Evidential Pluralism in the Social Sciences

Front matter

PHILOSOPHY AND METHOD IN THE SOCIAL SCIENCES

390 1UNOY

Evidential Pluralism in the Social Sciences

Contents

List of Figures x List of Tables xi Preface xii Acknowledgements XV PART I Philosophical Framework 1 1 Evidential Pluralism 3

ȘI Whatis Evidential Pluralism? 3

$2 Wh is Evidential Pluralism Plausible? 12

PART II Consequences and Concerns 41 3 Evidence-Based Policy: EBP+ 43

ȘII EBM and EBP 43 $12 EBP+ Evaluation Procedures 45 $13 EBP+ in Comparison to Existing Approaches 58 viii

Contents

$14 $15

$16 $17

The Context of the Origins of Mixed Methods Research 65

Mixed Methods Research and its Philosophical Foundations 67

A Critical Analysis 70

Evidential Pluralism and Mixed Methods Research 75

$18 $19 $20 $21

PART III Particular Social Sciences The Problem of Sufficiency 79

The Problem of Necessity 81

The Problem of Causal Monism 83

The Problem of Defining Mechanisms 86

$22 $23

Causal Enquiry in Sociology 93 Sociologists’ Methodological Reflections on Causal Enquiry 96

$24 $25 $26 $27

Causal Enquiry in Economics 100

Benefits of Evidential Pluralism in Economics 104 Mechanisms and Theory in Economics 109

Causal and Methodological Pluralism in Economics 113

$28 $29

$30

The Need for Methodological Diversity in Political Science 115

Case Study: Resource Wealth and Violence in Rebellions 117

Understanding Causal Enquiry in Political Science 124

The Bifurcation Approach to Causation in the Law 132 The Bifurcation Approach and Evidential Pluralism 138

$33 Liability-Tracing Mechanisms 142 $34 Against Causal Autonomy in the Law 144 The Scope of Evidential Pluralism in the Social Sciences

$35 Across the Social Sciences 149 $36 Where We Stand 152

Bibliography Index

Contents ix

154 172

Figures

1.1 1.2

Evidential relationships for causal enquiry (Williamson, 2021a) Common cause C is a potential confounder, with respect to the relationship between A and B

Evidential relationships for extrapolating a causal claim (Williamson, 2019a)

Evidential relationships for assessing a causal claim in a target population, including extrapolation from ‘source studies’, 1.e., studies performed in a different source population (Williamson, 2019b)

An example of ‘ontology-oriented’ philosophical foundations An example of ‘axiology-oriented’ philosophical foundations A simplified representation of the mechanism of Sewell

A simplified representation of the unemployment-crime mechanism complex of Cantor and Land (1985, p. 321)

The classical rational addiction mechanism complex (Maziarz, 2021)

The rational addiction mechanism complex (Maziarz, 2021) The IRMOGV mechanism

Incidents of violence against civilians in Uganda’s civil war (Weinstein, 2007, p. 221)

Resource wealth and violence in civil wars, 1945-2000 (Weinstein, 2007, p. 307)

LNQA (Goertz and Haggard, 2022, p. 5)

Multi-method LNQA (Goertz and Haggard, 2022, p. 6) Liability-tracing mechanism of R v Pagett 1973 Liability-tracing mechanism for R v Williams 2011 Liability-tracing mechanism for R v Hughes 2011

22 71 72

108 108 118

123 129 130 142 144 144

Tables

Possible explanations of an observed correlation between

A and B (Williamson, 2019a)

Examples of mechanistic studies in the social sciences

The SUNY evidence hierarchy (SUNY, 2004)

The Oxford Centre for Evidence-Based Medicine evidence hierarchy for treatment benefits (OCEBM Levels of Evidence Working Group, 2011)

An evidence hierarchy related to the GRADE approach (Guyatt et al., 2015, p.15)

Examples of sources of evidence of mechanisms in medicine (Clarke et al., 2014)

Quality levels of evidence (Parkkinen et al., 2018)

Status of a claim (Parkkinen et al., 2018)

The main steps of the EBP+ approach to evaluating efficacy The main steps of the EBP+ approach to evaluating external validity The main steps of the EBP+ approach to evaluating effectiveness Associational models versus causal models (Moneta and Russo, 2014, p. 56)

The educational profile of demobilised soldiers in Mozambique’s civil war (Weinstein, 2007, p. 115)

