A Theory of Non-universal Laws

Abstract

Laws in the special sciences are usually regarded to be non-universal. A theory of laws in the special sciences faces two challenges. (I) According to Lange's dilemma, laws in the special sciences are either false or trivially true. (II) They have to meet the ‘requirement of relevance’, which is a way to require the non-accidentality of special science laws. I argue that both challenges can be met if one distinguishes four dimensions of (non-) universality. The upshot is that I argue for the following explication of special science laws: L is a special science law just if (1) L is a system law, (2) L is quasi-Newtonian, and (3) L is minimally invariant.

Acknowledgements

I would like to thank Alexander Bird, Luke Glynn, Andreas Hüttemann, Meinard Kuhlmann, James Ladyman, Tim Maudlin, Wolfgang Spohn, Michael Strevens, Emma Tobin, the members of my DFG Research Group Causation and Explanation, and two anonymous referees of this journal for considerably improving my paper by their stimulating comments.

Notes

Cf., for instance, Earman and Roberts 1999; Lange 2000; Earman, Roberts, and Smith 2002; Woodward 2003, 2007; Roberts 2004; Maudlin 2007; Strevens 2009.

Two terminological clarifications: (1) I will use ‘non-universal laws’ and ‘ceteris paribus laws’ interchangeably. (2) My focus is on law statements rather than on laws themselves—thus, my aim is not to argue for any particular metaphysical claim (such as a regularity view or a dispositionalist account).

Many of the problems I will discuss in the paper would be even trickier if one disagreed with Loewer (and others) on this point. Some philosophers (e.g., Cartwright 1983, 1989; Mumford 2004) believe that even fundamental physics deals (at least in part) with non-universal laws. If this were the case, the issue of non-universal laws might turn out to be even more pressing.

Lange mistakenly writes ‘exponentially’.

Interventionist theories of causation rely on lawish generalizations because, if a causal relation between cause A and effect B obtains, a manipulation of A leads to a change in B that can be described by an invariant generalization (i.e. an explanatory, not necessarily universal, lawlike generalization). Strictly speaking, interventionists insist that generalizations need not be universal. But they also regard this as a challenge: one has to provide a theory of non-universal laws. Cf. Woodward 2003; Woodward and Hitchcock 2003.

Note that Dowe thinks that conserved quantities are only contingently, not essentially, described by the actual conservation laws. In metaphysics of science, such view can be described as categorialism about conserved quantities (cf. Bird 2007, §3.1). However, at least in the actual world, laws (and especially conservation laws) seem to matter for the truth conditions of causal statements.

Leuridan (2010) argues—correctly, in my opinion—that received accounts of mechanisms and mechanistic explanation (such as Machamer, Darden, and Craver 2000; Glennan 2002; Bechtel and Abrahamsen 2005; Craver 2007) also rely on lawish generalizations.

Cf. Earman et al. 2002, 297f.; Woodward 2002, 303; Kincaid 2004; Roberts 2004. As noted above, Cartwright (1983, 1989) and Mumford (2004) dispute the claim that paradigmatic laws of physics conform to the received philosophical picture (e.g. in being universal). However, they do not deny that that laws in the special sciences are non-universal, have exceptions, etc.

Pietroski and Rey (1995, 92) argue that A and C are not ‘grue-like’.

Notable exceptions are Mitchell 2000 and Schurz 2002.

A useful way to spell out the third dimension of universality could be found in Loewer's use of ‘global’ (see the passage quoted in the introduction): ‘The dynamical laws of classical mechanics are complete and deterministic. Given the state at any time t they determine the state at any other time. The determination is global since the position and momentum of any particle at a time t + r is determined only by the global (i.e. the entire) state of that system at time t. That is, to know how any one particle moves at t + x one has to know something at each particle at t. The dynamical laws and a partial description of state at t (except in special cases) do not entail much about the state of the system at other times and, in particular, don't say much about what any particular particle will (was) doing at t + r’ (Loewer 2008, 155). In contrast with the laws of classical mechanics, a special science law (such as the law of supply) is non-global, incomplete, and, thus, seems to provide only a ‘partial description’ of the phenomenon it describes. Special science laws leave out other influences on the phenomenon i.e. circumstances that are not referred to by the law statement itself—as stated in the description of the third dimension of universality (cf. Pietroski and Rey 1995, 89).

