, “When Other Thingsaren’t Equal: Saving Ceteris Paribus Laws from Vacuity”,British Journal for the Philosophy of Science, 46:81–110.
• Pollock, J., 1986,Contemporary Theories of Knowledge,Maryland: Rowman & Littlefield.
• Price, H. and Corry, R. (eds.), 2007,Causation, Physics, andthe Constitution of Reality. Russell’s Republic Revisited,New York: Oxford University Press.
• Psillos, S., 2002,Causation and Explanation, Chesham:Acumen.
• Quine, W.V.O., 1960,Word and Object, Cambridge,Massachusetts: MIT Press.
• Reiss, J., 2008,Error in Economics, London:Routledge.
• Reutlinger, A., 2009, “Review ofMarkus Schrenk: TheMetaphysics of Ceteris Paribus Laws”,British Journalfor the Philosophy of Science, 60: 229–233.
• –––, 2011, “A Theory of Non-UniversalLaws”,International Studies in the Philosophy ofScience, 25: 97–117.
• –––, 2013, “Are the Generalizations ofBiology Historical?”, in A. Hüttemann and M. Kaiser (eds.),Explanation in the Special Sciences. The Case of Biology andHistory, Dordrecht: Springer, 131–154.
• –––, 2014, “Do statistical laws solve theproblem of provisos?”,Erkenntnis, 79:1759–1773.
• Reutlinger, A., and H. Koch, 2008, “MethodologicalIndividualism and the Epistemic Value of SocialMacro-Explanations”,Percipi, 2: 1–14.
• Reutlinger, A. and M. Unterhuber (eds.), 2014a,CeterisParibus Laws Revisited,Synthese (Supplement):79(10).
• Reutlinger, A. and M. Unterhuber, 2014b, “Thinking aboutNon-Universal Laws”,Erkenntnis, 79: 1703-1713.
• Roberts, J., 2004, “There are No Laws in the SocialSciences”, inContemporary Debates in the Philosophy ofScience, C. Hitchcock (ed.), Oxford: Blackwell, pp.168–185.
• Roberts, J. T., 2014, “CP-law statements as vague,self-referential, self-locating, statistical, and perfectly inorder”,Erkenntnis 79, 1775–1786.
• Robbins, L., 1935/2008, “The Nature and Significance ofEconomic Science”, inThe Philosophy of Economics. AnAnthology, D. Hausman (ed.), New York: Cambridge UniversityPress, 2nd Edition, pp. 73–99.
• Rosenberg, A., 1992,Economics – Mathematical Politicsor Science of Diminishing Returns?, Chicago: University ofChicago Press.
• Ross, W.D., 1930,The Right and the Good, Oxford: OxfordUniversity Press.
• Ross, D., 2008, “Ontic Structural Realism andEconomics”,Philosophy of Science, 75:732–743.
• Rupert, R., 2007, “Realization, Completers, and CeterisParibus Laws in Psychology”,British Journal for thePhilosophy of Science, 58: 1–11.
• Samuelson, P., 1955,Economics. Introductory Analysis,McGraw-Hill.
• –––, 1958,Foundations of EconomicAnalysis, Cambridge, Massachusetts: Harvard UniversityPress.
• Samuelson, P., and W. Nordhaus, 1985,Economics. InternationalStudent Edition, Singapore: McGraw-Hill.
• Schiffer, S., 1991, “Ceteris Paribus Laws”,Mind, 100: 1–17.
• Schlicht, E., 1985,Isolation in Economics, Berlin/NewYork: Springer.
• Scholz, O. R., 2016,Verstehen und Rationalität.Untersuchungen zu den Grundlagen von Hermeneutik undSprachphilosophie, Frankfurt: Klostermann.
• Schrenk, M., 2007a,The Metaphysics of Ceteris ParibusLaws, Frankfurt: Ontos.
• –––, 2007b, “Can Capacities Rescue us fromCeteris Paribus Laws?”, inDispositions and CausalPowers, M. Kistler and B. Gnassounou (eds.), Aldershot: Ashgate,pp. 221–247.
• –––, 2014, “Better best systems and theissue of cp-laws”,Erkenntnis, 79: 1787-1799.
