scientific explanation: towards a neo-deductivist account
TRANSCRIPT
Scientific Explanation: Towards a Neo-Deductivist Account
By
Martin King
A Thesis
presented to
The University of Guelph
In partial fulfilment of requirements
for the degree of
Doctor of Philosophy
in
Philosophy
Guelph, Ontario, Canada
© Martin King, May, 2016
ABSTRACT
SCIENTIFIC EXPLANATION: TOWARDS A NEO-DEDUCTIVIST ACCOUNT
Martin King Advisor:
University of Guelph Andrew Wayne
This thesis is an investigation of philosophical accounts of scientific explanation. It is centered
around the question: how do scientific models explain? I argue that the model-based deductivist
account I propose in Chapter 5 is a viable and promising candidate for a successful account of
scientific explanation. This thesis shows that other structural, causal, and deductivist accounts
face significant challenges or are not reflective of the practice of scientific explanation.
In the first chapter, I introduce the concept of scientific explanation and review the goals
of a philosophical account of scientific explanation. In the second chapter, I explore the role of
idealized models in scientific explanation. In the remainder of Chapter 2, and in Chapters 3 and
4, I critically review literature on three of the main approaches to scientific explanation:
structural, causal, and deductivist respectively. Some of the main results of these investigations
are that: i) an account of explanation should include a broader range of models than a strictly
causal account, to more accurately reflect the explanatory practices of science; ii) a variety of
models can be explanatory in a given system, but that causal interpretations of these models can
be problematic; iii) and that highly-idealized models can be explanatory but that neither
structural nor causal accounts can fully capture why this is so.
Taking these results into account, in the final chapter, I present the most significant
contribution of the dissertation, which is the integrated model account of explanation. The
account relaxes local constraints on allowable features in explanatory models and instead
introduces a global constraint on the model’s relation to theory. It is a model-based deductivist
account that stipulates four criteria for explanation: the explanandum is deductively entailed by
the explanans; the statements in the explanans are true of a scientific model; the model shows on
what the explanandum depends; and the model is integrated with a global theory of science.
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To my father, who would have been so proud.
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ACKNOWLEDGMENTS
I express my sincerest thanks to my advisor, Dr. Andrew Wayne, for his continuous support and
encouragement over the years. Without his knowledge, insight, and enthusiastic dedication to his
students, this project would never have been completed. In fact, without him, I would have never
even undertaken this project. I owe him a great debt of gratitude for the effect his tutelage has
had on me, my research, and my future successes.
I would also like to thank the members of my advisory committee, Dr. Stefan Linquist and Dr.
Jessica Wilson, for their challenging comments and questions, as well as for their help and
guidance in the development of my project. I am also grateful for the expertise of Dr. Juha
Saatsi, whose input and careful reading of my work made for great discussion and will be
beneficial for my future work.
I am indebted to my fellow graduate student colleagues for the late-night philosophical
discussions over whisky and beer, and for their resolve and good spirits that made the last five
years of being overworked and underpaid, not only tolerable, but enjoyable.
Enfin et surtout, un grand merci à ma mère pour tous la patience et l’encouragement pendant ma
longue carrière d’étudient, and to David, Robin, and Kayla for the profound effect they have on
every aspect of my life.
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Table of Contents
Acknowledgments ..................................................................................................................... iv
Chapter 1. The Nature of Scientific Explanation ............................................................. 1
Introduction .................................................................................................................. 1
Some Kinds of Explanation ......................................................................................... 4
1.2.1. Non-Explanations ........................................................................................... 7
1.2.2. Token and Type Explanations ........................................................................ 7
Explanation and Understanding ................................................................................... 8
1.3.1. Pragmatics, Explananda, and Why-Questions ............................................. 11
Goals of an Account of Explanation .......................................................................... 15
1.4.1. Descriptivism and Normativity .................................................................... 15
1.4.2. Threshold and Depth .................................................................................... 17
1.4.3. Scope ............................................................................................................ 18
1.4.4. Accurate Representation .............................................................................. 19
Conclusion ................................................................................................................. 20
Chapter 2. Idealization and Structural Accounts of Explanation ................................... 21
Introduction ................................................................................................................ 21
Models and Idealization ............................................................................................. 22
2.2.1. What is idealization? .................................................................................... 23
2.2.2. Laws and Idealized Models .......................................................................... 26
2.2.3. Idealization and Explanation ........................................................................ 31
Modelling and Explanation ........................................................................................ 33
2.3.1. Desiderata ..................................................................................................... 33
Idealization in Physics ............................................................................................... 37
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2.4.1. Structural Model Explanations ..................................................................... 38
2.4.2. Two Approaches for Assessing Structure .................................................... 47
2.4.3. E3: The Justificatory Step ............................................................................ 54
2.4.4. Heuristics and Explanations ......................................................................... 58
Conclusion and Discussion ........................................................................................ 60
Chapter 3. Causal Accounts of Explanation................................................................... 62
Introduction ................................................................................................................ 62
Scientific Pluralism .................................................................................................... 63
3.2.1. Age-Polytheism in Eusocial Insect Colonies ............................................... 65
3.2.2. Three Problems with the Integrative Pluralism Approach ........................... 68
Causal Interventionism .............................................................................................. 70
3.3.1. Invariance and Intervention .......................................................................... 72
3.3.2. Circularity ..................................................................................................... 74
3.3.3. Causal Realism ............................................................................................. 75
Emergence and Reductionism .................................................................................... 79
3.4.1. Physicalism and Supervenience ................................................................... 80
3.4.2. Exclusion Arguments ................................................................................... 81
3.4.3. Intervention and Emergent Causation .......................................................... 87
Kairetic Explanation .................................................................................................. 91
3.5.1. Abstracting and Optimizing ......................................................................... 92
3.5.2. Idealization and Causal Realism .................................................................. 93
3.5.3. Concerns about the Kairetic Account ........................................................... 95
Conclusion and Additional Concerns ........................................................................ 96
Chapter 4. Deductivist Explanation ............................................................................. 101
Introduction .............................................................................................................. 101
The D-N Account ..................................................................................................... 102
4.2.1. Is it Necessary? ........................................................................................... 104
4.2.2. Is it Sufficient? ........................................................................................... 107
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Unificationism .......................................................................................................... 109
4.3.1. Unificationist Solutions to D-N Problems ................................................. 112
4.3.2. Challenges to Unificationism ..................................................................... 115
Conclusion and the Current State of Deductivism ................................................... 116
Chapter 5. Model-based Deductivism .......................................................................... 119
Introduction .............................................................................................................. 119
A Model-based Deductivist Account ....................................................................... 120
What is a Model? ..................................................................................................... 123
5.3.1. A Simple Model of the Fixed-length Pendulum ........................................ 124
Counterfactuals ........................................................................................................ 126
5.4.1. Same-object Counterfactuals ...................................................................... 127
5.4.2. Truth Conditions for Counterfactuals ........................................................ 128
The Simple Pendulum Revisited .............................................................................. 129
A Global Constraint on Explanation ........................................................................ 131
5.6.1. Some Aspects of Theoretical Integration ................................................... 134
5.6.2. Prediction without Explanation .................................................................. 136
The Integrated Model Account of Explanation ........................................................ 139
Empiricism, Emergence, and Reduction .................................................................. 142
Conclusion and Limitations of the Account ............................................................ 144
Conclusion 148
References 152
1
Chapter 1. The Nature of Scientific Explanation
Introduction
Scientific explanation has been a central topic in the philosophy of science since Carl Hempel
and Paul Oppenheim published “Studies in the Logic of Explanation” in 1948. Explanation
became a large part of 20th century philosophy in part because it is intimately related to many
different issues and important questions, including “what is a law of nature?”, “what makes
something scientific?”, “what is the nature of causation?”, and many more. In the decades
following, many philosophers of science either promoted or criticized various aspects of the
Deductive-Nomological account of explanation that Hempel and Oppenheim originally proposed.
For Hempel and Oppenheim, an explanation is a deductive argument featuring two parts,
where that which is to be explained, the explanandum, is deductively entailed by a set of
sentences that do the explaining, the explanans. The explanatory information for Hempel and
Oppenheim comes from deriving a desired explanandum phenomenon or pattern from a set of
true sentences containing a law, or laws, of nature. The subsumption of a phenomenon under a
law of nature allows us to expect, predict, and explain its occurrence. This formulation of
explanation allows one to answer explanation-seeking why-questions, like “why is the sky
blue?” and also questions about regularities themselves, questions such as “why do Kepler’s and
Galileo’s laws hold?” In the same way, it is the derivational structure of the explanation that
allows the explanandum to be a law, that is derived as a special case of another more general
law, in this case Newton’s law of gravitation.
Many of the philosophers since Hempel and Oppenheim have focused on causal
strategies for capturing explanation (Anscombe, 1969; Lewis, 1973; Salmon, 1984; Dowe, 2000;
Woodward, 2003). These causal accounts all employ quite different strategies for identifying
genuine causal dependency relations, but they share the idea that explaining a phenomenon is
achieved by giving its causes. The most important task for a causal account of explanation then
is to conceptualize causation in such a way as to align causal judgments with our explanatory
judgments, which may otherwise be at odds. For instance, certain models that are not generally
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thought of as causal may be thought of as explanatory. Causal approaches are popular in the
literature and considered to be very promising, though there are some issues of contention
surrounding the notion of causation. In Chapter 3, I present three causal accounts in some detail.
I first examine complexity, causal pluralism, and Sandra Mitchell’s account of explanation. The
pluralist solution to issues of complexity is to identify causes in models at various levels of
investigation, including high-level idealized models. I point out some shortcomings of taking this
approach and argue that her conclusions about the needs for causal pluralism are unwarranted.
The remaining accounts in Chapter 3 belong to Michael Strevens and Jim Woodward
respectively. The problems for Woodward’s account surround the fact that his manipulationist
criteria identify high-level causes and low-level causes in the same system, which invites
concerns about overdetermination and downward causation. I explore possible solutions via
some of the literature on non-reductive physicalism, which I review in 3.4. In the end, I argue
that Woodward’s position is properly emergentist and cannot benefit from defenses of non-
reductive physicalism. After this, I present Michael Strevens’ view of depth and his kairetic
account of explanation. Strevens attempts to show how a physicalist metaphysics can allow for
explanations at non-fundamental levels by preferring a degree of generality to total accuracy. I
argue that this approach is not subject to the same problems as Woodward’s, but is too
unrealistic to reflect explanatory practices or to be implemented.
Before I turn to causal explanation however, I take up structural accounts in the second
chapter, because it is in this context that I can best examine the role of idealization and modelling
in explanation. Proponents of structural explanation have taken approaches similar to causal
accounts but focused on the higher-level structural relations rather than the causal relations
(Worrall, 1989; McArthur, 2003; Bokulich, 2008). This allows these structural approaches to
identify a different class of model as explanatory, whose structure, it is argued, is the most
important factor in explaining the target system’s behaviour. The main benefit of structural
accounts is that they are able to count as explanatory models that do not accurately represent the
real causes of a system, and thus expand the scope of available explanations. I begin Chapter 2
by going over modelling and idealization and their roles in explanation. Idealization is a major
concern for explanation, and especially for causal and other representational accounts. The
immediate worry is that if an account identifies explanation by real causal dependency relations,
then it is not clear how to accommodate false assumptions, non-existing entities, abstractions,
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and otherwise non-representing models. Of course, there are various proposed solutions to this,
and many places to draw the line for what counts as an explanation. In Chapter 2, I explore one
potential solution, which is to focus on structure. Alisa Bokulich proposes an account that
attempts to emphasize the explanatory role that idealizations play in explanation. I argue that the
structuralist account she proposes favours explanations at the fundamental level and thus is not
capable of capturing the way that non-representing models explain.
In the fourth chapter, I return in detail to the deductivist approaches of Hempel and
Oppenheim and Philip Kitcher (Kitcher, 1981, 1989). I present the traditional Deductive-
Nomological account of scientific explanation and some responses to it, including some work on
statistical explanation. I review some challenges to this account, and then present Kitcher’s
unificationism and its proposed solutions in 4.3. Ultimately Kitcher’s defense is not satisfactory,
but the problems for deductivism are not essential or insurmountable. By availing myself of the
resources and outlining some remaining problems for the deductivist approach, I can set the
stage to offer an account that is aimed at improving the current state of deductivist explanation.
These results of these investigations serve to inform the features of the neo-deductivist
account I propose in Chapter 5. The account takes explanations to be arguments that derive the
explanandum from statements about certain kinds of scientific model. The account makes use of
many insights from other accounts, including: the importance of prominently featuring
idealization and counterfactual information; allowing for non-representing models; having a
high-threshold for explanation; requiring the deductive derivation of the explanandum from the
explanans; and much else. The account is intended to circumvent some of the concerns raised
about structural and causal accounts, and to be more relevant to explanatory practice than the D-
N account and more tractable than Kitcher’s unificationism.
The remaining sections of this chapter will deal with the nature of scientific explanation
in order to prepare the ground for later discussions. I begin this chapter by first analyzing various
kinds of explanation and focusing the discussion in the following chapters on the right kinds of
explanation. Then, I examine the role of understanding in explanation, as well as the role of
pragmatics of question-asking and the activity of explaining. The last section of this chapter goes
over some possible goals of an account of explanation, in order to set the standards for what is
and is not to be achieved, what is inside or outside of its scope, and what is or is not desirable for
an account of explanation.
4
Some Kinds of Explanation
It is important to keep in mind that the notion of scientific explanation naturally contrasts itself
with three other classes: those that are scientific, but not explanations; explanations that are not
scientific, and those that are neither (see Table 1-1).
Table 1-1
Scientific & an Explanation Scientific & Not an Explanation
Not Scientific & an Explanation Not Scientific & Not an Explanation
The present work is only intended to focus on the top left cell. In order to better elucidate this
distinction, the following section will discuss some of the various kinds of explanation and make
clear what the appropriate kinds are.
Explanation is a very broad term. There are at least a few distinct kinds of explanation,
most of which are not scientific. Sometimes, what counts as an explanation can be little more
than providing a reasonable reason – a motivation to think that a statement is an answer to a why
question. For instance, in certain circumstances, the question “why is Jones not home?’ can be
answered satisfactorily by the statement “he was out of milk”. These kinds of explanations are
what we might call common-sense explanations, because the statement(s) alone do not
necessitate the explanandum, and some sort of common-sense “law” needs to be used to join the
two and motivate the explanation: milk is a staple, and you should replace it when it runs out. If
this “law” were not common sense, the answer would effectively cease to be explanatory.
Explanations such as this merely provide good reason for thinking that the answer given is the
true explanation. The success of these explanations is due in part to the commonality of these
kinds of inferences (say, from being out of milk to going to the store). These kinds of
explanation are doubtless very common, but they are not scientific explanations.
One reason that they are not scientific is that they often involve human motivations or
intentions. When it comes to these kinds of explanation, it is unreasonable to expect there to be a
single explanation or cause for the fact. If someone asks Mary why she went to university, there
are likely to be many reasons which could all serve as good explanations. Her answer to the
question might depend on who is asking, what she has recently been thinking about, or what she
considers most important in the decision at the time she is asked. What counts as a good
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explanation here is almost entirely contextually dependent. I hope to avoid discussing
explanations that make use of motivations.
Another reason is that these common-sense answers do not necessitate the explanandum.
Because of this, one is never sure if they are correct explanations – there is an essential
ambiguity. Imagine a case where even if it is true that Jones is out of milk, the real reason Jones
is not home is because he is meeting a friend at the pub. The fact that he is out of milk is true,
and gives reason to expect his not being at home, but it is not in fact the correct explanation for
his not being at home. We can further complicate this scenario by adding instead that Jones did
go out to grab milk, but stopped at the pub on the way, making him late. In one sense, he is not
home because he went out to get milk, but in another sense, he is not home because he stopped at
the pub. In this new scenario, there are multiple factors that make it unclear that the answer
“because he went to get milk” is a good explanation. Many such examples have been raised in
the literature on explanation, but I think it is important to bracket off these explanations to focus
on those that are properly in the domain of science.
Other kinds of explanation have a closer relation to scientific explanation, and some
argue are a genuine part of it. There are what are known as teleological arguments or functional
arguments, which cite ends as explanations. Such an explanation would maintain, for instance,
that a trait’s function explains its existence or its continuation in a population. Whether these
arguments are distinct and ineliminable from other explanations in biology and other sciences is
a matter of some contention (Brandon, 1981; Sober, 1984; Reeve & Sherman, 1993; Godfrey-
Smith, 1994). These explanations fit well with human behaviour which has intentions that are
clearly future-oriented, but they are also popular in evolutionary biology, especially before
Darwin and often cited as evidence of intelligent design. Teleological explanations will not be
directly dealt with in this thesis.
In what follows, the kinds of explanation that will be focused on are scientific
explanations and not common-sense, or everyday, explanations. To explain something
scientifically is more than to make an everyday explanation about something that has scientific
content. This is something that is often passed over, and I think is something that is responsible
for many of the problematic counterexamples in the explanation literature. The connection from
explanans to explanandum must be more than merely an accepted inference. What I take a proper
scientific explanation to do is to demonstrate that a well-articulated explanandum is to be
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expected on the basis of a certain kinds of scientific model. A full elaboration of this will have to
wait until further chapters.
Following in the deductivist tradition, I take explanations to be arguments. Other
accounts have taken explanations to be causal stories, explanation-acts, inferences, models, or
whatever. Treating explanations as arguments has some benefits, such as that one can formulate
arguments about models or featuring causal relations. For instance, on this view, to say that C
explains E because C causes E, is really to say that there is an explanatory derivation of E
featuring C and ‘C causes E’. A simple causal explanation can be recast in the form of an
argument. This deductivist approach is open to what are considered causal explanations, but is
not limited by this. The breadth of allowable varieties of explanation is a distinct benefit of the
deductivist approach that will be elaborated on in Chapters 4 and 5. Thus, on this picture, causal
explanations are just one type of explanation among many. It is worth noting that because
explanations are arguments, models themselves do not do the explaining. Rather, to say that a
model is explanatory is to say that it is capable of supporting explanatory derivations. The
relations between models and explanations and the role of models in explanations will be
explored in Chapters 2 and 5.
The purpose of distinguishing these kinds of explanation is not to deny that they are
explanations or to claim that they are not as explanatory as scientific explanations, but rather the
limit the scope of the present investigation to those explanations that concern scientific models.
These models are abstract objects that are constructed for the purposes of explaining some
behaviour, pattern, or property of a target system. They often feature quantitative generalizations
that serve to make reliable predictions about the system. It has been increasingly recognized that
accounts of explanation ought to focus on models, rather than merely on arguments, or diagrams,
or speech acts, though this is not uncontentious. Even among model-based accounts, others have
argued that a scientific explanation consists in identifying the relevant causes of a phenomenon,
or in revealing the structural properties of a system that are responsible for the manifestation of
the phenomenon. I will review some of these other accounts of explanation in the following
chapters. Most of the accounts that will be discussed in detail, including the account proposed in
Chapter 5, all take models to be important sources of information for scientific explanation.
Reasons why this is taken to be a central aspect of scientific explanation will be discussed in
even more detail in Chapter 2.
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1.2.1. Non-Explanations
Conversely, there are also things that fit into the category of being scientific but not explanatory.
And so there is more to science than explaining, which even though it is perhaps obvious, bears
mentioning. Of course, it would beg the question to start an analysis of explanation by deciding
which cases are explanatory beforehand. While it is contentious whether certain models are
explanatory, there are some practices that do not aim to offer explanations. For instance, there
are aspects of science that primarily involve the collection and organization of data. Some of this
has been derogatively called “stamp-collecting”, but it can be an important aspect of scientific
research. Some research that falls into this might include the collection of astronomical
observations, the cataloguing of species, or the human genome project, which is aimed in part at
determining the sequence of base pairs of human DNA, but does not explain the emergence or
continuation of human traits or characteristics. This kind of information could potentially be used
to support explanation, but that is not necessarily the aim of these practices.
There is a different kind of scientific practice, and the last that I will mention, which aims
to create predictive models, like the models at work in meteorology, which make use of very
complicated algorithms that can serve, so legend goes, to predict the weather (Parker, 2011).
These are also known as data models, or phenomenological models. I maintain that to say that
these models of numerical weather prediction actually explain why, for instance, a storm
occurred, requires more than to simply say that it has predicted or forecasted it. The account I
propose in Chapter 5 aims to correctly identify why these predictive models are not explanatory,
unlike other accounts of explanation that have taken explanation and prediction to be two sides
of the same coin. The difficulty in identifying these non-explanatory models is that modelling is
often a very complicated process, and so a model merely built up from data and a model that is
idealized but genuinely explanatory can be difficult to distinguish.
1.2.2. Token and Type Explanations
The token-type distinction is an important one in many contexts, including in terms of scientific
explanation. Types refer to classes and tokens to the instantiations of those classes. A token
explanation is an explanation of a particular occurrence or phenomenon, while a type explanation
is an explanation of why a class of phenomena occurs. These explanations are often answers to
questions of the form why does/did this event happen? (known as why-questions). For instance,
a token explanation would explain why my plant died when I moved it from one part of a room
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to another. A type explanation might explain why similar plants tend to die in similar situations.
Primarily, accounts of explanation have focused on token explanations, and handled type
explanations as a derivative. However, some have argued that type explanations are actually
more interesting, more important, and must be handled separately.
Robert Batterman focuses on universality as the object of explanation, rather than on
individual events (Batterman, 2002b). He hopes to capture a distinct kind of explanation that
relies not on mechanisms or difference making, but in the story about why a class of systems all
exhibit the same large-scale behaviour. This is done in showing that the details are irrelevant to
that behaviour by looking at the explanatory role of what are called minimal models. These
minimal models are caricatures of real systems and represent their physical aspects in almost no
accurate way. The models are used to answer questions concerning the universal behaviour, and
not the individual behaviour of a target system.
He separates why-questions into two distinct kinds: type 1 and type 2. Type 1 questions
ask about specific instances of patterns, and type 2 questions ask why patterns remain stable
under various changes. Answers to type 1 questions do not answer type 2 questions and vice
versa. While mechanistic explanations may suffice for type-1 questions, they will not answer
type-2 questions, and for this one needs minimal models. Minimal models work because the
large-scale behaviour is unaffected by the underlying microstructure (Batterman & Rice, 2014).
What the explanandum is constrains what model, or what kind of model, can explain it. These
distinctions are important to keep in mind when considering what kind of model would best
explain a given explanandum phenomenon.
Explanation and Understanding
One thing that most accounts of explanation often have in common is the notion that explanation
is that which grants understanding. Understanding, then, is what might be considered a goal or an
essential product of a successful explanation. In some accounts, the understanding generated by
explanation is a by-product, and not something that needs to be analyzed along with it (De Regt,
2013). While others have argued that explanation cannot be analyzed without it (Khalifa, 2012).
The idea that explanation grants understanding is an intuitive and appealing way to characterize
explanation, and one that cannot be abandoned completely. In the following the section, I briefly
explore this relation and some of the work that has been done on it. Understanding might be
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intimately related to explanation, but I think that an analysis of explanation can be fruitful
without also being an analysis of understanding.
According to Hempel, in an explanation “the argument shows that, given the particular
circumstances and the laws in question, the occurrence of the phenomenon was to be expected;
and it is in this sense that the explanation enables us to understand why the phenomenon
occurred” (Hempel, 1965b, p. 337). The idea that being able to predict something because it
follows from certain conditions and a law of nature (covering law) is called nomic expectability.
The idea that this also provides understanding is an aspect of Hempel’s account that is
emphasized by Michael Friedman and by Kitcher, whose unificationism is looked at in detail in
4.3 (Friedman, 1974; Kitcher, 1981, 1985, 1989). The basic idea of unificationism, which is
quite compelling, is that the more unified is our knowledge of the world, the more we understand
it. Our understanding increases with a reduction in mystery and a reduction in brute facts. A
prime example of this is the unification of Galilean laws of free fall concerning objects on Earth,
and Kepler’s laws of celestial mechanics. When the two were theoretically unified in Newtonian
mechanics, and shown to be derivable from the same law of universal gravitation, our knowledge
of the world is increased. Aside from increasing the scope of our knowledge, the fact that two
disparate sciences were now united, increases our understanding of the world and is a testament
to the explanatory power of the unifying theory.
Some have denied the strong connection between unification and understanding, such as
Salmon (1984) and Barnes (1992), who argue that history is full of examples to the contrary.
They opt instead for a causal theory of understanding, which is discussed in Chapter 3. Salmon
seeks to modify Hempel’s idea of nomic systematicity, whereby a phenomenon or regularity is
explained if it is shown to fit into a nomic nexus. Salmon argues that a phenomenon is explained
if it can be fitted into a causal nexus (1984, p. 19). On this view, understanding why a
phenomenon is the case involves delineating its relevant causes. This view is shared by many,
and is also intuitively appealing. Proponents of the view also claim that it is able to corroborate
the intuitions about unification, because what is really happening on this picture as we move
from Galileo and Kepler to Newton is not a reduction in types of phenomena, but an
identification of their causal bases.
For Woodward, explanatory knowledge is knowledge that allows the manipulation of
causal systems (Woodward, 2003). Woodward argues that the reason we should care about and
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have an account of causal explanation is largely rooted in the idea that we have an almost innate
drive to understand and manipulate the causal structure of the world; it is not mere intellectual
curiosity. One can identify real causal dependencies even without knowing laws, or being able to
systematize, or unify our knowledge. This, for Woodward shows that our aim is not merely to
predict with descriptive knowledge, but to be able to control the world with explanatory
knowledge.
Michael Strevens makes understanding a central part of his account of explanation
(Strevens, 2008). Understanding is needed to distinguish different depths of explanation and
accommodate higher-level explanations, which are more general. Only explanations that are
exhaustive and provide background information can confer full exhaustive understanding. Other
explanations featuring “black boxes” of bracketed-off mechanisms can only confer qualified
understanding. His account takes as a central problem how we understand and explain high-level
phenomena in a world of only fundamental-level causes.
Alisa Bokulich has yet another view on the relation between explanation and
understanding, which will be given in more detail in 2.4 (Bokulich, 2008). It is her view as well
that nomic expectability is not sufficient for understanding. Understanding is given in the
physical insight that one has into the behaviour of a system. This physical insight is intended to
highlight the graspability of employing classical methods in quantum systems, whose solutions
she claims are purely numerical. This physical insight can be determined on her account by the
amount of counterfactual information that a model can provide about the behaviour of a system.
The role of understanding on this account aims to prevent a notion of explanatory depth from
being reductive in order to preserve the explanatory autonomy of irreducible models.
Some have argued that explanation cannot be as distanced from understanding and
communication as others pretend (Bromberger, 1966; van Fraassen, 1980; Potochnik, 2011). On
these accounts, explanation is an act involving communicators and this is often problematically
ignored in accounts of explanation. Some of these issues will be addressed in the following
section on the pragmatics of explanation.
While understanding may be an integral part of explanation, it is not sufficient for an
account of explanation to simply say that that which explains is that which increases our
understanding. There are some concerns with reducing explanation to studies of understanding.
The most obvious worry is that the issues of understanding are at least as contentious as those of
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explanation. To reframe problems in terms of understanding is unlikely to provide clarity and, I
think, more likely to obscure them. A project that serves to reduce explanation will miss out on
any insights that are gained in a study of explanation alone, which I think can be many. The most
promising strategy is to largely bracket issues of understanding to make up ground on the
epistemic issues of scientific explanation and explanatory knowledge. This coincides with what
Kareem Khalifa has argued about the relation between explanation and understanding: “Any
philosophically relevant ideas about scientific understanding can be captured by philosophical
ideas about the epistemology of scientific explanation without loss” (Khalifa, 2012, p. 17).
Khalifa looks at well-developed work on scientific understanding and finds that the idea that
understanding is distinct from and primary to explanation has not been successfully made and
further that the successes to be had are better solved by the explanation literature.
Separating explanation and understanding has the further implication that someone’s
ability to understand an explanation is not necessary. In a classroom setting, a professor may
lecture to students about how a particular model can be used to explain some interesting
phenomenon. It would seem absurd to claim that this is not an explanation if one of the students
cannot understand it. I claim that it ought to count as a genuine scientific explanation even if
none of the students are able to grasp it. Subjective notions of understanding should not limit
what counts as a scientific explanation. Accordingly, it can be decided that a model is capable of
supporting explanation independently of whether anyone understands or fully grasps the
explanation.
1.3.1. Pragmatics, Explananda, and Why-Questions
The common sense explanations seen in Section 1.2, involve fitting the facts to a reasonable
story in such a way that everything hangs together. By creating a narrative that is partly filled in
by the explainer and partly filled in by the explainee, a compelling justification of why the
explanandum is the case emerges. Fitting a fact into a reasonable story can also be considered as
explaining that fact. This, for some, points to the importance of understanding a network of
causes and effects. To others it points to the idea that the context of an explanation is an essential
part of explanation, and that a linguistic or syntactic analysis will always be incomplete.
The pragmatics of explanation are quite important to any actual explanation-act, but I do
not think that all the problems of explanation can reduced to problems of pragmatics and solved
that way. Much of the contemporary work on pragmatics draws on work from Bas van Fraassen
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and others (van Fraassen, 1980; van Fraassen, Churchland, & Hooker, 1985). In order to expose
the importance of context in explanation, van Fraassen addresses the issues of why-questions. He
was not the first to work on this issue, but made substantial contributions.
Sylvain Bromberger proposes a linguistic approach to explanation that pays close
attention to the importance of why-questions and their effect on explanation (Bromberger, 1966,
1982). He notes that most philosophers dealing with explanation focus on a particular subfamily
or subfamilies of questions that can be considered explainable: some on what causes or what is
the mechanism for questions, others on how possible questions, or according to what law
questions. Each of these demands a different explanation, and so a successful explanation of one
is not necessarily a successful explanation of another. And further, not all why-questions admit
of explanations.
Van Fraassen offers an account of the pragmatics of explanation that is capable of further
distinguishing explanation-seeking why-questions (van Fraassen, 1980). He considers the
question “why did Adam eat the apple?” The very same why-question can be construed in
different ways, he argues, each asking for a different answer:
1. Why did Adam (and not someone else) eat the apple?
2. Why did Adam eat (instead of throw away) the apple?
3. Why did Adam eat the apple (instead of something else)?
This reveals something that Bromberger’s analysis was not fine-grained enough to capture.
When spelling out the question with emphasis one can see that there is a different contrast class
involved in each formulation, which I have taken the liberty of adding in parentheses. For van
Fraassen, this points to a serious problem in explanation which is that context determines
relevance. An explanation is then only an explanation in context and not independently. Even if
one has the information needed to answer a question, “that information is, in and by itself, not an
explanation: just as a person cannot be said to be older, or a neighbor, except in relation to
others” (p.130). Van Fraassen argues that an explanation is always given in context and a
question is always asked with respect to a contrast class. As such, explanation is not the same
kind of thing as description, but is rather a three term relation, between fact, theory, and context.
For van Fraassen, the right answer to an explanation-seeking why-question, cannot be
determined without context.
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Another related view is given by Peter Achinstein (1984). Like Bromberger he places a
strong emphasis on linguistic analysis. He goes further into what he means by understanding and
offers an account of what kind of questions explanations answer. He first notes that an
explanation can be both a product and a process. It is, in Austin’s terms, a performative word
(Austin, 1955). Achinstein relies on a prior evidential relation between explanans and
explanandum for explanations to be true. So rather than only the explanans being required to be
true, the relation between the explanans and the explanandum must also be empirically true. This
is intended to be able to avoid problems of relevance that vex the D-N account (4.2.2). For
Achinstein, then, what counts as an explanation depends on the mechanisms in the world, and is
not determinable a priori. He also introduces the idea of a set of instructions for explanations that
specify relations to listeners. His view inevitably relies on empirical facts about psychological
conditions and senses of intellectual satisfaction, and because of this he finds no possible set of
instructions that satisfy specifying scientific explanations.
Robert Brandom champions a kind of inferentialism about language, but also about
explanation (Brandom, 1994, 2007). Broadly, it is the position that holds that inferences establish
the meaning of expressions. This is contrary to denotationalism, which holds denotative
meanings as primary. In terms of explanation, the conclusions one can draw is influenced not by
deductive logic, but by the available shared evidence, shared knowledge, and the intended
purposes. This is because what counts as reasons is context-dependent in just these ways. An
explanation can render the explanandum more credible, by either increasing its plausibility
(producing a warrant) or fruitfulness (increase ratio of successful to unsuccessful inferences, like
explanations by unification). This view has been taken up and expanded upon in recent literature
by others (Donato Rodríguez & Zamora Bonilla, 2009; Reiss, 2012).
Accounts such as these that focus on language and communication are not meant to be
accounts of specifically scientific explanations. And I believe this is a major reason why context
is seen to play such an important role. The situations in which one has only a why-question to
answer are different than those where one has a well-defined explanandum phenomenon and a
history and community of accepted explanatory practices and related previously explained
phenomena. In many ways, when one is looking to provide a scientific explanation, one already
has some knowledge of the context and an idea as to what will count as a good explanation; it is
not hopelessly contextual.
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This discussion about pragmatics serves to illustrate the importance of disambiguating
why-questions for providing a successful explanation. I think the important lessons from van
Fraassen and others can be translated into lessons about explananda. A why-question is not
exactly the same as an explanandum, but it is related. It is more productive to recognize an
ambiguous why-question, like “why did Adam eat the apple?” and either ask for more
information, or admit that there is not enough information to provide a good explanation. If one
is sensitive enough to what makes a good and bad explanandum then much can be gained from
an analysis of explanation that does not entirely rely on the pragmatics and context. I think it is
very unlikely that one can reduce issues of scientific explanation to issues of communication and
pragmatics, but the latter can inform the former.
To see how this can work, consider the well-known case of Mayor John who develops
paresis (Scriven, 1959; Carroll, 1999). This example is intended as a counterexample to
Hempel’s account of statistical explanation, which will be reviewed in 4.2.1. Knowledge at the
time the example was put forth suggested that one can only develop this by having untreated
syphilis. Even if one has untreated syphilis it is still quite unlikely that one will develop paresis.
So when asked “why did the mayor come down with paresis?” a reasonable answer seems to be
that he had untreated syphilis. What this really is is a how-possible explanation with a poorly
worded why-question. The explanation is answering the question of how it was even possible
that the mayor developed paresis. In this case, it is an adequate explanation. However, if the
explanandum phenomenon were different and one was asked “why, given that the mayor had
untreated syphilis did he come down with paresis even though others who have the same do
not?” then this no longer serves as a good explanation. There simply is not enough information
provided to adequately answer the question. The same can be said for explanations of why
someone develops lung cancer given that they smoke. It is important to recognize that why-
questions can be ambiguous, and that only having an unambiguous why-question allows for a
satisfying analysis of explanation. My interpretation of this example is a sign of my commitment
to the deductive nature of explanation, which is expanded on in Chapters 4 and 5.
This raises a similar point about the answerability of explananda: complex explananda
may not support explanations. If one’s explanandum phenomenon is “the behaviour” of a large,
complex system, then it is not always clear what will count as a satisfactory explanation. If one’s
goal is to accurately represent the interactions of a certain class of micro-constituents, then a
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representational model, or set of models, would suffice. If one wants to capture only the macro-
level behaviour of the system, then a tractable, non-representing model will be preferable. But it
is not fair to assess the success of an explanation irrespective of explanatory goals. One cannot
demand an explanation without clear expectations of what can count as a successful explanation.
Another important trait of explananda is that they are statements that are true or
approximately true of a real world system. This is an idea I return to in examining explanatory
derivations in 5.2. The truth or approximate truth of the explanandum can differentiate a good
from a bad explanation. If a robust scientific model makes a great deal of predictions about the
behaviour of a system, but is wildly inaccurate, then no matter what kind of restrictions are
placed on the model, it cannot be explanatory of the behaviour. A model needs to make fairly
accurate predictions in order to be said to support explanations. It is difficult to understand what
kind of explanation would result from a false or impossible explanandum, such as “why does the
Sun revolve around the Earth?” It may seem obvious but it bears remarking that if the
explanandum is false and not even approximately true, then there is no explanation because there
is no actual phenomenon or pattern to explain. Some explanations concern the behaviours of
models themselves. In such cases, the explananda are descriptions of the model and not a target
system, but what I have said still applies.
Goals of an Account of Explanation
In order to judge how successful an account of explanation is, it is important to establish what
the goals of an account should be. Outside of contributing to understanding, which was discussed
above, there are a few things that many have argued should be goals of an account of
explanation. There are other things that are not seen as necessary for explanation, but are
desirable. This section will speak to some of these.
1.4.1. Descriptivism and Normativity
One mark of a successful account of explanation is that it is informed by and largely agrees with
the consensus of the scientific community regarding explanation. It would be a failing for an
account of explanation to have vastly different judgments about explanation than scientists do. If
it did, there would not be much sense in calling it explanation, rather than an assessment of some
other scientific value. This means that good models from successful theories ought to be counted
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as explanatory on a good account, but models from false theories or data models should be
debarred.
