Durham Research Online Deposited in DRO: 27 July 2016 Version of attached �le: Published Version Peer-review status of attached �le: Peer-reviewed Citation for published item: Cartwright, N. (1997) 'Models : the blueprints for laws.', Philosophy of science (supplement)., 64 . S292-S303. 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Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 http://dro.dur.ac.uk http://www.dur.ac.uk http://dx.doi.org/10.1086/392608 http://dro.dur.ac.uk/19364/ http://dro.dur.ac.uk/policies/usepolicy.pdf http://dro.dur.ac.uk Models: The Blueprints for Laws Nancy Cartwrighttl London School of Economics and Political Science In this paper the claim that laws of nature are to be understood as claims about what necessarily or reliably happens is disputed. Laws can characterize what happens in a reliable way, but they do not do this easily. We do not have laws for everything occur- ring in the world, but only for those situations where what happens in nature is rep- resented by a model: models are blueprints for nomological machines, which in turn give rise to laws. An example from economics shows, in particular, how we use-and how we need to use-models to get probabilistic laws. 1. Three Theses About Models And Laws. Margaret Morrison has taught us to think of models as mediators. They mediate between our various parcels of general and specific scientific knowledge and the world that that knowledge is about. Here I want to explain one of the principal mediating roles that models serve. Models show us, I shall argue, where laws of nature come from and how we can produce new ones. This way of putting the claim is tied to the standard, so-called 'empiricist', account of what laws are, the account that tells us that laws describe what regularly and reliably happens. If that is what we mean by a 'law' in science, then laws are few and far between-and that reflects the fact that what they are supposed to represent is scarce. It takes very special arrangements, properly shielded, repeatedly started up, and running without hitch, to give rise to a law; it takes what I call a 'nomological machine'. My claim then is that models serve as blue- prints for nomological machines. There are three separate theses involved in this claim. The first is that tDepartment of Philosophy, Logic and Scientific Method, London School of Econom- ics and Political Science, Houghton Street, London WC2 2AE, UK. {Research for this paper was supported by the project "Modelling in Physics and Eco- nomics" at the LSE. Towfa Shomar was a great aid in both the research and production of the paper. Philosophy of Science, 64 (Proceedings) pp. S292-S303. 0031-8248/97/64supp-0027$0.00 Copyright 1997 by the Philosophy of Science Association. All rights reserved. S292 http://www.jstor.org/page/info/about/policies/terms.jsp MODELS: THE BLUEPRINTS FOR LAWS the general scientific knowledge that we use to construct models is not knowledge of laws. This is a familiar thesis from me. I began my career by arguing that the laws of physics lie (Cartwright 1983). That was on the assumption that what we call laws in physics really are laws in the sense I grew up with: claims about what necessarily or reliably happens. Ever since then I have been looking for an alternative philosophical ac- count of laws closer to the way law claims are expressed and more re- sponsive to the way they are used, an account that would give them a more reputable status. I shall not pursue this first thesis now. The second thesis is my chief focus here. It is hard to get a law in nature. One of the principal functions that models serve is to represent those very special circumstances where laws arise. This is not a new thesis either. I have been building up the case for it already in a number of places (Cartwright 1994, 1995, 1996, 1997). But the focus of my discussion so far has been deterministic and causal laws. Here I shall try to show how we use-and how we need to use-models to get probabilistic laws. The third thesis is that there are no laws to be represented outside the highly structured arrangements that are well characterized as nomological machines. I shall lay out this thesis briefly in ?3 and ?4 in order to highlight how important models are. Models matter because they represent for us just those peculiar situations where nature is reliable. 2. Where Probabilities Come From. Ian Hacking, in Logic of Statistical Inference, taught that probabilities are characterized relative to chance set-ups and do not make sense without them. My discussion is an elab- oration of his claim. A chance set-up is a nomological machine for probabilistic laws, and our description of it is a model that works in the same way as a model for deterministic laws (like the Copernican model of the planetary system that gives rise to Kepler's laws). A sit- uation must be like the model both positively and negatively-it must have all the relevant characteristics featured in the model and it must have no significant interventions to prevent it operating as envisaged before we can expect repeated trials to give rise to events appropriately described by the corresponding probability. 