• Schumpeter, J.A., 1954,History of Economic Analysis,Oxford: Oxford University Press.
• Schurz, G., 1998, “Probabilistic Semantics forDelgrande’s Conditional Logic and a Counterexample to hisDefault Logic”,Artificial Intelligence, 102(1):81–95.
• –––, 2001a, “Pietroski and Rey onCeteris Paribus Laws”,British Journal forPhilosophy of Science, 52: 359–370.
• –––, 2001b, “What isNormal? AnEvolution Theoretic Foundation of Normic Laws and their Relation toStatistical Normality”,Philosophy of Science, 28:476–497.
• –––, 2002, “Ceteris Paribus Laws:Classification and Deconstruction”, inCeteris ParibusLaws, J. Earman,et al. (eds.),Erkenntnis, 52(Special Issue): 351–372.
• –––, 2004, “Normic Laws, NonmonotonicReasoning, and the Unity of Science”, in:Logic,Epistemology, and the Unity of Science, S. Rahman,etal. (eds.), Dordrecht: Kluwer, pp. 181–211.
• –––, 2005, “Non-monotonic Reasoning froman Evolutionary Viewpoint”,Synthese, 146(1):37–51.
• –––, 2014, “Ceteris paribus and ceterisrectis laws. Content and causal role”,Erkenntnis, 79:1801–1817.
• Scriven, M., 1959, “Truisms as Grounds for HistoricalExplanations”, inTheories of History, P. Gardiner(ed.), New York: The Free Press.
• Silverberg, A., 1996, “Psychological Laws and Non-MonotonicLogic”,Erkenntnis, 44: 199–224.
• Smith, S., 2002, “Violated Laws,Ceteris ParibusClauses, and Capacities”,Synthese, 130(2):235–264.
• Sober, E., 1997, “Two Outbreaks of Lawlessness in RecentPhilosophy of Biology”,Philosophy of Science, 64:432–444.
• Spirtes, P., C. Glymour, and R. Scheines, 2000,Causation,Prediction and Search, New York: Springer.
• Spohn, W., 1997, “Begründungen a priori – oder:ein frischer Blick auf Dispositionsprädikate”, inDasweite Spektrum der Analytischen Philosophie. Festschrift fürFranz von Kutschera, W. Lenzen (ed.), Berlin: de Gruyter, pp.323–345
• –––, 2002, “Laws, Ceteris Paribusconditions, and the Dynamics of Belief”, inCeteris Paribuslaws, J. Earman,et al. (eds.),Erkenntnis, 52(Special Issue): 373–394.
• –––, 2012,The Laws of Belief. RankingTheory and its Philosophical Applications, Oxford: OxfordUniversity Press.
• –––, 2014, “The Epistemic Account ofCeteris Paribus Conditions”,European Journal for Philosophyof Science, 4(3): 385–408.
• Steel, D., 2007,Across the Boundaries. Extrapolation inBiology and Social Science, New York: Oxford UniversityPress.
• Strevens, M., 2003,Bigger than Chaos, Cambridge, MA:Harvard University Press.
• –––, 2008,Depth, Cambridge, MA:Harvard University Press.
• –––, 2012, “Ceteris Paribus Hedges: CausalVoodoo That Works”,Journal of Philosophy, 109(11):652–675.
• –––, 2014, “High-level exceptionsexplained”,Erkenntnis, 79: 1819–1832.
• Strößner, C., 2015, “Normality and Majority:Towards a Statistical Understanding of Normality Statements”,Erkenntnis, 80(4): 793–809.
• Tobin, E., 2005,On the Disunity of the Sciences and CeterisParibus Laws, Dissertation, Trinity College Dublin.
• Unterhuber, M., 2014, “Do ceteris paribus laws exist? Aregularity-based best system analysis”,Erkenntnis, 79:1833–1847.
• Unterhuber, M., and Schurz, G., 2013, “The new Tweetypuzzle: Arguments against monistic Bayesian approaches in epistemologyand cognitive science”,Synthese, 190:1407–1435.