There seems to be two kinds of approaches one can take towards this aspect of
explanation: descriptive and normative. The obvious problem with descriptivism is that the
community could be mistaken and there are no independent assessments of the community’s
claims. There is a further concern that the community cannot be rigorously defined as a set and
surely not all scientists in a field agree. It is important not to rely exclusively on claims by
scientists about whether or not something is explanatory. Scientists are not always concerned
with providing rigorous explanatory derivations that we like to see in contemporary philosophy
of science. And of course, an account of explanation need not reflect every claim from scientists
who claim to explain some behaviour or phenomenon. I think that an account of explanation
should not simply agree with the judgments of scientists. To have an account of explanation that
says that what is explanatory is exactly what a community of scientists says is explanatory, is to
have no account of explanation at all; it essentially leaves the notion unanalyzed. This is
something I will come back to in 2.4.3 and 3.3.2. It is of course important to have an idea what
this consensus is, but it does not provide any independent reasons for claiming that something is
explanatory.
On the other hand, a normative account attempts to articulate independent standards
according to which valid explanatory judgments can be made. The main worry with this is that
this can be descriptively inadequate and epistemically suspect if it is only accountable to its own
independent evaluation. This becomes something of a balancing act as it is important to reflect
the practice of scientific explanation, but to not simply have a descriptive account. There is an
aspect of normativity involved in accounts of explanation. Perhaps a good approach is to attempt
to strike a balance; to have an account that captures the best explanatory practices, or to be
“partially revisionary” as Woodward calls it (3.3.2). This way one can have well-founded
reasons for excluding some claims about explanation, such as that they do not meet the necessary
criteria that are all met by prime and obvious examples of successful explanation.
This continuity with explanatory judgments also runs in the other dimension, in that an
account of scientific explanation ought to be continuous with an everyday sense of explanation.
It would be unreasonable to accept as successful an account of scientific explanation that made
no mention of knowledge or understanding, or which ran contrary to all our shared judgments,
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insofar as we have them, about what explains what. I take from this is the idea that it is
legitimate and reasonable to rely on our judgments about explanation as guides, but not to solely
rely on them. It is also important not to automatically take conclusions about everyday
explanations as also applying to scientific explanations. There is no need to attempt to cover all
kinds of explanation in a single account. This can be done while also saying more than simply
that explanation is contextually determined. Independent criteria can and should be established.
Some might say that a significant marker of the success of an account of explanation is
that it can make sense of our past judgments about explanations. This is not normally seen as a
necessary goal of explanation, but it is perhaps a desideratum. One can be more confident in the
success of an account of explanation if it can also give insight into why past theories were
preferred over others. This kind of independent assessment of explanatory merit and theory
choice has of course been strongly problematized by Thomas Kuhn and others (Kuhn, 1962).
However, if an account of explanation can make sense of some past judgments about
explanation, then all the better.
1.4.2. Threshold and Depth
There are different methods of determining what counts as an explanation. Some accounts
construct a threshold to determine what is explanatory. This is often done by establishing a set of
necessary criteria for explanation, and allowing that any model (or theory, or argument,
depending on the account) that satisfies the criteria is considered explanatory. This method
makes room for a pluralist notion of explanation, where models from different levels of
investigation can all be considered explanatory if they meet the criteria.
Other accounts have proposed means of evaluating the relative explanatory merits of
competing explanations, by means of something like explanatory depth. Strevens, for instance,
proposes a measure of depth that can be used to determine which explanations are the deepest,
but also not necessarily the most fundamental (Strevens, 2008). Philip Kitcher’s unificationist
account has winner-take-all conceptions of explanation, in which only the most unified theory is
considered explanatory (Kitcher, 1981). This allows Kitcher to debar certain non-explanations,
but has drawbacks of its own, which will be covered in 4.3.2.
These two methods are not exclusive. In fact, Woodward offers an account of explanation
that makes use of both (Woodward, 2003; Woodward & Hitchcock, 2003b). There is a threshold
above which a model is explanatory, but there is a range of relative explanatory depth above that
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threshold. Having this hybrid approach helps to make sense of intuitions about explanation that
reasonably could find two models to be explanatory, but also show that one provides a better or
deeper explanation.
I do not think that having a method for comparing relative explanatory merit is necessary.
If an account is capable of supporting correct judgments about what counts as a genuine
explanation, then it ought to be considered successful. There is little reason to think that an
account that successfully characterizes explanation needs to also double as an account of
explanatory depth or explanatory power. Similarly, it seems reasonable to think that an account
of explanatory power or depth can be given independently of an account of explanation.
However, if an account has only a method of comparing relative explanatory merit, then it is
incapable of supporting claims that a model explains a certain phenomenon. Rather, it could only
support claims that a model explains a certain phenomenon better than another model does. This
is far less satisfying. The other obvious problem with this approach alone is that it will count all
models to be explanatory to some degree, which runs contrary to out intuitions about
explanation, and is likely just not true. This will be expanded on in 2.4.2.
It seems that a hybrid account is the most ambitious and if successful, would achieve
more. In the account I propose in Chapter 5, I argue that there should be a high threshold for
explanation. This will highlight the best explanatory practices and will establish normative
claims as to what a good explanation ought to do. I also allow that multiple models can be
explanatory and offer some suggestions as to how one could determine the relative explanatory
merit among those models that satisfy the necessary criteria.
1.4.3. Scope
The goals of what an account of explanation ought to accomplish have been growing steadily
since this was first analyzed by Hempel and Oppenheim (1965a). When Aspects of Scientific
Explanation was written, there was an effort to focus on a tractable project that could be
successfully analyzed. When the focus remains small, a sizeable contribution can be made to our
understanding of how science explains particular phenomena and generalities. The focus had
changed for Kitcher in Explanatory Unification (1981). What Kitcher undertook was an analysis
of the explanatory power of theories. He provided a formal system to capture the intuitive idea
that a global theory of science is explanatory to the degree that it unifies scientific knowledge. It
seems that problems arising from Kitcher’s account are due at least in part to the fact that the
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goal he set for his account of explanation was very unlikely to succeed. The idea that a single
dimension can reflect what makes a theory explanatory across all scientific theories is
unfounded, and it is also not a necessary component of an account of explanation (Wayne, 2016).
There are good reasons to think that there is unlikely to be a completely general account of
explanation. Diez, Khalifa, and Leuridan have argued against Nickel’s stance that there can be
domain invariant constraints on explanation (Nickel, 2010). They argue that “the current
emphasis on domain-specific explanations seems to be justified, and philosophers interested in
explanation should feel little pressure to seek some underlying unity in explanations across
domains as disparate as physics and ethics” (Díez, Khalifa, & Leuridan, 2013, p. 395). I will
return to this in a discussion of contextual nature of explanation and explanatory judgments in
5.6 and 5.7.
It is important not to demand that an account of explanation do everything and cover all
domains. In particular, one thing the account I propose in Chapter 5 will not aim to do is to
capture everything that can be considered an explanation, such as everyday explanations,
teleological explanations, and explanations featuring motivations and intentions. It is an explicit
part of the strategy to focus on a tractable subtopic of explanation, viz. to investigate how certain
scientific models explain their target systems.
1.4.4. Accurate Representation
It is reasonable to consider a central goal of explanation to be the accurate representation of a
target system. This can be accomplished either by revealing its mechanism, capturing its causal
dependencies, or its structural relations. I will go over the benefits and limits of representational
accounts of explanation in 5.6. By contrast, the account presented in Chapter 5 does not require
that the model contain accurate representations of the target system. However, unlike the
statements in the explanans, the explanandum is a description of the phenomenon to be
explained. As such, the model is constrained in that the explanans must deductively entail
explanandum. I will expand on this in 5.7. This openness regarding the features of the model
allows for explanations to make use of models that do not accurately represent, even
approximately. The reason behind this is not simply to include more as explanatory, but more
accurately reflect what is considered explanatory in science, and to less problematically capture
how it is that those models explain. The increasing scope of goals of an account of explanation is
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a problem in a different sense, but expanding the scope of explanatory idealization will help to
match an account of explanation to the explanatory practices of science.
Conclusion
There is much to say about the relation between explanation and understanding, but an account
of explanation need not fully analyze this. What is important is to draw a connection between a
good scientific explanation and an unanalyzed notion of understanding that it grants. There is a
subjective and psychological side to understanding that is not helpful in explicating explanation,
and so it is best to take on a tractable project and focus on one or the other. It is important for an
account of explanation to take pragmatic considerations into account, but pragmatic concerns do
not exhaust what can be said about scientific explanation. If an account is sensitive to the
concreteness and specificity of the explanans, then many problematic cases of explanation can be
dealt with.
I have outlined what I believe are reasonable goals for an account of explanation: an
account should have a high threshold and largely reflect the explanatory judgments of a scientific
community; it may have a measure of depth to further capture our judgments about which
models explain better or deeper; it should not be so broad as to be intractable, but should be
broad enough to capture a wide range of explanations accepted in practice.
At present, the literature on explanation is heavily focused on causal approaches.
Woodward’s account is very appealing to many. It lays out both a threshold for explanation and
allows for a range of explanations above that. It is well-motivated by manipulationist ideas about
how we understand causes, and what it means to explain something. Others, like Strevens, have
also favoured causal accounts, but of a different kind. Strevens is concerned with generality and
higher-level explanations. He develops his account in part to capture idealized models and
provide a measure of explanatory depth that does not favour explanations at the most
fundamental level. In response to these accounts and others, some like Bokulich, have proposed
structural approaches to explanation in order to capture the central role that idealization plays in
explanation and to allow for a wider range of explanatory models. The following chapter
examines the role of idealization in model-based explanations and assesses the success of
Bokulich’s structural account in capturing the way that highly-idealized models explain.
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Chapter 2. Idealization and Structural Accounts of Explanation
Introduction
This chapter will first introduce the concept of idealization and the inevitable role it plays in the
practice of modelling in terms of realism and representation. I use this discussion of idealization
also as a means of introducing some concerns that have been raised about distinguishing laws
from accidental generalizations. These concerns have been used to problematize covering law
accounts of explanation. But I think that these potential problems can be circumvented with a
model-based account. I then present McMullin’s distinction between idealized models, which
approximately represent, and highly-idealized models, which do not. I follow others in
maintaining that highly-idealized models can be genuinely explanatory, and part of the
motivation of my account is to capture this. Section 2.3 focuses on idealization in the practice of
modelling and the different aims involved in model construction. Modelling, as a pragmatic and
goal-directed practice, has implications for the use of idealizations in explanation.
After introducing idealization, modelling, and highly-idealized models, I turn to examine
an account of explanation that is sensitive to the issues these raise for explanation. In 2.4, I
present Bokulich’s account of structural model explanations that is designed to capture the way
that highly-idealized models explain. On her account, some models of semiclassical mechanics
are explanatory because they capture the structural dynamics of quantum systems. However, I
argue that structure is incapable of capturing the explanatory role of highly-idealized models,
because it either counts all models as explanatory, or shows preference for models that
accurately represent (M. King, 2015). In either case, I aim to show that a structural approach to
explanation cannot capture the way that highly-idealized models explain. The problem is that her
account, even though structural, is still representational, and so the better the representation, the
deeper the explanation. This is not in itself problematic, but it does fail in the aim of showing
how models of semiclassical mechanics and other highly-idealized models are explanatory. This
leaves that task open to a non-representative account, like the one I present in Chapter 5.
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Models and Idealization
The focus in the literature on scientific explanation in philosophy has shifted in recent years
towards model-based approaches. The idea that there are simple and true laws of nature has met
with considerable objections from philosophers such as Nancy Cartwright (1983), Paul Teller
(2001), and numerous others. This has made a strictly Hempelian D-N-style explanation largely
irrelevant to the explanatory practices of science (Hempel & Oppenheim, 1948). The practice of
explanation does not merely involve subsuming particular events under laws of nature. It is
increasingly recognized that science across the disciplines is to some degree a patchwork of
scientific models, with different methods, strategies, and with varying degrees of successful
prediction and explanation. And so accounts of scientific explanation have reflected this change
of perspective and model-based approaches have flourished in the explanation literature
(Woodward, 2003; Craver, 2006; Bokulich, 2008).
To talk about idealization, it is important to be clear on what is here meant by a model.
Paul Teller has argued in favour of a broad conception of a model (Teller, 2001). It is impossible,
he argues, to give necessary conditions for a particular representation to be considered a model.
A model is a model because it is chosen to represent something, and there simply are no intrinsic
features of a model. However, I think it is useful to consider at least some characteristics of
scientific models, even though there may be no general criteria for the similarity that a model
must bear to the system it represents. The models of interest to me (ones that feature in scientific
explanations) are not merely sets of propositions, or accurate representations of a system or
phenomenon, but quantitative and capable of reliably reproducing or predicting the desired
behaviour. Models are the result of a scientific modelling processes, involving experimentation,
measurement, testing against prediction, calculation, idealization, and often theory. There are
important explanatory differences between the various kinds of models used in science. A
narrower view of what constitutes a scientific model is, I think, advantageous.
A model may be either concrete as in the case of a physical object, like a model ship, or a
map, or it may be abstract like a set of statements, diagrams, and mathematical equations and
parameters. A single equation on its own is not enough to constitute a scientific model in the
sense employed here, though the term model is often employed this way. Historically, some
philosophers have looked at models as linguistic objects; as sets of statements. Recently others
have taken models to be artefacts, or tools of surrogative reasoning. Zamora-Bonilla and Donato-
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Rodriguez follow Knuuttila in this approach (Knuuttila, 2005; Donato Rodríguez & Zamora
Bonilla, 2009). On their view, models are not only sources of knowledge about the world, but are
used for the inferences drawn from the systems they are about. I return to this idea later in the
section.
The models of interest to me in talking about scientific explanation are abstract objects
used to gain knowledge about the world. A model is an idealized version of a real-world system
(often called a target system) designed with a purpose; to bring to light some relation or capture
some behaviour. While it is not a perfectly accurate version of the real world, it is related to it in
various ways. This relation is often given in terms of approximation, abstraction, similarity, or
isomorphism. What this process of idealization consists in and how idealized models relate to the
real world is a matter of no small debate and is the focus of this section.
2.2.1. What is idealization?
Idealization in the context of modelling is the process of approximating a system in order to
facilitate calculation, to bring a certain relation to light, or to render it easier to understand.
Models are not only sources of knowledge about the world, but are used for the inferences drawn
regarding the systems they represent. Zamora-Bonilla and Donato-Rodriguez take an inferential
approach to models and explanation that I will not pursue very closely, however the process they
describe is largely representative of a variety of views (Donato Rodríguez & Zamora Bonilla,
2009). They separate the process they refer to as surrogative reasoning into three steps:
1) Interpret the physical system and make inferences about the model from propositions
about the empirical system.
2) Make formal inferences within the model, either deductive, inductive, counterfactual, etc.
3) Retranslate (interpret) the conclusions into language about the empirical system.
For them, the steps are successful if the inferences made are valid and what determines this is the
contextual standards of a discipline or community of competent language users, a notion they
take from Brandom and the inferentialist tradition (Brandom, 1994, 2007). They argue that there
is no algorithm for constructed models, not even in theory. They will always involve background
knowledge, analogy, intuition, metaphors, empirical facts, etc.
Idealizations are false assumptions about the target system, such as that there is no air
resistance, the gas has an infinite number of molecules, the carrying capacity is constant, that an
object is a perfect sphere, or a point particle, or that liquid in a pipe exhibits perfectly laminar
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flow, and so on. They can be very helpful in providing mathematical representations of systems
that might otherwise be hopelessly intractable. This process might involve leaving details out of
a model in order to simplify it, or distorting existing data to eliminate noise that has no effect on
bringing about the phenomenon; a process often referred to as abstraction. In many cases, the
behaviour or phenomenon might be obscured without the idealization. But idealizations, even
though false, are not simply applicable to ideal worlds; they can generate knowledge about the
real world. Ernan McMullin, who wrote an important piece on idealization, reminds us that it is
not an “escape from the intractable irregularity of the real world into the intelligible order of the
Form, but to make use of this order in an attempt to grasp the real world from which the
idealization takes its origin,” (McMullin, 1985). Models are idealized in order to have real-world
application.
McMullin reaches back as far as Plato and Aristotle to make the point that the truth of the
mathematical representation of nature has always been an issue of debate. For Aristotle,
mathematics was a reliable mode of analysis of nature and not merely a study of a realm outside
of nature. One could be justified in using mathematics to study the quantitative aspects of the
real world. Plato on the other hand had argued that matter made a proper or complete realization
of geometry impossible.
When Galileo performed his experiments, he made assumptions that were false about
elements of the model. He assumed that a ball he was rolling was a perfect sphere, and that a
plane was perfectly flat, or exactly inclined. But he was not making claims only about the
behaviour of perfect spheres. In doing so, and applying the mathematical relations he derived to
the real world, he was in essence arguing that the difference between the perfect sphere of the
model and the ball he was using was negligible to the behaviour he was modelling. Of course,
when the difference is not sufficiently small the predictions of the model will begin to diverge
from actual measurements. McMullin also points out that models in contemporary science are far
more complex than justifying the application of a simplified geometrical scenario to a system
that closely resembles it. In the Galilean idealization that McMullin is characterizing, one can, at
least in principle, retrieve the original real-world system by de-idealizing the model; by adding
detail back into the idealized model one can generate a model arbitrarily close to the actual
system. These kinds of idealization are known as Galilean. This de-idealization helps to
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demonstrate why the model, though idealized and not literally true of the world, applies in a
particular case. In order to see this, we will turn to the ideal gas law.
2.2.1.1. The Ideal Gas Law
In 1738, Bernoulli, following Boyle and others, had supposed that gases are composed of an
enormous number of very rapidly moving particles. The compressibility of gases and its relation
to the pressure a gas exerts on a vessel, led him to theorize that pressure was the result of the
particles’ collisions with the vessel walls in a mostly empty space. Boyle’s picture of gases as
particles that act as springs exerting pressure on each other was widely accepted in Bernoulli’s
day and Boyle’s equation,
(1) 𝑃 = 𝐹(𝑇)
𝑉,
for relating pressure, volume, and temperature, where F(T) is a function on temperature and the
amount of gas, was also known. But Bernoulli’s idea that pressure is a result of collisions was
new, and allowed important progress in the development of the kinetic theory of gases.
This idea easily leads to Boyle’s law. If the average speed of the particles remains
constant, then the number of collisions on the container walls would be proportional to the
density of the gas, and so pressure would be inversely proportional to volume: 𝑃 ∝ 1 𝑉⁄ . The
speed of the particles was assumed to increase with an increase in temperature, and so Boyle’s
law falls out of Bernoulli’s kinetic picture. But he was also able to add that an increase in the
number of particles would increase the number of collisions and so a constant was multiplied to
the temperature function to give
(2) 𝑃𝑉 = 𝑛𝑓(𝑇),
where the pressure P and volume V are equal to the mols of the gas multiplied by a function of
the temperature, f(T). An equation of state one immediately recognizes as being just a few steps
short of the ideal gas law, 𝑃𝑉 = 𝑛𝑅𝑇.
The kinetic theory of gases presents a picture of a fictitious gas in which the particles are
thinly dispersed point-masses that do not interact with one another and have elastic collisions
with vessel walls, which is measured as pressure. We know that this is quite false of any actual
gas. Gas molecules take up space, some quite a lot, and certainly interact with one another and
26
dissipate energy on vessel walls. But the idealization is still applicable to the real world. Within a
certain range of temperature and pressure, the ideal gas law is capable of providing results
accurate to those experimentally measured. The reason this only works well within a certain
range is due to the nature of the assumptions. For instance, the assumption that the particles are
point masses begins to be problematic when the molecular size becomes significant relative to
intermolecular distances, and real gases will begin to condense at low temperatures.
The idealizations involved in this model are just the Galilean idealizations mentioned by
McMullin. What this means is that details of the real gas can be built back into ideal model, and,
in a sense, one can retrieve the real-world system via de-idealization. In fact, collisions were
introduced to the kinetic theory of gases by Maxwell and Boltzman, by the use of the concept of
a statistical average. The kinetic theory of gases and its idealizations are a prime example of an
idealized model, and one can easily see that it is not a special case.
Many arguments have been made regarding the unavoidability of idealization and non-
optimal models given facts about complexity and irreducibility, and the growing acceptance of
the impossibility of completely accurately representing most real-world systems. But not all
idealizations are of the Galilean sort. Non-Galilean idealizations are incapable of being de-
idealized in this way. A model featuring non-Galilean idealizations is often called a highly-
idealized model. Robert Batterman has described these models as having “controllable”
idealizations, in that the idealizations of the system are justified theoretically (Batterman, 2005,
p. 235). Bokulich and others, such as Batterman, argue that these models can be genuinely
explanatory and offer accounts of explanation to capture how. I review such an attempt by
looking at Bokulich’s case study of semiclassical mechanics in 2.4.
2.2.2. Laws and Idealized Models
There are many reasons for maintaining that idealized models are capable of supporting
explanations, even though they do not contain laws of nature. As mentioned, idealized models
make use of false assumptions, but they are essential for scientific explanation. The D-N account
requires that the explanandum be a logical consequence of the explanans and contain a law of
nature. However, fundamental laws of nature, as Nancy Cartwright points out, are very rare
indeed (1983). And further, in some cases the fundamental theory is incapable of generating the
best explanation, as is argued by Sandra Mitchell (2003). This subsection will outline these
arguments for the need for idealized models, and the following subsection (2.2.3) will go over
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arguments concerning not just the need for idealization and high-level, or non-fundamental,
models, but for the essential role of idealization in explanation.
2.2.2.1. Laws and Lies
Cartwright’s arguments regarding the status of laws of nature are well-rehearsed in the
philosophy of science, but her argument bears review here. This discussion will not support
Cartwright’s own conclusions that laws are false and non-explanatory, but rather that idealization
and compromise are integral parts of the practice of modelling and also the explanations in
which the models feature.
In How the Laws of Physics Lie, Cartwright argues that the entrenched, or traditional,
view that the laws of nature state facts is mistaken (1983). She calls this view the facticity view
of laws. She cites J.J.C. Smart as saying that biology has no genuine laws, but in fact, she argues,
neither does physics. Cartwright is a kind of anti-realist towards laws, but not in a
straightforward manner. Cartwright finds explanation and truth to be at odds everywhere. Her
argument revolves around the point that even laws of nature cannot accurately describe the
behaviour of systems to which they are applicable, because the application of each law assumes
that there are no other forces at play. Forces are ubiquitous in nature and so the actual movement
of two bodies is never fully given by laws of mechanics, for instance.
If one considers the universal law of gravitation, one will find that it never fully describes
the total force between two bodies. For charged bodies, this law and Coulomb’s law interact to
give the final resultant force. But Coulomb’s law does not work for massive bodies, and at very
small scales the electric charge far out-performs the gravitational force on massive bodies. So of
course the laws are to be considered ceteris paribus: if there are no other forces than gravity
acting on two bodies, they would be fully describable by the universal law of gravitation alone
(Hüttemann, 2014). Cartwright claims that this is no help in real-world scenarios, because the
antecedent is never satisfied. Curious, given that it is obviously well-known that this law and
other similar laws are in fact incredibly useful in real-world scenarios. When the other forces
acting on bodies are small, the law can accurately describe the behaviour of simple systems. One
can know why and when the law is reliable by looking at the assumptions involved: that the
masses are point particles, that other forces are negligible to the behaviour, and so on. When
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these idealizations become very far from the truth, the regularities cease to hold and explanations
fail.
Much of Cartwright’s argument relies on the notion that an explanation is given in
subsumption under covering-laws. If one holds to a strict covering-law account of explanation
where all phenomena neatly fall out of the laws of nature, then her argument poses more
significant problems. Covering laws are rare in scientific explanation. As was mentioned, in
recent literature the focus has shifted from law-based explanation to model-based explanations.
The fact that the direct and simple application of laws of nature is perhaps never realized, serves
as a reminder that the models built in science are not exact and universally applicable, but are
rather designed with aims that make a compromise with other desiderata.
2.2.2.2. Complexity
Sandra Mitchell has argued in favour of scientific pluralism, largely on the grounds that the
world is simply too complex to be represented at a single level of study (2003). She argues for a
particular kind of pluralism, in which multiple levels of models can be integrated to form a single
explanation of a phenomenon: what she calls integrative pluralism. In this section, only her
arguments from complexity will be examined, as they make a strong case for the explanatory
power of idealized and non-fundamental, or higher-level, models.
Mitchell argues that complexity is a critical tool in understanding the nature and limits of
scientific diversity (Mitchell, 2003). Attention to different components and different degrees of
abstraction are appropriate to our various explanatory goals and conceptual and computational
abilities. An examination of complex systems, Mitchell argues, can inform us about the limits
scientific reduction and explanation. By the term complex system, she means a system in which
the micro-details are so many, or take place on such a long time scale, that the computation of
the evolution of the system becomes intractable. Mitchell distinguishes between three kinds of
complexity: a) constitutive complexity, where the sheer number and organization of the system’s
components demand a higher-level model. She has in mind here the case of eusocial insect
colony organization, where models of the lower-level interactions are insufficient to bring about
the behaviour observed at the colony level. In general, the problem is that the observed macro-
level properties of some systems cannot be determined from models of, and what is known
about, micro-level interactions; b) dynamic complexity, where the time scale and number of
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interactions of the system’s components preclude full calculation. This is a familiar kind of
complexity featured in chaotic systems and A-Life simulations. Even simple deterministic
systems, left long enough, can become chaotic and unpredictable; and c) evolved diversity which
is a historical indeterminacy brought by the fact that one cannot know the particular history that
brought about the state of a complex system. Mitchell’s example here is the evolutionary history
of species. There is not enough information to determine the causal history of evolution. There is
no telling why certain traits were selected and others that could have succeeded as well or better
were not.
For Mitchell, all this goes to show that biology and the sciences in general ought to
represent the complexity and diversity of its subject matter by abandoning the idea of laws of
nature and recognizing the pragmatics of scientific modelling. Instead of merely claiming that
there are no laws, or that they are inapplicable in real scenarios, she plots all laws in a 3-
dimensional graph, measuring stability, strength, and degree of abstraction (Mitchell, 2000). The
values of scientific law are continuous and not binary. Mitchell makes this claim about laws as a
way to support the legitimacy of biological generalizations by placing on a spectrum,
generalizations such as the law of the conservation of mass-energy, Mendel’s laws of
inheritance, and the metallic composition of the coins in Nelson Goodman’s pockets. In earlier
work, she argued that the pragmatic function of the law was more important than whether it was
exceptionless (Mitchell, 1997). How and why the law applies under which circumstances is what
is important.
The search for quantitative generalizations in biology is not fruitless, but at the same
time, we must be cautious about claiming that all of these are laws of nature merely separated by
degrees. Mitchell has argued cogently that the binary view of laws as either contingent or
universal is either mistaken or unuseful, and that the generalizations employed in science are
often quite contingent and specialized. Scientific models feature just these generalizations, with
degrees of stability, strength, and abstraction. They are not universal, but designed for specific
purposes, limited in application, and restricted by tractability and complexity. Mitchell is right
that these generalizations are not perfect laws of nature.
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2.2.2.3. Perfect Models
McMullin’s discussion of Galilean idealization involves the assumption that one can actually get
arbitrarily close to the real world by de-idealizing models. Consider an idealized model like the
simple pendulum, whose motion is partially described by the following equation,
(3) 𝑇 ∝ √𝐿
𝑔,
where T is the time, L the length of the string, and g the acceleration due to gravity. This
equation gets at the relation between the length of the string and period of oscillation (under
certain conditions), specifically by ignoring details like the damping effect of air resistance, the
mass of the string, elasticity, the minute change in gravity as the pendulum swings up and down,
and so on. It is an ideal pendulum that accurately describes no pendulum in the world. One can
take this model, or rather its modern version, and add corrections to account for the presence of
air, the mass of the string, shape of the bob, and so on. One can build detail back in and get
closer and closer to the real world pendulum, i.e. to a perfect model of the pendulum.
However, the idea that there even is a perfect model, toward which any scientific model
aims, has been criticized. Paul Teller has cogently argued in “Twilight of the Perfect Model
Model” that there is little reason to think that models could ever be perfect (Teller, 2001). The
main thrust of his argument comes from concerns about providing a general account of similarity
and approximate truth. The problem for many anti-realists is “to identify the relevant similarity
between situations, on the one hand the actual situation, and on the other some non-actual
idealized simplification of the way the world really is, what is being called a model” (Teller,
2001, pp. 403-404). The problem only arises when the idealized situation is considered in
linguistic terms. When theories are seen as a collection of models and the relevant interests of
similarity are fixed, then one can see that there is a sufficiently similar structure. The demand of
the anti-realist for a definition of closeness to truth simpliciter is misguided. Whether a model
exhibits similar behaviour to the system will depend on such things as the details of situation and
the aims of the modeller. The similarity between the model and the real world is largely
dependent on what feature or features of the system the scientist is aiming to model. There are no
general criteria for a perfect model; models are compromised from perfection and designed with
aims in mind. The aims of modelling will be looked at in more detail in section 2.3.
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2.2.3. Idealization and Explanation
Idealization is unavoidable in science, but it is not merely unavoidable: it is an important aspect
of modelling and explanation. This subsection tries to highlight that idealization not just a
necessity that must be dealt with, but integral to what an explanation is. The organized,
simplified, and related information that idealized models provide is key to the model’s being
explanatory. Its importance is that it helps us understand by highlighting or capturing salient
features of systems, relative to our explanatory interests. The remainder of this section will be
spent showing that many of the most successful accounts of explanation attempt to get at higher-
level, and highly-idealized, explanations, but will not go into detail on how. Details about their
attempts and their success will have to wait until chapter 3.
2.2.3.1. Causal Processes
James Woodward criticises certain causal-mechanical approaches to explanation, such as
Salmon’s, that identify cause in physical processes and interactions (Woodward, 1989).
Salmon’s causal mechanical account of explanation states that explanation involves showing
how an occurrence fits into the causal network of the world (Salmon, 1984). For Salmon,
causation has three aspects: causal interactions, causal processes, and conjunctive forks. Causal
processes are characterized by the transmission of a mark or a structure, such as the movement of
a free particle. A causal interaction is when one causal process intersects and modifies another,
such as the collision of two particles. Conjunctive forks are correlations among spatio-temporally
separated effects, explained by separate causal processes deriving from a common cause.
Salmon’s conception of causation seems to fit best with simple physical systems governed by
classical mechanics, but complications arise when we move away from such systems. Woodward
points out that this would lead to denying the explanatory power of quantum mechanics and
other systems that make no appeal to underlying causal processes.
This also applies to any higher-level models. For example, when explaining the
expansion of a gas in a balloon when heated, one does not need to focus on the causal histories of
each molecule and say that the behaviour of the system is somehow the sum of these. Instead one
focuses on the general assumptions regarding the forces and then deriving and solving the
Maxwell-Boltzman equation for the system. One can then show the various behaviours of the
system that follow from this. An account of explanation that focuses on processes will inevitably
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fail to include higher-level and highly idealized models. Woodward’s argument is that even if
this were possible, it would not be explanatory. The actual causal history of the molecules gives
no explanatory information on why the balloon expanded when the gas was heated. The best
kind of explanation is a higher-level model relating properties such as temperature, volume, and
pressure.
Michael Strevens also recognizes the need for higher-level explanations (Strevens, 2008).
He aims to provide an account of explanation that begins with the total causal picture of the
world, and abstracts away irrelevant details until only difference making causes are left. His
account of explanation has two aspects: lower-level causes and higher-level relevancy criteria.
Strevens’ aim is to use an aspect of unificationism to solve one of the major problems of causal
accounts: specifying the causes that are relevant to an explanandum phenomenon. He is using
unification in order to pick out not the most unified of theories, but which causes are the
difference makers. By using this process he will be able to count higher-level models as
explanatory by linking them through a process of optimization to the world’s fundamental causes
(Strevens, 2004).
As an example of how a higher-level, and non-veridical, model can be more explanatory
than a more fundamental model, Strevens compares possible explanations of Boyle’s law. He
begins with a textbook explanation that makes use of many idealizations, such as that the
collisions with container walls are completely elastic, and that molecules do not collide with one
another. It gives a decent explanation, but makes some false assumptions. He next goes over the
complete description from modern kinetic theory, including the influence of the molecules on
one another at a distance, and intermolecular collisions. He then goes over the process of
eliminating the aspects of the complete description that do not make a difference to the Boylean
behaviour of the gas. It is only in this way that one can understand Boyle’s law: the relations that
are highlighted in that law are the difference makers for the explanandum. Thus, the textbook
explanation is the best. It is not the most veridical, but it has only difference makers. He
supposes that the assumption that there are no collisions is actually the way of communicating
that it makes no difference to the behaviour to suppose that there are none. Idealization, for
Strevens, shows what is irrelevant. Idealized models are preferable to veridical models in a few
ways: they highlight the irrelevance of certain factors; they are much simpler; they can remain
effective predictors as long as the idealizations are reasonably faithful.
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As was mentioned above, many have argued that non-Galilean idealizations featured in
highly-idealized models ought not to preclude explanation (Rueger, 2001; Batterman, 2002b;
Bokulich, 2008; Batterman, 2009; Wayne, 2011; Jansson, 2014a, 2014b; Reutlinger, 2014;
Saatsi, 2014; Wayne, 2014; Rice, 2015). The argument is often made on the basis of the
importance of these idealizations in the scientific practice of explanation. If highly-idealized
models are not explanatory, then scientists must be quite mistaken about explanation. It is widely
recognized that models featuring non-Galilean idealizations are pervasive and can support
explanation. These idealizations can range from two-dimensional agitators and lattice gas
automata to the perfect rationality of consumers and constant populations. There is a push for
philosophical accounts of explanation to handle this. Even though specific approaches that allow
for these are not without concerns. So far in this chapter we have focused on characterizing
idealized models in the context of an account of explanation. The following section will
elaborate on the aims of modelling mentioned earlier in order to preface the discussion of
Bokulich’s structural model-explanations.
Modelling and Explanation
This section aims to highlight the strong role that modelling plays in explanatory arguments.
Specifically, I look at the desiderata involved and show that they often pull in opposite
directions, and the model that one constructs is dependent on the aims of modeller, or
explanation seeker. This is to highlight the pragmatic and imperfect nature of models, and to
reinforce the idea that models can be constructed for specific explanatory purposes, which I
make use of in 5.4.
2.3.1. Desiderata
The aims of the modeller and the behaviour of interest in the system does much to determine
what information is included in the model. If precision and accuracy are the main desiderata for
the model, then maximizing details is a useful strategy. This is the kind of modelling that takes
place in highly detailed computer simulations of particular systems comprised of multiple
individual models. In biology, this can be used to model ecosystems while taking into account
models that reflect the changing effects of sunlight, temperature, air pollutants, water pollutants,
and various interacting species populations (Mitchell, 2003). Of course, maximizing details of a
34
particular system is likely to make it less generally applicable, and less tractable. And so for
instance, generality is compromised when maximizing realistic detail.
Sometimes, however, capturing the desired behaviour involves ignoring details. In order to
focus on a specific behaviour, the investigator might look at the system at a higher level of
inquiry. The term level has different meanings in various sciences. In biology, it might be used to
differentiate between the models of molecular biology, cellular biology, models of individual
behaviour, and models of group behaviour. In physics, it might mean the difference between the
study of a gas in fluid mechanics, and the same gas at a small enough scale that intermolecular
forces affect its behaviour. For instance, as Teller mentions, if one is interested in modelling
waves in water on the order of one meter in length or more, then one can safely ignore the effect
of surface tension (Teller, 2001). However, if one is looking for ripples only millimeters in
length, then surface tension will play a large role.
Physical scale alone is not the determiner of level either. Temporal scales can differ
enough to generate very different models, some effective at short time intervals, and some to
describe the long term behaviour. For instance, the equations governing the long-timescale
motion of an oscillator are quite different from those governing its short-timescale behaviour.
This is primarily because the model of the long-timescale behaviour does not exhibit the same
periodic motion. When the number of oscillations is small the model exhibits near-harmonic
oscillation, but when the number increases, the behaviour is dominated by the rate in the change
of amplitude (Wayne, 2012). The level at which the model is constructed is related to the
behaviour that is being modeled. Some models are designed to capture behaviour at scales or
values where the models of fundamental theories no longer apply. These higher-level models are
often said to capture the system’s emergent behaviour or properties, though when a behaviour or
property becomes properly emergent is a matter of no small debate (3.4).
But there are additional dimensions to modelling as well. Sometimes the aim of the
modeller is to generate a maximally simple model, or to capture the large-scale structure of quite
different systems. This model may retain very little of the system’s details, only enough to
generate the desired behaviour or to reveal a pattern it shares with different systems. This could
be favoured for its calculational tractability, or for revealing a bare-bones mechanism. Simple
models will lose out on accuracy, but many models could have the accuracy added back in, at
least in principle, as was seen in Section 2.2.1. Simple models have the added benefit of being
35
very general. Simple models that exhibit the same behaviour may apply to systems with radically
different components, and this is an interesting phenomenon in its own right, one looked at in
depth by Bob Batterman (Batterman, 2002b).