Two types of but-for tests and their variants for actual causes (Moore, 2011, 2019)

Three types of but-for tests and their variants for proximate causes (Moore, 2011, 2019)

Preface

Evidential Pluralism is a new philosophical account of causal enquiry. It espouses two kinds of pluralism. The first, object pluralism, holds that in order to establish a causal claim, one normally needs to establish two things: that the putative cause and effect are appropriately correlated, and that there is an appropriate mechanism complex linking the putative cause and effect which can account for the extent of the observed correlation. The second, study pluralism, holds that in order to assess a causal claim, one normally needs to assess relevant association studies and mechanistic studies where available. Association studies are studies that test whether the putative cause and effect are correlated, while mechanistic studies test for hypothesised features of a mechanism complex linking the putative cause and effect.

In this book, we argue that Evidential Pluralism can be fruitfully applied to the social sciences. This application is fruitful in three respects: Evidential Pluralism can explain and validate successful examples of causal enquiry; it can help us understand the general structure of causal enquiry; and it can help to inform practice in the social sciences. With regard to informing practice, Evidential Pluralism can help researchers ascertain how best to establish a causal claim of interest, and it can help evaluators assess causal claims that have been put forward by other researchers.

These arguments appeal to a mixture of philosophical theory building and close analysis of practice in the social sciences. We begin with the theory building. Evidential Pluralism was originally proposed in the context of the biomedical sciences. In Chapter 1, we develop a general account of Evidential Pluralism that is not tied to any particular domain of application. We devote some space to carefully stating Evidential Pluralism, explaining the relevant terminology and trying to avoid potential misconceptions. We also motivate the theory and sketch its application to the biomedical sciences. While Evidential Pluralism is an epistemological theory—concerned with how to establish and assess causal claims—we outline one particular philosophical theory of the nature of causality that coheres well with Evidential Pluralism.

In Chapter 2, we explore some views that might be thought of as historical precursors of Evidential Pluralism, including the approaches of Claude Bernard, W.F.R. Weldon and John Goldthorpe. We also note some differences between

Preface xiii

Evidential Pluralism and analytic sociology, and some differences between Evidential Pluralism and Roy Bhaskar’s critical realism.

The book then turns to methodological developments in the social sciences that are motivated by Evidential Pluralism. In Chapter 3, we argue that Evidential Pluralism leads to a new account of evidence-based policy assessment, which we call EBP+. EBP+ provides the capability to assess mechanistic studies alongside the association studies that are the bread and butter of present-day evidence-based policy assessment. This can lead to better-informed judgements of the effectiveness of social interventions, and thus to better social policy. We compare this new account to related approaches, such as realist evaluation.

In Chapter 4, we argue that Evidential Pluralism provides new philosophical foundations for mixed methods research in the social sciences. Mixed methods research mixes quantitative and qualitative research methods and data. After providing an account of the context within which mixed methods research emerged, we explore the question of its philosophical foundations. We sketch standard approaches to the foundations of mixed methods research and note certain limitations to these standard approaches. We then argue that the account of causal enquiry at the heart of Evidential Pluralism requires a thorough consideration of both quantitative and qualitative methods and so provides strong motivation for mixed methods research in causal enquiry. Evidential Pluralism also provides practical guidance on how to integrate quantitative and qualitative studies.

Chapter 5 responds to four potential objections to Evidential Pluralism. The first two are objections to object pluralism: an objection that establishing correlation and mechanism is not sufficient for establishing causation, and an objection that establishing correlation and mechanism is not necessary for establishing causation. Next, we consider an objection based on causal pluralism— the view that there are multiple concepts of cause in use in the social sciences. Finally, we respond to concerns about how to define mechanisms in the social sciences.

While Parts I and II of the book (Chapters 1 to 5) are pitched at a very general level so as to apply right across the social sciences, Part III (Chapters 6 to 10) seeks to address considerations that are specific to particular social sciences.

Chapter 6 illustrates the advantages of applying Evidential Pluralism to sociology. We show that Evidential Pluralism can shed light on the use of evidence in causal enquiry in sociology by means of two examples: one involving the connection between socioeconomic status and health, and one concerning the link between family background and educational attainment. We also show that Evidential Pluralism generalises and motivates certain approaches to the methodology of causal enquiry in sociology, including that of Morgan and Winship.