• Wachbroit, R., 1994, “Normality as a BiologicalConcept”,Philosophy of Science, 61:579–591.
• Ward, B., 2007, “The Natural Kind Analysis of CeterisParibus Law Statements”,Philosophical Topics, 35(1/2):359–380.
• –––, 2009, “Cartwright, Forces, andCeteris Paribus Laws”,Southwest Philosophy Review, 25(1): 55–62.
• Weber, M., 1906, “Objektive Möglichkeit undadäquate Verursachung in der historischenKausalbetrachtung”, in M. Weber, 1991,Schriften zurWissenschaftslehre, Stuttgart: Reclam, pp. 102–131.
• Weisberg, M., 2013,Simulation and Similarity, New York:Oxford University Press.
• Weslake, B., 2014, “Statistical mechanicalimperialism”, in A. Wilson (ed.),Asymmetries of chance andtime, Oxford: Oxford University Press, 241–257.
• Wheeler, B., 2018,Idealization and the Laws of Nature,Cham: Springer.
• Whitaker, J. K., 2008, “Ceteris Paribus”, inTheNew Palgrave Dictionary of Economics, S. N. Durlauf and L. E.Blume (eds.), London: Palgrave Macmillan, 2nd Edition.
• Wilhelm, I., 2022, “Tractability and Laws”,Synthese, 200(4): 1–17.
• Williams, M., 1995,Unnatural Doubts. Epistemological Realismand the Basis of Skepticism, Princeton: Princeton UniversityPress.
• –––, 2001,Problems of Knowledge. A CriticalIntroduction to Epistemology, Oxford: Oxford UniversityPress.
• Wimsatt, W., 2007,Re-Engineering Philosophy for LimitedBeings, Cambridge, MA: Harvard University Press.
• Woodward, J., 2000, “Explanation and Invariance in theSpecial Sciences”,British Journal for the Philosophy ofScience, 51: 197–254.
• –––, 2002, “There is no such thing as aceteris paribus law”, inCeteris Paribus laws,J. Earman,et al. (eds.),Erkenntnis, 52 (SpecialIssue): 303–328.
• –––, 2003,Making Things Happen,Oxford: Oxford University Press.
• –––, 2010, “Causation in biology:Stability, specificity, and the choice of levels ofexplanation”,Biology and Philosophy, 25:287–318.
• Woodward, J., and C. Hitchcock, 2003 [EG1], “ExplanatoryGeneralizations, Part I: A Counterfactual Account”,Noûs, 37(1): 1–24. [For EG2, see Hitchcock &Woodward 2003.]
• Woolridge, J. A., 2009,Introductory Econometrics, Mason:South Western Cengage Learning.
• Wright, A., S., 2017, “Fresnel’s laws, ceterisparibus”,Studies in History and Philosophy of Science,64: 38–52.

Academic Tools

	How to cite this entry.
	Preview the PDF version of this entry at theFriends of the SEP Society.
	Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
	Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

[Please contact the authors with suggestions.]

Related Entries

a priori justification and knowledge |belief, formal representations of |causation: and manipulability |causation: counterfactual theories of |causation: probabilistic |dispositions |economics: philosophy of |laws of nature |logic: non-monotonic |reasoning: moral |scientific explanation |structural realism

Acknowledgments

We would like to thank Alexander Bird, Carl Craver, MatthiasHösch, Beate Krickel, Meinard Kuhlmann, Marc Lange, ChrysostomosMantzavinos, Margaret Schabas, Markus Schrenk, Rudolf Schüssler,Wolfgang Spohn, Michael Strevens, Emma Tobin, the members of our DFGresearch group (Causation, Laws, Dispositions, and Explanations at theIntersection of Science and Metaphysics), and an anonymous referee forhelpful comments that greatly improved our manuscript.