An example of such a minimal model might be the Lotka-Volterra model of predator-
prey interactions. This is the simplest model of predation. It can be written as the following pair
of equations describing the change in prey and predator populations:
(4) �̇� = 𝑏𝐻 − 𝑠𝐻𝑃; �̇� = −𝑑𝑃 + 𝑒𝑠𝐻𝑃,
where H and P are the prey and predator population, respectively, b is birthrate of prey, d the
deathrate of predators, s the searching efficiency, and e the efficiency with which extra food is
turned into predators. This model describes an ideal system with only two populations, and
ignores all realistic details about them. Not only this, but it makes clearly false assumptions
regarding the process of turning food into predators. This model as it stands does not describe
any particular population very accurately, but it can be refined with more details that are
particular to certain systems, and can then be used to approximate the near-cyclic fluctuations in
populations. This can be done by replacing exponential growth in the absence of predators with
two-term logistic growth expressions,
(5) �̇� = (𝑏𝐻 − 𝑟𝐻2) − 𝑠𝐻𝑃,
and the same can be done to represent alternative food sources,
(6) �̇� = (𝑘𝑃 − 𝑑𝑃2) + 𝑒𝑠𝐻𝑃.
A third population can be added to the food chain and the equation can be tailored in various
other ways. A model that favoured a more realistic representation or prediction of a particular
system would take more information into account, such as the competition between individuals
of the same species; the relation between predator consumption and prey density; and the
efficiency at which new food is turned into extra predators, and so on. The simplest way to take
some of this into account is to introduce carrying capacities for both species and a saturation
effect where the birthrate tends to a finite limit at high density. A more detailed model might
lend itself to a different interpretation of the observed cyclic behaviour: for instance, it could be a
result of a small habitat, one which is in the order of magnitude of the mean displacements of
individuals. Highly detailed and accurate models can be constructed, but this sometimes implies
36
that one must sacrifice the generality and tractability that comes with a simple model. (4) is very
minimal and very general, and attempts only to show the general behaviour of a large number of
possible predator-prey systems. Reasonably, even this most minimal model could support
explanations. It is reasonable to question whether this model can explain features of a population
that do not share its idealizations (Colyvan et al., 2009). It might not explain a particular
population’s changes very well, but it could be employed to explain the similarity of cyclic
population changes across very different systems. That is, it could reasonably be used to explain
why different systems exhibit the similar behaviours. Its aims and its explanatory potential are
very different than those of a model that attempts to maximize precision. It is not uncontentious
that this model ought to be considered explanatory, but I think it is reasonable that it could
explain certain explananda. I will return to this in 5.6.2.
Yet another strategy is to generate predictive accuracy, even at the cost of realistic
representation. These models may include what are called black boxes, which are mathematical
equations that are capable of approximating the desired quantitative result, but bear little
similarity to the system being modelled (Weisberg, 2007). These models can be very useful in
cases where high predictive accuracy is the main desiderata. Black boxes can stand in for
unknown mechanisms and processes and state only input and output. They are silent on the
causal or structural features of the system. Though Michael Strevens makes a case for certain
black boxes being explanatory when buttressed with the support of a framework, generally black
boxes are not considered explanatory (Strevens, 2008). Black boxes can be considered heuristic
devices that may simplify calculation or provide a numerical solution, with little to no
information about the process or mechanism at work.
As the previous sections showed, many models and strategies that are employed in
science are used only as heuristics. This means that the models are of practical use, but are not
intended to explain, or accurately represent the system. There are many models of particular
systems that do not aim to explain them. Some models focus on achieving high predictive
accuracy at the cost of realistic representation, employing black boxes. Some models are merely
calculational tools, whose use in the practice of science is entirely pragmatic, but others are
thought to actually explain the phenomena or system they are modelling. If one does not wish to
say that all models are explanatory, then one needs criteria for which models are going to count
as explanatory, and which are phenomenological, or merely predictively successful. I will return
37
to the relation between heuristics and explanation following a close examination of Bokulich’s
case study of semi-classical mechanics.
Idealization in Physics
This section will examine the potential for a structural model account to capture autonomy, the
idea that not the all good explanations are given from models of fundamental physics. This
section will look at Alisa Bokulich’s work on the central role that idealization has to play in
explanation (Bokulich, 2008, 2011, 2012). Bokulich claims that certain highly-idealized models
are explanatory, even though they are not considered explanatory by causal, mechanistic, or
covering law accounts of explanation. She calls these kinds of explanations structural model
explanations and argues that the structural similarity between the model and the target system
can play the role of determining whether or not that model is explanatory (Bokulich, 2008, p.
145). She formulates her account as structural in part to capture models that are not explanatory,
for instance, on Woodward’s manipulationist account (Woodward, 2003). She aims to expand
the store of explanatory models to include those that do not accurately represent - those that
model a physical system by means of fictitious entities or processes, what she calls explanatory
fictions.
This section examines Bokulich’s account to assess the success of her formulation of the
structural criterion, and also to make the more general claim that there are problems with
maintaining that a structural criterion can capture the way that highly-idealized models explain.
In order to argue this, I will first examine Bokulich’s account as given in (Bokulich, 2008, 2011,
2012) in 2.4.1. I then demonstrate the application of this account in explaining the phenomenon
of quantum wavefunction scarring, which is a pattern in probability distribution in certain
quantum systems. Bokulich argues that this quantum phenomenon is more deeply explained by
appropriating concepts and models from semiclassical periodic orbit theory, than it is by
employing quantum mechanical models. In order to evaluate this concern, I look at two
reasonable approaches to measuring or assessing a model’s structural information, and I argue
that neither approach is ultimately satisfactory. The simplest approach of just measuring
structure proves at worst impossible and at best arbitrary, and the comparative approach, while
succeeding in debarring the Ptolemaic explanation, fails to find the highly-idealized model
explanatory (M. King, 2015). The criterion either wrongly identifies all models as explanatory,
38
or prefers models from fundamental theory. Either way, it cannot capture the way that highly-
idealized models explain. Section 2.4.3, the non-structural aspects of the account are challenged
for their strong role in distinguishing explanatory from non-explanatory models. I close the
section by discussing heuristics and articulating what I believe to be the crux of the problem and
what can be taken away from the investigation.
2.4.1. Structural Model Explanations
This subsection examines Bokulich’s structural model account of explanation as laid out in
(Bokulich, 2008) and incorporates the responses and clarifications made in (Bokulich, 2011,
2012). Bokulich’s account of explanation relies in part on work done by James Woodward, in
particular the notion of depth he develops along with Christopher Hitchcock, which I go over
first (Woodward & Hitchcock, 2003a, 2003b). This notion is rather important for Bokulich as it
forms the basis of the structural criterion of her account. I then show how her structural account
aims to capture the explanatory value of semiclassical models, which are essentially classical
models that can be used to approximate quantum systems. I do this by looking at the
phenomenon of quantum wavefunction scarring and demonstrating how it satisfies the account’s
criteria. This shows both how a successful structural explanation is intended to proceed and
motivate some of the intuitions we have about highly-idealized models being explanatory.
On Woodward’s account, causality is framed in terms of manipulability relations rather
than in terms of causal mechanisms or physical interactions (Woodward, 2003; Woodward &
Hitchcock, 2003b). Causal relationships, he claims, are out there in the world, but they can be
well described in the reliable variable dependency relations of models. Explanation is the activity
of gaining information about these causal relations by discovering through intervention which
dependency relations are strongly invariant. The counterfactual dependency of these relations
gives us important information that provides explanatory depth. This is information that answers
what-if-things-had-been-different questions, or w-questions. Thus, the range of questions that
counterfactual dependence answers is related to the explanatory power of that causal relation.
This is because it is important to see “what sort of difference it would have made for the
explanandum if the factors cited in the explanans had been different in various possible ways”
(Woodward, 2003, p. 11).
Bokulich rejects the causal approach and favours a structural interpretation of
counterfactual information, or depth. In fact, she aims to give an account that can capture the
39
explanatory value of the highly-idealized, non-causal models that are not captured by
Woodward’s account. She claims that semiclassical models are not deemed explanatory on a
causal account because the entities involved (electron trajectories) are fictional and have can no
real causal power. As will be shown in more detail below, the morphologies (scarring patterns)
of the quantum systems of interest correlate strongly with the particular periodic, or repeating,
orbits of semiclassical mechanics, but the orbits cannot be said to cause the wavefunction
distributions, even though there is a reliable dependency relation. Bokulich argues that none of
the three main types of accounts of explanation (causal, covering law, or mechanistic) can
capture the way semiclassical models explain quantum phenomena, and offers her own account
of structural model explanations as a supplement. This account highlights the structural
similarities between the real world system and the idealized or fictional model. Bokulich argues
that structural model explanations are ones in which there is a pattern of counterfactual
dependence among the variables of the model, which can be measured in terms of w-questions,
and that this dependence is a consequence of the structural features of the target system
(Bokulich, 2008, p. 145).
In developing her account, Bokulich draws on a suggestion made by Margaret Morrison
that explanation has to do with structural dependencies (Morrison, 1999). Similar ideas have
been developed by John Worrall, James Ladyman, and others (Worrall, 1989; Ladyman, 1998;
French & Ladyman, 2003; Esfeld & Lam, 2008), but Bokulich’s account does not draw heavily
on these. Rather, Bokulich offers three general requirements for a structural model explanation,
which I have paraphrased and enumerated as follows:
E1. The explanation makes reference to a scientific model, M.
This first criterion specifies that the explanation is a model explanation and not a covering law or
mechanistic explanation. The structural aspect of the structural model explanation comes from
the second criterion, which says:
E2. M is explanatory by showing how there is a pattern of counterfactual dependence of
the relevant features of the target system on the structures represented in M.
This is intended to determine which models are genuinely explanatory by ensuring that an
explanatory model bears a close structural similarity to the counterfactual structure of the target
40
system. This structural ‘isomorphism’, as she calls it, is given an objective measure in terms of
w-questions (Bokulich, 2008, p. 145). The final criterion states:
E3. There must be a justification that specifies the domain of the application of M.
This is what she refers to as the justificatory step, intended to specify “where and to what extent
the model can be treated as an adequate representation of the world” (Bokulich, 2008, p. 146).
The bulk of what follows will assess the success of the structural criterion, but some concerns
about E3 will be raised in 2.4.3.
The worry for E2 is that all models exhibit a pattern of counterfactual dependency, and
that therefore E2, satisfied by all models, is an idle wheel. Bokulich herself does not think that
structure can distinguish explanatory from non-explanatory models, and instead relies on E3.
But, if E2 is an idle wheel and E3 is little more than a judgment about what is an adequate
representation in contemporary science, then the notion of explanation is unanalyzed – what is
explanatory is what is considered explanatory by the current state of science. However, I think
that there is a more promising avenue for structure: I argue that the criterion can in fact show
preference for certain models over others when employed in a comparative approach. However, I
also argue that this is unhelpful for Bokulich as the criterion shows preference for models of
fundamental theory over those of highly-idealized models. I return to these approaches in 2.4.2,
but now let us turn to the case study of structural model explanations in practice.
Bokulich applies her criteria for explanation to some cases of semiclassical mechanics as
part of a larger project of reconceiving the intertheoretic division between the quantum and the
classical. She argues that semiclassical mechanics can be genuinely explanatory of certain
quantum systems even though they are either deemed non-explanatory or fall outside of the
range of other accounts of explanation. The reason seems to be that the models of semiclassical
mechanics are non-Galilean, or highly idealized. To reiterate, idealizations that are non-Galilean
cannot be de-idealized to smoothly approach the real-world system. Many of these systems
identified by Batterman have singular limits which preclude this possibility. These models then
lack the approximate representation that is traditionally thought to justify their use in
explanation. However, as mentioned above, Batterman, Bokulich, and others argue that this does
not preclude explanation.
41
Models of semiclassical mechanics are highly idealized in that it is not possible to
recover the quantum models by removing approximations and de-idealizing. If the semiclassical
models have explanatory power, it cannot be due to an underlying causal mechanism of which
they are an approximate representation recoverable via de-idealization. Bokulich thinks that
semiclassical models of quantum wavefunction scarring are precisely the kinds of structural
explanations that Woodward’s account cannot consider explanatory. This is why she allows that
the justification of the application of the model to quantum phenomena (E3) can be top-down
from theory, rather than bottom-up where it would be smoothly recovered in Galilean
idealization. For semiclassical mechanics there is no smooth approximation, but there are top-
down theories that can be used to model quantum features in classical terms.
Batterman was the first to argue that semiclassical appeals to classical structures in
quantum phenomena at the asymptotic limit between the two can be explanatorily important
(Batterman, 1992, 2002b). Bokulich claims that structural explanations are actually quite popular
in mechanics, where appeals to structural restrictions can account for certain aspects of systems.
She argues that semiclassical mechanics can be an important interpretive and explanatory tool
for certain quantum phenomena, specifically in the subfield of quantum chaos. Classical chaos is
found in a great number of systems in which there is an extreme sensitivity to initial conditions,
such that an immeasurably small difference in two initial conditions may result in an exponential
divergence between them. Of course, this kind of extreme sensitivity to initial position and
velocity has no part in quantum theory, but quantum models that also describe these systems are
expected to exhibit something like chaos themselves. According to Bokulich and Batterman, one
expects to find a correlate of classical chaos in quantum systems. Due to the agreement between
quantum and classical predictions at the appropriate limit, there ought to be quantum systems
that underlie classically chaotic systems as well (Batterman, 1992, pp. 51-52; Bokulich, 2008).
One of Bokulich’s strongest examples is that of quantum wavefunction scarring in
systems known as quantum billiards. These are systems where the wavefunction of a particle
inside a stadium-shaped enclosure exhibits unusual patterns, which are called scars. Studies of
these quantum billiard systems by means of semiclassical mechanics and cellular automata have
revealed that there is an unusual accumulation of the wavefunction density along the trajectories
that would be periodic orbits in a classical system (C. King, 2009). The strong correlation
between the classical orbits and the observed quantum phenomenon makes these systems useful
42
for studying the quantum systems underlying classical chaotic systems. Bokulich argues that
these semiclassical approaches can be genuinely explanatory of the quantum scarring
phenomenon.
Work on these stadium billiards was introduced by Leonid Bunimovich (Bunimovich,
1974). In the classical billiard systems, a stadium shaped enclosed space is inhabited by a free-
moving particle whose trajectory is mapped. The boundary is defined by two semicircles
connected by parallel lines, and the particle suffers specular reflections with the boundary. The
shape of the enclosure has been since shown to be chaotic no matter how short its parallel
segments are and to exhibit other interesting properties (Bunimovich, 1979; Bleher, Ott, &
Grebogi, 1989). One implication of these results is that a mapped trajectory of the particle
generally displays an irregular pattern (Figure 1). This irregular pattern eventually leads to a
uniform distribution of trajectories throughout the space.
Figure 1 A typical example of a classical chaotic trajectory of a particle in a stadium shaped enclosure
(Stöckmann, 2010).
However, the chaos of this system is intermittent, as there are certain special initial
conditions that lead to periodic orbits in which the motion of the particle constantly repeats itself.
There are certain starting positions and velocities that will not result in a uniformly distributed
stadium, but exhibit a simple pattern of repeated motion. This pattern can occur in different
shapes including a rectangle, a vee, and a bow tie, among others, and are called periodic orbits
(Figure 2).
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Figure 2 Examples of periodic orbits in a stadium enclosure (Heller, 1986).
These periodic, or bouncing ball, orbits exhibit a “stickiness” on nearby chaotic
trajectories. The nearby trajectories, though not exactly periodic, exhibit quasi-regular behaviour
for long periods of time. When mapped out, this stickiness results in an accumulation of
trajectories in the near vicinity of the periodic orbits, due to the vanishingly small local
Lyapunov exponent, which gives the strength of the scar (Dettman & Georgiou, 2011). This
property of stickiness is also exhibited in other shapes of dynamical billiards such as in drive belt
shapes, where the semicircles are of different sizes, and in various mushroom shapes, such as the
one seen in Figure 3.
Figure 3 A mushroom shaped billiard system that clearly exhibits intermittent chaos. There is stickiness in
the vicinity of the periodic orbits and a region of diffuse chaotic trajectories elsewhere (Dettman & Georgiou,
2010).
What is very interesting about these systems is that because there are no trajectories in
the quantum analogs of these systems, one would expect to be unable to distinguish the chaotic
trajectories and the periodic orbits. Without orbit theory and sensitivity to position and velocity,
there is no obvious reason to expect these particular strong patterns in the quantum wavefunction
44
density. The interesting fact is that the quantum scarring phenomena actually converge on the
classically stable periodic orbits. It is almost as if the quantum system is sensitive to which
classical trajectories are periodic orbits and which are chaotic – a distinction that relies on the
details of the particular trajectory.
The structural model explanation claim is that the phenomenon of quantum wavefunction
scarring is best explained with the semiclassical model of the particle’s behaviour. On the
semiclassical account, the shape and size of the enclosure leads to certain periodic orbits being
favoured and exhibiting stickiness on nearby trajectories. And so as one changes the shape, the
allowed periodic orbits change in predictable ways, and the measured quantum wavefunction
distribution also changes accordingly. The semiclassical model is highly idealized and non-
representing. The use of electron trajectories, which are false of the quantum system, is justified
for Bokulich by means of Gutzwiller’s periodic orbit theory, which is a method of approximating
the density of quantum states from classical periodic trajectories. Gutzwiller’s theory specifies
how the behaviour of a Gaussian wavepacket (x,0) can serve as accurate solutions to the time-
dependent Schrödinger equation, and thus how the allowed classical periodic orbits correspond
to the accumulation of wavefunction density (phase interference) observed as the scarring
phenomenon (Heller, 1984). By considering the autocorrelation function of a Gaussian
wavepacket, (t)|(0), on a phase space point associated with a periodic orbit, one can see an
increase as the wavepacket overlaps with its initial state. The Fourier transform of this function
can be used to calculate the quantum spectrum. If the wavepacket is not initially on a phase space
point associated with periodic orbit there will be no pattern of increase in the autocorrelation as it
propagates, and hence no significant accumulation of the wavefunction in that region (Bokulich,
2008, pp. 128-129).
Bokulich does not argue that quantum mechanics cannot predict the scarring
phenomenon. In the quantum analog, the scarring phenomena can be reproduced by means of the
dynamics of the time-dependent Schrödinger equation with Dirichlet boundary conditions, so the
function vanishes at the walls. Rather than a classical bouncing ball in a stadium, the model
features a wave packet propagated through an infinite potential well with a stadium shaped
boundary. In a similar manner to the simpler rectangular infinite well, one can show that the
system will lead to ordered wavefunctions, and exhibit a phase interference pattern observed as
the scarring phenomenon (Figure 4). As the system evolves, the wavefunction settles on a
45
periodic solution. The scarring occurs when the probability amplitudes overlap in certain areas as
the wavepacket propagates throughout the space and reflects off the boundary (C. King, 2009).1
Figure 4 Amplitude contour maps of eigenstates that display a strong correspondence with the periodic orbits
(Heller, 1986).
Bokulich is primarily interested in the explanatory potential of the semiclassical model.
She argues that classical periodic orbits, though fictions – false of the quantum system – are
explanatorily relevant to the phenomenon of quantum wavefunction scarring. By falsely
assuming that the particle travels along a classical continuous space and time trajectory, one
correctly expects to find certain scarring patterns in quantum billiard systems, which one would
not expect prima facie in a quantum system. She argues that this example is a case of bona fide
structural model explanation. This example is not an outlier case, but one of many in which
Bokulich reaches the same conclusion about explanation; including the conductance peaks of
ballistic quantum dots, the orbits of Bohr’s model of the atom, and the resonance peaks of the
Rydberg electrons (Bokulich, 2008).
For Bokulich, these examples suggest that there is a “dynamical structural continuity”
between the classical and quantum theories. Because of this she argues that semiclassical
1 For more information about the quantum models, simulations of the scarring phenomenon, and ergodic and unique
ergodic properties of classical and quantum billiards, see (McDonald & Kaufman, 1979; Heller, 1984, 1986;
Gutzwiller, 1990; Tomsovic & Heller, 1993; L. Kaplan & Heller, 1999; Tao, 2007; C. King, 2009; Dettman &
Georgiou, 2010)
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fictions, in this case the classical periodic orbit theory applied to a quantum particle, can serve to
give counterfactual information about the quantum system. The closed orbits are not real, in the
sense that the particle is not actually travelling in a classically defined orbit with position and
velocity. Bokulich does not want to argue that the trajectories are real, but rather that they are a
special kind of fiction that is also explanatory: “These closed and periodic classical orbits can be
said to explain features of the spectral resonances and scarring insofar as they provide a
semiclassical model of these phenomena” (Bokulich, 2008, p. 140). The dependence of the
scarring phenomenon on the classical orbits conveys physical insight, or structural information,
on the underlying quantum dynamics.
Of course, in order for Bokulich to claim that there is a genuine explanation here, the
semiclassical model must satisfy her three criteria E1-3. And it can be quickly shown that it does.
The explanation makes reference to a scientific model, Gutzwiller’s periodic orbit theory, and so
it satisfies E1. The semiclassical model exhibits a strong counterfactual dependence of the
probability density in the billiard systems on the particular periodic orbits. It satisfies E2, the
structural criterion, because one is able to say how the wavefunction morphology inside the
stadium would change if the periodic orbit had been different, or if the shape of the stadium had
been changed. This semiclassical explanation is also justified in being applied to this domain
(E3) because Gutzwiller’s periodic orbit theory specifies how to model features of quantum
dynamics with classical trajectories – how to get real-world information from the information in
the model. So for Bokulich, this semiclassical model qualifies as explanatory.
As we have seen, Bokulich does not claim that quantum mechanics alone cannot predict
these scarring phenomena. Rather, her claim is that its explanations are deficient because they do
not provide as much counterfactual information about the system, which gives us physical
insight into the system and grants understanding. In order to get a more concrete sense of the
counterfactual information a model gives about the system, she makes use of w-questions and
Woodward and Hitchcock’s notion of explanatory depth, mentioned earlier. The more w-
questions a model answers, the more structural information it gives, the deeper the explanation it
provides (Bokulich, 2008, p. 152). Bokulich argues not only that the semiclassical models are
explanatory, but that the semiclassical models actually provide deeper explanations than the fully
quantum ones: “More importantly, the semiclassical models provide more information about the
structure of the quantum dynamics than do the fully quantum calculations. That is, the
47
semiclassical model allows one to answer a wider variety of w-questions, about how the system
would behave if certain parameters were changed…” (Bokulich, 2008, p. 154). Reference to the
classical structures is eliminable in that one can make simulations that exhibit scarring
phenomena from a quantum model. Nonetheless, Bokulich argues that “without knowledge of
the classical orbits, our understanding of the quantum spectra and wavefunction morphologies is
incomplete” (Bokulich, 2008, p. 154).
For Woodward and Hitchcock, the range of w-questions that a model can reliably answer
about a phenomenon is directly related to the model’s range of invariance. Brad Westlake has
identified many distinct measures of invariance in their account (Weslake, 2010). For instance, a
model can be more invariant if it is more accurate within a certain range; invariant under a wider,
or more continuous range of interventions; invariant under a wider range of different ways the
interventions may be performed; or under a wider range of background conditions. “What they
have in common is that they provide the resources to describe a greater range of true
counterfactuals concerning the possible changes to the system in question – that is, to answer
more w-questions…” (Weslake, 2010, p. 278). All of these different kinds of invariance are
relevant to the structural information Bokulich is looking for in E2.
Bokulich claims that the semiclassical model of wavefunction scarring gives counterfactual
information about the quantum system, and further that “there can be situations in which less
fundamental theories can provide deeper explanations than more fundamental theories”
(Bokulich, 2008, p. 153). Given that there are local models framed entirely in quantum terms that
can predict these scarring phenomena, if Bokulich wants to argue that the semiclassical model
provides deeper explanations than the fully quantum one, then she has to show that the
semiclassical model can answer a wider range of w-questions, in the full sense described above.
2.4.2. Two Approaches for Assessing Structure
I have shown how Bokulich’s account aims to capture the way highly-idealized models like
those of semiclassical mechanics explain, and I now turn to examine a serious concern about the
claim. In order to make the best case for a structural criterion, I present two possible ways of
assessing structure: a direct and straightforward measure of the number of answers to w-
questions (or w-answers) with a certain threshold for explanation, and an indirect measure for
comparing the different classes of w-answers that two competing models provide. I aim to show
that both avenues for assessing structure are unsatisfactory. The first approach proves impossible
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and leaves E2 an idle wheel. The second, when it is possible, does not side in favour of the
highly-idealized model, which it was intended to do. Further, I suggest that this is not something
particular to w-answers but applies generally to any similar measure for structure. Now, let me
first turn to the original worry about structure, before presenting the more promising view.
In a review of (Bokulich, 2008), Belot and Jansson express a concern that once the
account of structural model explanations allows for such fictions as classical trajectories in
quantum systems, it will be unable to reject models that are widely considered non-explanatory,
such as the models of Ptolemaic astronomy. They ask, “what is to stop you from viewing the
Ptolemaic model of the solar system as giving an adequate structural model explanation of this
phenomenon? Indeed, an appeal to the Ptolemaic model on this question would appear to satisfy
all four requirements for a structural model explanation” (Belot & Jansson, 2010, pp. 82, 83).
The worry is that once she opens the door up to explanatory fictions her criteria are not strong
enough to debar non-explanatory fictions, such as planetary epicycles. Bokulich (2012) is
explicit in wanting to allow for the fictitious electron trajectories in quantum billiard systems, but
not the fictitious epicycles of Ptolemaic astronomy. As is well known, the Ptolemaic model of
the solar system makes use of epicycles, on which there is a strong counterfactual dependence of
the apparent retrograde motion of the planets across the night sky as seen from Earth. At first
glance, Belot and Jansson are right to worry that epicycles and electron trajectories both satisfy
E2, but Bokulich argues that her account is capable of admitting one and not the other.
Both of the models satisfy E1, insofar as they reference scientific models: the geocentric
model of the solar system and the semiclassical model of the quantum billiards. The models also
seem to satisfy E2. They are counterfactually reliable under a range of conditions. The Ptolemaic
system has trigonometric tables of chords used for calculations, and these give us counterfactual
information about the visible solar system, and as Bokulich showed, the semiclassical model
gives counterfactual information about the scarring patterns. In fact, according to Bokulich, w-
questions are unlikely to be able to distinguish semiclassical explanations from Ptolemaic ones.
What she argues instead is that “the difference between explanatory and nonexplanatory models
is determined by something like a contextual relevance relation set by the current state of
scientific knowledge,” (Bokulich, 2012, p. 733). Thus, it is only on E3 that a distinction can be
made. In order to examine this claim, I will first assess the models according to the structural
criterion E2, and move to the third criterion E3, in 2.4.3.
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The structural aspect of Bokulich’s account (E2) requires that the “model must explain
the explanandum by showing how there is a pattern of counterfactual dependence of the relevant
features of the target system on the structures represented in the model” (Bokulich, 2008, p.
146). Unlike how it is framed in Woodward and Hitchcock, for Bokulich there is no need to
insist that the counterfactual dependency is representative of causal relations. Rather, she claims
what is being exhibited is the structural dynamics of the system. In order to most accurately
assess the satisfaction of E2, one needs to actually measure the w-answers that a model can
provide.
Unfortunately, obtaining a measure of the number of w-questions a certain model can
answer is not straightforward. Let us examine one approach, which is simply to count the
number of w-questions answered by a model and if the number is sufficiently large then the
model is explanatory. However, two problems with this immediately arise. First, the Ptolemaic
system, for instance, has methods of calculating the positions of the bodies of the solar system
for any given day, for any place on Earth, including not just positions in the night sky, but
eclipses, solstices, equinoxes, and so on. Importantly, these bodies have cycles and epicycles that
are geometrically continuous. This means that the model provides an infinite number of w-
answers, as there are an infinite number of points on the lines of the spherical trigonometry. And
so the number of w-questions answered by even a non-explanatory model is infinite. One could
make a stronger case for this approach by articulating a method which looks at the size of the
parameter space, the domain of values allowed by the model. This might allow one to get more
varied information about the amount of counterfactual information a model provides. This is an
interesting approach, however, even if one were equipped with a good quantitative measure of
the structural information a model provides, the second problem remains: there is no principled
way of determining exactly how many w-answers or what size of measure would count as
“sufficient” for explanation. Without some larger framework for determining exactly the
minimum size, a proposed threshold seems arbitrary.
And so this approach fails, as all models would equally satisfy E2, and thus the entirety
of judging whether something is explanatory falls on E1 and E3, which without E2, amounts to
nothing more than reflecting current judgments about explanation. In this case, there is no
analysis of explanation and no reason to think of these as structural explanations. However, I
think there is a more promising comparative approach. While Bokulich does not explicitly frame
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w-questions in a comparative way, I suggest that the Ptolemaic model does answer fewer w-
questions than, for example, the Newtonian model, and it can be debarred in that fashion. As we
have seen, a quantitative method for counting w-answers is not possible, and so a simple
quantitative comparison cannot be made either. Bokulich suggests examining the classes of w-
questions, and I believe that there is a sense in which an intuitive comparison of the classes of w-
questions is possible. Consider a relative formulation of the structural criterion E2*, which states
that M1 provides deeper explanations than M2 if M1 gives more counterfactual information than
M2, which is given in terms of classes of w-questions.
Because there is a lot of overlap in the information one gets from the Ptolemaic and
Newtonian models of the solar system, an argument could be made that the Ptolemaic model has
a narrower scope. In fact, because the Newtonian model can give all the w-answers that the
Ptolemaic model provides, but also a lot of additional w-answers as well, the Ptolemaic w-
answers are a subset. For instance, one can get reliable w-answers about how the orbits would be
different if the planets or the Sun had different masses, or if the planets were different distances
from the Sun, and so on. But as Woodward and Hitchcock were careful to point out, there is
more to explanatory depth than scope. In this case, the Newtonian model is dynamic, which
many argue gives a richer, deeper account of the planetary motions. It is not simply a kinematic
model that describes behaviours and matches observations, it makes use of forces and torques, to
give the causes of things. It seems that intuitively, the Newtonian model answers more different
kinds of w-questions more accurately. It gives stable and robust counterfactual information. The
Ptolemaic model does not fare well in this kind of comparison, so perhaps a comparative
structural criterion can be capable of distinguishing explanatory from non-explanatory models.
This could provide a defense for this structural approach from criticisms like those raised by
Belot and Jansson.
If this comparative framework works for Ptolemaic astronomy, is it also capable of
showing that, as Bokulich argues, the semiclassical model provides a deeper explanation of the
scarring phenomenon than the fully quantum model featuring the Schrödinger equation? Well,
when one returns to the semiclassical model and attempts to compare the w-answers with those
provided by the local quantum model, the comparison does not seem to lead to the conclusion
that the highly-idealized model provides deeper explanations.
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The semiclassical model can give counterfactual information about the distribution of
probability densities in the enclosure in straightforward way. There is a certain range of
questions that can be answered about the dependence of the wavefunction morphology on the
classical orbits. There is a striking correlation between the trajectories predicted by the classical
theory and the observed phenomenon. The semiclassical model is indifferent to the details and
particulars of the quantum dynamics and allows certain features like scarring phenomena to be
highlighted. Information about why particular scarring patterns, as seen in Figure 4, occur is
given by the semiclassical model, so the argument would go, because it is easily capable of
accounting for the chaotic and the particular periodic trajectories, and can show how the
quantum scarring would change if things (the periodic orbits in Figure 3) had been different. It
does seem that here too, “rather than obscuring the genuine mechanisms at work, this
idealization actually brings them into focus” (Batterman, 1992, p. 64). So it seems plausible that
there could be a class of w-questions that are more deeply, or at least more intuitively, answered
by the highly-idealized model.
Bokulich argues that the non-fundamental model can provider deeper explanations. But
the semiclassical model is not the only model that accounts for the scarring phenomena. The
Schrödinger equation can be used in a quantum model to achieve all the results that are obtained
in the semiclassical model. In fact, that is how we know the semiclassical model is successful.
As Bokulich freely admits: “one can “deduce” the phenomenon of wavefunction scarring by
numerically solving the Schrödinger equation,” the problem, she argues, is that “such an
explanation fails to provide adequate understanding of this phenomenon” (Bokulich, 2008, p.
151). Understanding, for Bokulich, is given by the physical insight, or information in terms of w-
questions that a model answers. And so, this means that the non-fundamental model provides
more w-answers. However, Weslake points out that the w-question notion of explanatory depth
favours more fundamental generalizations: “The fundamental laws are those generalizations that
are maximally accurate, robust, continuous, stabile, insensitive, and portable” (Weslake, 2010, p.
278). This is what many assume to be the case and is what Woodward and Hitchcock assume
when they explored the notions of depth and counterfactual information (Woodward &
Hitchcock, 2003b).
It is worth unpacking this assumption. What is crucial is that the quantum model can
account for all the same scarring phenomena as the semiclassical model. Of course, the quantum
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model cannot answer specific, classically-framed w-questions about trajectories or Lyapunov
exponents, because the concepts are unavailable to the quantum model. But, while the
fundamental theory loses w-questions in terms of trajectories, it gains “corresponding” w-
answers framed in terms of wave packet propagation. The quantum model is capable of giving
counterfactual information (w-answers) about the morphology of the scarring patterns.
But the local quantum model is also able to provide very deep explanations of these
phenomena, because as a model of a fundamental theory it can provide so much counterfactual
information. My contention is that the semiclassical to quantum relation of w-answers is the
same subset relation as that between Ptolemaic and Newtonian w-answers. Though this
comparative approach is not strictly quantitative, intuitively, the quantum model can be seen to
give at least as many w-answers as the semiclassical model, given that whenever the
semiclassical model can give information about wavefunction morphology, so can the quantum
model. However, the quantum model can also provide additional counterfactual information
about a variety of w-questions that can be answered by the quantum dynamics of the system in
terms unavailable to the semiclassical model. The semiclassical method is an approximation,
which is poor in certain conditions and fails in others. While a quantitative analysis of the
closeness of the semiclassical approximation is more properly suited to a physics paper, it is
clear the equations of the fundamental model that are more accurate and more strongly invariant,
in the many senses outlined above. An approximation can be very useful, but it does not contain
more information.
However, this comparative approach has an interesting corollary. If there were a domain of
phenomena in which the more fundamental theory could not derive the desired results, or
reproduce the phenomena, then the best available explanation would be given by the less
fundamental theory – which is to say that hope is not lost for this approach in capturing the way
some highly-idealized models explain. In this particular case however, Bokulich admits that a
quantum model can account for the scarring phenomena described by the semiclassical model.
And so it turns out that even though the classical trajectories can answer interesting w-questions
about the particular morphologies of the wavefunction scarring, models from the more
fundamental theory will always win out in terms of w-questions when they can account for the
same phenomena, because they can answer at least as many classes of w-questions.
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Someone working from Woodward’s account might accept that these highly-idealized
models are explanatory, but less explanatory than models of more fundamental theories that offer
much deeper explanations, as long as they satisfy his criteria for explanation by exhibiting a
strong enough degree of invariance under a range of interventions. However, this will not work
for Bokulich in this case, because it will not favour the models of semiclassical mechanics over
those of quantum mechanics because they are both capable of accounting for the scarring
phenomena. It is important for Bokulich, not only that semiclassical models be explanatory, but
that they actually provide deeper explanations of some quantum phenomena than the fully
quantum explanations.
While this measure of structure sides in favour of models of more fundamental theories
when they can predict the same phenomenon, when they cannot, an intuitive sense of which
model answers more classes of w-questions does not seem to have any bearing on the
explanatory depth of one model or the other. For example, if one compares a semiclassical model
with a Ptolemaic one, regardless of how the w-answer balance tips, the structural criterion still
has no real bearing on whether the models of semiclassical mechanics are themselves
explanatory. When there is no overlap in the domain of the models, the comparison is not
helpful. I contend that even if there were a quantitative way to measure the structural similarity
using something other than classes of w-questions, these problems remain. Imagine it were
possible to give a compressed scalar rating of all the complex representations of structural
similarity, given by a complicated process of calculations and perhaps insights from measure
theory. Now suppose that the Ptolemaic model of the solar system was given a rating of 4 and
the semiclassical model of quantum wavefunction scarring a generous 8. Even though it received
a higher rating than the Ptolemaic model, it is still reasonable to ask “does the semiclassical
model explain quantum wavefunction scarring?” And so it does not seem that there can be any
way that such a comparative framework can provide a general criterion for explanation. This
problem remains whether one uses w-answers or some other representational measure of
structure. It is only when both models can reproduce the same phenomenon that a meaningful
comparison can be made, but when it can be made, it does not favour the highly-idealized model.
The main worry for a structural criterion for explanation is that some measure of
structural similarity can be given to almost any model, no matter how inaccurately it represents,
or how little structural information it provides. And so if one wishes to debar the worst of these
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then either a threshold must be drawn, or a comparison must be made. For these semiclassical
models, a threshold proves unworkable. The comparison, when possible, sides in the favour of
the models of the more fundamental theory and not the highly-idealized model. Further, this
comparison is only helpful for models with overlapping domains, and leaves unanswered the
question of whether a particular model explains its target phenomenon. It does not seem possible
to conclude that the semiclassical model provides deeper explanations.
