Shades of Grey: Granularity, Pragmatics, and Non-Causal Explanation

Abstract

Implicit contextual factors mean that the boundary between causal and non-causal explanation is not as neat as one might hope: as the phenomenon to be explained is given descriptions with varying degrees of granularity, the nature of the favored explanation alternates between causal and non-causal. While it is not surprising that different descriptions of the same phenomenon should favor different explanations, it is puzzling why re-describing the phenomenon should make any difference for the causal nature of the favored explanation. I argue that this is a problem for the ontic framework of causal and non-causal explanation, and instead propose a pragmatic-modal account of causal and non-causal explanation. This account has the added advantage of dissolving several important disagreements concerning the status of non-causal explanation.

Full text

Shades of Grey:
Granularity, Pragmatics,
and Non-Causal Explanation
Hugh Desmond
Katholieke Universiteit Leuven
Implicit contextual factors mean that the boundary between causal and non-
causal explanation is not as neat as one might hope: as the phenomenon to be
explained is given descriptions with varying degrees of granularity, the nature
of the favored explanation alternates between causal and non-causal. While it
is not surprising that different descriptions of the same phenomenon should
favor different explanations, it is puzzling why re-describing the phenomenon
should make any difference for the causal nature of the favored explanation.
I argue that this is a problem for the ontic framework of causal and non-
causal explanation, and instead propose a pragmatic-modal account of causal
and non-causal explanation. This account has the added advantage of dis-
solving several important disagreements concerning the status of non-causal
explanation.
1. Introduction
Two obstacles seem to preclude any agreement on how causal explanations
should be delimited from non-causal explanations. The first concerns the
very definition of causal explanation, which determines how the boundary
between causal and non-causal explanation is drawn. Even though most
adhere to a relatively narrow definition of causal explanation and thus
allow for non-causal explanations (Woodward 2003; Craver 2014), it re-
mains possible to adopt a very wide definition of causal explanation and
thus to argue that all purported examples of non-causal scientific expla-
nation are in fact causal (e.g., Skow 2014).
The second obstacle, relatively overlooked by comparison, concerns the
pragmatics of causal and non-causal explanations. In a perceptive paper,
Holly Andersen points out that judging an explanation to be causal seems
to be very sensitive to the way in which the explanandum is described
Perspectives on Science 2019, vol. 27, no. 1
© 2019 by The Massachusetts Institute of Technology doi:10.1162/posc_a_00300
68

(Andersen 2016). If one varies the grain with which an explanandum is
described, previously paradigmatic examples of non-causal explanation
no longer seem unproblematically non-causal.
Few if any philosophers would entirely deny the importance of prag-
matics. The importance of implicit contextual factors for the analysis
of scientific explanations has long been recognized (going back to e.g.,
Hempel 1942, or Hempel and Oppenheim 1948). However, a separate
question arises as to why fine-graining an explanandum should have any
effect on our assessment of the causal nature of an explanation—and how
precisely this occurs.
If one adopts a purely ontic view of explanation, this may seem like a
non-question. According to the ontic view, an explanans explains in virtue
of picking out objective causal (or non-causal, as the case might be) depen-
dencies (Salmon 1989; Woodward 2003; Strevens 2008; Craver 2014).
Thus, to fine-grain an explanandum is simply to describe the correspond-
ing phenomenon with increased precision, which in turn helps prevent
confusion as to what explanatory dependencies are being picked out.
Re-describing examples of causal/non-causal explanation leads to differ-
ent assessments of their causal nature, simply because different causal/
non-causal dependencies are being picked out.
Yet, lurking beneath the surface are regularities that such an ontic view
is oblivious to. Our judgment of the causal nature of explanation is very
sensitive to slight re-descriptions of the phenomenon—as Andersen notes,
“slight reformulations of the explanandum will change the relevant expla-
nation(s) from causal to non-causal and back again”(Andersen 2016, p. 2).
The problem I wish to draw attention to is why this should be the case
at all. After all, strictly speaking, a re-description of a given explanandum
actually involves considering a different explanandum, and this expla-
nandum may in turn have many potential explanations. Some of these
will pick out causal dependencies, and others non-causal dependencies;
some of these explanations will be good, and others bad. Let us assume
going forward that there is always a single “best”explanation (this in fact
tends to be assumed in the examples discussed in the literature), so that a
single explanans can be associated with a given explanandum. This (best)
explanation can be called the associated explanation of a given expla-
nandum. Why should re-describing an explanandum affect the causal
nature of the associated explanation at all?
1
Why should, for instance,
1. Just to emphasize, speaking of a re-description (and in a more technical sense: a
fine-graining) of an explanandum as an operation that “affects”or “changes”the causal na-
ture of the associated explanation is a shorthand for the precise but convoluted description
of the connection between re-description and the causal nature of an explanation above.
69Perspectives on Science