A variable X (in the terminology of statistics and causal modeling) is a function X:D→ ran(X), with a domain D of possible outcomes, and the range ran(X) of possible values of X. For quantitative variables X, ran(X) is usually taken to be the set of real numbers (cf. Pearl 2000; Eagle 2010, §0.9). For example, temperature is represented by a variable T that has several possible values, such as T = 30.65°. However, in the debate on causation philosophers often use qualitative, binary variables with ran(X) = {0; 1}—whether a binary variable takes one of its values is taken to represent whether or not a certain type of event occurs (cf. Hitchcock 2001). On notation: capital letters, such as X, Y, …, denote variables; lower-case letters, such as x, y, …, denote values of variables; the proposition that X has a certain value x is expressed by a statement of the form X = x, i.e. X = x is a statement about an event-type (cf. Woodward 2003).

Note that the concept of a system in Schurz's sense seems to coincide at least with the use of the concept system in the literature on mechanistic explanation in the life sciences (cf. Machamer, Darden, and Craver 2000; Glennan 2002; Bechtel and Abrahamsen 2005; Craver 2007). Mechanists usually conceive a system to be composed of interacting parts. Schurz (2002, §5) seems to agree with this characterization of a system when he discusses examples of biological systems.

One could object that even if the restricted reading were the favoured reading it would not be clear why the corresponding universal statement should be true. Even the universal statement (quantifying only over commodities) is vulnerable to Lange's dilemma and the requirement of relevance. The lesson I think we should learn from this result is that the responses to these two challenges have to given with respect to be the third and fourth dimensions of non-universality.

An anonymous referee has pointed out that one might want to dispute the claim that even the fundamental laws do not apply to everything (contra Schurz 2002; Hüttemann 2007). He or she argues that the fundamental laws, for instance, do not apply to angels and numbers. However, I think that, even if this were the case, we could preserve universality₂ for the fundamental laws by exactly the same strategy which I just used for preserving universality₂ for lawish statements in the special sciences. Further, my arguments do not have to rely on the characterization of fundamental physical laws which Schurz and Hüttemann provide.

A typical example is provided by causal models in econometrics: according to these models, the causal influence of a variable in isolation is described by a single structural equation. Each one of those single equations might be called ‘inertial law’. However, the whole causal model (i.e. a set of equations) provides an overall output resulting from the interaction of various causal factors (cf. Cartwright 1989, §4.5; Pearl 2000, ch. 5).

Thanks to Tim Maudlin and Michael Strevens for suggesting this amendment.

Similar approaches such as Lange's (2000, 103; 2009, 29) and Mitchell's (2000) stability theories as well as Ladyman and Ross's (2007) ‘real pattern’ approach might also work for my argument.

My approach does not differ from Hitchcock and Woodward's invariance theory concerning the non-reductive feature of the explication of the concept of a lawish generalization. Both explications are non-reductive, because they use causal and nomological concepts in the explicans. I agree with Hitchcock and Woodward that the non-reductive character of an explication is unproblematic as long as the explication is not viciously circular. For a more detailed defense of non-reductive explication see Woodward (2003, 103f.; 2008, 203f.), Strevens (2008, 186), Reutlinger (forthcoming).

For instance, Strevens (2007, 2008) argues that interventions lead to an infinite regress problem; and Reutlinger (2011) argues that the modal character of possible interventions (i.e. logical possibility, as assumed by Woodward 2003, 131f.) leads to severe trouble when merely logically possible (and physically impossible) interventions figure in counterfactuals—as interventionists claim.

As a referee pointed out, invariance theories usually refer to counterfactual situation in which the factors governed are undisturbed (i.e. the counterfactual situation involves elements of idealization and abstraction) in order to state the truth-conditions of law statements. A methodological question immediately comes to mind: how can one test these statements? How do scientists actually test statements of this kind in practice? Various authors have addressed these questions (cf. Kincaid 1996; Cartwright 2002; Steel 2007; Reiss 2008; Hüttemann forthcoming).

Cf. Woodward (2003, 2008); Woodward and Hitchcock (2003) for a more detailed account of this invariantist strategy to distinguish lawish statements from accidentally true generalizations. Reutlinger, Schurz, and Hüttemann (2011) provide an easily accessible survey to invariance theories.

Additional information

Notes on contributors

Alexander Reutlinger

Alexander Reutlinger is at the Department of Philosophy, University of Cologne.