This does not mean that depth is incompatible with autonomy. Weslake, for instance,
provides a different abstractive account of depth, which he argues is more promising for
preserving autonomy (Weslake 2010). An assessment of Weslake’s arguments or of the
possibility of incorporating different notions of depth into a structural account, fall outside the
scope of this paper. It is also possible that one not choose w-questions to form the basis of a
measure of structure. Others have attempted to capture explanatory depth by other means
(Schupbach and Sprenger 2011; Strevens 2008; Weslake 2010), but few if any have offered
concrete measures of structure. One of the aims of offering a structural account is to preserve
autonomy of higher level explanations, and to allow some highly-idealized and non-causal
models, such as those of semiclassical mechanics, to be considered explanatory. I admit that
some notion of depth might be able to preserve explanatory autonomy, but a notion of structure
that depends on isomorphism or counterfactual information will favour fundamental models and
confront the difficulties of the approaches I have presented.
2.4.3. E3: The Justificatory Step
Bokulich’s solution to the problem posed by Belot and Jansson was to debar Ptolemaic
epicycles, not with the structural criterion E2, but with the justificatory criterion E3. And so in
this section I will analyze this criterion of Bokulich’s account and raise some concerns about it
playing the major role in distinguishing explanatory from non-explanatory fictions.
This justificatory step has three aspects, which I have taken the liberty of enumerating as
J1-3. J1 is a contextual relevance relation set by the contemporary state of science, which
ensures that scenarios like falling barometers causing storms are simply not even candidate
explanations. The justification also involves an articulation of the domain of applicability, J2,
wherein it is an adequate representation of the system. To satisfy this there must be either a top-
down or bottom-up justification of the model’s use, as I described above. Lastly, and closely
related is J3, a translation key of sorts that allows information about the model to be translated
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into conclusions about the real system. There must be some reason why information gained in
the model is applicable to conclusions about the world. For example, Gutzwiller’s periodic orbit
theory specifies how the semiclassical trace formula is related to the actual observed
morphologies in the quantum stadium billiard (Bokulich, 2012, p. 736). E3 taken together is
something like the explanatory standards of current science. It ensures that the model makes
reference to the right kinds of accepted entities, states, and processes, and that the relation
between the model and the real system is not merely accidental. For Bokulich, it is on E3 that the
distinction between explanatory and non-explanatory lies.
It is only on the third criterion, E3, that the Ptolemaic models will be debarred according
to Bokulich. The geocentric model and its epicycles are not at all adequate representations of the
real structure of solar system, as determined by the current state of scientific knowledge, and so
not deemed relevant to the explanation of planetary motion (Bokulich, 2012, p. 735).
Explanatory fictions “represent real entities, processes, or structures in the world, while [non-
explanatory ones] represent nothing at all (Bokulich, 2012, p. 734). She wants to allow for
fictions to be explanatory, but only fictions that count as adequate representations. In the context
of these two examples, the Ptolemaic model is non-explanatory because the orbits are not
adequately representative of the real structure of planetary motion: “given the relevance relation
set by the state of contemporary science, epicycles are irrelevant to the explanation of retrograde
motion. This is not simply because they are fictional but, rather, because they fail to be an
adequate fictional representation of the real structure of our solar system,” whereas “the classical
periodic orbits are able to capture, in their fictional representation, real features of the quantum
dynamics…” (Bokulich, 2012, p. 735). So the representational inadequacy is what debars
Ptolemaic epicycles.
In her response to Belot and Jansson, she says: “although the range of w-questions that a
phenomenological model can answer will typically be more limited, scope alone cannot
distinguish between explanatory and phenomenological models” (Bokulich, 2012, p. 733). She
offers instead the idea that the current state of scientific knowledge precludes the possibility of
Ptolemaic epicycles being counted as explanatory, in the same way that it ought to preclude
falling barometers causing storms – neither satisfy J1. It was shown in the previous section that
E2* also debars both Ptolemaic epicycles and semiclassical models. Now it can be seen that E3
debars Ptolemaic epicycles but not, according to Bokulich, semiclassical models. Semiclassical
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models are not so obviously inadequate as to be excluded from the explanatory store, like
shadows causing flagpole heights, and so they satisfy J1. For J2 and J3, the semiclassical models
of interest employ Gutzwiller’s periodic orbit theory to justify their application and provide a
means of getting real-world information from the fictional model. And so the semiclassical
models seem to satisfy E3 as a whole and are justified in being used in these systems exhibiting
quantum chaos, even though it was shown that they did not satisfy the comparative formulation
of E2.
In the remainder of this section, I will raise three worries about E3 and about this kind of
criterion as the main deciding factor for explanation. The first worry is that even though she
insists that electron trajectories in semiclassical models are explanatory and Ptolemaic epicycles
are not, it is not clear that, in distinction from epicycles, classical electron trajectories are
representative of the true electron dynamics, of the real structure of the quantum systems, as she
claims (Bokulich, 2012). Part of the requirements of E3 is that entities and processes of the
model are considered by scientists to be potentially relevant to the explanation (J1). Earlier, I
conceded that the semiclassical models should not be dismissed from potential explanations
outright, but this does not imply the positive claim that they do capture real quantum structures.
Consider the fact that the predictive success of semiclassical models is rather unexpected. This is
so precisely because they are not true descriptions of the systems. It may be that there is a certain
range of counterfactual information about the systems’ morphologies that can be gathered, but it
is not readily understood why it is that the dependency relation holds. Given only the full
semiclassical explanation, it is still a bit mysterious why the quantum effect would be dependent
on the classical trajectories. If one is able to derive this phenomenon and render it expectable on
a fully quantum picture, that mystery would disappear. This seems to suggest that the real
structure of the system is only given in a fully quantum picture, in the same way that the
numerical coincidences of Ptolemaic calculations are revealed by more fundamental theories.
The second worry is that because this is supposedly a structural explanation, a lot should
depend on the satisfaction or degree of satisfaction of E2, as Bokulich’s own formulation
implies. But, this does not seem to be the case. E2 is not capable of doing the work of showing
how a model is explanatory, since on one interpretation all models are explanatory and on the
other it debarred both Ptolemaic models and those of semiclassical mechanics in favour of
models with broader scopes from more fundamental theories. Due to this, E3 has to do most of
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the heavy lifting. However, if E3 is largely responsible for maintaining a threshold for
explanation, then there is not much of a sense in which these three criteria taken together are
independent criteria for explanation. The deciding factor is what satisfies E3, i.e. what is
consistent with that currently considered to be explanatory in science. The structural criterion
that was intended to pick out which models were genuinely explanatory by showing whether the
models exhibited the relevant structural properties of the system failed to do so. In order to make
a strong case that semiclassical mechanics can provide structural model explanations, the
structure that allegedly links the models to the real-world system should determine that. In a
structural explanation, the structural criterion should not be an idle wheel.
The last related concern is not only that E2 should distinguish explanatory from non-
explanatory in a structural explanation, but that E3 is too context sensitive to do this. It seems as
though E3 could be determined, or estimated, with structural information. If one wanted to assess
the adequacy of a model’s depiction of reality, determine whether its relation was numerological,
or correlational, and know if the model’s information is applicable to the real world system, then
its ability to give a wide range of reliable counterfactual information about that system seems a
reasonable measure. This information is something that the model can provide on Woodward’s
account, because it is explicitly manipulationist. But because Bokulich does away with the causal
interventions and only imports the notion of explanatory depth, this must be added on as a
separate criterion and loses objectivity. On Bokulich’s account, there can be no interventions to
separate the correlational from the causal, instead it falls on the scientific community to decide if
it is adequate. E3 is not meant to employ the measure of w-questions – it is not a measure of
structural similarity, but a criterion for ensuring that the model is not known to be
phenomenological. The criterion is context sensitive and particular to the details of the model
and the current views in science regarding what explains and what accurately represents. What
counts as an adequate fictional representation (J1) is a moving target, and not likely to be
unanimously agreed on across a discipline.
But even if this were widely agreed upon, there is something missing in this kind of
justification – a degree of normativity. And that even if some scientists, or a majority, do find
these models to be explanatory, there is more to a philosophical account of explanation than
capturing that. The judgments of scientists regarding explanation is important, but it should not
be the only aspect of an account of explanation. An account of explanation should not be merely
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descriptive, but provide independent criteria capable of assessing putative explanations, as I
argued in 1.4.1. There are benefits to maintaining a high threshold for explanation. It allows the
account to not simply capture the current use of explanation, but to highlight the best explanatory
practices of science. Having an account of explanation that claims that what is explanatory is
what is considered explanatory by a relevant scientific community says very little precisely
because it lacks normativity.
2.4.4. Heuristics and Explanations
Heuristics are powerful tools of science. A model that is heuristic is one that makes calculations
easier or provides a short-cut kind of reasoning. In psychology, a heuristic can be a kind of rule
of thumb or mnemonic. A model that is heuristically valuable can be one that is simple enough to
provide approximate answers with little computational load. Or it can be a model that can reveal
new avenues for research (Hartmann, 1996). A heuristic model, might allow researchers to learn
more about the behaviour or system at hand and discover new, related phenomena as well.
Heuristics are strategies or methods that can be very useful, generally accessible, and widely,
though loosely, applicable. The importance of heuristics in philosophy and scientific discovery
has been noted since at least Popper (1959).
Throughout her work, Bokulich argues for two points that should be considered
separately: 1) classical approaches in quantum mechanics are not only fruitful, but expose a
complex relation between the two that is more than a simple reduction of one to the other.
Regarding the influence of heuristics in the development of quantum mechanics she says that the
correspondences between the two theories “…are continuing to play a role in modern
semiclassical research. Understanding the heuristic ground of these correspondences can also
help us recognize that intertheory relations are not static, but rather are evolving, and are
continuing to be developed and extended in new ways” (Bokulich, 2008, p. 172). This
conclusion is in support of her interstructuralist position on the relation of quantum and classical
theory, and well summed up here:
Unlike Heisenberg and Pauli who thought new insights into physics would come only
through working with our most fundamental physical theory, semiclassical theories –
along with Dirac – think that classical mechanics still has something more to teach us.
More specifically, one of the most important insights of semiclassical mechanics is that
many new discoveries about quantum mechanics can be made by exploring its relation to
classical mechanics.
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(Bokulich, 2008, pp. 174-175)
Semiclassical mechanics has proved useful and fruitful in recent physics and might still,
but I maintain that this is not enough for it to count as explanatory. Bokulich is also sensitive to
this distinction, which is why she makes a second point directly related to the topic of
explanation which is that: 2) semiclassical mechanics and their fictitious electron orbits can in
fact explain certain quantum phenomena. The first may be convincingly argued for, but I have
shown that in the end, the second relies on nothing other than the models’ current accepted use in
science, as evidenced by testimonials. However, even though scientists may make claims of
explanation in proposing a new method, discovering a mechanism, or more accurately predicting
the behaviour of a system, this does not mean that their collected use of the term is coherent,
useful, or philosophically rigorous. I contend that often when scientists make claims about
explanation and explanatory value, they are actually talking about heuristic value; these
testimonial speak largely to the fruitfulness of semiclassical mechanics.
And indeed, semiclassical mechanics is a fruitful research avenue and it is intuitively
powerful. It allows us to picture and grasp systems that we should not be able to picture, and
frame them in familiar terms. And quite astonishingly, it can give us simple and reliable
counterfactual information about certain quantum systems. On the contrary, Ptolemaic epicycles
are no longer a fruitful avenue of research and their ability to predict seems quite accidental,
almost numerological. When Bokulich argues for the explanatory power of semiclassical
mechanics she concludes from the work of Wintgen, Richter, and Tanner (1992), as well as
others, that it is more than a tool or a method for generating simple, reliable predictions.
Bokulich cites physicists as saying that semiclassical descriptions are desirable because the full
quantum mechanical calculations are cumbersome and elaborate and that the “simple
interpretation of classical and semiclassical methods assists in illuminating the structure of
solutions” (Wintgen et al., 1992, p. 19). Here we see that it is in getting “the structure of
solutions” that the semiclassical methods are most useful. Batterman has argued along similar
lines citing the work of W.H. Miller: “Semiclassical theory plays an interpretive role; that is, it
provides an understanding of the nature of quantum effects in chemical phenomena, such as
interference effects in product state distributions and tunnelling corrections to rate constants for
chemical reactions” (Miller, 1986). These quotations are clearly in favour of the cognitive value
of semiclassical mechanics, and some explicitly state that semiclassical mechanics are
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explanatory. However, it is important to remember that scientists are unlikely to have in mind a
rigorous and philosophically robust notion explanation, complete with independent criteria.
There are a few reasons for interpreting these testimonials as actually being in favour of
the heuristic value of semiclassical mechanics. First, many of the reasons cited for considering it
explanatory, such as fruitfulness, simplicity, and understanding, more directly support its being
heuristic. The explanatory interpretation is also no longer supported by independent arguments
about the semiclassical models’ structural isomorphism, since I have shown that E2 is inert. It is
not as though these quotes corroborate the findings of a structural model account; they are now
its sole evidence. It is a consequence of the account that I present in Chapter 5 that these
semiclassical models lack a necessary component for explanation, which makes them at best
heuristic.
Bokulich hopes that by expanding on her criteria for explanation and introducing an
appeal to the current state of science she can maintain this distinction. It is true that current
science has turned its back on geocentric models of the solar system, and the Ptolemaic model
has been found to be empirically wanting. Semiclassical mechanics is new by any standards and
new results are published in reputable scientific journals. But more than this is needed is make
the case that one is explanatory and the other is not.
Bokulich wants to place semiclassical mechanics in a special place that is neither full-
blown realism about semiclassical mechanics, nor mere instrumentalism. Semiclassical
mechanics are certainly more relevant to the current state of science than Ptolemaic epicycles
because they are heuristically valuable in providing frameworks for investigating and calculating
quantum systems. I maintain that the value that semiclassical mechanics has is merely heuristic
and not explanatory. Where semiclassical explanations fit her criteria, quantum explanations fit
better, and thus she has not provided an account that can fully capture the way that highly-
idealized models explain.
Conclusion and Discussion
Bokulich has taken bold steps forward in arguing that w-questions can be used to measure
structural similarity. However, this measure proves rather difficult to determine. Neither of the
two approaches I outlined is capable of concluding that the highly-idealized semiclassical model
is explanatory. Even non-explanatory models provide an infinite number of w-answers, and even
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when employing measure theory, a given threshold seems arbitrary. When comparing, one finds
that a fundamental model wins out in terms of classes of w-answers. Either way, the account
does not capture the way that highly-idealized models explain.
Because the contribution of E2 is negligible to whether a highly-idealized model is
explanatory, the entirety of judging explanation falls on E1 and E3. Indeed, Bokulich herself
does not appeal to structure in order to debar the Ptolemaic explanation, rather she argues that
the model fails to be explanatory because it simply does not qualify as an adequate fictional
representation of the solar system in contemporary science (it does not satisfy E3). The relevance
relation set by the contemporary state of science has precluded explanations of planetary motion
featuring epicycles, barometers causing storms, and shadows causing flagpole heights, but not
necessarily the models of semiclassical mechanics. E3 only reflects our current judgments about
whether a model is explanatory and does not make any claims about why the model ought to be
considered explanatory. E3’s descriptive nature takes away the normative aspect that an account
of explanation ought to have, and has traditionally aimed for. Rather, it is that E3 has to play
such a strong role in judgments about which models are explanatory in light of what I have
shown about E2. I do not think that E1 and E3 alone can provide an adequate account of how
highly-idealized models explain.
Highly-idealized models are common in science and as other have argued (Batterman,
2002a; Wayne, 2011; Rice, 2012, 2013; Batterman & Rice, 2014), there is reason to consider
them explanatory. Developing an account of explanation that more accurately reflects the
explanatory practices of science is a next major step in the philosophy of science. Bokulich tries
to do so by providing a quantitative measure for a structural criterion but it ends up not working
in her favour. She attempts to preserve the autonomy of semiclassical, and perhaps other highly-
idealized explanations, but her notion of structure is representational and cashed out in a measure
of explanatory depth that prefers more fundamental generalizations with strongly invariant
mechanical laws.
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Chapter 3. Causal Accounts of Explanation
Introduction
This chapter turns to examine causal accounts of explanation. In 2.2, I made the case that higher-
level models can be explanatory by citing arguments made by Woodward against the causal-
mechanical account provided by Salmon and the arguments made by Mitchell regarding the
complexity of biological phenomena (Salmon, 1984; Woodward, 1989; Mitchell, 2003). This
third chapter examines how the particular accounts of Woodward, Strevens, and Mitchell
propose to treat these high-level explanations. In 3.2, I present Mitchell’s account of integrative
pluralism and its proposed solution for explanation given the complexity of biology. Her view of
explanation is that multiple models of a complex system are combined, or integrated, in order to
form an explanation. According to Mitchell, there are facts about the sheer complexity of biology
that preclude any traditional kind of explanation. She makes a case for her integrative pluralism
in part by looking at the self-organization of eusocial insect colonies. I examine her case study to
demonstrate her arguments against other forms of pluralism. In the end, I argue that Mitchell
provides no real framework for performing the integrations she proposes and that the detail and
particularity required for modelling the kinds of complex phenomena she examines actually
preclude the possibility of explanation. Her example of the detailed pluralistic model of the Lake
Erie ecosystem focuses on simulating its behaviour and not explaining why it behaves the way it
does.
In 3.3, I lay out how Woodward’s manipulationist account proposes to handle explanation at
multiple levels. I argue that the possibility of there being models on multiple levels that pass his
criteria for manipulationist causation implies emergent causation, and that therefore there can be
many incompatible causal models all operating in the same real world system. In 3.4, I present
the worry that having multiple models representing the real causal dependency relations brings
with it all the problems of emergentism, including downward causation and overdetermination.
These problems are examined by looking at Kim’s causal exclusion argument and its application
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to interventionist causation. I then briefly look at the reception of Kim’s arguments and defenses
of non-reductive physicalism (NRP) by Jessica Wilson and others.
An alternative strategy to emergent causation is to favour explanatory reductionism or a kind
of non-reductive physicalism. In 3.5, I review Michael Strevens’ kairetic account of explanation.
Strevens offers an account of depth that is capable of counting high-level models as explanatory
in certain cases, while maintaining that the real causes are all operating at a fundamental level. I
argue that this method avoids any problems of emergentism generated by other causal accounts
but has other drawbacks. I claim that Strevens’ account does not reflect the actual practice of
scientific explanation and is for all but toy examples not implementable.
I finish this chapter by recapitulating the results of this investigation and reviewing
additional concerns that have been raised for causal accounts of explanation in general. The
central aim of this chapter is to present a variety of popular accounts of causal explanation and
show that they face serious challenges. This does not entail that causal approaches cannot
succeed or that the problems are insurmountable. However, exposing these concerns about causal
accounts is a key result in the overall dissertation, which aims to show that a neo-deductivist
approach is more promising.
Scientific Pluralism
Sandra Mitchell explicitly states that pluralism in science is an unsurprising fact (Mitchell,
2003). It is reflected everywhere in the models and methods that are being advanced in science.
It is not unreasonable to ask why explanations of one world would be so diverse. Kuhn said that
it was a mark of the adolescence of science, but Mitchell notes that the diversity and
specialization of science is only increasing. She attributes the diversity, not to the immaturity of
the study, but to the complexity of the subject matter. This is partly what led Feyerabend to
advocate his epistemological anarchy (Feyerabend, 1975). His extreme position has opened up a
more modest middle ground for many kinds of pluralism or disunity accounts of science, such as
those provided by Cartwright, Longino, Harding, Dupré and others (Cartwright, 1983; Dupré,
1983; Harding, 1986; Longino, 1990; Davies, 1996).
Arguments from complexity are common in the literature of the philosophy of biology on
scientific pluralism. As was mentioned in 2.2.2, Cartwright developed a view of scientific anti-
realism based on arguments that it is impossible for laws to perfectly represent any real-world
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scenario (Cartwright, 1983, 1999). Cartwright concludes that laws are not true and exceptionless
as we ordinarily think of them, but function more like simulacra. Explanations on her account
make use of models, which employ laws that range over abstract objects in the model, but not the
actual phenomena. The objects in the model do not have stable properties, but rather capacities to
act in certain conditions. This is what counterfactual cases of the model represent. For
Cartwright, explanations involve a complexity of causal interactions which simulate real-world
scenarios.
Mitchell forms her integrative or critical pluralism specifically in response to this
scientific pluralism, in particular to account for the complexity of biological phenomena. She
argues that her integrative form of pluralism can explain the behaviours of complex systems of
biology better than other pluralist accounts. By the term ‘complex system’ she means a system in
which the micro-details are so many, or take place on such a long time-scale, that the
computation of the evolution of the system becomes intractable. In pluralist arguments from
complexity, there is an assumption that the impossibility of a complete lower-level description
requires a multi-level causal pluralism. Many have argued that this would preclude the
possibility of any deductive-nomological or base-level causal explanation of certain complex
phenomena (Dupré, 1983; Hüttemann, 2004; Love, 2012). I accept that multiple models can be
explanatory of a given target system, but deny the causal interpretation of these models, because
it is both unhelpful, and potentially problematic.
Mitchell argues that science, and biology in particular, is riddled with multilevel and
multicomponent systems that are incapable of being captured by anything but causal pluralism.
An account of scientific explanation, she says, must represent that fact. She argues that models at
different levels are compatible and can be integrated to form a single pluralistic explanation of a
complex phenomenon or system. She argues that the sheer number and variety of biological
models and methods belies the assumption that we live in a world that can be completely unified
and reduced. She claims that the diversity and variety of life is devalued in reductionism which
seeks to reduce the diversity of explanations, to a privileged set of laws. For reductionists, the
unity is found in a metaphysical and methodological monism. But, she points out that the
reductions of chemistry to physics, or biology to chemistry has never been fully realized.
Mitchell does not think that reductionism or global unification is even a desirable goal of
science:
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The complexity of the various subjects studied by science and the limitations of our
representations of acquired knowledge jointly entail an integrative, pluralistic model of
science. It is a model that allows for but does not require reduction. It is a model that
recognizes that pluralist ontologies and methodologies are required to give an accurate
picture of science as currently understood and practiced.
(Mitchell, 2003, p. 2)
Thus, she begins by assuming the fact of pluralism, then asking what kind it is, and what kind of
pluralist account best represents it.
In her earlier work, she marks the distinction between compatible and competitive views of
scientific pluralism (Mitchell, 2002b). Competitive views of pluralism hold that different
explanations conflict and a better one will win out and gradually move science towards unity.
Compatible views hold that the pluralism in science is not just a way to unity, but a benefit in
itself. This includes such views as the classification of levels of analysis and involves the
division of questions according to their own framework, as argued by Sherman (1988). Mitchell
thinks this latter view is better, but not enough, as it does not represent the interaction between
levels. She argues for this by means of a case study of eusocial insects, the conclusions of which
lead her to formulate a third integrative kind of pluralism. Next, I will present the case study and
critically examine Mitchell’s conclusions. I find that her approach stays true to the practice of
scientific modelling, but presents little in the way of an account of explanation. Further, the case
study does not support her conclusions as strongly as she claims.
3.2.1. Age-Polytheism in Eusocial Insect Colonies
Page and Mitchell (1991) run simulations of bee colonies using individual units with simulated
genetic variation and which interact with one another. When they were given stimulus they
would self-organize and specialization emerged spontaneously and a response to colony needs
was observed. This self-organization is what is called “age polytheism,” which divides tasks
non-randomly among individuals in the colony. The colony adjusts the proportion of workers in
each caste in correspondence to both internal and external factors. The standard interpretation of
this phenomenon has been the adaptationist one, which looks at past fitness of this strategy
compared to alternatives in order to explain its emergence (Bourke & Franks, 1995). According
to Mitchell, this ignores the mechanism by which the strategy is first adopted, something central
to the explanation of its emergence as an adaptation.
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The levels at which the bee colony can be analyzed are the cellular level, the individual
level, and the colony level. The debate among philosophers of biology has revolved around
which level of explanans should be prioritized in explaining evolved traits and behaviours. Some
have favoured reductive pictures of evolution, centered on the gene as the major player
(Dawkins, 1989), while others have resisted this, claiming that some explananda require group-
level selection to be explained (Brandon, 1978; Sober & Wilson, 1998; Bersgstrom, 2002; Boyd
& Richerson, 2002). Sherman has argued for a ‘levels of analysis’ approach which places the
appropriate level of selection at the level of the explanandum phenomenon (Sherman, 1988).
Here colony-level explanations are given for colony-level phenomena, and individual-level
explanations for individual level phenomena, and so on. Mitchell wants to resist even this. She
claims that there is only one true explanation of the phenomenon of age-polytheism in honey bee
colonies and it is one which allows for the interplay of models at all the various levels. The same
phenomenon in an ant colony or in some other complex system, will have a different single
integrative explanation. This is because “there is only one causal history that, in fact, has
generated the phenomenon to be explained,” which is itself “a combination of genetic, learning,
and architectural causal components,” (Mitchell, 2003, p. 216). The true explanation is local,
piecemeal, and an integration of partial solutions.
She says that three models have surfaced that challenge any explanation of this
phenomenon which does not cite multiple levels of causes. Along with her own work with
Robert Page (Page & Mitchell, 1991), she cites the work of Tofts and Franks, who propose a
separate algorithm in which foraging for work is sufficient to form a basis for a division of
labour (Tofts & Franks, 1992). In their study, a distribution of workers was organized to satisfy
the colony needs given simple rules regarding workload. Deneubourg et al. offered yet another
model for self-organization where the individual learning and forgetting is sufficient to generate
castes (1987). Mitchell says that these models suggest the phenomenon of colony level
organization is not necessarily determined by the genetic blueprint of individuals. Models of
genetic diversity, uniformity of work, and learning algorithms are all capable of generating self-
organization in these simulations. She claims that these studies promote her account in particular
because of its ability to treat these models explaining the phenomenon as compatible. On her
integrative pluralism, the models all pick out partial causes actually involved in the emergence of
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age polytheism. The explanation of its emergence then is a pluralistic one involving all these
models.
In response to a possible reductive explanation, Mitchell argues that if the pattern
generated in self-organization arises necessarily from properties of individuals, then this pattern
would have emerged everywhere and there would have been no variation between colonies to
conflict with each other. “Yet,” she says, “the evolutionary explanation of the origin of division
of labour appeals to colony-level selection for energy efficiency and thus must, by definition,
presuppose a history of heritable variation between colonies with such a pattern and colonies
without it” (Mitchell, 2003, p. 214). She concludes from this that evolutionary and
developmental levels interact. Developmental explanations might limit the possible solutions and
the variations for natural selection to choose between, or they might discover structural
necessities or universals, in which case natural selection is not the sole agent in bringing about
the present trait.
She also argues that if Sherman is correct in his levels of analysis approach to explaining
this adaptation of self-organization, then the different models of self-organization should be
mutually exclusive, but Mitchell does not think this is the case. She thinks this because each
model is an abstraction of a particular cause at the expense of others that, therefore, all models
are in fact acting in a real world scenario. For her, the separate models do not compete because
they describe “nonoverlapping ideal worlds” (Mitchell, 2003, p. 216); the worlds in which the
idealized models are true. In the case of the self-organization models she examines, each focuses
on particular idealized aspects of system: one models only genetic diversity, another only
learning diversity, and another only architectural diversity. She maintains that there is only one
causal history, and one causal explanation, that is true of any concrete situation, but it is a
pluralistic one composed the various causes identified in each of the specific idealized models.
Each complete explanation is piecemeal, local, and unique. She takes her advantage over
Sherman to be that the ideal worlds described by these models do not conflict and so they can be
integrated into a pluralistic explanation. By claiming that the models describe non-overlapping
ideal worlds, her account aims to make room for colony-level, individual-level, and genetic-level
selection in the explanation of age-polytheism.
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3.2.2. Three Problems with the Integrative Pluralism Approach
The first problem with this approach is that contrary to her conclusions, the studies she cites
actually seem to show the possibility of self-organization from individuals without colony-level
causes. These simulations and others that can be easily performed with artificial life (A-life)
games show that nothing more than information at the individual level is required to entail the
emergence of self-organization. Each of the factors in the studies are sufficient to generate age-
polytheism. What Mitchell and Page should have tried to show in order to argue for colony-level
selection is to demonstrate that no amount of genetic diversity was sufficient to reproduce the
observed caste system, and that what was actually needed was a colony-level directive. If age-
polytheism required rules about worker distribution this might have evidenced the claim that the
base-level models are insufficient to produce the observed behaviour. It would have been better
still if the simulation of the desired behaviour had required that many directives operate
simultaneously at the levels of genetic variation, individual behaviour, and colony distribution
requirements.
The second problem is that her integrative solution to the difficulties of modelling
complex phenomena does not provide much, if any, explanatory information. This problem can
be seen in a different example she uses involving the ecosystem behaviour of Lake Erie. In a
case such as this, she claims that an explanation of the whole system’s behaviour has to take into
account many different models and spatial simulations describing the advance of zebra mussels,
the levels of phosphorous in sediment, fish harvesting, the change in solar radiation, temperature,
predation, and so on (Mitchell, 2003). All these models and simulations are constructed using
data specific to that particular lake. The particular models used will be informed by research into
that particular ecosystem and are unlikely to accurately describe any other. The final product of
an integrative explanation will be a collection of such models and simulations and be quite
distinct from any other integrative explanation. The scope or generality of such an explanation is
essentially one system. The set of models, or pluralistic model, that Mitchell aims at ranges only
over a single scenario.
The tension between accuracy and generality reflects the difference between models and
simulations. The value of simulations in scientific practice is clear, but lies mainly in the analysis
of particular cases, and not necessarily concerned with explaining phenomena. What is missing
in simulations is information regarding why the ecosystem, for instance, behaves the way it does.
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Her account seems to suggest that highly detailed simulations are the best possible explanations
of these complex systems. But these simulations are made for accurate predictions, and do not
contribute to our understanding of either the system in question or other similar systems. There is
no information about why it works. The ways in which all the various factors are integrated are
unlikely to be global or algorithmic and more likely to be particular and concrete; because the
models of these complex phenomena are so specialized, there are not likely to be general rules
for explaining complex phenomena. It is not clear that her integrative pluralism is actually
providing an account of explanation rather than merely describing the challenges of modelling
complex systems.
The real problem this points to is that all models are explanatory on her account.
Integrative pluralism is so inclusive that no scientific model or simulation is non-explanatory. It
is part of her aim to remain true to the practice of science, but as I have argued, one aspect of a
successful account of explanation is that it ought to set some reasonably high threshold for what
counts as an explanation. Proposing an account of scientific explanation that makes room for all
models of a system to be incorporated into a pluralist explanation is too inclusive. Some models
are not explanatory, and this account fails to reveal this.
The third concern is that she claims that all three models for age-polytheism can be
integrated to explain the real concrete case, but provides no framework or method for doing so.
One of the merits of her account compared to other pluralist accounts was that it did not treat the
various levels of models at play as competing for priority of explanatory relevance. All the
models were considered partial, picking out aspects of a single pluralistic causal explanation. But
simply saying that the models can be integrated does not guarantee that they can or that this is
generally the case. The particular models describing aspects of an ecosystem’s behaviour may
not be compatible, in the sense that their predictions could conceivably differ. That the many
models capable of describing a system can all be brought together without conflict is not
something that can be decided a priori.
Mitchell’s work in this area is mostly focused on making her pluralism distinct from
others’, not on arguing for causal pluralism as a whole. That causal pluralism is a fact is an
assumption she starts off with. She takes for granted that explanation is causal and so concludes
from her investigations into the complexity of biological phenomena that some kind of pluralist
causation is correct. In fact, because of the problems with her approach, as well as those that will
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be examined more closely in the following sections, I maintain that it is better to conclude that
science makes use of multiple explanatory models, rather than multiple levels of causes. Mitchell
provides no actual framework for performing the kind of integrations she proposes, and there are
reasons to doubt the possibility of this seamless integration. It may be that for complex systems,
such as Great Lake ecosystems, the most accurate modelling techniques will run multiple
simulations. It is very unlikely that a small set of equations will be discovered to govern its
behaviour, but this does not imply that a loosely grouped set of simulations and models is
explanatory.
Causal Interventionism
In this section, I will present James Woodward’s account of scientific explanation and review
some worries I have about this account concerning its potentially circular notion of causation and
an ambiguity with respect to commitments of causal realism. Causal realism is at the heart of
many of the issues with causal accounts of explanation and discussion of potential problems here
will transition to a review of emergence and reduction in the following section.
Woodward rejects law-based accounts of explanation in favour of a manipulationist
account. While he notes that Hempel’s D-N account captures many explanatory relations, there
are many explanations that do not feature laws, but are nonetheless explanatory. This is because,
he argues, they give counterfactual information about a system which allows us to understand
how to manipulate it. The reason the D-N account is as successful as it is, is because good D-N
explanations also give us this counterfactual information in lawlike relations. Thus, his account
attempts to demonstrate why a broader range of models ought to be considered explanatory.
Woodward asks us to consider the explanation of the magnitude of the electric field
created by a long wire with a positive charge uniformly distributed along its length. A standard
textbook explanation “proceeds by modelling the wire as divided into a large number of small
segments, each of which acts as a point charge of magnitude dq. Each makes a contribution dE to
the total field E in accord with a differential form of Coulomb’s law:
(7) 𝑑𝐸 = (1
4𝜋𝜀0) (
𝑑𝑞
𝑠2 ),
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where s is the distance from the charge to an arbitrary point in the field. Integrating over these
contributions yields the result that the field is at right angles to the wire and its intensity is given
by
(8) 𝐸 = (1
2𝜋𝜀0) (
𝜆
𝑟),
where r is the perpendicular distance to the wire and 𝜆 the charge density along the wire”
(Woodward & Hitchcock, 2003a, pp. 3-4). Here Coulomb’s law plays a vital role in the
deductive derivation. But it can also give w-answers, and this is the real reason that the law is
explanatory, though it is not terribly clear in this case. Instead imagine a case from the structural
equations literature. We want to determine the extent to which the amount of water X1 and
fertilizer X2 influences plant height Y. We can use this linear regression equation:
(9) 𝑌 = 𝑎1𝑋1 + 𝑎2𝑋2 + 𝑈,
where 𝑎1 and 𝑎2 are fixed coefficients and U is an error term. Here, even if this gives
information about general causal relations, it falls short of the requirements of laws. It will fail to
hold at large values of 𝑋1 and 𝑋2, it does not account for background conditions that may cause it
to fail, and even in perfect conditions it may never perfectly describe the system. Here, because
the model presents relevant counterfactual information about changes to the system, the model
can be explanatory even if there are no laws of nature. For Woodward, a derivation in the D-N
account only plays an explanatory role insofar as it also gives counterfactual information.
Rather than physical interactions, or causal mechanisms, Woodward favours a causal
approach to explanation that focuses on counterfactual information derived from reliable
manipulations. The ability to explain comes not from tracing and delineating causal histories, but
from the ability of the generalization to answer w-questions, as seen in 2.4.1. Woodward
criticizes Wesley Salmon’s causal-mechanical approach to causation, in which causes are
identified by the transmission of information, or marks, because it does not account for
explanations at higher levels (Woodward, 1989). He uses as an example an explanation of the
changes in pressure given changes in temperature, which feature the ideal gas law. He argues
that Salmon’s account requires tracing the trajectories of individual molecules. Not only is this
impossible, but even if it were done, it still would not give us a good explanation of why the
phenomenon occurred the way it did; it would give us no information about why this happens in
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general and why other changes occurred in the gas’s temperature and pressure. The standard
explanation, and the one proposed by Woodward, proceeds by making assumptions about the
distribution of the molecules, their forces, and collisions, in order to describe the general
behaviour of the gas. For Woodward, physical interactions will not suffice as a basis for
causation, though some doubt that its rejection is this simple (Ney, 2009).
3.3.1. Invariance and Intervention
Woodward’s proposal is to identify causes as evidenced by certain reliable changes in the
variables of a model (Woodward, 2003). This account of causation involves the notions of
invariance and intervention, notions that stem from the work of Judea Pearl and others (Spirtes,
Glymour, & Scheines, 1993; Meek & Glymour, 1994; Pearl, 2000). Suppose that X and Y are
variables. If an intervention I is performed on X for the purpose of changing the value of Y in
such a way that the change in Y is due only to the change in X, then X can be said to be a cause of
Y. The relation between X and Y is invariant if the relation holds under a range of interventions
on X. This domain of invariance then specifies the strength of the causal relation and explains
the causal dependence of Y on X. Any domain of invariance implies explanatory power. This
power comes from the ability to derive information regarding what would have happened if
things had been different. In this way, invariance under interventions serves to distinguish causal
relations from accidental generalizations, which would not display this invariance.
Woodward introduces the notion of a testing variable, to make sure that there is a range
of possible values for Y. This ensures that it must be possible for the explanandum to be different
than it is. It must be possible to turn off a light bulb by flicking a switch, so that it is not the case
that one explains a light being off by the fact that the switch is down while the light is also
broken. One can contrast this with the D-N account, and notice that the problem in certain
derivations is that the generalization gives no information about on what the explanandum
depends. Testing interventions allow one to debar unhelpful generalizations, which give no w-
answers. This prevents those generalizations such as “all men who regularly take birth control
pills will not become pregnant” and “all hexed salt dissolves in water,” because the other testing
interventions, where the salt is not hexed and Jones does not take birth control, do not have
different results.