In Chapter 7, we turn to economics. Again, we begin by considering two examples: the link between the legalisation of abortion in the USA in the 1970s and the subsequent decline in the crime rates in the 1990s, and the link between unemployment and crime. These two examples highlight the roles of association studies and mechanistic studies in causal enquiry in economics. We argue that Evidential Pluralism can help us to understand the structure of causal enquiry in xiv Preface economics, and discuss in more detail the role of mechanisms and of theory in economics. Finally, we address concerns about causal pluralism in economics.

Chapter 8 discusses Evidential Pluralism in political science. We see that Evidential Pluralism can account for the need for a diversity of methods in political science, and we examine a case study concerning the role of resources in shaping strategies of violence in rebellions. We discuss causal pluralism in political science and show how Evidential Pluralism coheres well with process tracing, which is a well entrenched method for causal enquiry in political science, as well as multi-method large-N qualitative analysis, which is a newly emerging approach.

Chapter 9 considers law. Law is often taken to presuppose a concept of cause that is autonomous from that used in philosophy: this is partly because causal claims in the law are constrained by legal rules about liability, while causal claims elsewhere are not. We argue that while Evidential Pluralism applies directly to our usual notion of cause, a slightly modified version of Evidential Pluralism can shed light on causation in the law. This modification requires considering what we call ‘liability-tracing mechanisms’ in place of regular mechanisms.

Chapter 10 suggests that Evidential Pluralism also applies to other social sciences, including anthropology, psychology, and demography, for example. Thus Evidential Pluralism has broad scope in the social sciences. We close the book by making some general comments about the overall approach and by suggesting some potential avenues for further research.

We hope that this book provides a good example of what might be called epistemology-driven philosophy of science. Much work in the philosophy of social science is driven by metaphysics or conceptual analysis—by views about the nature of social reality or about general concepts employed in the social sciences. That is not the only way to proceed. The approach that underpins this book is one which focuses on epistemological questions surrounding how to establish, confirm and assess causal claims in the social sciences, in order to develop a theory of causal enquiry. This account of causal enquiry, if successful, can then be viewed as a constraint on the metaphysics of causality or on an analysis of the concept of cause. If a theory of the nature or concept of cause can validate and explain a successful account of causal enquiry, so much the better for that theory. On the other hand, if the theory is incompatible with the successful account of enquiry, then it can be viewed as challenged, or undermined, by scientific practice. Exactly which theories are compatible with Evidential Pluralism is an interesting question for further research.

Acknowledgements

We are grateful to the Leverhulme Trust for supporting the research project Evidential pluralism in the social sciences, from which this book grew. The book has benefited enormously from discussions with the two other project members, Samuel D. Taylor and Alexandra Trofimov. We are also indebted to Graeme A. Forbes, Brian Kett, Hugh Robertson-Ritchie and Michael Wilde for very helpful comments on an earlier draft. We are grateful to Svein Anders Noer Lie for his kind hospitality in hosting a workshop on Evidential Pluralism in Tromsø, and dedicate the cover image to him and his colleagues there.

We would like to thank Neil Jordan, Gemma Rogers, Alice Salt, Phil Hutchinson and all the team at Routledge for their work in bringing this book to fruition.

Some text previously appeared in the following published works:

Shan, Yafeng, and Jon Williamson. 2021. “Applying Evidential Pluralism to the Social Sciences.” European Journal for Philosophy of Science 11(4): 96.

Shan, Yafeng. 2022. “Philosophical Foundations of Mixed Methods Research.” Philosophy Compass 17(1): e12804.

Part I Philosophical Framework

§1 What is Evidential Pluralism?

Evidential Pluralism is an account of causal enquiry. Throughout this book, we take causal enquiry to include both establishing causal claims and assessing causal claims. Evidential Pluralism can be expressed as follows:

Evidential Pluralism. In order to establish a causal claim, one normally needs to establish two propositions: that the putative cause and effect are appropriately correlated, and that there is some mechanism complex involving the putative cause which is responsible for the putative effect and which can account for the extent of the correlation. So, in order to assess a causal claim, one normally needs to assess relevant association studies and mechanistic studies where available.