2.1. An Example from Wesley Salmon. I begin with an example fa- miliar to philosophers of science: Wesley Salmon's argument that causes can decrease as well as increase the probability of their effects (Salmon 1971). Salmon considered two causes of a given effect, one highly effective, the other much less so. When the highly effective cause is present, he imagined, the less effective one is absent, and vice versa. S293 http://www.jstor.org/page/info/about/policies/terms.jsp NANCY CARTWRIGHT Thus the probability of the effect goes down when the less effective cause is introduced even though it is undoubtedly the cause of the effect in question. That is the story in outline, but exactly what must be the case to guarantee that the probabilities are a) well defined and b) that they fall within appropriate ranges to ensure the desired inequality (Prob (effect/less effective cause) < Prob (effect/- less effective cause))? For that we need an arena-a closed container, then a mechanism for ensuring both that there is a fixed joint probability (or range of fixed probabilities with a fixed probability for mixing) for the presence and absence of the two kinds of causes in the container, and that under these probabilities there is sufficient anticorrelation, given the levels of effectiveness, to guarantee the decrease in probability. There must be no other source of the effect in the container or introduced with either of the elements. There must be nothing present in correlation with either of the causes that annihilates the effect as it is produced. Etc. Etc. Figure 1 is a model of the kind of arrangement that is required: a model for a chance set-up or a probability-generating machine. Salmon himself used radioactive materials as causes, the effect being the pres- ence or absence of a decay product. (Figure 1.) My experiment is de- signed by Towfic Shomar, who chose different causes to make the de- sign simple.' The point is that without an arrangement like the one modeled (or some other very specific arrangement of appropriate de- sign) there are no probabilities to be had; with a sloppy design, or no design at all, Salmon's claims cannot be exemplified. 2.2. How Probability Theory Attaches to the World. Turning from this specific example, we can ask, "In general how do probabilities attach to the world?" The answer is via models, just as on my account the laws of quantum mechanics apply to concrete situations, and on Ronald Giere's (1988), those of classical mechanics. Assigning a prob- ability to a situation is like assigning a force function or a Hamiltonian. Set distributions are associated with set descriptions. The distributions listed in the table of contents of Harry E. McAllister's (1975) Elements of Business and Economic Statistics, shown in Figure 2, are typical. (Compare, for instance, Kyburg 1969, Mulholland and Jones 1968, or Robinson 1985.) Further familiar distributions appear later: the t-dis- tribution, the Chi-square distribution and the F-distribution. 1. Focusing on the need for a design like Shomar's for Salmon's original choice of radioactive materials shows how odd the so-called quantum probabilities are. They are not real probabilities for events that happen, or happen on 'measurement', for mea- surement itself is a chance set-up and the probabilities to be expected will depend jointly on the quantum state and the structure of the set-up. S294 http://www.jstor.org/page/info/about/policies/terms.jsp MODELS: THE BLUEPRINTS FOR LAWS alpha C Alpha - Source Figure 1A Figure 1B Figure 1C Alpha - Source Figure 1. Consider a radioactive source with a half life such that an alpha particle is on average radiated once every 15 minutes. The source is installed in a cylindrical container opened to a spherical chamber containing a proton. If the radioactive material radiates an alpha particle, the expulsion forces between the alpha particle and the proton push the proton out into the cylindrical box at the other side of the source (see Figure 1A). If the alpha particle is influenced by magnetic field, it will travel through the path toward exit C (as in Figure 1B). Assume that a magnetic field going into the page is turned on in the chamber for 15 minutes and is cut off for 15 minutes. At the moment the magnetic field is off it will cause the upper cylinder to become positivily charged and that will force the proton back into the chamber (as in Figure 1C). We can assume that the positive charge at the upper cylinder will influence the system for no more than half a minute, allowing the proton influenced by the alpha particle to enter. So, we have the following probabilities: Let c = df the presence of an alpha particle, e = df the proton is forced into the top cylinder, m = df magnetic field in the chamber. Then we have P (e/c) > P (e/-c) P (e/c) is very high (- 0.9) P (c &m) = 0 P (e/m) is very low (- 0.1) So we can conclude that P (e/m) < P (e/ - m), because P (e/ - m) = P (e/c) even though m causes e. S295 http://www.jstor.org/page/info/about/policies/terms.jsp NANCY CARTWRIGHT Ch. 6 Probability Distributions 6.1 Introduction 6.2 The Hypergeometric Probability Distributions 6.3 The Binomial Probability Distributions 6.4 The Poisson Probability Distributions Parameters of the Poisson Use of the Poisson Expected Gain of the Poisson Probability Distributions 6.5 The Normal Probability Distributions Converting other Normal Distributions to the Statistical Normal Distribution Applications of Converted Normal Distributions 6.6 The Exponential Poisson Distributions 6.7 