Woodward’s manipulationist theory holds that to make the claim that X causes Y, where
X and Y are variables, is to say that there is some possible manipulation of X that can be used as a
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strategy for changing the value of Y. “Causal claims tell us not that one property is associated or
necessitates another, but rather that certain changes in the value of a variable will produce
associated changes in the value of another variable” (Woodward, 2003, p. 112). An intervention
on one variable is used to discover its relation to another variable. Formally, this is given as
follows:
I1. I causes X
I2. I acts as a switch for all the other variables that cause X.
I3. Any directed path from I to Y goes through X
I4. I is independent of any variable Z that causes Y and that is on a directed path
that does not go through X.
I5. I does not alter the relationship between Y and any of its causes Z that are not on
any directed path from X to Y.
(Woodward, 2003, pp. 98-99)
It is important to note that he explicitly makes use of the concept of cause in I1 in formulating
his concept of intervention, which itself is intended to discover causes between X and Y. He
acknowledges this and thinks that it is unproblematic and necessary, as it would be problematic
to give a fully reductive, or eliminative, account of causation. This results in a kind of circularity,
which I will speak to in 3.3.2. His account is rather a way to discover which dependency
relations are truly causal in virtue of the underlying metaphysics of causation, by looking at the
counterfactuals that generalizations support.
Of course not all counterfactuals are going to describe causal relations. He distinguishes
between interventionist and non-interventionist counterfactuals. Traditionally, counterfactuals
are seen to be what he calls other-object counterfactuals, which give information about what
would be the case for a different object. His same-object, or intervention, counterfactuals refer to
hypothetical changes to the same object. Basically, an explanatory counterfactual will tell us
what would happen to the value of Y, say where 𝑋 = 𝑥1, 𝑌 = 𝑦1, if X were manipulated, rather
than what value Y would have in some other system where 𝑋 = 𝑥2. This distinguishes
information about how a system responds to manipulations, from information given by a mere
regularity. Only the relations which are invariant under the changes of these same-object
counterfactuals support explanations.
Knowledge about these counterfactuals comes from brute facts about the real causes in
the world. Explanation is the activity of gaining information about these causal relations by
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discovering through intervention which dependency relations are invariant. “No causal
difference without a difference in manipulability relations, and no difference in manipulability
relations without a causal difference” (p. 61). Woodward is committed to the idea that if there is
an invariant dependency relation between two variables then they are causally related, even if
this is contrary to popular belief.
The notion of invariance is that there are true counterfactuals about interventions on X,
and the ensuing values of Y. Invariance is a notion tied to explanatory depth for Woodward
(Woodward & Hitchcock, 2003b). As discussed earlier (2.4.2), Brad Weslake has outlined the
various ways depth, or relative invariance can be established in the interventionist framework
(Weslake, 2010). He notes that a regularity is more invariant than another if it is: more accurate
within a specific range; more robust, in being more accurate under a wider range of
interventions; invariant under a more continuous range of interventions; invariant under a wider
range of ways in which interventions are performed; and invariant under a wider range of
background conditions. Essentially, when there is a stronger the connection between the
variables, the generality can provide a deeper explanation.
3.3.2. Circularity
As mentioned above, Woodward’s account of causation is non-reductive, as he calls it. His
account does not attempt to eliminate the notion of causation by replacing it with the concepts of
invariance and intervention: “Because the notion of an intervention is already a causal notion, it
follows that one cannot use it to explain what it is for a relationship to be causal in terms of
concepts that are themselves noncausal” (p. 104).
The intervention requirement I1 involves knowledge that I causes X, but it is used in
determining whether there is a causal relation between X and Y. This sounds suspicious, but
Woodward does not think that this is problematic. He notes that the causal information needed to
characterize the notion of an intervention is only information about the relation between I and X,
but not X and Y. It is information about other causal relationships than that between X and Y
that is evidence for the claim that X causes Y. He acknowledges the fact that he has imported a
notion of causation into his concept of intervention. However, he thinks it would be problematic
to give a fully reductive account of cause, where causes are stipulated in the method by which
one comes to know them. He claims that if a reductive account were attempted one would be
attempting to derive causal claims from correlational information, something notoriously
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problematic, if not impossible. Causal claims contain more information than there is in
correlational information (Woodward, 2003, p. 106). Woodward remains silent on what this
information really is, on the metaphysical truth-makers for causal claims.
Woodward claims that his account is neither viciously circular, nor trivial. It attempts to
elucidate the relations between these interconnected terms in the context of scientific
explanation, which is the practice of gaining information about these genuine causal relations.
He maintains that if his definitions constrain the properties, then they are not viciously circular.
His project is epistemological, then, rather than metaphysical, insofar as it remains silent on the
metaphysics of causal connections and focuses on elucidating what information causal claims are
giving us.
Even if it is acknowledged by Woodward, I believe there is a cause for concern. It is
difficult to see how one can gain any causal knowledge if one begins with knowledge of causal
relations. In order to make the claim that X causes Y, one must know that I causes X. But the
way that one gets information about causal relations, say between I and X, is by knowing that
there are interventions I' on I that make reliable changes in the value of X. This in turn is
garnered by knowing that there are interventions I'' on I' that make reliable changes in the value
of I, and so on. There is an infinite regress that results in never getting new information about
causal dependency relations. The worry is that the account only works if one assumes one has
genuine knowledge of causes, say between I and X, to begin with. In order to set up a directed
graph, one must have knowledge of all the causal relations among the variables. In which case,
the purpose of gaining knowledge about causal relations is trivialized. This point ties in to a
more general worry about his commitments to causal realism and the normativity of the project,
which I turn to next.
3.3.3. Causal Realism
Behind all causal accounts of explanation is the idea that giving an explanation involves
identifying the relevant causes. Citing real causes is an effective method for debarring non-
explanatory relations, like backtracking counterfactuals, cases of symmetry, and other non-
explanatory generalizations or derivations. Because of this, it is important that such an account
be committed to the metaphysical stance of causal realism, even accounts that focus on
epistemological issues, like Woodward’s.
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Woodward distinguishes his own manipulationist account from regularity accounts of
causation. Regularity accounts are focused on establishing regular connections for the purposes
of prediction, but say little to nothing about why the regularity itself holds. They fail to
distinguish between cases where the regularity is causal and cases where it is not, according to
Woodward. Manipulationist accounts are designed specifically to handle such cases. It is built
into the framework of manipulationist causation that there is a stronger realism about causes that
allows for this distinction. Further, unlike other manipulationist accounts, Woodward does not
think there is a serious way in which these causal claims are merely projections of our agency.
Causal relations exist independently.
It is a presupposition of her deliberation that if it is possible to change Y by intervening
on X, then there must be an independently existing, invariant relationship between X and
Y that the agent makes use of when she changes X and, in doing so, changes Y – a
relationship that would exist and have whatever characteristics it has even if the agent
were unable to manipulate X or chose not to manipulate X or did not exist.
(Woodward, 2003, p. 119)
For Woodward, the fact that the manipulation is invariant is due to the real causal dependency
relation, whatever it may be, that underlies the reliable manipulation. The truth-makers for causal
claims are the prior, independently existing, objective differences between causal and
correlational relations. Even though his manipulationist account does not focus on specifying the
metaphysical relations between cause and effect, it does rely on there being such relations.
However, Woodward claims that a benefit of the manipulationist account is that it can
focus on the instrumental success of manipulations in making its causal claims. Woodward
maintains that he can be more or less uncommitted to any particular interpretation of the
processes and mechanisms involved in causal models. He does not require that there be a
continuous causal process – his theory “assigns a more limited significance to correctness at the
level of fundamental ontology.” As he puts it, “a theory might, for example, correctly capture the
dependency relations between a certain set of measured quantities and, hence, qualify as
explanatory, even if it says nothing about or makes mistaken claims about intervening processes
or mechanisms” (Woodward, 2003, pp. 223-224). This is reflected in his argument against
Salmon, which is not simply that conserved quantity accounts of causation do not capture what
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we consider to be causal relations, but that they fail to include the explanatory phenomenological
generalizations such as those of thermodynamics.
Even though Woodward maintains that “what matters for purposes of causal explanation
is what the real dependency relations in the world actually are” (Woodward, 2003, p. 202), it has
been argued that the causal aspect comes apart from his counterfactual account of explanation.
Saatsi and Pexton argue that if counterfactual information alone can provide a basis for
explanation, then there is little explanatory role for the causal interpretation of this information
(Saatsi & Pexton, 2012). I suggest that this exposes an ambiguity in Woodward’s account
between the requirement of causal realism and the ability of his counterfactual account to
misrepresent or be mistaken about causal claims. Causal realism is necessary when, for instance,
Woodward puts his account to the task of debunking the case of explaining the length of the
pendulum by its period. The way that his account does this is by tracing “this explanatory
asymmetry to an underlying physical asymmetry in the roles played by the length and the
period” (2003, p. 197). It is true that the length in this model is related to the period via a
generalization, but the fact remains that there are no physical manipulations on the period that
will change the length. Because of this, it cannot be featured this way in an explanation. This
difference is reflected in the common sense judgment about not being able to change the length
through manipulations of period. Our knowledge of causes debars certain relations. I think that it
is not possible to simply jettison the causal interpretation of the account; it is needed to debar
certain non-explanatory cases, such as this symmetry.
This raises an interesting issue about normativity for Woodward. Woodward makes a
strong case against Salmon and others that high-level models, such as those of thermodynamics,
are explanatory in their own right. There are many models that are capable of reliably capturing
counterfactual dependencies, but many of these are not commonly held to be causal. These are
the explanatory models that make mistaken or false claims about intervening processes or
mechanisms. Woodward’s account is “partially revisionary.” It aims to “make recommendations
about what one ought to mean by various causal and explanatory claims, rather than just
attempting to describe how we use those claims” (Woodward, 2003, p. 7). And so, his account
will find certain phenomenological generalizations to be causal and explanatory, even contrary to
popular belief. One might be concerned that the revisionary aspect of the project will misidentify
certain counterfactual relations as causal. In fact, it seems as though if any equations are reliably
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invariant according to the manipulationist criteria, then they would be given a causal
interpretation. This would include quantum models that support manipulationist counterfactuals
about the reliable changes to variables and other highly-idealized models that are non-causal like
those of semiclassical mechanics.
However, because the project is only partially revisionary, it is not clear that one needs to
be committed to any particular revision. One could use the very same project to declare that
quantum models are outside the scope of the account. This is because in order to demonstrate
that a model satisfies the manipulability criteria, one needs to know I1, that ‘I causes X’ (3.3.1).
Because I does not cause X in the non-causal case, the account makes no misidentification.
There is a drawback to this in that it precludes a wide range of models that make explanatory use
of principles, rules, or are otherwise thought to be non-causal. For example, the use of entropy in
thermodynamics, stable states in dynamical systems, and any laws of coexistence would fall
outside the scope of the account. Woodward is quite clear that there are non-causal explanations
and other limitations to his account, and so perhaps the most charitable way to understand this is
to see the account as having a more limited scope and being less revisionary. Some might think
that this is a small price to pay, but one of the main motivations behind the account in Chapter 5
is to broaden the scope of possible explanations to precisely these kinds of models and
idealizations.
What the circularity demonstrates is that there is an ambiguity with respect to
Woodward’s commitment to causal realism. The concern is that it is not clear when the project is
describing or recommending. The project as a whole is rather ambiguous concerning about this,
but in certain passages it is clear that models of macroeconomics and thermodynamics are found
to be causal and explanatory. In which case, perhaps the model is more revisionary than
descriptive and really only precludes models from satisfying I1 when they are obviously non-
causal. In fact, this seems to be what Woodward claims. Given a single real-world system,
Woodward’s account would identify as explanatory and causal a number of models that describe
its various behaviours at various levels. Again, consider the ideal gas law. One can use the
formula to approximate the behaviour of gases under certain conditions. Woodward’s account
identifies the best explanation of some high-level phenomena as being at a higher level than the
molecular level, but it would also identify causes at the molecular level, and as described by
models of fluid mechanics, given that these too would satisfy manipulationist criteria. Woodward
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does not claim that all causes are actually fundamental. As mentioned above, he is explicit that
there are models that are explanatory and causal that are non-fundamental, like the ideal gas law.
There is an important sense in which models at high levels of generalization are describing
different causes than models at a base level. High-level generalizations reflect independent
causal facts that are not simply parasitic on base-level causal stories. As such, Woodward seems
to be committed to a kind of emergent causation. There are some concerns that have been raised
about the metaphysical implications of emergent causes. And it is to this which I next turn.
Emergence and Reductionism
This section will present and review some of the literature on reduction and emergence and
attempt to apply it to discussions of causation and scientific explanation. A full treatment of this
is not feasible in this context, but I hope to clarify some issues about Woodward concerning
emergent causation. After introducing emergence and physicalism, I present Jaegwon Kim’s
argument against emergence and non-reductive physicalism as well as Jessica Wilson’s analysis
of it in 3.4.2. Wilson shows that Kim’s conclusion is not necessary, and in fact leaves open the
possibility of a kind of non-reductive physicalism. In 3.4.3, I look at List and Menzies’ defence
of high-level causation in the context of a counterfactual account of causation. I aim to show that
neither of these defences are applicable to Woodward’s account because his account is
emergentist. An account that is explicitly non-emergentist, like Michael Strevens’, is then
reviewed in the following section, 3.5.
The literature on emergence goes back at least as far as C.D. Broad and Samuel
Alexander, but this discussion will involve more contemporary contributions (Alexander, 1920;
Broad, 1925). There have been many articulations of what constitutes emergence and whether
emergence is best understood as epistemological or ontological. Epistemological views see
emergence as describing the limits on human knowledge of complex systems and come in
various forms: predictive, which says that emergent properties are features which cannot be
predicted from the pre-emergent stage even given complete knowledge of laws and features; and
irreducible-pattern, which says that emergent properties and laws are true laws which govern
special sciences which are irreducible to physical theory for conceptual reasons (Fodor, 1974).
Some theorists focus on diachronic relations between pre- and post-complexity, and others study
synchronic patterns at different levels (Bedau & Humphreys, 2008).
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Popper offers an argument for the existence of emergent properties by focusing on the
unpredictability of systems. For him unpredictability is a sign of emergence (Popper & Eccles,
1977). Mark Bedeau’s weak emergence defines a weakly emergent state as one that could be
derived from knowledge of the system’s microdynamics and external conditions, but only by
simulating all the interactions (Bedau, 1997). He formulates this in order to deal with chaos,
which relies on very tiny differences generating unpredictability. This leads to a kind of
emergence in principle for any realistic observer. Andy Clark also takes into account dynamical
systems theory, but focuses more on cognitive science, and argues that emergent phenomena are
those best understood by changing values of collective variables (Clark, 1997). Batterman by
contrast connects emergence to inter-theoretic reduction. He shows that these reductions are
rarely smooth in actual science. What often happens is that there are singular limits for reduction
wherein one model is reduced by taking elements of the other. These special cases are emergent
for Batterman. However, Andrew Wayne argues that there are systems like the van der Pol
nonlinear oscillator whose high-level behaviour can be adequately explained across these
singular limits which suggests that the behaviour is not genuinely emergent (Wayne, 2012).
Ontological views see the physical world as entirely constituted by physical structures,
but composite structures are not always mere aggregates of simples. At each stratum there is a
new kind of property, complete with new causal powers, novel entities, or laws which connect
the complex physical structure to the emergent features. This kind of view has numerous
positions2. Paul Humphreys argues in favour of a metaphysical approach he calls fusion, wherein
entities can become fused and cease to function as separate entities, and gain novel properties
and causal powers (Humphreys, 1997). Jessica Wilson argues that emergence comes from
novelty in terms of a sets of causal powers (Wilson, 2011b). This allows her to argue that higher
levels do not compete with lower levels when their sets of powers are proper subsets of
fundamental casual powers. I will return to this in 3.4.2 when I turn to exclusion arguments.
3.4.1. Physicalism and Supervenience
Physicalism is a position that essentially denies that there is any genuine emergence. A popular
way to look at physicalism is to talk about supervenience. Supervenience is a relation that holds
2 For a comprehensive taxonomy of positions in the emergence literature, see (Wilson, 2011a)
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between two properties, such that changes in one necessitate changes in the other, but not vice
versa. Imagine the properties of being red, and that of being a specific hue of red, scarlet. An
object that is scarlet is red. However, it can also be non-scarlet and still be red, but if it is non-
red, then it must be non-scarlet. In this case, the property of being red supervenes on the property
of being scarlet.
David Lewis introduces the idea of physicalism by analogy to a printer (Lewis, 1986).
The physical matter, he says, is like the dots of ink and the social and political aspects of the
world are the patterns in the ink. The patterns supervene on the dots – the image cannot be
different unless the physical arrangement of dots is changed, while the reverse is not true. Lewis’
physicalist claim is that a physical duplicate of our world is a duplicate simpliciter. What it
means to say that everything is physical is to say that any physical duplicate of our world would
be identical to it in every respect, because all non-physical properties supervene on the physical.
This is a position that has been taken up by Frank Jackson (Jackson, 1998).
Physicalism comes in two distinct varieties, reductive and non-reductive. Reductionism
has also been popular in the philosophy of science with people like Nagel (Nagel, 1961). Nagel
argued that a theory was reducible to another if one could be deduced from the other with bridge
laws. There have been strong reactions against this kind of approach, and since his time,
physicalists have moved away from it, because it seems incapable of handling the problems of
multiple realizability, the idea that the same high-level property (or behaviour, etc.) can be
instantiated in different physical properties, states, or events (Fodor, 1975; Putnam, 1988). Many
hold that a kind of non-reductive physicalism is the least problematic on the scale from strong
emergence and substance dualism to reductive physicalism.
3.4.2. Exclusion Arguments
I stated earlier that Mitchell’s and Woodward’s positions entail realist commitments to multiple
levels of causes and that there were concerns about this emergent causation. The concern, as
raised by Kim, is that emergent properties are identified by their novel causal powers, but that
because they must always be instantiated by their realizers, they are better explained by the (non-
novel) causes of their realizers. Thus, because they both require novel causal powers, yet cannot
have them, emergence is problematic.
Kim notes that in a supervenience relation, an emergent E occurs only when the base-
level realizers of E are instantiated. If one knows that neural state N occurs, and that emergent
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pain state E occurs, and that N is the realizer of E, then the presence of N explains the occurrence
of E. One can also predict the occurrence or non-occurrence of E given knowledge of its
emergent base. Because he is working from an assumption that realizers are known he is quite
confident that “all our explanatory demands can be met” (Kim, 1999, p. 13). However, this claim
is not on trail.
There is a further ontological consideration. Kim puts forward what he has called the
causal inheritance principle, which says that the causal powers of an emergent that has been
functionally reduced are identical with the causal powers of the realizer. If this is accepted then
one must ask what the status of E is, and why it should be considered distinct. He finds three
options. First, one can maintain that the emergents are irreducible to multiple realizers. Second,
one can identify the emergent with a disjunction of realizers. Third, one can simply give up the
idea that E is a genuine property and only recognize it as a useful concept. The second two are
both reductions of E. To the first, he responds that the realizers themselves must have distinct
causal powers and so the multiply realizable properties must be distinct. All this, he says, points
to the fact that E is unfit to play a role in science. No one would insist on the existence of
emergents if they had no role to play in explanation or prediction in science. Their causal
efficacy is their most important feature. For Woodward, as will be shown in the following
subsection, the right kind of invariable dependency relations at higher levels identify causes at
that level. He is not a reductionist about causes or entities.
Kim shows that there are three kinds of causation that could occur among emergents and
realizers: same level, downward, and upward (Kim, 1999, p. 19). He argues that upward and
same level causation among emergents implies the possibility of downward causation. The
argument proceeds as follows.
Suppose there is a property M at level L which causes property M+ at L+. Given that M+
is an emergent property, it has an emergent base at L, call it M*. The question of what caused the
occurrence of M+ has two solutions and only one reasonable one, viz. 2.
1. L+ M+ 2. L+ M+
L M M* L M M*
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Here the slim dark arrows represent causal relations and the black arrows represent the
emergence relation. The original scenario in 1 is far better explained by 2, since every
occurrence of M+ must be instantiated by its realizer. The only way M causes M+ is by causing
its realizer, M* which also must be caused by M. Upward causation is only possible if same level
causation is possible. The same can be argued for same level causation presupposing downward
causation.
To see this, suppose again there is a property M at L which causes M*. M* itself has a
basal realizer M- at L-. If we ask again how M* was brought about, there are two answers and
only one reasonable one, viz. 4.
3. L M M* 4. L M M*
L- M- L- M-
Again, M can only cause M* by causing its realizer M-. This can all be generalized such that to
cause a property one must cause its basal realizer – what Kim calls ‘the principle of downward
causation’. The next step in the argument is to problematize downward causation, because he has
shown that if emergents cannot have downward causation, they cannot have causal powers at all.
Kim asks why the putative cause of M cannot always be the cause of its emergent base P.
Consider case 4 above. His argument is that it can always be better explained by a cause of its
emergent base as shown below (5.).
5. L M M*
L- P P*
Here we can see that the emergent property M is entirely dispensable in bringing about M*. “If
emergent properties exist, they are causally, and hence explanatorily, inert and therefore largely
useless for the purpose of causal/explanatory theories” (p. 33). This points to a problem that has
far-ranging consequences for any kind of non-reductive position. However, it may not be
applicable to Woodward’s account, which makes use of an entirely different notion of causation.
On Woodward’s account, in order to discover an invariant relation between X and Y, one
must fix the other causal pathways such that the only effect on Y is X. If X and Y are emergents,
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irreducible higher-level variables (like M and M* above), then there can be no manipulation on X
in order to see if it is a cause of Y, because one cannot fix the emergent base of X due to the very
nature of the supervenience relation. The Kim diagrams are part causal mapping/part
supervenience mapping, but the relation between the realizer and emergent is not causal, but an
“automatic change”. Woodward argues that there is a meaningful sense in which one should
draw a causal arrow between supervening properties M and M* (Woodward, 2011). This is
because it is possible that P causes P*, and yet their emergent properties are not causally linked.
Thus, there is additional causal information that can be gained by saying that M causes M* over
and above saying that P causes P*. Contrary to Kim’s conclusion, Woodward thinks that
manipulationism is open to those emergent causes.
There might be additional information in high-level causes but it does not necessarily
mean that that is unproblematic. And, of course, not everyone is convinced by Kim’s argument.
Some have argued that the argument does not lead to Kim’s conclusion, and others that it only
applies to certain interpretations of causation. To show how one might argue that it is
unproblematic for counterfactual accounts of causation, I will turn to a defence mounted by List
and Menzies in 3.4.3. First, I look at an examination of Kim’s argument by Jessica Wilson,
which effects a defence of non-reductive physicalism by means of a metaphysical constraint on
causal powers.
Kim’s argument has been challenged by many philosophers who advocate some kind of
non-reductive physicalism or want to defend high-level causation. According to Jessica Wilson,
the problem articulated by Kim is that there is no satisfying answer to the question: how can
special science entities have real causal powers given their dependence on lower-level entities
(Wilson, 2011a). She identifies six premises that lead to the problem:
1. Dependence. Special science features depend on low-level physically acceptable
features.
2. Reality. Both features are real
3. Efficacy. Special science features are efficacious.
4. Distinctness. Special science features are distinct.
5. Causal Closure. Every low-level effect has a low-level cause.
6. Non-overdetermination. Effects are not causally overdetermined.
Kim’s argument shows that these premises are inconsistent. Based on Kim’s commitments and
assumptions, he takes this to entail the denial of P4, and thus takes the argument to lead to
reductive physicalism. However, Wilson notes that other premises may be rejected in its place,
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since all that was shown was that together they are incompatible. One could deny P1 and support
substance dualism; deny P2 and support eliminativism; deny P3 and support epiphenomenalism;
deny P5 and support strong emergence; or deny P6 and support non-reductive physicalism.
Wilson focuses on the latter two positions. Strong emergentists deny the causal closure of
the physical, which means that at least some high-level entities have the power to produce an
effect which low-level entities do not – a position taken, for instance, by Sandra Mitchell. The
powers of the high-level feature must satisfy this new power condition. Thus the movement of
reaching for a water glass when thirsty would not have a purely lower-level physical cause, but
come in part from the property of being thirsty. The emergent property of being thirsty can do
one thing that the physical realizers cannot.
The NRP claim is that special science features are real and distinct, but stand in an
intimate relation that, while not identity, is close enough to avoid overdetermination. A number
of such relations have been proposed, such as functional realization, part-whole relation, and
determinable-determinate relation. These all rely on what Wilson calls “the subset condition on
powers: Token higher-level feature S has, on a given occasion, a non-empty proper subset of the
token powers of the token lower-level feature P on which S synchronically depends, on that
occasion” (Wilson, 2011a, p. 263). This condition both avoids overdetermination and conforms
to physicalism. Wilson argues that all forms of emergent dependence conform to either strong or
weak emergence. According to strong emergence, the higher-level entities have at least one
causal power that the lower-level entities do not have. NRP is associated rather with a kind of
weak emergence, where the upper-level entity has a proper subset of the powers of the lower-
level, such as that it can be counted a distinct entity but still physically acceptable.
The ability to explain a higher-level property in terms of a physically acceptable property
is not sufficient to demonstrate its physical acceptability, because this may not speak to its
having independent causal powers, which is the distinguishing feature of an emergent property
(Wilson, 1999, p. 42). What Wilson proposes is a constraint on causal powers, CCP, that a causal
power associated with a supervenient property is numerically identical with a causal power
associated with its base property. Higher-level entities need not have a novel causal power in
order to be autonomous, they only need have a distinct power set, even if it is a proper subset.
This is because an entity is determined by the set of causal powers it possesses. What this allows
is that there can be distinct entities (as determined by non-identical sets of powers) without
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maintaining that there is a novel (emergent) causal power. In this way, a strong case can be made
that there are physically acceptable higher-level entities and that NRP does not necessarily slide
into problematic emergentism or reductionism.
However, there is a worry that the discourse of metaphysics and that of model-based
scientific explanation are quite distinct. The way that explanatory models are is not necessarily
the way that the world is. This is something that I showed in Chapter 2. In order for such a
defense to work for Woodward, he would need to be committed to a claim that the causal powers
of entities featured in explanatory models, as identified by manipulationism, stand in this proper
subset relation to each other. Another way to put this would be that physicalism and the
constraint on causal powers must hold for features of explanatory models. No argument to this
effect is given by Woodward or Wilson, and it is not obviously true.
Further, there is at least one reason to suggest that it is not always the case: some
explanatory models make use of entities that do not properly exist and employ properties that are
nowhere instantiated, e.g. perfect spheres, frictionless planes, infinite populations, etc. For
instance, Mitchell maintains that the various explanatory models at work in a complex system
describe “non-overlapping ideal worlds.” It is not clear in what sense the causal powers of
entities in a group-level model could be a subset of those in an individual-level model, and that
the powers of entities of an individual-level model are subsets of those at the genetic-level.
As we also saw in the case of semiclassical mechanics in Chapter 2, classical properties
cannot be de-idealized to quantum properties. Properties like having velocity and position are not
subsets of quantum properties like having a spin quantum number. Wilson’s concern is not with
properties in general but with causal powers. The constraint on causal powers means that the
powers belonging to entities in explanatory classical models would have to be proper subsets of
the powers of entities in explanatory quantum models. The lack of causal powers at the quantum
level shows that features of semiclassical models at least do not satisfy the constraint on causal
powers.
Generally, it is difficult to see how there could be known or demonstrable proper subset
relations between features of models where there are no known regular limits or other mapping
relations. Because Woodward’s account holds that manipulation relations among the variables of
high-level models describe causes that are non-fundamental, he is unlikely benefit from this
defense of NRP. This is something I expand upon in the following section as well. It is important
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to note, that NRP has not been problematized here, and may well turn out to be true of the world.
However, the entities and dependency relations of explanatory models need not reflect a
consistent metaphysical position – that requires an additional claim of physicalism about models.
The claim that the real entities of the world are physically acceptable and that the claim that the
entities featured in explanatory models are physically acceptable are distinct. If the constraint on
causal powers was shown to be true about explanatory models, then it seems that problems of
high-level causation would disappear. But a convincing argument for this has not been offered.
There is no obvious reason why physicalism and the constraint on causal powers ought to hold
for the features of explanatory models, and good reason to think that in some cases it does not. In
the following subsection, I turn to a different kind of defense that is geared towards
counterfactual accounts of explanation and may turn out to be more useful for Woodward.
3.4.3. Intervention and Emergent Causation
In order to see how a counterfactual account could be defended against Kim’s argument, I will
look at one such defence mounted by List and Menzies (List & Menzies, 2009). Woodward
distances his notion of causation from the classic causal-mechanical view of Salmon. But
because of this, it is not clear that Kim’s causal exclusion argument applies to his interventionist
notion of causation, which is quite distinct from the one Kim problematizes. List and Menzies
attempt to make use of a concept of causation as difference making in order to show that the
causal exclusion principle only works for some formulations of cause. They argue that causal
exclusion actually supports the causal autonomy of certain higher-level properties. However,
similar to the application of subsets in the previous subsection, I think that it is not obvious that
the features of explanatory models always stand in the right supervenience relation to one
another. I argue that the defense against the exclusion argument fails to save Woodward’s
account because it too does not apply.
List and Menzies interpret Kim’s argument slightly differently than Wilson. They claim
that non-reductive physicalism has three theses: The properties of the special sciences are not
reducible to physics, but multiply realizable by them; these properties supervene on physical
properties; these properties are the causes and effects of other properties. Kim’s argument
proceeds by showing how the first two of these contradict the third. It makes use of the exclusion
principle: if a property F is causally sufficient for some effect G then no distinct property F* that
supervenes on F can be a cause of the effect G. Simply put, problems of overdetermination are
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avoided because only supervening properties are excluded. List and Menzies have two main
claims in this 2009 paper. First, that the truth of this principle is contingent, and second, that
when true, can actually support causal autonomy at the higher levels. They opt for an account of
cause as different making, which aims to include contrastive, counterfactual, and interventionist
accounts of causation, such as Woodward’s.
A central assumption in this argument is that a cause should be proportional to its effect,
something they draw from Stephen Yablo (1992). They argue that in many cases the difference
maker in the occurrence of a phenomenon is the supervening property and not the realizer. Given
the case of a pigeon who is trained to peck at red targets for food, the redness of a target is the
important factor in pigeon pecking, rather than the particular hue of red (Yablo, 1992). In fact,
the hue is not sufficient to count as a cause, or the pigeon would not peck at any red colour, but
only a particular hue, which it does not.
List and Menzies look at two counterfactuals to apply the idea of causal exclusion to a
difference making account of causation.
1a Target is red □→ pigeon pecks
1b Target is not red □→ pigeon does not peck
1a and 1b are true, as outlined by the description of the thought experiment. But, consider the
following two counterfactuals about specific hues of red:
2a Target is crimson □→ pigeon pecks
2b Target is not crimson □→ pigeon does not peck
2a and 2b are not both true despite the closest world being where the target is still some shade of
red where the pigeon would peck. The supervening property is clearly the difference maker. The
counterfactual that demonstrates the difference is the one that identifies the relevant cause. And
so, on this account of causation, there can be high-level causes without an underlying causal
relation. In effect, the excluded cause is the realizer. Kim did not consider causation to be cashed
out in counterfactual terms of any kind, but in a primitive notion of production or generation, and
this, they argue is why high-level causation seemed to be problematic.
On Woodward’s account, wherever there are the manipulability relations, there are causal
relations and vice versa. What this means is that if one can reliably manipulate M in order to
change M*, then M is a cause of M*. It is also true that if one can reliably manipulate P in order
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to change P*, then P is a cause of P*. Thus, the Kim-style diagram for Woodward would take the
following shape (assuming the properties are related via supervenience relations).
6. L M M*
L- P P*
What this implies is that unlike the strategy taken by List and Menzies (or Sherman (1988)), for
Woodward, there can be genuinely emergent causes.
As was shown in 3.3, Woodward’s notion of causation is different than Kim’s, but it is
also different from List and Menzies notion. It does not operate by selecting the difference
maker. It can allow for the higher-level generalization to be explanatory, while not having to
deny that there are causes also at the system’s micro-level. He puts forward no principle of the
proportionality of causes that selects a single level of causal operation. Because manipulationism
identifies both high-level and low-level generalizations as causal and explanatory, then
Woodward, like Mitchell, is committed to emergent causation. This position makes realist claims
about both the lower and higher levels; for Woodward, any level where a generalization passes
his criteria for invariance under interventions makes claims about real causes
On Woodward’s account, and on Mitchell’s, there are many models and levels of models
which exhibit causal dependencies. The relations between features of models may not be that of
supervenience. The models might be competing models on the same level, or irreducible in terms
of time scale or spatial scale. Just as in the case of the proper subset relation, there is no reason
given to think that the entities and causal properties of a model at higher levels will necessarily
supervene on those of models at lower levels. There are surely some cases where this relation is
clear, as in the simple examples often cited. But it has been argued that in other cases there are
no such relations, for example concerning entangled particles, and other non-classical properties
(Karakostas, 2009). I am not committed to this claim, but only wish to point out that in the case
of explanatory models, the argument has not been presented that they will stand in the
appropriate supervenience relations. And further, that it is far from obvious that the properties of
various idealized, and highly-idealized, models will stand in a determinant-determinate relation
or a relation of parts and wholes.
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Nonetheless, Woodward’s and Mitchell’s accounts identify explanations and causes at
multiple levels. I think that there are some reasons to be concerned about emergent causation that
implicitly claims that models track causes. I assume that there are explanations at multiple levels
and that this is not itself problematic. It seems like there are a few options to make sense of this:
1) Claim they are all real causes; 2) Privilege a certain level as real causes; or 3) remain silent on
whether they exhibit real causal dependency relations or not. Each of these has some
implications, some more undesirable and problematic than others. List and Menzies opt for the
second position, as does Strevens, which will be shown in the following section. Woodward’s
and Mitchell’s strategy is to take the first option, but this involves demonstrating that emergence
is unproblematic, or that some other defence is applicable. I advocate the third option to remain
silent on whether the dependency relations among the variables of explanatory models exhibit
causal dependencies. As was discussed in 2.1-2.3, different models can be used for different
explanatory purposes, but there is nothing in this alone that necessitates that there also be
different causal dependencies in the world that correspond to the relations among the variables of
the models. Giving explanatory models at multiple levels a causal interpretation is at best
unnecessary and at worst problematic. But, if one is to remain silent on whether the dependency
relations in explanatory models are causal or not, then one has to provide another means of
identifying genuine explanations, which is what I offer in Chapter 5. There are predictive
divergences and conceptual inconsistencies among models, but this need not preclude their being
explanatory. By favouring a non-representative account of explanation, it is much easier to make
sense of the fact that explanatory models make use of entities and relations that may not be
physically acceptable.
None of this is to demonstrate that causal accounts cannot succeed or that non-reductive
physicalism is untenable. Rather the aim is to demonstrate that there are some challenges that
make a causal approach less attractive. And nor is this meant to be an exhaustive treatment.
There remains a great deal of work to be done regarding supervenience relations and scientific
models, which is outside the aims and scope of this dissertation. Woodward’s account requires
knowledge of causes in order get further information about causal dependency relations, and may
never get off the ground. It is also committed to causal emergentism, which invites problems of
overdetermination and downward causation. Defenses mounted to preserve either NRP or high-
level explanations are not going to work for Woodward, whose account identifies genuinely
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emergent causes. Wilson’s and List and Menzies’ defences of NRP may be strong defenses of a
physicalist conception of the world, but it is not obviously true of explanatory models. Let me
now turn to Strevens’ account which may be more compatible with these defenses because of its
explicit rejection of emergent causal facts.
Kairetic Explanation
This section looks at an account of causal explanation that is compatible with non-reductive
physicalism and specifically geared towards capturing the explanatory role of non-fundamental
models. Michael Strevens’ kairetic account of explanation begins with a basis in causal
explanation, but proposes an account of depth that can prefer higher-level models (Strevens,
2004). Strevens’ aim in this project is to use unificationism to accomplish one of the major goals
of causal accounts, which is to specify the relevant causes of a given effect, such as to constitute
an explanation of that effect. He is using unification in order to pick out not the most unified
theories, but which causes are the difference makers. In this, it is similar to the position presented
by List and Menzies, though the focus is not on counterfactuals.
Strevens claims that any of the common accounts of causation are sufficient to give
causal asymmetry, and so he does not want to specify the details of the metaphysics of causal
relations and will instead focus on relevancy criteria. He thinks this approach can ignore the
problems (and potential benefits) of metaphysical realism. He does not even argue that causation
is either reducible or not, whether explanations feature laws or not, whether all causation is local,
or forward in time. The only strong claim he thinks he needs to make is that there is nothing over
and above fundamental-level causation: “Causal emergentism has no place in the causal
approach to explanation” (Strevens, 2008, p. 35). There are no independent high-level causal
facts. As such, fundamental-level causal influences can explain everything that can be explained
causally. Because of this, his account faces different problems than Woodward’s or Mitchell’s
does. In 3.5.1 and 3.5.2, I present his take on idealization and abstraction and how a notion of
depth is supposed to preserve high-level explanation on this picture. In 3.5.3, I present some
objections that have been raised to this account concerning how to weigh his criteria, the process
of abstraction, and the unrealistic process of forming an explanation.