In this section, we explain in detail what Evidential Pluralism says. In the next section, §2, we explain why Evidential Pluralism is plausible. Russo and Williamson (2007, $$ 1-4) put forward the thesis that establishing causation in the health sciences requires evidence of mechanisms in addition to evidence of probabilistic dependencies. Evidential Pluralism is a development of this thesis, and we shall sketch its use in the health sciences in §3. The Russo-Williamson thesis was clarified and developed by Gillies (2011); Illari (2011); Clarke et al. (2014); Gillies (2019); Williamson (2019a), amongst others. We discuss some points of clarification in §4. Evidential Pluralism is a theory about the epistemology of causality— about how we can identify causal relationships. However, Russo and Williamson (2007) used this theory to argue for a particular account of the nature of causality, namely epistemic causality. We introduce epistemic causality in §5. In this book, however, we do not commit to any particular account of the nature of causality— we leave this question open. This book is primarily about the epistemology of causality in the social sciences.

Let us begin by clarifying what Evidential Pluralism says. This is best done with the help of a diagram, Figure 1.1.

A is a cause of B a s There is a mechanism A is correlated with B linking A to B 13 H2 a2 a! Specific mechanism hypotheses

Jn

Association studies Mechanistic studies (measure A and B (evidence of features together) of the mechanism)

Figure 1.1 Evidential relationships for causal enquiry (Williamson, 202 1a).

At the top of this diagram is the causal claim under investigation. Evidence for the claim that A is a cause of B is provided by evidence for two further claims: that A and B are appropriately correlated and that they are linked by a suitable mechanism. This part of the diagram corresponds to the first component of Evidential Pluralism:

In order to establish a causal claim, one normally needs to establish two propositions: that the putative cause and effect are appropriately correlated, and that there is some mechanism complex involving the putative cause which is responsible for the putative effect and which can account for the extent of the correlation.

We shall call this component of Evidential Pluralism object pluralism, because it specifies two objects of evidence—two claims that we need evidence for (Ilari, 2011). Let us consider in more detail some of the terms that occur in the statement of object pluralism.

‘Establish’. A proposition is established just when standards are met for treating the proposition itself as evidence, to be used to help assess further propositions. !

1 This leaves open the question of what constitutes evidence. An agent’s evidence has variously been analysed as her knowledge, or her full beliefs, or those of her degrees of belief which are set by observation, or her information, or what she rationally grants (Williamson, 2015). Fortunately we do not need to settle this very controversial philosophical question here. For our purposes, all that is required is an account of when a putative item of evidence might be relevant to a causal claim. Evidential Pluralism provides such an account, as we will see.

Whether a proposition is established depends on prior evidence. Prior evidence must warrant not only sufficiently high confidence in the truth of the proposition, but also high confidence that further evidence will not call the proposition into question, i.e., that confidence in the proposition will remain sufficiently high in the light of new evidence. We can refer to these as the ‘threshold’ and ‘stability’ conditions for establishing (Williamson, 2022).

One can distinguish establishing as an evidential relation from establishing as an act. As an evidential relation, an evidence base establishes a proposition when the threshold and stability conditions are met. As an act, an agent establishes a proposition when her evidence base establishes the proposition and she takes the proposition to be evident on those grounds. The act of establishing does not need to be a conscious act. Moreover, several propositions can be established by a single act: for example, causation, correlation and mechanism might all be established at once, if Evidential Pluralism is correct. An agent is ‘in a position’ to establish a proposition if her evidence base establishes the proposition. One can then construe Evidential Pluralism in terms of either the act or the evidential relation. In the former case, Evidential Pluralism offers practical advice for someone intending to establish causation: for you to establish a causal claim, you need to gather evidence in order to be in a position to establish both correlation and mechanism. Under the evidential relation construal, Evidential Pluralism is impersonal: it says that the evidence base establishes causation just when it establishes correlation and mechanism.

Establishing requires meeting a high epistemological standard. In particular, establishing a causal claim should be distinguished from acting in accord with a causal claim as a precautionary measure. In certain cases in which a proposed intervention has a clear benefit and a relatively low cost, or if failing to act has a high cost, it may be appropriate to initiate the intervention even when its effectiveness has not been established, so that benefits can be reaped in case it turns out to be effective.

Although establishing requires meeting a high epistemological standard, it is fallible. One’s prior evidence can be systematically misleading, making a proposition very plausible and making it very plausible that confidence in the proposition will not significantly decrease in the light of new evidence, even though the proposition is in fact false. If the threshold and stability conditions are sufficiently demanding, such cases will be very rare, which reduces the need to revisit previously established propositions. On the other hand, if the conditions are too demanding then they will too rarely be met and enquiry will stall. The need to avoid falsity must be balanced against the need to establish truth.