Approximation with Substitute Distributions The Binomial as an Approximation to the Hypergeometric The Poisson as an Approximation to the Hypergeometric and the Binomial The Normal Curve Approximation to the Hypergeometric and the Binomial An Overall Comparison of Approximation Results Figure 2. Harry E. McAllister, Elements of Business and Economic Statistics. Wiley, NY: 1975. As in physics, where the description of a situation that appears in the mediating model must be specially prepared before the situation can be fitted to the theory (e.g., once you call something a harmonic oscillator, then mechanics can get a grip on it), so too in probability theory. As we know, the description of events as independent and as equally likely or of samples as random are key. We can illustrate with the simple binomial distribution, which McAllister describes this way: "A large class of problems applicable to situations in business and economics deals with events that can be classified according to two possible outcomes ... If, in addition to having two possible outcomes, the outcome of one particular trial is independent of other trials and if the probability ... is constant for other trials, the binomial proba- bility distribution can be applied" (McAllister 1975, 111). Again as in physics, in learning probability theory we are taught a handful of typical or paradigmatic examples to which, ceterisparibus, the prepared descriptions of the model may be applied (e.g., vibrating strings, pendula, and the modes of electromagnetic fields may all be treated as harmonic oscillators), so probability theory too has its stock examples that show what kinds of more concrete descriptions are likely to support the theoretical descriptions that must be satisfied before the theory can apply. "Uses of the Poisson distribution," McAllister in- S296 http://www.jstor.org/page/info/about/policies/terms.jsp MODELS: THE BLUEPRINTS FOR LAWS forms us, ". .. include the broad area of theory involving random ar- rivals such as customers at a drive-in bank, customers at a check-out counter and telephone calls coming into a particular switchboard" (1975, 120). Mulholland and Jones repeat the example of telephone calls in a given period, adding particle emission and the number of minute particles in one milliliter of fluid, as well as a caution: "But there must be a random distribution. If the objects have a tendency to cluster, e.g., larvae eggs, then the Poisson distribution is not applica- ble" (1968, 167). The hypergeometric distribution tends to have three kinds of illus- trations: defective items (especially in industrial quality control), cards (especially bridge hands), and fish sampling (without replacement of course). And so forth. In each case a given distribution will apply only to situations that have certain very specific-and, generally, highly theoretical-features. Because the requisite features are so theoretical, it is best to supply whatever advice possible about what kinds of situ- ations are likely to possess these features. But these are just rough indications and it is the features themselves that matter: situations that have them-and have no further features that foil them-should give rise to the corresponding probability; and without them, we get no probabilities at all. 2.3. An Economics Example. So far we have looked at the chance set-ups with well-known arrangements of characteristics that feature in probability theory and the corresponding distributions that they give rise to. I would like now to look at an example from an empirical science, in particular economics. Most economic models are geared to produce totally regular behavior, represented, on the standard account, by deterministic laws. My example here is of a model designed to guar- antee that a probabilistic law obtains. The paper we will look at is titled "Loss of Skill during Unemploy- ment and the Persistence of Unemployment Shocks" by Christopher Pissarides (1992). I choose it because, out of a series of employment search models in which the number of jobs available depends on work- ers' skills and search intensities, Pissarides' is the first to derive results of the kind I shall describe about the probabilities of unemployment in a simple way. The idea investigated in the paper is that loss of skill during unemployment leads to less job creation by employers which leads to continuing unemployment. The method is to produce a model in which ft = the probability of a worker getting a job at period t S297 http://www.jstor.org/page/info/about/policies/terms.jsp NANCY CARTWRIGHT (i) depends on the probability of getting a job at the previous period (ft-1) if there is skill loss during unemployment-i.e., shows persistence; and (ii) does not depend on f/t- if not. The model supposes that there is such a probability and puts a num- ber of constraints on it in order to derive a further constraint on its dynamics: (i) aft/Oft_1 > O, given skill loss (ii) dft/Oft_1 = 0, given no loss of skill. The point for us is to notice how finely tuned the details of the model plus the constraints on the probability must be in order to fix even a well-defined constraint on the dynamics of ft, let alone ft itself. The model is for two overlapping generations each in the job market for two periods only: workers come in generations, and jobs are avail- able for one period only so that at the end of each period every worker is, at least for the moment, unemployed. 