On Strevens’ account, explanation is a two-step process of isolating difference makers.
The first step is to start from the causal web of an occurrence E and remove all causal influences
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that are not necessary to bring about E. The second step is to remove all things that are not
relevant to the logical entailment of E by the set of circumstances and laws. What is left is what
he considers to be the causal difference makers for E. The role of logical entailment here is only
to represent the causal factors, which is why it is the second step. If it were the first step, non-
causal dependencies would be allowed. He finds that in this way one can debar claims about the
gravitational influence of Mars being causally relevant, for instance, to Strevens’ much-loved
case of the death of Rasputin. The goal is to separate what makes a difference to the physical
system from what makes a difference to the explanandum.
His account can, he claims, identify all the difference makers for a particular event. The
eliminative procedure is to remove as many parts of a causal model M that entails event E
without invalidating the entailment of E. This is followed by an optimizing procedure which
constructs sets of statements, or models, to form a standalone explanation (more on this in 3.5.1).
He takes a causal model to be a set of veridical and deterministic causal statements about
the world that entails E. An atomic model may be thought of as picking out a single link or a
length in a long causal chain. A composite model contains two or more atomic models strung
together. For Strevens, explanations are still deductive entailments, and what makes them causal
is that they make use of causal laws whose content is determined by the metaphysics of causal
influence. There are no independent high-level causal facts, and so fundamental-level causal
influences can explain everything that can be explained causally.
3.5.1. Abstracting and Optimizing
The first step of his account is to eliminate causal influences that are irrelevant. Picking out
difference makers is the process of identifying which causal factors are relevant to the
occurrence of E, by looking at what plays an essential role in the causal entailment of E. If C
cannot be removed from the causal model without eliminating E, then C is a difference maker.
He claims that the kairetic account can solve many of the problems known to face causal
accounts.
The process he refers to as abstracting is that of ignoring the details of a model. This is
the process by which a mechanism is substituted for a black box making no reference to the
internal causal process. His example involves explaining why a window broke when a
cannonball was thrown at it. The fact that the projectile weighed exactly 10kg is not important to
the explanation of why the window broke. What is important is that the ball was rather heavy,
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say more than half a kg. Strevens gives us the conditions under which he considers a model to be
an abstraction of another:
A. One model M is an abstraction of another M' if its propositions are entailed by the
detailed model M' and if all the causal influences of M are described by M'.
In this case, it follows that if the cannonball is 10 kg, that it is more than half a kilogram. What
makes a difference is that the cannonball is either above or below a certain threshold, not that it
is or is not exactly 10kg. By abstracting in this manner one can get to the difference making
cause, what Strevens calls a kernel.
A model can also be too abstract. In order to prevent the optimization from favouring
radically and uselessly disjunctive models, he imposes a cohesion requirement on the process. A
model is cohesive only if all of its realizers possess the same causal elements; if it is causally
contiguous at the fundamental level. The kairetic account is constrained on both sides by
ignoring causal detail and requiring cohesion among the model’s realizers. This is what Strevens
finds so promising in this approach, the balance between abstractions and causal realism. An
abstract model is best unless it violates the cohesion requirement. This happens in cases where
the model is radically multiply-realizable.
3.5.2. Idealization and Causal Realism
As was mentioned in the last section, causal realism plays a strong role in causal accounts of
explanation, and the kairetic account is no exception: “no causal account of explanation –
certainly not the Kairetic account – allows nonveridical models to explain” (p. 297). Thus,
contrary to Bokulich, myself, and many others, he explicitly rejects that highly-idealized models
can support explanations. Models may intentionally misrepresent elements of the causal
mechanism, but Strevens’ kairetic account “demands that factors be omitted in a way that does
not compromise the veridicality of the model” (p. 298). He examines the case of Boyle’s Law,
which assumes that molecules in a gas do not collide, even though they surely do. Strevens must
justify this falsity.
Strevens does not take the stance that idealizations are better on pragmatic grounds. For
this strategy, idealizations are compromises of a perfect explanation, which would be entirely
veridical. This downplays the importance of idealization in explanation. For Strevens,
idealization makes an explanation better by conveying only essential information. He goes as far
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as to claim that an idealizing explanation is always better than its veridical counterpart. It is
better because it does not present all causes as equal, but rather highlights the difference makers
that are key to good explanations. He characterizes an idealization as the following: It is false;
the false claim fills out details left unspecified; and the details are filled in with zeros or infinites
in order to eliminate them, because they do not matter (p. 318). He is a physicalist, but he is not
reductive with respect to explanation.
Take for instance Boyle’s law relating pressure and volume,
(10) 𝑃 ∝1
𝑉.
A textbook explanation of gas behaviour that makes use of the law features many idealizations,
such as that the collisions with container walls are completely elastic, and that molecules do not
collide with one another. It gives a decent explanation, but makes mostly false assumptions about
the nature of the gas. There is another common explanation involving a more complete
description from modern kinetic theory, including the influence of the molecules on one another
at a distance, and allowing for intermolecular collisions. Strevens’ claim is that one can eliminate
details of the kinetic model that make no difference to the Boylean behavior of the gas and
recover the relation found in (10). It is only in this way that one can understand Boyle’s law and
how it can be explanatory; the relations that are highlighted in that law are the difference makers
for Boylean behaviour. Thus, the textbook explanation is the best; it is not the most veridical, but
it has only difference makers.
Idealized models are favourable to veridical models in a few ways: they highlight the
irrelevance of certain factors; they are much simpler; they are effective predictors as long as the
idealization is reasonably faithful. That a kairetic explanation is always at bottom a physical
explanation does not imply that kairetic explanatory models must describe the trajectories of
particles. Idealized models can be effectively employed where the omissions they make coincide
with the non-difference makers of the kairetic explanation. Strevens justifies the explanatory
value of idealizations by claiming that they are only distortions of non-difference makers. If the
range of values a variable can take make no difference to the explanatory target, then the
idealization is justified.
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3.5.3. Concerns about the Kairetic Account
The kairetic account attempts to strike a balance between the fundamental causes in nature and
the explanatory value of idealized models. Strevens’ account is complex and nuanced and worthy
of in-depth discussion on a variety of topics. I will restrict this to a few concerns that seem most
worrisome for the account. For Strevens, the first step to an explanation is to begin with total
causal information. One must ask, in what circumstance is this the first step to forming an
explanation? One can use Strevens’ account to justify judgments about higher-level explanations
if they are abstractions of a complete picture of the base-level causes of a system. But this
ignores the fact that when modelling a high-level behavior, a scientist will not begin by looking
at the fundamental level of interactions and idealize away non-difference making processes. His
account is essentially a justification of textbook explanations and singular event causal
explanations in the face of more accurate competitors, but it does not reflect the practice of
scientific explanation or modelling outside of this pedagogical context.
Echoing concerns mentioned in 3.4, the scope of this account is quite limited. Very few
models are just abstractions of base-level causes. He maintains that it is consistent to allow for
the ideal gas law, because it is an abstraction of physical laws described in the molecular model
of a gas. But his notion of abstraction (A) is quite stringent. It involves a requirement that an
abstract model is entailed by a lower-level model and that all of its causal powers are described
by the lower-level model. This precludes any case in which this kind of reduction has not been
performed, including highly-idealized models and models with multiply realizable properties. In
order for a high-level model to be explanatory, it must be able to be mapped somehow onto the
real causal mechanism, or “distilled” from it, as Strevens says. These models may be
explanatory, but I and others maintain that this is a small subset of the set of explanatory models.
What I view as the most serious problem for the kairetic account is that the process is
very detached from the scientific practice of explaining. This is an objection raised by others
who have noted that his oft-cited examples are far from those considered scientifically adequate
(Hartmann & Schupbach, 2010; Levy, 2011). He focuses on what I have called common sense
explanations, and on maintaining an account of causation that is strongly continuous with an
everyday sense of what it means to explain. He refers throughout the book to an anecdotal case
study of the death Rasputin. He uses this to explore issues such as overdetermination, pre-
emption, and more. He takes these results as having direct implications for his account of
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scientific explanation. The accounts limited scope and distance from the explanatory practice
science make it a bit too ideal.
It is Strevens’ aim to formulate an account that allows for trade-offs between the
generality gained in the optimizing procedure and the cost to cohesion, but Potochnik argues that
there is no reason to always prefer the most general model (Potochnik, 2011). He finds it
unproblematic to trade off cohesion and accuracy for generality across the board. If the cost is
low, he says it is mandated to make the trade-off. But Potochnik notes that it is not always the
case that the most general explanation is the deepest. On some occasions a scientist might prefer
models that highlight subtle perturbations and explore fine-grained dynamics. There is no
objective standard for determining these trade-offs. Hartmann and Schupbach also argue that a
great deal of work remains to be done to fill out the concepts of accuracy, cohesion, and
generality, such that they and the account might be meaningfully applied (2010). They also
mention that the limited scope of Strevens’ account is slightly ironic. There are much more
general accounts of explanation and given its preference for trading off accuracy in favour of
generality, its limited scope almost seems to “mandate its own rejection.”
Strevens attempts to formulate an account that captures how high-level models can be
explanatory. In the end, it is very limited in scope, provides very few measures for implementing
and trading off its desiderata, and is not reflective of explanatory practice.
Conclusion and Additional Concerns
Mitchell, Woodward, and Strevens recognize the need for higher-level explanations and have
formulated causal accounts of explanation that attempt to include some higher-level models as
explanatory. Mitchell’s solution of integrative pluralism is hardly a solution at all. It provides no
framework for performing the integration of multiple levels of models into a singular multi-level
causal explanation. And further, even if it did, it advocates that a system-specific explanation is
going to be the best explanation of a complex system, but such system-specific models tell us
very little about why the system behaves similarly or dissimilarly to other systems. By including
all models that can be used to describe a system as explanatory and veridically tracking partial
causes, there is no actual threshold for explanation: all models qualify. If we want to maintain
that some models are mere phenomenological generalizations, then some reasonable threshold
for which models count as explanatory ought to be set.
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Causes on Woodward’s account are determined by the reliable variable dependencies of
models. This allows for models at levels other than that of fundamental physics to be considered
explanatory. This strategy relies on a commitment to causal realism that provides underlying
metaphysical truth-makers to support causal claims and debar non-causal ones. But his account is
at least partially revisionary and so determines to an extent what is considered causal. As such,
he has an ambiguous stance towards causal metaphysics. His account does find many high-level
generalizations to be causal and explanatory, which threatens to invite the problems of
emergentism. Unfortunately, defenses of NRP and high-level explanations, such as those given
by Wilson and List and Menzies, cannot be used to rescue Woodward’s account. This is because
Woodward is committed to independent high-level causal facts. Further, the literature on
emergence and physicalism and the literature on models and scientific explanation talk past one
another to a large extent. Deciding which models are genuinely explanatory need not coincide
with physicalism, for instance, when one takes a non-representative approach, the properties and
features of explanatory models are under no obligation to be physically acceptable, even for a
physicalist. A lot of work remains to be done concerning where supervenience relations might
hold between features of models, and where not. The present chapter could only survey some of
the various positions and possibilities.
I have expressed suspicions that supervenience relations and proper subsets are not
ubiquitous among explanatory models. However, it is interesting to note that in a case where
there are no supervenience relations, then the problem as articulated by Kim does not even apply.
The argument is aimed at exposing problems of overdetermination and downward causation for
non-reductive or emergent accounts that employ supervenience. So, in the cases where there are
no such relations, this is not an issue. It is possible that this could present a viable way of
circumventing Kim-style arguments. I was not able to explore this avenue here. What this
suggests is that perhaps the features of explanatory models are not best understood as exhibiting
genuine causal relations, which reflects the limited scope of causal accounts.
The strategy taken by Strevens is to focus on the explanatory relevance relation among all
the fundamental causes in a system, thus denying any high-level causal facts. By beginning with
a complete causal story and swapping out processes for black boxes, only relevant causal
difference makers remain, and thus abstractions and certain kinds of idealization play the role of
highlighting important explanatory information. But the two-step process for explanation is
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largely fictional and does not reflect any actual explanatory practice. The account seems to have
a different aim than to capture scientific explanation. It shows how one could justify preferring
abstract models to more detailed models, and how harmless idealizations allow important
information to be highlighted.
It is worth reiterating my concern with the limited scope of causal accounts, which has
been emphasized throughout this dissertation and by many others. Deductivist approaches to
explanation promise to permit a wider variety of models as explanatory, including those that
feature non-Galilean idealizations, laws and principles, and other causally unacceptable features.
For instance, non-reductive systems where long time-scale behaviour cannot be smoothly
approached by models of short time-scale behaviour, and systems where a high-level
explanandum behaviour cannot be explained with the use of low-level models would not be
precluded. There are many cases of idealized models which are not representative of target
physical systems. All of Batterman’s asymptotic explanations of universal behaviour do not
qualify as causal explanations (Batterman, 2002b, 2002a). Further, cases for non-causal
explanation will be presented in reviewing Hempel and Kitcher in the following chapter.
There are some additional reasons that I think warrant being skeptical about the promise
of causal accounts of explanation. These are not arguments that are meant to seriously
problematize the causal accounts examined in this chapter. I am merely raising some flags on
issues that I think would benefit from further investigation and development. My main concerns
stem from commitments to causal realism. This has implications for the kinds of idealizations,
features, and relations that can be permitted in explanatory models. This in turn severely limits
the scope of such accounts and is potentially problematic.
It seems that causal accounts of model-based explanation must be committed to some
degree of realism about explanatory models. The relations in explanatory models need to
describe real causes or encode their causal information in some way. Let us assume that
explanatory models track real causes and that propositions about these models are approximately
true. I have two reservations about this being an accurate way to look at explanatory models. The
first is that the various models that operate at various levels in a real-world system diverge in
their predictions of its behaviours. Some are more accurate, others more general, some are only
accurate at short time-scales, others only at long, and so on. They may focus on different
explananda, but if they concern the same real-world system, then they are all tracking real causes
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in the system. This works fine where the predictions are all in line. But it seems that if the
predictions of two or models diverge significantly, then they cannot both be tracking real causes.
This realism about explanatory models also has implications for the features of these
models. And so my second concern is that the features and the entities of explanatory models are
incompatible, in the sense that propositions describing the models are contradictory. This can be
seen in the explanations of light diffraction in particle and wave theories of light, and the
statistical mechanical and kinetic explanations of the behaviour of gases and fluids. It seems odd
to maintain that the entities in this real-world system are both fluids with no discrete part and are
mere particles, or that the air has a damping effect and does not. Again, I find these two
scenarios puzzling, but I can develop them no further here. It is also worth mentioning that this
would not be puzzling at all if one’s account maintains only fundamental-level causes, because
there is no need for realism about high-level causal models, but this is not the position taken by
Woodward and Mitchell.
Strevens’ account in particular highlights a different concern about the prospects of
causal explanation in general. Strevens relies on a known web of causal influence, among which
difference makers are selected for a particular explanandum, but this starting point is in serious
trouble if there are no causes at the fundamental level. Fundamental physics (i.e. quantum
mechanics) is non-causal and does not entail the propositions of classical physics, nor describe
its causal influences. Taking this seriously would preclude all classical explanations, because no
classical model counts as an abstraction of a fundamental model. In which case, if one cannot
abstract smoothly up from the base level then there are no high-level explanations. This goes
beyond the epistemological and methodological concern raised above. It is widely understood
that at the level of fundamental physics there are no classical causal processes. If one takes
Strevens’ requirement on abstraction literally, it is only applicable in a classical world where the
lowest causes are kinetic interactions between atoms or molecules. And thus, there is no place
for the fundamental causes to enter.
One last concern is a very general epistemological one. I have mentioned it as a drawback
that Woodward’s and Strevens’ accounts begin from correct causal knowledge: knowing that I
causes X; knowing how to arrange a directed graph; and beginning from a complete picture of
fundamental causes. The reservation about beginning from this point is that it arouses a
perennial, epistemological concern stemming from Hume about how one gets there in the first
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place. Kitcher has noted that philosophers concerned with explanation once strongly avoided
invoking causation. To borrow his words, appeals to causal knowledge were seen as appeals
“that would make life so much easier if only they could be made” (1989, p. 460). In the wake of
logical empiricism, invoking causes has become commonplace. But, the concern that remains is
not that it offends empiricist sensibilities. The concern is that we cannot justify our inferences to
causal claims and so we begin with them. I sometimes see these starting points as a kind of
conditional: if one has knowledge of causes, then many of the issues surrounding causal
explanation disappear. Indeed.
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Chapter 4. Deductivist Explanation
Introduction
This chapter will review the tradition of deductivist accounts of explanation to survey their
successes and failures in order to inform the account I propose in Chapter 5. Hempel and
Oppenheim began the tradition of analyzing scientific explanation and offering a metatheory
about what an explanation does and what it ought to do (Hempel & Oppenheim, 1948). Their
deductive-nomological (D-N) account has met with many objections. Some argue that the
account is not necessary, because there are many kinds of important scientific explanations that
are outside the scope of the account, such as explanations that make use of statistical regularities.
Others have argued that the account is not sufficient for explanation, because many non-
explanatory derivations meet the D-N criteria. These criticisms led philosophers to turn to a
stronger notion of causation as a mean of debarring certain of these now well-known
counterexamples. The D-N account was formed around the idea that explanation was derivation
– that an explanation should take the form of an argument which derives the explanandum from a
set of sentences that contain at least one general law (Hempel & Oppenheim, 1948).
An explanation functions like a syllogism. One has a law or set of laws as the major; as
the minor, there are sentences of particular facts about the antecedent conditions that are
subsumed by the law or laws; and we find the explanandum as the deductive conclusion. These
explanations are intended to be strongly relevant to the explanandum, since they do more than
simply give good grounds for phenomena, they logically entail them. Questions of the sort ‘how
did this phenomenon happen?’ are regarded as ‘according to what general laws and by what
antecedent conditions does the phenomenon occur?” This account applies not only to particular
phenomena, but can also be applied to explain regularities. This occurs in the same way, i.e., by
subsuming one law under a more general one. It is in this way that one can account for the truth
of Galileo’s laws, since they can be deduced from Newton’s laws of motion.
Section 4.2 begins by reconstructing Hempel and Oppenheim’s D-N account and then
reviewing some of the challenges that have been issued regarding its necessity and sufficiency.
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This involves reviewing the role of statistical generalizations in explanation, and reviewing some
counterexample cases. In 4.3, I turn to Kitcher’s account of explanation, which sought to
supplement the D-N account with a formal account of the explanatory power of theories by
unification. After looking at Kitcher’s proposed solutions to the D-N’s original problems in
section 4.3.1, I review some objections that have been raised for unificationism. In 4.4, I then
present an overview of the current state of deductivist accounts, looking at what challenges
remain to be adequately dealt with and in which direction some solutions may lie. The results of
this directly inform the model-based deductivist account I propose in Chapter 5.
The D-N Account
Hempel and Oppenheim begin by distinguishing explanation from description by distinguishing
knowledge how from knowledge why. When put in these terms it is not unforeseeable that some
would question whether science can indeed offer explanations over and above descriptions.
Hempel and Oppenheim set out to characterize the kinds of deductive arguments that could be
said to do such explaining. The D-N model they outlined was the first serious attempt at showing
that science is concerned with explanations.
Hempel and Oppenheim analyse explanation into the explanans and the explanandum.
The latter is the phenomenon to be explained and the former is what does the explaining.
Explanans are subdivided into two: the antecedent conditions and the law statements. Together,
these jointly entail the explanandum. The derivation serves to give explanatory information
about the explanandum’s occurrence by showing under what conditions and according to which
scientific law or laws it was to be expected. In order to qualify as an explanation, the derivation
must satisfy certain conditions of adequacy R1-4 as follows (Hempel & Oppenheim, 1948, pp.
137-138):
R1. The explanandum must be a logical consequence of the explanans.
R2. The explanans must contain general laws.
R3. The explanans must have empirical content.
R4. The sentences of the explanans must be true.
They give the following explication, where T is the law statement, and C is the antecedent
conditions:
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<T, C> is a potential explanans of singular sentence E (explanandum) only if:
1) T is essentially general and C is singular, and
2) E is derivable from T and C jointly, but not from C alone.
They are careful to note that this is not a definition but a necessary condition. It lacks
sufficiency, which would allow any statement to be joined to a theory and constitute an
explanation. In order to prevent some false explanations via the quirks of truth-functional logic,
they add the following restriction that
3) T must be compatible with some basic class of sentences which has C but not
E as a consequent, i.e., T must be verifiable without reference to E, in order to
avoid circularity.
(Salmon, 1989, pp. 20-21)
The explanandum also need not be a single occurrence but could be a kind, thus giving the
explanation universality. This would take the form of ‘in all cases of kind F, conditions of kind G
are realized.’ But not all statements of this form are true laws of nature. For example, Kepler’s
and Galileo’s laws are only considered approximations. Statements concerning restricted cases
would also display the same form but fail to be true universals, because they are only accidental
generalizations. The difference between accidental and true generalizations is not easy to
articulate. Goodman finds it in the ability of laws to support counterfactual and subjunctive
conditionals about potential instances (Goodman, 1973). But any universal statement can only be
counted as a law if it is implied by the accepted scientific theories at the time, and will not
qualify as law if it precludes hypothetical occurrences that an accepted theory finds possible.
In order for explanation, only laws of nature and not accidental generalizations can be
featured in the derivation. True laws are those that express real empirical regularities. It might be
true that all the coins in my pockets are quarters, but it is a law that all gases expand when
heated. Distinguishing between true laws and accidental generalizations has been problematized,
as reviewed in 2.2.2. Hempel and Oppenheim introduced a formal language in which to
formulate laws. It is essentially a standard first order calculus with no open statements. Hempel
resisted formulating a general account of laws. He did not think it was necessary as long as we
can recognize and agree on whether a generalization is a law. We need not know why a
generalization is a law, as long as it is one. Even though Hempel would not provide a definition
of a law, he outlined four characteristics. A law must be (1) universal, (2) unrestricted in scope,
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(3) not referring to particulars, and (4) contain only purely qualitative predicates. This
characterization of law has been criticized as too stringent and not necessary for explanation by
Woodward, Mitchell, and others (Achinstein, 1971; Mitchell, 1997; Lawton, 1999; Mitchell,
2000; Woodward, 2000; Frisch, 2014). A laxer characterization of explanatory regularities would
open the door to more kinds of explanation, including those citing probabilistic causes, such as
‘smoking causes cancer’. The literature since Hempel has been moving in the direction of a more
inclusive conception of law or explanatory generalization.
Hempel argued that causal explanations can be formulated in the character of a D-N
explanation, but that not all D-N explanations are causal. For instance, Kepler’s laws of motion
can be explanatorily derived from Newtonian mechanics, but this is not in virtue of Kepler’s
laws being caused by Newtonian mechanics. Thus, the D-N account is capable of supporting
causal explanations, but is more general. This empiricism implied a regularity account of
causation, which enabled Hempel to respond to certain criticisms about the D-N but also
generated others, as will be shown in the following subsections.
4.2.1. Is it Necessary?
The D-N account has met with numerous concerns that chiefly fall into two categories: concerns
that it is not necessary for explanation (it is too narrow), and concerns that it is not sufficient for
explanation (it is too broad). Together these are taken to show that satisfying the D-N criteria is
not really relevant to capturing explanations. It is a self-acknowledged limitation of the D-N
account that it is not necessary for explanation; there are many kinds of explanation that do not
meet these criteria. However, some hold that this is a serious limitation of the account, because
there are important kinds of explanation that ought to be included, like causal explanations that
make no reference to laws, and statistical explanations.
Michael Scriven argued that many explanations are causal and make no reference to laws
(Scriven, 1962). For instance, his claim is that a statement like ‘the impact of my knee on the
desk caused the inkwell to tip over’ is an explanation of the tipped-over inkwell. This is often
referred to as a singular causal explanation, and notably, it makes no explicit reference to a law.
How is the D-N account to handle such simple and common cases? Hempel’s response is to
argue that the use of ‘cause’ in that sentence is indicative of a causal regularity that links knee
impacts on desks under certain conditions with the tipping over of inkwells (1965a, p. 423).
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Thus, what makes the single sentence causal statement and explanation of the event is that it, at
least implicitly, shares the D-N form in virtue of a causal regularity.
Explanations featuring statistical regularities are another such example. Statistical
explanation is outside of the scope of the D-N account, because it is not deductive. It is
nonetheless prevalent and important in science, such as in the predictive use of radioactive decay
and in the basic laws of genetics. Because statistical regularities are not universal and necessary
laws, it is always possible that they be undermined by new information. When the statistical law
fails to predict, it has no explanatory significance for the case at hand. The fact that a statistical
explanation could have the same true premises and yet would yield both favourable and
unfavourable predictions is what Hempel calls “the ambiguity of statistical explanation”
(Hempel, 1965a, p. 382). Hempel was pressured by critics into extending the D-N account to
include statistical or probabilistic explanations.
4.2.1.1. Statistical Explanation
Hempel distinguishes two types of statistical explanation, the deductive and the inductive. The
deductive-statistical (D-S) performs much like the D-N, but it deduces one statistical uniformity
from a more general statistical law. The inductive-statistical (I-S) involves the subsumption of
events under statistical laws. This kind of inference offers explanation where it finds a high
degree of probability to an event. For instance, if one wants to know why they failed to roll three
sixes on three dice, the high probability of this failure is explanatorily relevant. Thus, while not
guaranteeing the explanandum, a degree of rational expectability can still be conferred to the
explanandum, given the high probability in the explanans. In certain cases, it definitely seems as
though there are genuine statistical explanations.
Nicholas Rescher was one who made a plea for the inclusion of statistical explanations
(M. King, 2014). Hempel responded the same year with an inclusive account (M. King, 2015).
These explanations also take the covering law form of following from laws, but from statistical
laws. They, therefore, cannot be arrived at with deductive certainty, but with at most high
probability. It usually takes the form of:
P(G|F) = r
Fb
====== [r]
Gb
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The idea behind these explanations is that if one can give a statistical law that says that an event
G is highly likely given another event F, and one looks at an example of an F event, then one has
an explanation of the G event (Hempel, 1965a, pp. 383-384). While this does grant predictive
value, it fails to necessitate G’s occurrence. This is simply because the same explanans could be
true even if the G event had not taken place. The problem is that one can generate an acceptable
derivation with either outcome.
Inductive logic has no weakening principle like deductive logic. The addition of a new
and contradictory piece of evidence can conclusively refute a previous strongly favored inductive
argument. Thus, there is a principle of total evidence, which makes the claims into true
accidental general statements, which cannot be refuted by new evidence. But if the conclusion of
such an induction is included among the premises, then the argument is not inductive at all, but
deductive. So Hempel had to look for a less stringent requirement than that of total evidence.
This he called the requirement of maximal specificity, which requires that the conditions reflect
all the relevant information about the specific situation. Hempel holds that it is high-probability
regularities (𝑟 > .5), are the ones that can support explanations. Hempel maintains that
explanations confer nomic expectability, even given that the covering law can either be statistical
or universal.
The ambiguity of I-S and other problems led philosophers to offer alternative accounts of
statistical explanation. Salmon raises several criticisms of the I-S account. He argues that it is
unnecessary for explanation, because it is unable to handle cases where the probability is low. If
the probability of an event occurring is 1%, then for Hempel it is not explanatory, even if it is the
only known explanation. This is what was seen in the case of the mayor who develops paresis in
1.3.1. Salmon also argues that it is not sufficient, because derivations meet the I-S requirements,
but are not explanations. For example, consider the case where John has a cold and is taking
vitamin C. There is a statistical relation between the taking of vitamin C and a cold’s
disappearance in a week. Because colds generally clear up in a week anyways, the fact that he is
taking vitamin C ought not count as an explanation of the cold’s disappearance. Salmon’s
solution to the problem is to argue that what is important is a change in the probability of one’s
getting well in a week. Salmon provides an account of causation which replaces high probability
of the explanandum with the statistical relevance of the explanans on the explanandum. Salmon’s
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strategy required that he compare a prior and posterior probability, and thus establish that the
result is due to the intervening cause.
Kitcher’s strategy is to reject the idea of statistical explanation altogether. On Kitcher’s
view, it is a mistake to think that individual occurrences are explained by statistical regularities.
Instead one merely offers a deductive explanation of the probability itself, like a D-S
explanation. Kitcher’s position is that all explanations are deductive; what he acknowledges as
deductive chauvinism. This position will be examined more closely in 4.3.1, when considering
further counterexamples to the D-N and explanations in quantum mechanics.
4.2.2. Is it Sufficient?
Some have argued that the D-N account is not sufficient for explanation, which is to say that a
derivation might meet all requirements of a D-N explanation, yet still not be an explanation. This
is more problematic then there being good explanations that are outside the scope of the D-N
account. The worries about the sufficiency of the D-N account come in two main varieties. There
are those that expose the symmetrical nature of derivations to generate non-explanation. These
cases run an explanatory derivation in reverse, while still satisfying Hempel’s criteria. There are
also cases that expose the irrelevance of the generalization in necessitating the explanandum.
Together the explanans are sufficient to guarantee the explanandum, but are irrelevant to its
actually being the case.
Many argue that problems of symmetry arise when the explanans and explanandum do
not stand in the right causal relation to each other. Generally, the problem is that one can derive
the explanandum E, by means of a general law L, and initial conditions C, and meet the
requirements of the D-N account. However, in some instances, one can also derive C from L and
E and meet the requirements of the D-N account, yet this is not a good explanation. Let us briefly
consider the counterexample of the flagpole, as raised by Bromberger (1966).
In this example, there is a flagpole that is casting a shadow on the ground (Fig. 4.1). With
the D-N account, one can explain the particular length of the shadow, s, by the initial conditions
of the height of the flagpole, h, the angle of the Sun θ, and electromagnetic laws about the
straight-line propagation of light L. However, one can use the same laws coupled with the length
of a flagpole’s shadow to explain its height. This is an incorrect explanation, because clearly this
is not the reason why the flagpole is the height it is.
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(Fig. 4.1)
This is not a particularly unusual case. Another example that have been raised concerns the
derivation of a simple pendulum’s length from its period. The mathematical regularity allows us
to know the length if we know the period, but this is not an explanation of why it is the length
that it is. The regularity alone does not possess information about the causal structure, or give
counterfactual information, which could potentially debar these kinds of cases. These cases
reflect the symmetric nature of law-like generalizations and the covering-law nature of D-N
explanation. The D-N account is not sensitive to the asymmetry that a good explanation
sometimes demands.
Hempel maintains that no general account of laws is necessary for the success of the D-N
account, but this relies on clarity about what counts as a scientific law. A brand of
counterexample is to employ generalizations that are ostensively laws, but that fail to support
explanations. Some argue that one of the most vexing problems for the D-N account is this
characterization of law sentences. A particular problem arises when the generalization used in
the derivation is true but irrelevant. One ends up with a sound argument, but one that does not
explain why the conclusion is true. One can think of this as a kind of epistemic luck, reminiscent
of Gettier cases of knowledge. One ends up with good grounds for believing an explanation, and
the explanans and explanandum are true, but the explanandum is not true in virtue of the
explanans. This happens where the generalization is true, but it is not relevant to the occurrence
of the explanandum.
A famous counterexample to demonstrate the irrelevancy that can obtain in a D-N
derivation is the hexed salt example (Kyburg, 1965). It is presumably a true law that all table salt
that has been hexed with the wand of a witch will dissolve in water. Thus, one can putatively
𝜃 s
s
h
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explain the dissolution of a sample of hexed salt by citing the law that all hexed salt dissolves in
water. Citing that the reason that some salt dissolved was because it was hexed and all hexed salt
dissolves satisfies the D-N criteria, but is clearly not an explanation of why the salt dissolves.
Another example is that Jones takes birth control pills and it is true that taking birth control pills
will prevent pregnancy. This ought to imply that, on the D-N account, the reason why Jones does
not get pregnant is because he is taking birth control pills. This is obviously not a good
explanation of why the explanandum is the case.
These examples strike me as cases of common place explanations that are not intended to
be covered by an account of scientific explanation. Yet this is not a response that Hempel gives.
This is reflective of the idea that there is strong continuity between ordinary and scientific
explanation, such that results like this have consequences for accounts of scientific explanation. I
stated in Chapter 1 that covering common-sense explanations is not a condition for a successful
account of scientific explanation. However, these cases of irrelevancy seem to demonstrate that
nomic expectability is at most an answer about how we know something to be the case, but not
why it actually is the case. Many see the problem as the account’s inability to track causation.
The D-N’s regularity account of causation does not strictly lead through sound argument to the
fact that P together with Q caused r, but only states that it conforms to a regularity. If one finds
that the satisfaction of the D-N criteria is not sufficient to generate an explanation, then one
might offer an alternative causal account, or one might ask what additional criteria might help.
Philip Kitcher’s holds that nomic expectability plus unification is the answer.
Unificationism
Along with logical empiricism’s official account of explanation (D-N/I-S) was the idea that
scientific explanation has been achieved with the goal of unification in mind. Science aims at
maximum explanations with the minimum possible theoretical concepts and assumptions. For
Hempel and Oppenheim, an explanation is given in subsuming particular phenomena under
general theories, because theories have systematic power. For Hempel, it was important that the
theory featuring in the explanation can make “systematic connections among the data of our
experiences, so as to make possible the derivation of some of that data from others” (Hempel &
Oppenheim, 1948, p. 164). The ability of some theories to derive large amounts of data from a
small amount of initial information speaks to their explanatory power. The official account, as it
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is given by Hempel, is criticized as being too stringent, and far too liberal in accommodating
intuitively non-explanatory derivations. What is worse, according to Kitcher, our ability to derive
a “description of a phenomenon from a set of premises containing a law seems quite tangential to
our understanding of the phenomenon” (Kitcher, 1981, pp. 508-509). Nomic expectability is not
always clearly connected to our understanding of why; it is not enough to guarantee explanation.
Hempel’s account is fraught with difficulty, but there is an ‘unofficial account’ of logical
empiricism that makes use of this concept of unification, and it is this that Kitcher will develop.
As Hempel himself says, “what scientific explanation, especially theoretical explanation, aims at
is... an objective kind of insight that is achieved by a systematic unification, by exhibiting the
phenomena as manifestations of common, underlying structures and processes that conform to
specific, testable, basic principles” (Hempel, 1966, p. 83). This story involving unification,
Kitcher argues, can be much more easily connected with our understanding than mere nomic
expectability. Kitcher’s approach, which follows in this tradition, is to assess the worth of
explanations by their unification with in a systematic picture of the order of nature (Kitcher,
1981). To explain is to show that a sentence is appropriately related with the explanatory store of
scientific knowledge. Much of Kitcher’s account involves specifying precisely what this means.
Unificationism really started with Michael Friedman, who was the first to look deeply at
explaining not only single events, but regularities (Friedman, 1974). According to Friedman, the
subsumption of a regularity under another is what makes it explanatory. Reducing a regularity
replaces two facts or more with one. It is the unification, or integration, of one regularity under
another that provides understanding. The reduction of the total number of disparate phenomena
is a primary goal of science and this account of explanation stresses that. What this involves is
reducing the number of theories and regularities that are needed to account for the facts. Theories
that are the most unified in this way are explanatory.
Kitcher’s task is to show what it means to say that an explanatory theory is one which
best unifies knowledge. Friedman conceives of it as a trade-off between the minimization of
theses and the maximization of conclusions reached. For Kitcher, the degree to which a theory is
unified is determined by three criteria. The first is similar to Friedman’s idea that unification
involves a reduction in regularities. Instead of this, Kitcher proposes the idea of the reduction of
what he calls argument patterns. An argument pattern is a triplet consisting of a schematic
sentence with dummy letters that can be filled in, a set of filling instructions that specify how the
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dummy letters are to be replaced, and a classification that determines how a set of schematic
sentences can be arranged as premise and conclusion in the form of an argument. On Kitcher’s
unification, what is minimized is the number of patterns that a theory employs for deriving
conclusions. The second mark of unification is a theory’s empirical scope; the greater the
empirical scope, the more unified it is. A theory unifies our beliefs if a small number of
argument patterns can be used to derive a great range of conclusions. This makes room for
argument patterns to be very general, in fact, so general that clearly non-explanatory derivations
would qualify. Thus, he introduces the last central aspect of unification, viz., the stringency of
the argument patterns used. An argument pattern is said to be more stringent than another if there
are more restrictions on the arguments that instantiate it. Kitcher proposes that scientists are
concerned with stringent patterns that place restrictions on the substitution conditions for dummy
letters and on the logical structure imposed by classification. If one relaxes both conditions, the
pattern admits of more and more arguments, and conversely if one tightens restrictions on both,
then it admits of fewer and fewer. It can be seen that these criteria pull away from each other.
The fewer and the less stringent the argument patterns, the more likely the theory is to lead to a
wider range of conclusions, and vice versa.
For Kitcher, good explanations are instances of patterns that rank better along these
criteria than derivations that we consider bad explanations. These derivations are available
explanations, and so they are in what he calls the explanatory store. The set of argument patterns
that most unify a set of accepted sentences, K, is the explanatory store over K, which he denotes
as E(K).