This balance may differ from discipline to discipline and field to field. A younger field, such as social psychology, may attach less weight to avoiding falsity than an older field, such as particle physics, in the interest of building up a body of established propositions. (This kind of difference between fields is witnessed by their use of different p-values when establishing statistical hypotheses.)

As this book has broad scope, covering many different fields, we make no attempt to commit to specific standards for establishing.?

A proposition will be said to be ‘ruled out’ just when its negation is established. Thus the claim that 4 is a cause of B is ruled out when it is established that 4 is not a cause of B. In §12 we introduce a range of other possible status levels, intermediate between established and ruled out, that the available evidence may confer on a proposition.

‘Needs’. Evidential Pluralism is primarily a normative thesis, about what one ought to do in order to establish, or assess, causation. Whether particular instances of causal enquiry conform to its advice varies: not all practice is good practice. In Part II and Part III, we will examine a range of examples of causal enquiry to investigate the extent to which they can be thought of as conforming to Evidential Pluralism.

Next we turn to ‘correlation’ and ‘mechanism’. Importantly, Evidential Pluralism holds that establishing causation requires establishing the existence of a correlation and the existence of a mechanism, not necessarily the extent of the correlation, nor the details of the mechanism. In some cases, of course, establishing the extent of a correlation is a means to establishing its existence, and establishing the details of a mechanism is a means to establishing its existence, but these means are not the only means.

‘Correlation’. Evidence needs to establish that the putative cause and effect are ‘appropriately correlated’. Here, ‘appropriately correlated’ just means probabilistically dependent conditional on all potential confounders, where the probability distribution in question is relative to a specified population or reference class of individuals. Thus, if 4 is the putative cause variable, B the putative effect variable, and C is the set of potential confounder variables, one needs to establish that A and B are probabilistically dependent conditional on C, often written A Ji B |C.

Note that if the causal claim is single-case, i.e., if the reference class only contains a single individual or a single situation, then the probability function in question needs to be interpreted as single-case probability, such as single-case chance or rational degree of belief. See §4 and §30.2 for more discussion of single-case causal claims. On the other hand, if the causal claim is generic, i.e., repeatedly

2 There is one philosophical approach, knowledge-first epistemology, which deems it futile to attempt to provide specific standards for establishing a proposition. See Wilde (2021) for a discussion in relation to establishing mechanistic hypotheses. Note that the knowledge-first approach also takes establishing to be infallible, however, and holds that one cannot in general determine what one’s evidence is. This is quite a departure from the common usage of the terms ‘establishing’ and ‘evidence’ in scientific practice. In this book, we stick more closely to the standard usage of these terms.

3 ‘Correlated’ is often used in other senses, e.g., as meaning unconditionally probabilistically dependent, or unconditionally linearly dependent. These are not the senses in use here. On the other hand, much of what we say in the book also goes through when ‘correlated’ is used in senses other than probabilistic dependence conditional on all potential confounders.

instantiable, then the probability function can be interpreted as generic probability, such has limiting relative frequency or generic propensity.*

A confounder is a variable associated with both 4 and B, e.g., a common cause of A and B (Figure 1.2). One needs to establish probabilistic dependence conditional on confounders because an observed association between A and B might be attributable to their association with C, rather than attributable to 4 being a cause of B. The set of ‘potential’ confounders should include any variable that plausibly might be a confounder, given the available evidence of the area in question. The set of potential confounders includes variables that are established by current evidence to be confounders (e.g., the known common causes of A and B), as well as variables that are deemed confounders by more speculative mechanistic hypotheses. As with standards of establishing, standards of what counts as a potential confounder may vary from field to field. If too many variables are taken to be potential confounders, it will be very difficult to establish correlation. On the other hand, if too few variables are considered as potential confounders, establishing correlation will only weakly confirm causation. Thus, a balance needs to be struck.

Establishing correlation is non-trivial for two reasons. First, because it requires establishing a probabilistic dependence in the data-generating distribution, rather than simply in the distribution of a sample of observed outcomes. The method of sampling and size of sample can conspire to render an observed sample association a poor estimate of a correlation in the population at large. Second, the correlation claim invokes all potential confounders, and there can be very many of these.