'Short-term unemployed' re- fers to 'young' workers just entering the job market at a given time with skills acquired through training plus those employed, and thus practising their skills, in the previous period; 'long-term unemployed' refers to those from the older generation who were not employed in the previous period. The probability f, of a worker getting a job in the between-period search depends critically on x, the number of times a job and worker meet and are matched so that a hire would take place if the job and the worker were both available. By assumption, x at t is a determinate function of the number of jobs available at t (Jr) and the number of workers available at t (2L). Wages in the model are deter- mined by a static Nash bargain which in the situation dictates that the worker and employer share the output equally and guarantees that all matches of available workers and jobs lead to hiring. The central fea- tures of the first model are listed in Figure 3. Variations on the basic model that relax the assumptions that all workers search in the same way and thus have the same probability for a job match are developed in later sections of the paper. The details of the argument that matter to us can be summarized in three steps. (I follow Pissarides' numbering of formulas, but use primes on a number to indicate formulas not in the text but that follow in logical sequence the numbered formula.) A. A firm's expected profit, n,, from opening a job at t is (4) t, = [1+ ft-, + (1 - ft-)y] (L ft/Jt). where y = 1 represents no skill loss, y < 1 the opposite. S298 http://www.jstor.org/page/info/about/policies/terms.jsp MODELS: THE BLUEPRINTS FOR LAWS Loss-of-skill during unemployment: Model 1 1. Discrete time. 2. Two overlapping generations. a. Each of fixed size, L. b. Each generation is in the job market exactly two periods. 3. Each job lasts one period only and must be refilled at the beginning of every period. 4. The number of jobs, J, available at beginning of period t is endogenous. 5. Workers in each of their two life periods are either employed or unemployed. 6. a. Output for young workers and old, previously employed workers = 2. b. Output for old, previously unemployed workers = 2y, O-0 cy 1. (y< 1 represents skill loss during unemployment.) 7. Unemployed workers have 0 output, no utility, no income. (This is relevant to calculating wages and profits.) 8. In each period all workers and some jobs are available for matching. 9. Each job must be matched at the beginning of a period to be filled in that period. 10. In each period workers and jobs meet at most one partner. 11. The number of matches between a job and a worker is designated by x, where a. x is at least twice differentiable. b. dx > 0, d2x < 0. c. x is homogeneous of degree 1. d. x (0,2L) = x(Jt, 0) = 0. e. x (Jt,2L) < max (Jt,2L). 12. There is a probability that a worker meets a job at the beginning of period t, designated by f. a. ft does not depend on what a worker does nor on whether the worker is employed or unemployed. b. ft is a function only of Jt and L. 13. There is a probability that a job meets a worker at the beginning of period t. a. This probability is independent of what jobs do. b. This probability is a function only of Jt and L. 14. The cost of opening a job and securing the output described in 6 = 1/k (whether the job is filled or not). 15. Wages are determined by a Nash bargain. 16. Workers and employers optimize expected utility. Figure 3. It is crucial that ft-_ appears in this formula. It enters because the expected profit depends on the probability of a job meeting a short- and a long-term unemployed worker, which in turn depends on the number of long-term unemployed workers available and hence on the probability of employment at t - 1. S299 http://www.jstor.org/page/info/about/policies/terms.jsp NANCY CARTWRIGHT B. The number of jobs will adjust so that no firm can make a profit by opening one more, which, given that the cost of open- ing a job is l/k, leads to (5') , = 1lk and thus, using (4) to (7) J = Lk [l+ y+(l - y) ft-] f. C. As a part of the previous argument it also follows that Jt > x (Jt, 2L). In addition ft = min {x (Jt, 2L), Jt, 2L}/2L since the number of hires cannot be greater than the number of jobs or workers available nor the number of meetings that take place. Coupling these with the assumption that the ho- mogeneous function x is of degree 1 gives (8) ft = min {x (J/2L, 1), 1}. When f, = 1-full employment-there are no problems. So consider (8') ft = x (Jt/2L, 1). To do so, substitute (7) into (8') to get (9') ft = x [(k/2){1 + y + (1 - y) ft _}ft, 1] = x (0,1) letting(D = df(k/2){1 + y + (1 - y) ft-} ft. We are now in a position to draw the two sought-for conclusions, beginning with the second: (ii) The case where there is no skill loss during unemployment is represented by y = 1. (Short- and long-term workers are equally productive. See Assumption 6, Figure 3.) Then ft = x(kft, 1), from which we see that ft does not depend on ft- . Hence with no skill loss there is no unemployment persistence in this model. (i) When there is skill loss, y < 1. Differentiating (9') with respect to ft- in this case gives (11) [1 - {dx/dI} {k/2} {1 +y + (1 -y)ft-i}] [Oftlaft -] = (k/2)(1 - y)ft (dx/d