Kitcher points us to two prime cases where we can see the explanatory power of
unification at work: in the reception of Darwin’s theory of evolution, and in the wake of
Newton’s mechanical theory. Newton’s successes prompted others to take on an even more
ambitious enterprise called dynamic corpuscularianism, which sought to unify all the phenomena
of nature in a single framework. It encouraged Newtonians to construct corpuscular theories of
everything, including light, even in the absence of evidence. Its main appeal was the promise of
unification. The hope was that one kind of force law would suffice to describe all interactions in
the same way that gravitation was ruled by a single law. A small number of general patterns of
argument were sought to explain all of nature – this was the Newtonian ideal.
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Similarly, the attraction of Darwin’s theory was its ability to unify a host of biological
phenomena in a small number of common patterns. The theory presents a pattern to describe
imaginary examples that can be used to explain the features of any existing species. It offers
explanation-sketches by describing how particular traits may be advantageous and contribute to
the survival of the species. Kitcher claims that it was the argument pattern that made the theory
compelling and that is precisely what characterizes unification and explanation.
Kitcher addresses the problem of what he calls spurious unification. The problem is that
it is possible to derive for example, a law L, from L and an arbitrary conjunction with another
law B: L & B, therefore L. This self-explanation is clearly not what Kitcher has in mind, and
opens up the door to unifying our beliefs completely by this one simple argument pattern. His
answer to this is that such derivation may win in least number of argument patterns, but fails
when it comes to stringency, since it would allow any vocabulary to fill the dummy letters. Even
though one can strengthen this problem by artificially introducing restrictions on the pattern, the
accidental quality of the restrictions will never fail to provide argument patterns as one changes
the filling instructions for the pattern. By contrast, if one considers the Newtonian pattern, the
constraints are essential to it, and cannot be amended without destroying its stringency.
So after having found a way to distinguish the genuine from the spurious unification, he
makes this requirement explicit. If the filling instructions can be replaced to yield a new pattern
which allows the derivation of any sentence, then the unification is spurious. This new
requirement will also be able to decide against the unification of doctrinal arguments, which may
make claims to unification by such laws as ‘What God wants to be the case is the case”. Because
the restrictions of the filling instructions in this case are so liberal as to bar almost nothing, if
anything at all, the unification is clearly spurious. He assures us that this requirement is not out
of place, and is closely tied with the idea that explanations should be testable. If the argument
pattern unifies incredibly well, but makes no restrictions on its possible filling instructions and
which conclusions it is capable of accommodating, then it is spurious.
4.3.1. Unificationist Solutions to D-N Problems
Kitcher proposes that the unofficial account is capable of solving the three most challenging
problems that remain for the D-N account. Kitcher defends a position that is independent of his
account of unification, but which would help circumvent problems that arise for statistical
explanation. This position has been called “deductive chauvinism” by Salmon and Coffa. For
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Kitcher, though, it is a virtue and not a defect of his account. Kitcher says that “in a certain
sense, all explanation is deductive” (1989, p. 448). He maintains that the explanatory store
contains only deductive arguments, and so it actually prohibits the possibility of inductive
explanation. The objection might be raised that if D-S were sufficient to characterize
explanation, then Hempel would never have needed to talk about I-S explanation in the first
place. Kitcher thinks that the idea that there is a deductive explanation that is being replaced by
an epistemic probability is not sufficient to account for all cases of probabilistic explanation.
This can be clearly seen in the cases where there is no, or no likely, deductive story of particular
occurrences – for instance, in quantum mechanics.
Kitcher argues that there are two senses of ideal explanation. In the first sense, it is a
deductive derivation, but in the other, it is the best explanation the phenomena will admit.
Quantum mechanical explanations are ideal in the second, but not the first, sense. Consider a
case where there is a 0.9 probability that a barrier will reflect a particle, and given two instances,
one where a particle 𝑒1 is reflected and one 𝑒2 where it penetrates, an important question must be
asked to whether it is possible that there is an ideal explanation of both cases. Kitcher argues that
there is not. There can be no ideal explanatory account, because there is no information to
distinguish these cases. There is no account of why 𝑒1 was reflected and 𝑒2 tunnelled through.
Kitcher thinks that we mistake what it is that quantum mechanics explains. Quantum
mechanics can explain how things are possible by allowing them to be possible results, but it
does not explain particular instances. Probabilistic explanations in quantum mechanics take on
the form of D-S explanations – they explain probabilities about individual outcomes. So what is
explained is not why they occur, but why they occur with a certain probability. One has an
explanation when one has facts about barrier penetration for example, that are derived from a
generalization like the Schrodinger equation. It is easy to confuse a why-question with the how-
possible questions that quantum mechanics seems to address. The why-question it answers is
about why the probability is 0.9, it is a question of the general occurrence. Kitcher argues that the
need for I-S explanations comes from a confusion of the kinds of questions asked about quantum
mechanics. Kitcher’s solution is simple and radical: all statistical explanations of individual
events need to be spelled out in terms of deductive arguments about the probabilities, rather than
inductive arguments about the event itself.
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Unificationism proposes to solve the problems of asymmetry and irrelevance that
troubled the D-N account. Kitcher’s strategy is to compare the putative explanations that fit some
deductive argument pattern but are not explanatory to another possible explanation that also
derives the explanandum, and showing that the former does not best unify our set of accepted
statements, K. Thus, it is not explanatory; not a member of E(K) and not an explanation at all.
This marks an important aspect of unificationism, which is that only the theory that best unifies
our knowledge is explanatory. This is known as the winner-take-all conception of explanation,
which was mentioned in 1.4.2.
Even though one can derive the length of a simple pendulum by looking at its period, the
problem as Kitcher sees it, is that this is not what we normally take to be an explanation of the
dimensions of manmade bodies. Those we normally take to be explanations of artifacts, he calls
the “origin and development” type of explanations, as Kitcher calls them. If we consider the case
of the flagpole, it is surely true that some objects do not have shadows or cannot always lend
themselves to deriving facts about the object’s dimensions. What is one to do about explaining
the dimensions of these kinds of bodies? Adopt a separate and quite different argument pattern,
or pick the origin and development explanation that is capable of explaining both? The best
explanation for unificationism is not the counterexample case. It is the origin and development
pattern, which is more widely applicable in explanations of the dimensions of manmade bodies,
and thus more unifying.
The unificationist solution to the case of irrelevance is to once again compare the
troublesome case to a more reasonable explanation and show that it is not the most unifying.
Given the explanation that employs the fact that all hexed salt dissolves in water, what is one to
do about instances of the dissolution of unhexed salt? One could either maintain two separate
explanations, one for each case, or one could simply choose the explanation which is capable of
covering both cases, viz. that all salt is water soluble. The second option is clearly more unifying
and instantiates a pattern that is much more generally applied. He concludes then that
“unificationism has the resources to solve some traditional difficulties for theories of
explanation” (Kitcher, 1981, p. 526). While the counterexample cases might meet deductivist
criteria for nomic expectability, it does not instantiate the most unifying argument pattern.
Kitcher made several contributions to deductivist explanation. He developed an account
of explanation with different aims than Hempel’s. He sought to provide an account of how laws
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confer explanatory power, something on which D-N account remained silent. By developing the
unofficial view of logical empiricist explanation he was able to propose solutions to vexing
problems. In denying that inductive statistical explanations exist he is able to avoid the problems
of the ambiguity of I-S, without making use of a high-probability requirement or maximal
specificity. The comparative nature of unificationist explanation helps Kitcher determine that the
counterexample cases ought not be considered explanatory and thus defend deductivist
approaches to explanation. However, many have concerns about unificationism and these
proposed solutions, and it is to this that I turn next.
4.3.2. Challenges to Unificationism
There are a number of serious challenges to unificationism and to the proposed solutions to the
D-N counterexamples. I will bring up a few that I find to be most troubling and relevant to what
has been, and will be, discussed.
Some concede that Kitcher’s defense of the flagpole case works well enough, but that it is
not a general result. Barnes has noted that when one considers a time-symmetrical system like
the Newtonian mechanical description of the solar system, one finds that there are as many
retrodictive as predictive derivations (Barnes, 1992). The argument patterns for the retrodictions
are as unified as those of the predictions, and thus contrary to our judgments, they are equally
explanatory on the unificationist picture. Woodward has reinforced this criticism by showing that
the more general problem is that there are many kinds of unification and not all of them are
relevant to explanation (Woodward, 2003). Some unifications that count as explanatory for
Kitcher are no more than the application of a common mathematical formalism to different sorts
of phenomena. Woodward argues that “the mere fact that we can describe both the behavior of a
system of gravitating masses and the operation of an electric circuit by means of Lagrange’s
equations does not mean that we have “unified” gravity and electricity in any physically
interesting sense” (Woodward, 2003, p. 363). It could be argued that the argument patterns
involve more than merely making use of a set of equations, such that the patterns are only the
same for very non-stringent characterizations.
Woodward notes that this raises the question of how one is to trade off the criteria of
stringency, paucity, and scope, against one another. One theory is more explanatory than another
if it can derive a wider range of phenomena with fewer, more stringent argument patterns. But,
these criteria pull apart, and there’s no rule or procedure for how to weigh them against each
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other, even though this weighing is quite important. Kitcher’s solution to the flagpole
counterexample relied on a single origin and development pattern for explaining the dimensions
of objects, but this is a very non-stringent pattern. It is quite different from patterns used to
explain the dimensions of objects with biological or geological origins. In order to debar
counterexamples, a lot turns on “correctly” counting argument patterns and assessing stringency.
A different concern for unificationism is the winner-take-all conception of explanation.
Recall that according to Kitcher, only the most unifying pattern that derives the explanandum is
explanatory. This is an essential feature of the account that enables him to debar the
counterexample cases. However, it leads to counterintuitive judgments about certain derivations
being non-explanatory. One compares the degree of unification of two competing theories and
the one that unifies more is explanatory, and the other not at all. Many have argued that this does
not follow. I stated in Chapter 1 that it seems reasonable to hold that two derivations of a
phenomenon can be counted as explanatory. However, this is not tenable on the unificationist
view, because then the case of the hexed salt and the flagpole’s shadow pattern are also
explanatory. However, I do not believe that these problems are ineliminable for deductivist
accounts, as I will show in the following chapter.
Conclusion and the Current State of Deductivism
There are two main motivators for thinking that the D-N/I-S account can cover most scientific
explanations. First, because of its deductive structure, Hempel says that it provides an answer to
why the particular phenomenon occurred. It says why the result was expected and allows us to
understand its occurrence. The I-S model does not show expectation with certainty, but high
probability – they both share in conferring nomic expectability. Secondly, it only requires a
regularity theory of causation, namely, the laws it uses. Following in the empiricist tradition,
Hempel construed causation as the obtaining of regularities, but spoke no more about the
metaphysics of causation. A benefit of the account is that one could avoid talking about the
metaphysics of causation and scientific realism. Instead one could talk about entities and
dependency relations in terms of scientific explanation. This also more accurately reflects the
fact that many kinds of scientific explanations are outside the scope of causal accounts.
Kitcher proposes supplementing the D-N with a formal account of the explanatory power
of theories. The hope is that the extra criterion that theories that feature the deductive argument
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patterns in the explanatory store can debar three of the main objections to deductivist explanation
(statistical explanation, irrelevance, and asymmetry). Deductive chauvinism seems to provide a
means of circumventing the myriad problems that inductive explanations generate, many of
which I passed over. Moreover, making use of a comparative, or winner-take-all conception of
explanation, allows Kitcher to prefer standard explanations to the counterfactual cases, which are
less unifying. However, some serious problems remain for unificationism, and so it is not clear
that the current state of deductivism is very promising. However, I aim to show that prospects are
better than is commonly thought.
There are several lessons for deductivist approaches that can be taken from this
investigation. One is that counterexample cases seem to demonstrate that there is a difference
between explanation and prediction. Being able to make statistical inferences is not the same as
giving a statistical explanation, and perhaps making deductive inferences is not the same as
giving deductive explanations either. What is required is that if the explanation proceeds by
covering law, then there needs to be something about the regularity that guarantees its relevance
to the explanandum. This should avoid the problem of being able to derive the explanandum by
accident, so to speak. While nomic expectability is not enough, nomic expectability plus
unificationism is a step in the wrong direction.
Providing an account that is capable of handling counterexamples cases will also require
something beyond nomic expectability plus some other criteria. It ought to be possible to
preclude problematic symmetrical derivations and to constrain the relevance of the explanans for
the occurrence of the explanandum. The symmetry problem follows from the symmetrical nature
of the logical form of deduction and the problem of irrelevance follows from the covering law
conception. What these problems seem to require is to make use of facts about the empirical
content of the explanation. Purely syntactic restrictions are unlikely to be able to reflect the
asymmetry and the close relevance relations that some explanations require. This might go a
long way towards showing that a deductivist account can be sufficient for explanation.
I think that Kitcher was correct in his claim that deductive chauvinism is a virtue of his
account. It is also something that Hempel endorsed, to a lesser degree, about deductive
explanations. It is also important to note that this position is independent from unificationism.
One of the motivations for a deductivist account is that it opens up the range of possible
explanations to include those that are non-causal. Causal explanation is merely one kind of
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explanation. Being able to accommodate singular causal explanations and denying that there are
real statistical explanations removes perceived limitations on the scope of a deductivist account.
With deductive chauvinism, one can hopefully mitigate the counterexamples that showed that
deduction is unnecessary for explanation.
A pervasive criticism of deductivist explanation is that it is largely irrelevant to the
actual practice of scientific explanation. The way that explanations proceed is not by subsuming
particular instances under general laws. Even for Kitcher, individual explanations are still D-N
derivations. There is a trend in the philosophy of science to move towards explanations that
feature models. The importance of this is something I discussed in Chapters 1 and 2, and in the
following chapter, I hope to incorporate that scheme into a deductivist approach. The plan is to
synthesize the results of the previous investigations and show that there can be a promising and
relevant account of deductivist explanation.
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Chapter 5. Model-based Deductivism
Introduction
Having taken stock of the resources and the shortcomings of deductivist approaches, I am now in
a position to present a novel, model-based deductivist account of explanation. I start from the
position that takes explanations to be deductive arguments. On this view, the explanans is a set of
statements about a scientific model that is used to derive the explanandum, which is a statement
about, or a description of, a target phenomenon, pattern, or behaviour. I take scientific models to
be the objects of explanation, in part because I and many others feel that covering law accounts
are largely irrelevant to the actual explanatory practices of science. Many explanations offered
do not simply derive explananda from laws of nature. This is a very restrictive selection of the
kinds of explanations scientists are actually offering.
As was shown in Chapter 2, the recent literature on explanation has focused on model-
based accounts of explanation. Some accounts require that explanatory models reflect the real
causal or structural relations of a target system (Woodward, 2003; Bokulich, 2008; Strevens,
2008). In place of a causal or structural restriction, I propose that models that support
explanations are those that give appropriate counterfactual information and are integrated with an
established scientific theory. This integration is what gives the model its unifying power and
demonstrates its ability to explain. This requirement has the added benefit that it does not place
any representative or metaphysical restrictions (causal, structural, or otherwise) on the relations
of the model itself. This expands the scope of explanation to include non-representing, or highly-
idealized, models; a goal that has been argued for by many, mentioned in Chapter 2.
The goal of the chapter is to present an updated, model-based version of a deductivist
account and demonstrate its promise. A model-based approach allows the explanation to give
counterfactual information about the system, but does not require that the relations among its
variables represent real causal dependency relations. As a deductivist account, it both confers the
nomic expectability that Hempel desired and remains open to causal and non-causal
explanations. The incorporation of counterfactual information about the facts of the models helps
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to establish relevancy and asymmetry. The criteria as presented at this stage are still preliminary
and in need of being tested and refined in the crucible of detailed case studies.
The following section begins to build up an account from the basic deductivist criterion
that the explanans entails the explanandum. In 5.3, I revisit what a model is in this context and
begin to introduce some restrictions on what kinds of models can support explanations. In 5.4
and 5.5, I articulate the local criteria for explanation. I argue that an explanatory model must
support same-object counterfactuals about changes to the model. In 5.6, I discuss the strategy of
placing a global constraint on explanatory models such that they must be appropriately related to
a global theory of science. In 5.7, I present the account as it currently stands, and in 5.8, I put the
account to work in reviewing some of the case studies that have already been brought up in light
of the account I am proposing. I hope to motivate the deductivist approach, demonstrate its wide
scope, its ability to clarify thorny issues of explanation, and its agreement with our judgments
concerning explanation.
A Model-based Deductivist Account
Let me now propose the formal criteria for this account of scientific explanation. Let us begin
with the simplest and most central criterion of a deductivist account. This is the criterion of
deductive entailment, which is as follows:
D1. The explanandum must be a deductive consequence of the explanans.
This criterion is the backbone of deductivist explanation. This alone is enough to preclude many
putative explanations. Just like Kitcher’s deductive chauvinist position, I maintain that all
genuine scientific explanations are deductive. Many reasonable explanations are not deductive.
They may cite single-event causal stories, or accepted inferences, say from lightning to thunder.
My claim is not that these are not explanations, but that are not scientific explanations. This
suggests some degree of discontinuity between the two, which Hempel and Kitcher were hesitant
to admit. However, I maintain that many common sense explanations merely provide a good
reason to think that the explanans is the right explanation. Being out of milk does not necessitate
being out of the house, and therefore Jones’ being out of milk fails to scientifically explain why
he is not at home.
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I think that it would be difficult to have a successful account of scientific explanation that
allows the explanans to be true and the explanandum to be false. This is essentially the ambiguity
of I-S and echoes concerns about ambiguous explananda (4.2.1; 1.3.1). To see why I think that
inductive-statistical derivations are not explanations, let us take for example that we want to
know why a particular smoker develops cancer. A reasonable explanation is that she was a
smoker and smoking causes cancer. We can formulate this in a deductive argument:
1. Mary is a smoker.
2. Smoking causes cancer.
3. Therefore, Mary develops cancer.
However, this explanation may fail to hold for Bob, who is a smoker, but does not have cancer:
1. Bob is a smoker.
2. Smoking causes cancer.
3. Therefore, Bob develops cancer.
Here, the premises are true, as in the previous explanation, but the explanandum is false. A
reason for this is that the premises are not informative enough. It is not clear what it means to be
a smoker or how long or how much Bob and Mary smoked. Further, the second premise could
more precisely be stated as:
2*. Smoking increases the risk of developing cancer.
However, the argument, when substituted with this premise, now fails to hold deductively. The
explanandum that deductively follows is:
3*. Bob/Mary has an increased risk of developing cancer.
This explanandum now fails to establish the difference between Bob and Mary, both have an
increased risk. But this is quite reasonable, as there is no information provided to distinguish
between the two. There is no explanation of both cases. Based on this criterion, if one wants to
know why Mary develops cancer and Bob does not, one would need more information. Just as
Kitcher argued, I too claim that the single occurrence of this statistical relation cannot be
explained given the available information. My contention is that this probabilistic causal story is
not scientifically explanatory.
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While I believe this criterion can circumvent problems associated with causal and
inductive explanations, D1 alone will not guarantee explanation. Hempel proposed that only
certain kinds of regularities feature in explanatory derivations. Rather than follow suit with a
covering law account and perhaps attempt to specify lawlikeness, I will turn to a model-based
approach. Let me propose the second criterion:
D2. The statements in the explanans are true of a relevant scientific model, M.
This criterion refocuses the explanans to truth about models, rather than truth about the world. So
rather than requiring true laws of nature, this account requires that the generalization in the
explanans be true of a model. The truth of the explanans is considered to be necessary on many
accounts, but some, such as van Fraassen, have argued otherwise (van Fraassen, 1980).
There are some advantages to not requiring accurate representations. First, unlike other
deductivist accounts, lawlikeness is no longer an issue. As we saw in Chapter 4, one of the
criticisms levelled against the D-N account is that it relies on lawlike generalizations in the
explanans, but tells us little about how we can decide if a generalization is lawlike. Some who
have examined laws on causal grounds, focus instead on a spectrum of generalizations ranging
from accidental truths (statements about the coins in Goodman’s pockets) to truly universal and
exceptionless laws of nature (law of the conservation of mass-energy) (Mitchell, 2000, 2002a;
Woodward & Hitchcock, 2003b). Mitchell frames this in terms of realigning our concept of laws
with the practice of biology, while Woodward focuses on the degree of invariance at the heart of
explanatory generalizations. Both move away from the traditional concept of law and this
account also does, albeit in a different manner. This account requires that the generalization in
the explanans be true of a certain kind of model, and thus avoids having to distinguish between
laws and accidental generalizations, which has been problematized (Scriven, 1962; Goodman,
1973; Cartwright, 1994, 1997; Lange, 2002, 2004; Craver, 2006).
Second, idealization is no longer an obstacle to explanation. In fact, this allows for such
things as idealizations and abstractions to be central components of explanation. An issue for
many accounts of explanation is the reconciliation of idealized models with truth about the
world. As we saw in Chapter 2, when one recognizes that all models are to some degree
idealized, the requirement that the statements in explanans be strictly true cannot be satisfied.
Others claim that approximate representation is enough to justify a model’s application and
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satisfy any weak commitments to realism, but even that is not a factor here. The statements in the
explanans are true of the model, but no explicit limitations are placed on the representative
accuracy of the features of the model. Not requiring that explanatory models accurately represent
also expands of the scope of explanatory idealizations, by opening it up to non-Galilean, or
highly-idealized, models. As we saw in Chapters 1 and 2, there are persuasive reasons to think
that highly-idealized models can support explanations, but this is very difficult to reconcile with
representational accounts of explanation. With D2, non-representing models are not necessarily
debarred from supporting explanations.
It is important to mention again that, while the explanans need not accurately represent
the target system and need only be true of a scientific model, the explanandum must be an
approximately true statement about the target system. Unless the explanandum concerns the
behaviour of the model itself, in which case, it must be an approximately true statement about the
model. The explanandum is a description of the phenomenon to be explained and it cannot be
false. It is important that this indirectly restricts the explanans. This is because if the explanans is
not capable of deductively leading to a statement that accurately describes the explanandum
phenomenon, then there is no explanation.
Lastly, it employs no particular account of causation and makes no commitment of causal
realism. This avoids any problems or issues that may come with the metaphysics of emergent
causation. Many accounts hold that explanatory relations are those that describe or capture the
real causal relations in the world. This is a useful way to debar some non-explanatory relations,
like backtracking counterfactuals, but has implications for the accuracy of the statements in the
explanans and consequently limits the scope of explanatory models. Just as Hempel’s account
was designed to capture explanations that are causal as well as those that are not, this account
aims to be more broadly applicable than one with causal criteria.
What is a Model?
One of the most important developments in the literature on explanation since Hempel, is the
recognition that explanations deal with models. This change has come from the desire to reflect
the explanatory practices of science, which largely proceed from pragmatically-oriented models
rather than laws of nature.
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Many other accounts of explanation have focused on models, but it is either assumed or
argued that the models ought to be causal models. Some have downplayed the role of causation
itself and focused instead on mechanisms (Machamer, Darden, & Craver, 2000; Cat, 2005;
Darden, 2006; Weber, 2006; Craver, 2007, 2009, 2010; D. M. Kaplan & Craver, 2011; Knuuttila
& Loettgers, 2012; Schindler, 2013), or on structural models (Worrall, 1989; French &
Ladyman, 2003; McArthur, 2003; French & Saatsi, 2006; Bokulich, 2008). However, this
account will focus on models in a deductivist framework.
As was mentioned in 2.2, my approach treats models as objects. Some are physical
constructions, like a road map, a model ship, or a pendulum that sits on a desk, but the kinds
referenced in scientific explanations are abstract objects. This kind of model is not simply a set
of statements, but can be described by statements. Some of these statements feature idealizations
that are false of the world. Such as: “the plane is frictionless,” “there is no damping due to air
resistance,” “there is an infinite population,” and so on. These can be true of the model but false
of the target system. The model is that about which these idealizations are true.
A target system does not have a single corresponding model. Models are constructed and
constructed with a purpose. Usually this purpose is to bring to light some particular relation, the
effect of one variable change on another, for instance, between string length and period of
oscillation of a pendulum. The model will be designed to bring to light this relation, but not
necessarily any other (Weisberg, 2007). In constructing a model, the modeller must make trade-
offs. Some models are made to be as accurate as possible to all the actual components and
processes, to function as simulations; others to be maximally tractable, or have high predictive
accuracy within a limited range; others to be very general and widely applicable to different
systems to show why different materials exhibit the same behaviour, and so on. Which desiderata
are considered most important and how a balance is struck, will change from case to case.
Modelling desiderata are inevitably contextual. But in all cases, there is no single perfect model.
5.3.1. A Simple Model of the Fixed-length Pendulum
In order to clarify exactly what kinds of models are at issue, let us examine a simple example. If
we want to build a model to explain the period of the oscillation of a simple fixed-length
pendulum, we might construct it like this:
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1. There is a bob suspended from the end of a string fixed to a pivot.
2. The bob is a point mass that moves in two dimensions.
3. The length of the string is L.
4. The string is massless, fixed in length, and taut.
5. There is no loss of energy via air resistance or friction.
6. The only forces on the bob are the tensile strength of the string and
gravity.
7. The acceleration due to gravity is fixed at g = 9.8m/s2
8. The period of the oscillation of the pendulum, T, depends on L and g, such
that 𝑇 = 2𝜋√𝐿/𝑔.
9. The oscillation amplitude is small enough that sin 𝜃 ≈ 𝜃.
This is an ideal pendulum that accurately describes no real-world pendulum. Its motion is
actually given by this model. The motion of this pendulum can be related to a physical one in
approximate conditions, but that is not important at the moment. We are speaking of this abstract
object and these statements are true of this particular model. The model is very simple. Note that
there are no equations for angular acceleration, no Lagrangian, and no differential equations.
This is because this model is constructed to capture the dependency of the period of a
pendulum’s oscillation on its length and not, for example, to represent its movement.
If we want to explain a particular oscillation period phenomenon we can formulate a
deductive derivation of this, with explanans statements that are true of this model. For instance,
we might ask “Why is the period of oscillation of this simple pendulum 1s?” The obvious answer
is because the length of the string is a particular length, viz. 25cm. An outline of the explanatory
derivation might look something like this:
1. 𝑇 = 2𝜋√𝐿/𝑔
2. 𝐿 = 0.25𝑚
3. 𝑇 = 1𝑠
This is not only a derivation of a particular phenomenon. The model also gives us counterfactual
information about changes to the system, such as what the period would be if the length were
1m, or 10cm, or whatever. What this model tells you is that changing the length of the string will
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change the period, and it tells you by how much. This counterfactual information is essential to
an explanation-supporting model.
Counterfactuals
The D-N account makes use of causal laws as regularities, which can be exploited because the
derivation is allowed to run both ways. It is important to note that regularities alone do not
express counterfactual information. Without this information there is nothing to prevent
backwards-running non-explanations. And so, some restrictions outside of the derivation itself
are required for explanation. Counterfactual information about the dependency relations of the
model is relevant to explanation, something argued by Woodward and Hitchcock and others
(Lewis, 1979; Tooley, 2003). My proposal is that this counterfactual information is available in
the details of the model. Woodward and Hitchcock argue that an explanatory generalization “not
only shows that the explanandum was to be expected, given the initial conditions that actually
obtained, but it can also be used to show how explanandum would change if the initial and
boundary conditions were to change in various ways,” (2003a, p. 4).
The model referenced in the explanans must provide information about how the values of
the explanandum variable would change if there were changes to the values of the explanans.
Formally, the criterion can be stated as follows:
D3. The model M referenced by the explanans gives counterfactual information that
shows on what the explanandum depends.
This criterion distinguishes a model-based account from a covering law in an important way. The
regularities of the D-N account were able to be exploited precisely because there was no
requirement on counterfactual information.
This requirement also ensures that the explanandum phenomenon can actually change.
Consider again a case introduced by Salmon (Salmon, 1971). Jones can avoid becoming
pregnant by regularly taking birth control pills, and every man who regularly takes birth control
pills avoids pregnancy. The problem with an derivation such as this is that the generalization is
irrelevant to the reason why the explanandum is the case. However, such cases are debarred by
this criterion, since that kind of regularity gives incorrect counterfactual information about what
the explanandum depends on. In effect, this criterion is able to serve as a requirement for
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testability. If it is not possible for the explanandum to change (Jones becoming pregnant), then
the criterion is not met.
One could ask why it is that the ability to answer counterfactuals is so crucial for
explanation. Woodward ties the importance to knowledge of changes made by manipulations. On
this account, the importance comes in confidence in the reliability of explanatory models.
Models that are capable of supporting explanations are able to provide accurate, reliable
information about changes to the target system. Because the explanandum is approximately true
of the target system, and the counterfactual information in the model demonstrates on what the
explanandum depends, then the counterfactual information in the model has implications for its
application to the target system.
5.4.1. Same-object Counterfactuals
The account I propose borrows the notion of a same-object counterfactual from Woodward and
Hitchcock. This kind of counterfactuals is useful, in that it can provide a means to debar
backtracking counterfactuals. Requiring this kind of counterfactual information precludes the
possibility of exploiting the symmetry of a deductive derivation.
To reiterate from 3.3.1, Woodward and Hitchcock state that traditionally counterfactuals
are seen to be what they call other-object counterfactuals, which give information about what
would be the case for a different object to have different values for its variables. Same-object
counterfactuals refer to hypothetical changes on the same object. To illustrate the difference,
consider the following example of Galileo’s pendulum law, which explains why a pendulum of
length x has period y. According to the law, the following counterfactual is true: if this laptop
were a pendulum with length x, its period would be y. The law is sufficient to support such
other-object counterfactuals that involve changes in identity. This counterfactual however, tells
us nothing about how changes in the values of x would affect the values of y. If this
counterfactual concerned the very same object’s hypothetical values it might be framed as
follows: if a pendulum with length x and period y had its length adjusted to x', then its period
would be y'. This counterfactual gives information about what would happen to other features of
the model given hypothetical changes to the very same object. It is this information about the
possible changes to a model that satisfies the conditions of D3. It is a generalization’s ability to
support same-object counterfactuals that is relevant to its being explanatory.
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Of course, more than one model can support same-object counterfactuals for a single
target system. It is useful, but not necessary, for my account to provide or at least be compatible
with a method for comparing relative explanatory depth. Some philosophers want more than a
threshold in an account of explanation in order to match up with our intuitions about which
models are more explanatory than others. There is a further way that one can rank, or compare
different models that have already qualified as explanatory by meeting minimum requirements.
This could be done by the depth of the counterfactual relation, but I will not explore this
possibility here outside of what has already been said about w-questions and depth in Chapter 2.
However, it seems likely to me that there are many measures of explanatory value, not all
of which will be compatible. It is very reasonable to assume that certain pragmatic issues, such
as levels of abstraction and the context of knowledge and communication skills, that can have a
great deal to do the with the explanatory value of an explanation-act. And furthermore, these
likely will not all pull in the same direction. In different circumstances, some explanations can be
of more value if they are simpler and others if they are more detailed. A notion of explanatory
value then is unlikely to be a single unanimous measure.
Before looking in more depth, so to speak, at the pragmatics of explanation with respect
to models and model construction, it is worth saying a few words about the truth conditions of
counterfactuals and what exactly is supporting the counterfactuals that underwrite these model-
explanations.
5.4.2. Truth Conditions for Counterfactuals
Counterfactuals are different from conditionals in an important way. If one wants to know
whether a conditional is true, one can simply test for it. However, because counterfactuals are by
definition contrary to fact, there is no simple truth-functional way of determining whether they
are true or not. And yet, we have strong intuitions about the truth of some counterfactuals, and
use them in everyday reasoning.
A central tradition in the literature on counterfactuals stems from Robert Stalnaker’s and
David Lewis’ accounts of counterfactuals. This approach make use of possible world semantics
and work done by Saul Kripke (Stalnaker, 1968; Kripke, 1972; Lewis, 1973). Very briefly, this
approach is to look at the closeness of possible worlds to determine which counterfactuals are
true. By contrast, Woodward and Hitchcock look to the metaphysics of causation. According to
this view, there are brute facts about the causal relations in the world that can be used to
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determine the truth value of counterfactuals. We gain knowledge about these causal facts by
discovering relations that are invariant over a range of interventions.
A model is an abstract object. And in a similar vein, there are brute facts about a model.
There are objective facts about relevant counterfactual circumstances in a model. Statements
about a model can either be true or false or indeterminate. An explanatory model should be
described adequately enough to support same-object counterfactuals. What changes can be made
and what boundary conditions and parameters there are in place may need to be spelled out. The
truth of counterfactuals come from brute facts about this abstract object.
Because models are constructed purposefully and involve only certain aspects or relations
of a system, there are only certain relevant w-questions that can be answered. It is obvious that in
a model of sunspot activity there is no information about what would happen given changes in
the population of lemmings in Norway. In other cases, it is less obvious when there is no
information. In a simple model focused on the changes in barometer readings as storms
approach, there is no information about what would happen storm-wise if one fiddles with the
barometer. While a regularity alone might support this counterfactual, it does not support the
right kinds of counterfactuals about changes to the objects in the model.
The Simple Pendulum Revisited
Now that we have an understanding of same-object counterfactuals and the work they will be
doing in this account, let us return to the model of the simple pendulum. Recall that we were able
to explain what would happen to the period of a pendulum if its length were changed. Now let us
consider the symmetrical case, and say a particular string length is our explanandum
phenomenon. We might ask “Why is the length of the string in this fixed-length pendulum
25cm?” Here, the model has no obvious answer. There are likely many reasonable pragmatic and
circumstantial reasons for the exact length of string, but none that come from the model we
outlined above. There is nothing in this particular model to say what the length of the string
depends on. It might be possible to use the equation provided in the model to derive the length
given a period, but this is not enough for explanation. This does not show counterfactual
information about what the explanandum actually depends on.
What we have is a statement about the dependency relation of the period on the length.
In the model we have built above, there is no information about what would happen to the string
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length if we changed the period. Further, changing the period would involve adding a driving
force, pushing on the bob, changing the acceleration due to gravity, or something else which is
outside the parameters of the model. The model is not equipped to handle this kind of
counterfactual.
Of course, one can build a very general model in which the explanation runs both ways.
And one can also build a model in which this explanandum phenomenon seems perfectly
reasonable. But it will likely look different than this model. Grandfather clocks, like other
mechanical time-keeping devices, need to be calibrated. You can adjust the time that the clock
keeps by adjusting a small screw at the bottom of the bob, which raises or lowers it along the
rod. The height of the bob along the rod effectively changes the length of the rod, since the
portion of the rod that dangles below is massless in our model. If you want to build a model to
figure out what length of string (to keep the language consistent) you should have to keep proper
time with the right period, you might describe the model this way:
1. There is a bob suspended from the end of a string fixed to a pivot.
2. The bob is a point mass that moves in two dimensions.
3. There is no loss of energy via air resistance or friction.
4. The only forces on the bob are the tensile strength of the string and
gravity.
5. The acceleration due to gravity is fixed at g = 9.8m/s2.
6. The period of the oscillation of the pendulum is T.
7. The string is massless and its length is variable.
8. The length of the string L depends on T and g, such that 𝐿 = 𝑇2𝑔/𝜋2.
9. The oscillation amplitude is small enough that sin 𝜃 ≈ 𝜃.
The key difference in this model is that the dependency relation is spelled out in terms of
changing the length to fit a desired time. Now there is an explanation, an answer to our why
question. The length of the string is 25cm, because the desired period of oscillation is 1s. The
length of the string depends on what one desires as a value for T. We know what the length
depends on, T, and we know how, quantitatively, the dependency works, viz. by the
generalization 𝐿 = 𝑇2𝑔/4𝜋2. This is more than a simply changing the equation around: it
reflects the dependency relations stipulated by the model. What this model tells you, is that
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changing desired period will the change the length of the string, and by how much. Here,
whatever the length of the string is, it depends on what the period of its oscillation is. This
explanation no longer seems backwards.
What this shows is that there are brute facts about the model and about hypothetical
changes to the features of the model that make it different from a mere regularity that might
allow D-N-style counterexample derivations. The counterfactuals that would support
counterexample cases are not stipulated in the model. This is one of the main benefits of taking a
model-based, rather than covering law, approach to deductivism. However, it seems reasonable
to worry that if one can simply construct any model with certain counterfactuals but not others,
that the victory is rather hollow. In order to prevent this from seeming so arbitrary, I will next
introduce a global constraint on which same-object-counterfactual-supporting models can be
explanatory.
A Global Constraint on Explanation
Some models are phenomenological; they are merely representations, or systematized
collections, of data, like in the statistical modelling of regression analysis. These can be used to
examine the relations between sets of variables, make predictions, and can have considerable
heuristic value, but I maintain that they are not explanatory. This is a consequence of the
account, but also reflects a disagreement with Hempel about the similarity of prediction and
explanation and the adequacy of nomic expectability. One of the aims of the account is to be
sensitive to the difference between showing that we know something to be the case and
explaining why it is the case.
An obvious problem arises with the criterion that we have established so far. It is too easy
to just build dependency relations into a model and use it to support only the explanations you
want. Without further restrictions, a model can be built to support all manner of counterfactuals,
and of course, not all of these are explanatory. In order to maintain a high threshold for
explanation, there must be further restrictions on the kinds of models that support explanation.