On the other hand, establishing correlation, 1.e., establishing probabilistic dependence on all potential confounders, is not as difficult as establishing probabilistic dependence conditional on all possible confounders. The set of all possible confounders includes not only the potential confounders but also unforeseeable confounders: variables that, on the basis of current evidence, cannot be anticipated to be confounders. Establishing probabilistic dependence conditional on all possible confounders is an extremely demanding task. Evidential Pluralism does not require establishing this latter form of probabilistic dependence, although some other accounts of causal enquiry do. For example, probabilistic theories of causality tend to identify causality with correlation conditional on all confounders—not

Figure 1.2 Common cause C is a potential confounder, with respect to the relationship between A and B.

4 See Gillies (2000) for an introduction to interpretations of probability.

merely on those variables that might reasonably be anticipated to be confounders. It is hard to see how many standard methods can be used to establish causation under such an account. Indeed, a probabilistic account of causality can lead to scepticism about any causal claim that is supported by anything but the largest randomised controlled trials (RCTs). Evidential Pluralism is less demanding. One merit of Evidential Pluralism is that it shows how study designs other than RCTs can help to establish a causal claim. We will encounter many examples of the confirmatory value of other kinds of study design in this book.

To be clear about terminology, we shall use ‘observed correlation’ to refer to a correlation found in the sample data, ‘genuine correlation’ to refer to a correlation in the population from which the data are drawn and ‘established correlation’ to refer to a claimed genuine correlation that has met the threshold and stability conditions for establishing. Since establishing is fallible, the fact that a correlation is established does not guarantee that there is a genuine correlation, though it makes a genuine correlation very likely. We will use ‘association’ in a more general sense than ‘correlation’, to refer to any kind of probabilistic dependence of two variables—either unconditional probabilistic dependence or probabilistic dependence conditional on some further set of variables (e.g., a subset of potential confounders). Thus a correlation is an association between A and B conditional on the set of all potential confounders.

‘Mechanism’. We take a mechanism to consist of a complex-systems mechanism, a mechanistic process or some combination of the two. A complex-systems mechanism consists of entities and activities organised in such a way that they are responsible for some phenomenon to be explained (Machamer et al., 2000; Mari and Williamson, 2012). An example is the mechanism by which a stock exchange trades stocks. A mechanistic process is a spatiotemporally contiguous process along which a signal can be propagated (Reichenbach, 1956; Salmon, 1998). An example is an electronic signal being transmitted via air and cable from a smartphone to a stock exchange. A mechanism might also be composed of both these sorts of mechanisms: for example, a trading mechanism might include the complex-systems mechanism of the smartphone, the complex-systems mechanism by which the stock exchange trades stocks and the mechanistic processes linking the two.

With regard to Evidential Pluralism, it is not enough to establish the existence of some mechanism of action: it is also essential to establish that the influence of this mechanism is not negated by that of counteracting mechanisms. Thus one needs to consider the whole complex of mechanisms that link 4 and B and ask whether there is a mechanism of action, which when taken together with any counteracting and reinforcing mechanisms, can account for the extent of the observed correlation between A and B. Moreover, when establishing a generic causal claim it is not enough to show the purported mechanism merely exists in some individuals—it needs to be present in sufficiently many individuals to account for the extent of the observed correlation.

‘Normally’. Evidential Pluralism says that one ‘normally’ needs to establish both correlation and mechanism. This is because there are certain cases in which causality is apparently not accompanied by a correlation and there are also cases in which causality is apparently not accompanied by an underlying mechanism. If this is so, one cannot expect to establish both correlation and mechanism in these pathological cases.

In cases of overdetermination, where the cause does not raise the probability of the effect because the effect will happen anyway, the cause and effect need not be genuinely correlated. In many such cases, one can at least expect a counterfactual correlation: if things had been different in such a way that the effect would not have happened anyway—e.g., had a second, overdetermining cause been eliminated— then the cause and effect would indeed be correlated. One might think, then, that one ought to be able to establish a counterfactual correlation for any causal claim, if not an actual correlation. However, there are cases in which the cause of interest and a second, overdetermining cause are mutually exclusive, so that it is not possible both to eliminate the second cause and allow the first cause to vary so as to establish a correlation. For example, an unstable atom may decay to one of two mutually exclusive intermediary states, B and B’, on the way to a ground state C. Attaining either one of the intermediary states causes the particle to reach the ground state, even though there may well be no correlation, P(C|B) = P(C|B”) = P(C). One cannot simply eliminate B'and vary B here (see Williamson, 2009, $10). Therefore, even the demand for a counterfactual correlation may be too strong.