We are now in a place to put in place the last component of this account of explanation. There
are a few avenues one might take to impose the right restrictions on explanatory models in order
to reflect this distinction between models that predict and models that also explain.
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One could require that the model be approximately accurate to the real world system,
either in terms of causal processes, causal mechanisms, or structure. These strategies all require
accurate representation and allow for idealizations only when they are not harmful, i.e., when
they are Galilean (2.2.1). It is not impossible to reconcile representational approaches with
explanatory autonomy, but it has been agued that they favour reductive explanations given in
terms of fundamental theory, since they are the most accurate and invariant and robust and stable
and so on – they give the most accurate representations (Weslake, 2010). These are the kinds of
account that Batterman and Rice refer to as common features accounts (Batterman & Rice,
2014). Bokulich cogently argues that idealizations need to play a central role in explanation, and
not merely be tolerated when mostly harmless (approximately representative). Something
Bokulich makes very clear is that a representational approach loses out on a way to capture how
highly-idealized models explain. Her requirement for which idealizations, or fictions, can be
explanatory ultimately relies on a kind of representational view of structural realism: explanatory
models must be isomorphic to the systems they describe. Following Belot and others, I maintain
that a stronger criterion is needed.
Rather than imposing an additional local constraint on explanatory models, this account
introduces a global constraint. This means that instead of relying on structural or causal realism
in order to debar such models as being counted as explanatory, this theoretical approach requires
that explanatory models be integrated with a scientific theory. There needs to be information, a
set of assumptions, other models, or laws, justifying the idealizations in order to show why the
relations the model describes hold; why the system exhibits this behaviour. This information
explains why the model works when it does and why it fails when it does, which is vital to
understanding why the behaviour occurs the way it does. This also does the work of Alisa
Bokulich’s E3 criterion of specifying the domain of applicability (2.4.3), but also gives a
stronger requirement that helps to only capture explanatory models. This is the global
requirement of theoretical integration.
D4. M must be integrated with an independently explanatory scientific theory, T.
It is not enough that the model can faithfully reproduce the explanandum and give counterfactual
information, the model must also be appropriately related to a global scientific theory, such as
General Relativity, Quantum Mechanics, or the Theory of Evolution. The theory must have large
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empirical scope, which serves to unify otherwise disparate phenomena. The theory must also be
considered true, or mostly true, and explanatory by a relevant scientific community.
Instead of restricting the model on representational grounds, I propose that explanatory
models are a part of, or able to be integrated with, a successful and well-established scientific
theory. A motivation for this approach is the idea that an explanation is that which joins the
derivation of a new phenomenon to that which we already understand; it simplifies, organizes,
and relates phenomena and regularities, and so contributes to our understanding. An explanation
presents a new or unexplained phenomenon in a similar manner to already understood
phenomena, or in the context of an established body of scientific theory. Connecting a model to a
theory by performing an integration can broaden and deepen our understanding of an already
partially understood field.
Of course, not all theories are explanatory and not all theories are scientific. The theories
that give explanatory power to models are global scientific theories that feature models, make
use of shared assumptions, contain laws (or law-like statements), and are able to account for and
accurately predict a wide range of phenomena. They are global theories in that they are well-
established in science, broad in empirical scope, and widely believed by a relevant scientific
community to be explanatory. Prime examples are theories such as general relativity, Newtonian
mechanics, cellular biology, the kinetic theory of gases, the theory of evolution, and many
others.
A theoretical, rather than representational, approach allows for idealizations in a way that
a representational approach cannot. A representational approach is limited by requiring that
explanatory models feature only real entities and approximately real dependency relations. This
precludes highly-idealized models outright. Many representational models are of course
explanatory, but they are not explanatory because they accurately represent. It is rather because
they satisfy the criteria of this account, including because they are theoretically integrated, even
if this relation is not explicit. I contend that systems are modeled by theories and successful
explanatory models that seem to accurately represent causal relations are smuggling the
justifications and idealizations from some (likely Newtonian) theory.
As an example, Andrew Wayne (2015), echoing Hempel, argues that Galileo’s ideal
pendulum model is not explanatory in itself. Rather, it seems that it was mostly important
because it was a particular instance of Galileo’s general laws of motion. Historically, Galileo’s
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pendulum model served to base his studies of free-fall and is essentially a phenomenological
model. It gives some counterfactual information, and expresses a regularity. Wayne argues that
this suggests that its actual explanatory power comes from its integration into Newton’s general
mechanics, which could provide explanations of forces and the motions of bodies. The model’s
ability to satisfy Woodward’s requirements is not sufficient to guarantee its being explanatory,
regardless of whether it accurately maps onto the target system and answers counterfactuals.
Rather, it is the integration with a global theory with independent explanatory power that makes
the local model explanatory.
5.6.1. Some Aspects of Theoretical Integration
The relation between a local model and global theory can be very straightforward. In some cases,
the model features a generalization that is a central component of the theory and is capable of
deriving the explanandum phenomenon. This is seen in cases of textbook explanations, in which
the generalization or law is applied to a system and dictates its behaviour. The simplest case of
integration then is where the equations of the global theory are directly applicable. In these cases,
the systems themselves are probably rather idealized. Let us look briefly at a simple example.
Imagine we want to explain the rise in temperature when pressure is increased in a gas.
We might build a model containing the ideal gas law. One can use the formula to approximate
the behaviour of gases under certain conditions. The formula works most accurately at high
temperatures, low pressures, and for monatomic as opposed to molecular gases. The formula can
describe how an ideal gas behaves under certain changes. But it is only when it is seen in the
context of the kinetic theory of gases or statistical mechanics that it can be explanatory. This
comes forward in the idealizations included in the model. The theory is able to provide
justification for the idealizations that the particles are point masses, have elastic collisions, and
so on. The model is built according to the theory. The model’s idealizations show why the
regularity works best with monatomic gases at low pressure, for instance: the assumption that the
particles are point masses begins to be problematic when the molecular size becomes significant
relative to intermolecular distances. One can constrain the range of phenomena based on the
parameters included in the model. The theoretical background is what allows the formula to
provide explanatory information on the behaviour of the gas; information on why it behaves this
way when it does. The model featuring this generalization can be explanatory, but it is not in
virtue of the fact that it accurately represents or because it supervenes on the real fundamental-
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level causes. It is because the system can be modeled according to a theory. Again, a regularity
on its own is not explanatory.
However, in many cases, when the explanandum or the target system demands more
accurate models, the laws or fundamental equations of the scientific theory will no longer serve
to derive the explanandum phenomenon. In yet other cases, the theory’s equations may not be
directly solvable. In such cases, the integration becomes rather complex. I take it as a reasonable
assumption that the integration relation is quite different in various scientific disciplines. There is
a range of integration, from the straightforward pedagogical cases, like that shown above, to very
complex relations. In cases where the integration is complex, a modeller may need to employ
various mathematical methods for constructing a local model that is related to and consistent
with the global theory. What this means precisely will depend on the model and the theory. As
such, I cannot give a complete formal account of integration, but I can elaborate on some of the
methods and procedures that might be employed. One can think of integration as a process that
may feature one, a few, or several steps. Some of these steps may involve single perturbation
methods, renormalization groups, other mathematical techniques, and otherwise theoretically
justifying idealizations.
Wayne has argued that one of the factors that can differentiate a successful integration
from a failed one is that it makes certain idealizing assumptions unproblematic for the
underpinning theory (2016). He notes that in two competing putative explanations of the
phenomenon of gravitational waves, there is the same point-particle idealization, which is false
of the fundamental theory of general relativity. However, the post-Newtonian model has recently
been able to accommodate extended bodies, and thus partially discharge the assumption. The
competing derivation cannot discharge this assumption. In this case it is this theoretically-based
justification for the idealization that marks a distinction between an explanatory model and one
that is not.
A full exposition of all the various kinds of complex integration that might allow a model
to support explanations in different disciplines and subdisciplines is far beyond the scope of what
I can to accomplish here. However, since it is easy enough to see what kinds of models are
simply or deductively integrated with theory, it would be fruitful to see what kinds of models fail
to be integrated and thus fail to support explanations.
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5.6.2. Prediction without Explanation
Part of the support for the claim that integration is essential to explanation is that cases of non-
explanation lack theoretical integration. The hope is that the account of explanation can show
that models appropriately connected with these theories are explanatory and that two kinds of
models, even ones that are predictively successful, are not explanatory: those that are not
appropriately connected with theory; and those that are connected with theories that are not
explanatory. Phenomenological models, forecasting models, and others are of the former; models
of phlogiston theory, Ptolemaic astronomy and others are of the latter.
It will be useful to consider again the semiclassical model of quantum wavefunction
scarring from 2.4. It was shown that because it was still possible to derive the scarring
phenomena with a semiclassical model that there are good grounds for it being explanatory. The
model was counterfactually robust, though not as much as the local quantum model, and it fit
Bokulich’s criteria for explanation. The real problem, I claim, is not only that there is a better
explanation, but that the models of semiclassical mechanics are not integrated with a theory that
has independent explanatory power.
If one wants to claim that semiclassical mechanics is explanatory there are two
possibilities for showing this. Either semiclassical mechanics is just such an explanatory global
theory, or it is a method of integrating semiclassical models with quantum mechanics. If one
argues the former, however, the only way semiclassical mechanics can be explanatory is if
semiclassical mechanics were a well-established global theory of science. The two related
theories mentioned by Bokulich, closed-orbit theory and periodic-orbit theory, are not widely
held to be true, and have rather limited empirical scope. They make false claims about the
contributions of orbits to the quantum spectrum and are really only applicable in certain special
cases of quantum chaos. They are methods for approximating quantum calculations, but do not
have any real theoretical components. It is important to debar models of non-explanatory
theories. To use a rather extreme and unlikely example, if a model is appropriately connected to
astrology or the phlogiston theory of combustion, it is not therefore explanatory. There is an
important role to play by the relevant scientific community in determining what is an acceptable
and explanatory theory.
Perhaps it is best understood as attempting to integrate with quantum mechanics. What
the periodic orbit theory specifies is a way to approximate the quantum wavefunction density by
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taking the Fourier transform of the autocorrelation function of a Gaussian wavepacket. What it
essentially shows is that under certain conditions, its solutions are reliably approximate to the
full quantum calculations. The semiclassical approximations attempt to match the calculations of
quantum mechanics, and in this way show that the scarring phenomenon can be expected. This,
granted, is no small feat. However, there is no justification from the theory that justifies the
contrary to fact assumptions about the classical trajectories in the model and their effect on the
quantum spectrum. This is unlike the case examined by Wayne where the explanatory model can
discharge the assumptions that are false according to the theory. The false assumptions of the
semiclassical model cannot as yet be discharged. The quantum model on the other hand is
capable of showing that the phenomenon is to be expected without implementing assumptions
about classical trajectories and the effects of periodic orbits. It is not enough that there is reliable
prediction from the semiclassical model; justification needs to be given from the theory. It is
possible that there will one day be a way to refine semiclassical methods to discharge these
assumptions, but it would probably require more research about how the quantum to classical
transition takes place. This marks an important aspect of integration: it is relative to the state of
science.
While Hempel and Oppenheim argue that explanation entails prediction and vice versa, I
maintain that only the former holds. The reason for this follows from the idea that nomic
expectability is not sufficient for explanation. This can be seen by looking at cases where models
are predictively accurate, but not explanatory. Consider also the numerical models of circadian
rhythms. Circadian rhythms are biological processes that follow a 24-hour period. They are often
modeled as oscillators. It is possible to derive the right explanandum of say, a sleeping pattern,
with a mathematical model that fits the data. In fact, even some story can be developed about
exactly why this particular mathematical relation holds, but it is nonetheless non-explanatory.
Here there is no connection to a theory. It is a consequence of this criterion that such
phenomenological models are not explanatory.
The case of a Ptolemaic model of the solar system explaining planetary motion (2.4.2)
points to a different issue regarding the role of scientific theories in explanation. Systems are
modeled according to a theory, and it is certainly possible to model planetary motion in this way,
but this is not how a competent scientist today would actually proceed. It is important to have an
account that reflects how scientific explanation actually happens. However, if a scientist were
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committed to the truth or approximate truth of the Ptolemaic system and constructed such a
model to derive the explanandum, the derivation would still not count as explanatory on this
account. The reason this will not work out is that the Ptolemaic theory does not have independent
explanatory power. It is only predictively adequate insofar as it can be used to approximately
derive a limited range of phenomena. For reasons already mentioned, the conditions that make a
theory acceptable and explanatory is not something I want to focus on, but it can be clearly seen
that this theory does not have the empirical accuracy, nor the unifying and systematic value that
Newtonian mechanics or general relativity has. Further, while the debate between Ptolemy and
Copernicus was famously underdetermined, by contemporary standards the theory is now known
to be completely false and as a result has a seriously limited range of true counterfactuals that it
can support.
Mitchell’s case study of Lake Erie (3.2.2) has a rather different problem: it has no
specific explanandum. What is being asked for here in explaining the “behaviour” of an
ecosystem is unclear. This is what leads Mitchell to conclude that no single model, and
especially none at the fundamental level, can capture the target system’s behaviour. Had the
explanandum been a more precisely formulated phenomenon, such as the effect of zebra mussels
on the aquatic plant life, then a specific model could possibly have been built that derived the
result and provided counterfactual information. In the case of Lake Erie, as provided by Mitchell,
there is little sense as to what would constitute a good explanation of the behaviour of the
ecosystem. What this seems to be aimed at is a detailed simulation of various interacting
elements, and that is precisely what Mitchell claims provides the best explanation. But how this
constitutes an explanation on her account is not clear. In light of what has been said of this
account, that does not provide an explanation, but is rather phenomenological. Explanations are
to be had in the target system for various well-articulated explanandum phenomena.
The idea that combining multiple models can be explanatory something favoured by
Michael Weisberg, who also claims that it can be compatible with causal realism (Weisberg,
2007). This strategy is popular in dealing with highly complex phenomena. Each of the
assumptions made in particular models will be different and often incompatible. Practical trade-
offs are made to favour certain desiderata: generality, precision, simplicity, or accuracy. Levins
says the following about the conflicting assumptions of such models: “These conflicts are
irreconcilable. Therefore, the alternative approaches even of contending schools are part of a
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larger mixed strategy. But the conflict is about method, not nature, for the individual models,
while they are essential for understanding reality, should not be confused with that reality itself”
(Levins, 1968, p. 431). Wimsatt thinks that such highly-idealized models help us build truer
theories (Wimsatt, 1987). Some of these multiple-model idealizations, however are employed for
the sole purpose of predicting, such as the practice of weather forecasting. Here the accuracy or
realism of the idealizations is hardly a concern compared to their predictive abilities. If one looks
at the literature on complex models there is little in the way of a search for a single model, rather
what one finds is that there are many models with different desiderata. Levins and others are
realists with respect to these multiple model idealization, since they lead us to true theories “Our
truth is at the intersection of independent lies” (Levins, 1966, p. 423).
I maintain that the multiple-models solution is not really an explanation, but it does
reinforce something I have said about potentially explanatory models, viz. there is more than one
for any system. It is not reasonable to expect that there is only one explanatory model of a real-
world system. No model is perfect; models are constructed for certain purposes. With a clearly
defined and individuated explanandum phenomenon a good explanation featuring a particular
model can be explanatory (provided it meets certain criteria). The benefit of this account is that it
makes no demands that the relations described in the explanatory model be real causes. As such,
it is possible to consider multiple models to be explanatory, while avoiding commitments to
causal realism. In the end, Mitchell is right about the importance and explanatory value of
multiple models, but the assumption that these models are causal is unwarranted and potentially
problematic.
The Integrated Model Account of Explanation
Let us now assemble the criteria that we have established. We can take the criteria that have been
gathered to be sufficient to constitute an explanation, but not necessary for an explanation. As
mentioned as early as Chapter 1, there are many kinds of explanation and many senses of the
word and there is little hope that one account of explanation can point to some criteria that are
necessary for any explanation whatsoever. Rather, what I argue is that if the following criteria
are satisfied by a derivation, then it constitutes a scientific explanation.
D1. The explanandum must be a deductive consequence of the explanans.
D2. The statements in the explanans are true of a relevant scientific model, M.
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Representation
n
Deduction Idealization
D3. The model M referenced by the explanans gives counterfactual information that
shows on what the explanandum depends.
D4. M must be integrated with an independently explanatory scientific theory, T.
These criteria taken together are sufficient to capture good scientific explanations, though of
course, there are many other explanations and kinds of explanations which do not meet these
criteria. What we have in this account is a model-based deductivist explanation, whose models
are constrained by their integration with an accepted theory of science.
There are two aspects to this kind of explanation: local and global. The local aspect of
explanation concerns the local idealized model, the statements in the explanans that are true of
the local model, and the explanandum statement. As mentioned, sometimes the explanandum
phenomenon concerns the behaviour of the model itself. This is seen in cases of pedagogical
examples where only the behaviour of an ideal and abstract object is being accounted for. Where
there is a real-world target system, the explanandum is true, or approximately true of that system.
The following chart shows the relations that hold between the aspects of explanation.
Figure 5.1. A caricatured mapping of the relations between an explanation, the world, models, and a global
scientific theory.
Gobal Scientific Theory
Local Model Explanans
Target System Explanandum
Data Model
Description
Description
Integration
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Here we see the global scientific theory with systematic explanatory power at the top. Below this
is the local idealized model, which is related to it via the large arrow of integration. This
indicates that the model was constructed on the basis of the theory, or in a way that its
idealizations are theoretically acceptable. The construction of the model will depend on the
explanandum, and on various pragmatic concerns of the modeller, as well as on available data
from the target system. By contrast, we see that a data model is not connected via integration to
theory and primarily built from data. Realistically, the data model is not constructed from mere
raw data points, and is also refined over time by testing predictions and being modified. On the
right we see an explanation, separated into its two parts: the explanans and the explanandum.
The statements in the explanans are descriptions of the local model, but not necessarily of the
target system. This is a key difference in a non-representational approach. The statements in the
explanans deductively entail the explanandum statement, which itself is true, or approximately
true of the target system. Recall that, under this framework, saying that a model is explanatory is
to say that the model is capable of supporting explanatory arguments. I believe this account and
this simplified diagram reflect the way a range of scientific explanations proceed. An explanation
uses a model constructed according to a global theory to derive an explanandum phenomenon.
The explanatory power of a theory T is relevant to whether a model of that theory MT explains
its target phenomenon.
It is a reasonable question at this point to ask what it is that makes a global theory
explanatory. Unfortunately, the criteria for a good scientific theory can probably not be stated in
exact terms. What precisely constitutes a global explanatory theory is, I think, impossible to fully
articulate, because of the contextual, domain-specific, and paradigmatic aspects of scientific
theories. What counts as a successful theory changes over time and with respect to different
scientific communities, subdisciplines, and schools of thought. However, one characteristic of an
explanatory theory has been suggested above, and that is that it systematizes knowledge. This is
something that deductivists have always maintained, from Hempel to Feigl to Kitcher, and a
great many others besides. Kitcher attempted to elaborate on and formalize this unofficial
account of logical empiricism to explain how global theories are explanatory. Wayne has argued
that this is an aim that cannot be, and does not need to be, answered on a deductivist account
(Wayne, 2016). Following Hempel, and Kuhn to a degree, I think that it must be sufficient to
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point to examples of theories that are known to be explanatory, and examples of theories that are
known to not be (more on this in 5.6.2).
In terms of explanation, what counts as a theory with independent explanatory power is
determined not by strict criteria but by shared practices and standards of a scientific community.
It is only reasonable to allow that standards vary between communities, disciplines, fields, and at
different times. However, this is not mere descriptivism or relativism with respect to explanation,
as described in 1.4.1. There can still be normativity to claims about explanation. Rather, only the
kind of independent, syntactic justification that Kitcher was seeking is given up. What there is is
a network of mutually enforcing justifications that arguably give it more normative power than
the syntactic assessment of argument patterns on the unificationist view. The contextual
justification of explanatory judgments depends on known evidence, accepted standards, the goals
of investigation, and more.
One might be concerned that this is no different than Bokulich’s third criterion. Like
Bokulich, I also place some importance on the explanatory judgments and practices of a relevant
community of scientists, and I think to an extent, this is unavoidable. The role that this
contextual element plays in my account is quite different. Bokulich sees the structural criterion
as incapable of determining whether a model is explanatory. The role of distinguishing
explanatory from non-explanatory models is something she places solely on E3. I argued that
this makes the account descriptive, and rather uninformative. What I have offered is a high
threshold for explanation which can be assessed independently of the assessments of the
scientific community. That what counts as an global explanatory theory is contextually
determined reflects a fact. It also allows the account to appeal to a wider range of disciplines,
because it allows that the standards are, to a large degree, relative.
Empiricism, Emergence, and Reduction
This account, as it is not specifically causal, has the added benefit of being compatible with anti-
realism with respect to high-level causes and science itself. This account allows that properties
and entities in a model can be genuinely explanatory whether or not they accurately represent
features in the target system. The account I propose remains silent on the causal nature of the
relations of models that support same-object counterfactuals. The truth-makers for the
counterfactuals stem from brute facts about the model, but no further commitments are made.
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The account is still open to accepting as explanatory models that are widely considered to
accurately capture the causal (and/or structural) dependency relations in the world, as long as
they also satisfy its criteria. Importantly, on this account, an empiricist can say something about
explanation beyond issues of pragmatics.
Many accounts of explanation have focused on making high-level explanations palatable
to physicalist intuitions, and this often involves either showing how they can be consistent, or
showing how they can be reduced. It is often said that scientific explanations effect a reduction
of the puzzling to the familiar, but Hempel does not think this stands up (Hempel, 1966, p. 83). It
is not merely concerned with the psychological aspects of understanding that may help one grasp
what is going on. In fact, the at-homeness of explanation can lead to very unscientific theories
(anthropomorphic nature, e.g.). Science should not hesitate to do the opposite and go against
intuition if it is necessary. It is not the aim of scientific explanation to explain away the
everyday. Rather, it is to account for it. The kinetic theory of gases does not say that there are not
swarms of gases that change volumes; and atomic theory does not show that there are no tables
and chairs. They are not explained away, but remain objects and entities that can be legitimately
be used in scientific explanations.
The account I propose, along with many others, does not favour reductionism to a
problematic degree. Some have focused on levels of selection, others have made abstraction or
proportionality and explicit requirement (Sherman, 1988; Yablo, 1992; Strevens, 2008; List &
Menzies, 2009). Some even object to any preference for reductive explanations (Mitchell, 2003;
Grantham, 2004; Brigandt, 2011). However, I suggest that having a criterion of derivation (D1)
can also help to prevent problematic reductionism, of which many accuse those like Bickle
(Bickle, 1998, 2003; Churchland, 2004). If one allows explanations featuring highly-idealized
models, then there is little reason to focus on proportionality as a distinct criterion, as some do.
Consider a case where one wants to know why Joe sold his house. It is reasonable to think that a
high-level explanation is perfectly adequate. In fact, an explanation from fundamental physics is
not likely to ever be formulated. Recall that if one cannot derive the explanandum from the
explanans, then there is no explanation. In this sense, a deductive criterion can actually help
prevent harmful reductionism. Successful explanation of a macro-level phenomenon requires
either that the explanans contains macro-level terms, or has micro-level terms as well as
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information about the relationship between the micro- and macro-level terms. If it does not, then
it simply does not derive the explanandum and does not answer the why-question.
I think that this can prevent the account from being harmfully fundamentalist. I think that
this account and its approach to high-level explanation can be consistent with a wide variety of
metaphysical views. Unlike certain approaches that require emergent causation, this account
remains open to non-reductive physicalism. It also does not require any particular theory of
emergence to account for non-fundamental entities, such as Humphreys (1997). High-level
models are explanatory when they meet the requirements of the account, regardless of whether
there is another explanation featuring a model from a more fundamental theory. For instance, it
is possible to maintain that the ideal gas law can be explanatory when the explanandum regards,
say temperature. The ideal gas law can be de-idealized, and so in a straightforward way, one
knows why it is accurate within certain ranges of the values of its variables. One might argue that
the statistical mechanical explanation is better, or deeper, but on this account, the high-level
model is explanatory if it meets the criteria.
Lastly, it bears mentioning that the account remains open to being supplemented by a
measure of explanatory depth. The incorporation of counterfactual information into the account
likely makes it amenable to an account of w-questions, but possibly also some independent
measure. However, it is important to note that even if this account exhibits a preference for
reductive explanations, a high-level model can be not only explanatory, but also the best
available explanation when there is a high-level explanandum and no way to derive the
explanandum from a more fundamental level, as was mentioned in the case of semiclassical
mechanics in 2.4.2.
Conclusion and Limitations of the Account
This account is intended to reflect the results of the investigations in previous chapters. Firstly, it
aims to reflect the model-based approach to explanation in practice. This allows it to avoid
relying on laws of nature and opens it up to more than covering law explanations. Secondly, it is
deductivist and non-representational. This allows it to avoid any problems arising from the
metaphysics of emergent causation, while remaining compatible with metaphysical realism about
causation as well as empiricism and non-reductive physicalism. This also expands the scope of
explanatory idealization beyond the merely causal. This opens up the range of explanatory
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models to include high-level models that may or may not be causal, which better reflects
explanatory practices. However, the scope is not increased too broadly, because, thirdly, it
imposes both a local and global constraint to ensure that explanatory models are those that can
provide counterfactual information and are not merely predictively successful phenomenological
models. This allows one to set a high threshold for explanation, while still allowing the range of
possibly explanatory models to exceed that of other accounts.
The account is still largely conceptual at this stage. A lot of work remains to be done to
test it against scientific practice. No doubt this requires in-depth case studies. Future work will
attempt to explore how integration actually proceeds for different kinds of models. Case studies
will also show how the application of the account fares in various disciplines and with respect to
changes. As of now, it is an assumption that the restrictions on models are sufficient and that
they track our judgments about explanation reasonably well. It remains to be seen how the
criteria hold up against detailed case studies. I would like to acknowledge some possible
concerns and limitations to the account and to what was accomplished here.
There is a concern some might have that a theoretical approach is better suited to some
fields like physics, which have well established laws and theories. Not all disciplines are equally
theory-oriented. However, this account is aimed at increasing the scope of possible explanations
and I think the case can be made that this approach is general enough to capture explanations in
various sciences. One reason is that this approach is capable of capturing the way highly-
idealized models explain. Many have brought to light the fact that various branches of science
make extensive use of highly-idealized models that we would want to claim are explanatory,
even though they may not fit into a causal-explanation scheme. So while the emphasis on theory
may seem to limit disciplines, I think it can be understood as including causal and non-causal
explanations.
Another reason is that unlike other deductivist accounts of explanation, it does not focus
on universal and exceptionless laws of nature. This allows models from ecology and other
biological sciences to be capable of supporting explanations. Much as the work of Mitchell and
Woodward focused on degrees of invariance, stability, exceptionlessness, and universality, this
account can make use of models that support same-object counterfactuals. The generalizations in
other sciences are not precluded from explanation because they are not exceptionless and
universal. Further, the fact that there are discipline-relative standards of what counts as an
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explanation is recognized. These standards are unlikely to be the same across the sciences, and
on this account there is no independent measure of the explanatory power of theories.
Explanations in physics are not necessarily better than those in other disciplines. Some will find
it unsatisfying to simply leave the explanatory power of theories to the judgments of a relevant
scientific community, but I think that formally analysing that power may well be intractable.
Many have argued that explanations in the social sciences are necessarily local and
piecemeal, and that there is not likely to be any global constraints on explanation in areas as
diverse as psychology and economics (Kincaid, 1986, 2004). However, as we have seen,
explanation in the hard sciences are also local and piecemeal. In many cases, one uses the
equations of a fundamental theory as a kind of base with theoretical constraints in order to
construct a local model of limited applicability. Something very similar happens in population
biology. Here too, one never finds a system that is perfectly captured by a simple base model of
predator-prey interaction. What is done to explain the behaviour of a particular population is to
construct a local model based on a very highly-idealized base model, featuring something like
the Lotka-Volterra predator-prey equation, but heavily modified to the target.
In some disciplines, it is not the case that theories are successively replaced with better
ones, in which case it is difficult to see how integration with a theory is necessarily tied to
explanation. However, I think it still is, but what counts as theory here is often implicit, but also
involved in the model construction. I think it is possible that in some cases there is a theory
operating in the background, according to which the model is being constructed. It is little more
than conjecture at this point, but this account could reasonably provide insight into explanation
in non-fundamental sciences, and even the social sciences. Its high threshold can serve as a guide
for formulating better explanations. If one considers a non-explanatory model in higher-level
science, one might see that an aspect that makes it merely phenomenological is not its accuracy,
but the fact that it is built up from data with no regard to an explanatory theory. An explanatory
model in this context might be constrained on both sides by accurately matching observed
phenomena and reflecting global theories about the subject matter.
The thesis was also not able to deal with mechanistic explanations in any capacity. The
popularity of mechanistic accounts and mechanistic explanations in practice demands that this be
treated in detail. A further study of this would inform the fit of this account in disciplines outside
of physics.
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More is also needed in terms of data about scientific judgments concerning explanation.
In particular, the account makes use of scientists’ judgments about which models are considered
explanatory and what is generally thought of as causal. This dissertation acknowledges the
limitations in not beginning with this kind of data, and not also being a study of the sociology of
explanatory practices in science. This no doubt limits the precision and strength of claims about
explanation.
Outside of concerns about its general applicability, there is a genuine worry that a lot of
details remain to be filled in. The scope of the project at this early stage means that there are
quite a few promissory notes written in. For instance, there is much more that can be done to
determine more precisely what a global explanatory theory is. A number of questions remain
unanswered: are there certain necessary features of explanatory theories, or is it entirely
contextually determined? Where is the cut-off for exactly how broad in scope a theory has to be?
There are doubtless many other important question that will have to be relegated for future work.
Though the account is embryonic, untested, and in need to fleshing out, I hope to have
demonstrated in this chapter that the model-based deductivist approach is a promising one.
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Conclusion
I will here briefly review the results of this thesis and speak to the prospects of this account. In
the first chapter, I introduce the idea of scientific explanation and the tradition in the philosophy
of science going back to Hempel and Oppenheim. I begin by establishing what is the proper
domain of an account of scientific explanation. I do this by contrasting scientific explanations
with non-scientific explanations and scientific practices that are not explanatory. In 1.3, I argue
that pragmatic issues concerning understanding and communication are an important part of
explanation, but do not exhaust it. In 1.4, I review the goals and desiderata of an account of
explanation and argue that an account should maintain a reasonably high threshold for
explanation. And it should largely reflect the explanatory judgments of a scientific community,
but not be merely descriptive. I maintain that in order to remain tractable, an account of scientific
explanation should not attempt to also be an account of explanation in general, and also that an
account of scientific explanation can succeed without also being an account of explanatory
power or explanatory depth.
In Chapter 2, I review structural accounts of explanation, in particular the one given by
Alisa Bokulich. Structural accounts are intended in part to expand the scope of explanation by
allowing non-representing, or highly-idealized, models to be considered explanatory. I begin by
taking a stance on what scientific models are, viz. that they abstract objects, and introducing the
notions of idealization and idealized models and the problems they might present for
explanation. I find that arguments problematizing laws of nature and the representational nature
of explanatory models to be persuasive. I take this as a strong indication that a model-based
account of explanation is much more likely to be successful and reflective of the practice of
explanation. The remainder of the chapter is dedicated to reviewing Alisa Bokulich’s account of
structural model explanations A structural account of explanation maintains that an explanatory
model is one that captures the structure of the system it represents, regardless of whether it
accurately represents the entities or causes. I argue that structural accounts of explanation similar
to hers are faced with two possible choices: either they stipulate a threshold above which a model
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is explanatory, or they take a comparative approach for deciding the best structural explanation. I
conclude that neither is an attractive option: The first is arbitrary and the second shows
preference for representational models. Thus, I conclude that even though structuralists have
argued for expanding explanation to non-representing models, structural accounts of explanation
like Bokulich’s do not succeed in showing how non-representing models can be explanatory. I
take accounts like Bokulich’s to have demonstrated very cogently the need for an account of
explanation that can capture the way that highly-idealized models explain. This is a worthy goal
of an account of explanation, but the structural approach is at base representational and this gives
it preferences for accurate, representative models.
In Chapter 3, I turn my focus to causal accounts of explanation. I first review arguments
from Sandra Mitchell regarding the inherent complexity of biological systems and the
implications this has for explanation. Mitchell argues that the sheer scale, heterogeneity, and
time scales involved in biology gives rise to complexity and chaos which prevents a single causal
explanation from succeeding. She argues that explanation in biology is piecemeal and local and
not theoretical and law-based like physics. I critically review her arguments and find that her
case studies do not show what she takes them to show; that her integrative pluralism, while less
problematic than other pluralist accounts, offers little information about why these models are
explanatory; and lastly, that her account finds all models to be explanatory. I take this to show
that there are multiple models that can describe aspects of a system, which could all be
explanatory of various explananda. Where I part with Mitchell is in giving all these models a
causal interpretation.
I then present James Woodward’s account of manipulationist explanation in 3.3.
Woodward identifies causes as exhibited in the relations between the variables of models that are
invariant under a range of interventions. I note that this implies that there can be multiple models
at various levels of a system that all represent real causes. This is a metaphysical position similar
to Mitchell’s wherein there are high-level causal facts, a position referred to as emergent
causation. A worry emerges that this will fall prey to the arguments from Kim and others
proposing that emergence is inherently problematic. I then review these exclusion arguments and
their application to Woodward’s account. I find that while Wilson and List and Menzies can
mount defenses against exclusion arguments in favour of non-reductive physicalism, this fails to
apply to Woodward. Woodward maintains that there are high-level causal facts that are not
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reducible to a fundamental level. They are described by the invariant relations between variables
of a model, of which there are many in a physical system. This is in contrast to List and Menzies
who take causation to be fully counterfactual and proportional to its effects, and to Michael
Strevens who denies that there are high-level causal facts.
Strevens’ account is concerned with how to show preference for non-fundamental
explanation given that all causal facts are at the fundamental level. What he introduces is a
measure of explanatory depth that is capable of preferring high-level explanations. Like List and
Menzies, he too looks to difference making to establish this. Strevens’ account is a two-step
process of identifying all the causes of the explanandum and eliminating those that make no
difference to it. The second step is to remove all factors that do not logically entail the
explanandum. Strevens argues that this should provide fundamental-level causes for explanations
at multiple levels. I argue that this approach is very distinct from the practice of scientific
explanation which never simply begins from knowledge of total causes. I also take issue with the
idea that all high-level models are simply abstractions and idealizations of fundamental level
causes. This is an assumption that was problematized in Chapter 2 and would preclude all
highly-idealized models. Lastly, I mention that it is not clear what fundamental level causes are,
since the fundamental level of physics is quantum.
Chapter 4 presents the traditional deductivist accounts of explanation to inform the
model-based deductivism presented in Chapter 5. I begin by looking at Hempel and
Oppenheim’s account of D-N explanation, according to which explanations are deductions of
phenomena by means of initial conditions and laws of nature. I then explore the limitations of the
D-N account before I turn to Kitcher’s unificationism, which attempts to add unification to the
D-N account in order to solve many of the counterexamples that were raised. Kitcher’s idea was
that only the most unified set of argument patterns is explanatory. By demonstrating that the
counterexample cases could be debarred as not the most unified, he was able to propose a
solution that might defend deductivism. Unfortunately, there are concerns about Kitcher’s
unificationism. One concern is that Kitcher’s criteria are to be weighed against one another, but
there is framework for determining how to do this. Another concern is that the solutions to the D-
N counterexamples do not work as well as he hopes. Lastly, the winner-take-all conception of
explanation is necessary for Kitcher, but is highly unintuitive and does not follow from the
comparative assessment of theories. I conclude the chapter reviewing the current state of
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deductivism and presenting what the investigation of the deductivist account has provided me in
formulating a neo-deductivist account.
In the final chapter, I offer an account of scientific explanation that aims at capturing
explanatory models regardless of whether they accurately represent causes or structure. It further
does not require articulating laws of nature or restricting explanation to derivations therefrom. I
begin from the idea that explanations are arguments that feature certain scientific models, viz.
those that provide counterfactual information about the system and are related to a global theory
of science. I maintain that this account is more reflective of explanatory practice than other
accounts and that it is broadly compatible with metaphysical stances on causation as well as with
empiricism and non-reductive physicalism. It allows for the high-level models to be explanatory
independently of low-level models and fundamental causes, but does not require a theory of
emergence or physical acceptability. As such, there is a hope that this account will be broadly
applicable and largely reflective of the methods, practices, and scope of explanation in the
sciences and also palatable to a wide range of philosophical positions.
At this stage the account is rather conceptual and the criteria are still preliminary.
Although, it remains to be tested against various in-depth case studies, I hope to have
demonstrated that the prospects for a neo-deductivist account are better than the past and current
literature indicates. I hope that the work I have done motivates the idea that this kind of non-
representational approach to explanation reflects explanatory practice and that the particular
account I have outlined is a promising alternative to other popular accounts of explanation.
152
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