re-describing the bridges of Königsberg lead to different evaluations of the
causal nature of the associated explanation? Why could not re-description
reveal only a host of causal explanations, or only a host of non-causal
explanations?
If one were to compare a scientific explanation to an object in an old-
fashioned, grey-scale photograph (as in Andersen 2016), the re-description
is like a grain-focuser that reveals the grey object to be composed of
some slightly darker grains (e.g., non-causal explanations), and slightly
brighter grains (causal explanations). However, why should this be the
case? Why should zooming in not just reveal shades of the same mono-
tone grey? Or why could a scientific explanation not be like a digitized
photograph, where zooming in on a grey-scale object reveals a host of
perfectly white and perfectly black pixels? From a purely ontic view of
explanation, this complex interplay between granularity and our judgments
of the causal nature of explanation is more or less a mystery.
In this paper I will propose an account of how the granularity with
which explanandum and explanans are described affects our judgments of
the causal nature of the associated explanation. I call this the “pragmatic-
modal”view of the causal nature of explanation: it accounts for the relation
between granularity and (the judgment of) the causal nature of explanation
in terms of how often implicit contextual factors (the “pragmatic”)affect
the modal structure of the relation between explanans and explanandum.
After considering some examples in the first section, in the second
Idefine fine-graining as a set-theoretic operation involving the introduc-
tion of additional contrast classes (in continuity with, e.g., Van Fraassen
1980 or Hitchcock 1996). In the third section I draw on Lange’s concept
of modal strength (but bracket Lange’s metaphysics: see Lange 2009) to
show how the structure of the relation between explanans and expla-
nandum can be connected with judgments of the explanation as causal
or non-causal. In the fourth and final section I bring the two preceding
sections together, and account for the fundamental connection between
fine- and coarse-graining and the causal nature of explanation. I further
illustrate where this approach can be philosophically fruitful: dissolving
disagreements concerning the causal nature of paradigmatic cases (Euler’s
bridges, marbles in bowls, etc.), and dissolving (at least some) disagree-
ments concerning the definition of causal explanation.
2. The Role of Pragmatics: Some Observations
Does a mathematical model, such as the Lotka-Volterra model, constitute
a causal or non-causal explanation? This obviously depends on how it is
used, and, in particular, on what explanandum is targeted. For instance,
if the explanandum is a general feature of population dynamics, e.g., the
70 Granularity, Pragmatics, Non-Causal Explanation

simple fact of periodicity, then the Lotka-Volterra equations can be used
to mathematically deduce periodicity. However, if more specific outcome
states are taken as the explanandum, such as the frequencies of hare and
lynx in an actual population as they change over time, then additional
causal factors will likely have to be invoked. For instance: there may be
other marginal predators in the ecosystem preying on the hare; the hare
may have other competitors in its habitat; the lynx may be suffering from
a parasite that prevents the lynx population from reaching a certain
density. In this case the Lotka-Volterra model is used as an abstract causal
model that represents only some key causal processes while ignoring
others; for accurate predictions concerning actual populations, the Lotka-
Volterra model needs to be supplemented with additional causal factors.
This is one instance of how fine-graining the explanandum can change
the causal nature of the associated explanation. Andersen (2016) identifies
some other ways in which causal and non-causal explanations can com-
plement each other, but all do so by targeting slightly different aspects
of the same phenomenon.
It is interesting to note that fine-graining the explanantia can also change
the causal nature of the associated explanation. For instance, Graham
Nerlich (1979) considered the explanandum of a particle traveling along
a curved line even though no force is acting on it. The explanans lies in
the geometry of space, and because, as Nerlich argues, space is causally
inert and there are no other causes for the particle’s motion, the expla-
nation is non-causal.
While many might not agree with Nerlich’s assessment of the example
(for instance, in judging space to be causally inert he seems to assume a
process rather than a difference-making account of causation), it is inter-
esting to note how the causal nature of the associated explanation changes
when the explanantia are fine-grained. This is what seems to underlie the
strategy Bradford Skow takes in reassessing this example (Skow 2014,
pp. 451–2). He points out that relevant causal information remains hidden
in Nerlich’s description of the explanans: if the dynamics governing par-
ticle motion were different, for instance by causing the particle to spon-
taneously prefer one direction over another, then the particle path would
also be different. So, there are other causes for the particle’s motion beyond
the geometry of space. Skow concludes that the resulting explanation is in
fact causal.
Note that it would be inaccurate to state that Nerlich and Skow dis-
agree about the same explanation, since they are operating with different
explanantia. In Nerlich’s analysis, the assumption is that particle dynamics
cannot vary and hence does not feature in the explanans. In Skow’s anal-
ysis, particle dynamics is a factor that can vary, and one on which the
71Perspectives on Science

particle’s movement counterfactually depends. In Skow’s understanding of
causal explanation (roughly, an explanation of an event Eis causal when it
provides information about what could have prevented Efrom occurring),
this is sufficient to reassess the example as an instance of causal explanation.
Attention to granularity can also help make sense of the diverging
intuitions in two otherwise similar accounts of the role of mathematics in
scientific explanation (see Strevens forthcoming; Huneman ([2010] 2018).
Philippe Huneman argues that mathematical properties non-causally
explain by constraining the range of possible causal explanantia (e.g.,
causal histories, or causal mechanisms). This constraint thus explains a
certain global property of the explanandum, one that counterfactually
depends on the mathematical property. Michael Strevens likewise holds
that the explanatory function of mathematics is to constrain the set of
possible difference-makers but reaches an opposite conclusion: far from
constituting distinctively non-causal explanations, the use of mathematics
serves only to reach a deeper causal explanation, where we not only iden-
tify the difference-makers but also are able to explain why this and not
some other factor is a difference-maker. There is no distinctively non-causal
explanation, only different levels of causal explanation.
The divergence in assessment can be associated with an implicit differ-
ence in their description of the explanantia. To this end, consider the sub-
tle differences in how “the seven bridges of Königsberg problem”is
analyzed. In Strevens’account, causal explanation seeks to identify and
isolate the causal difference-makers from the causal web consisting of all
causal processes in the universe (Strevens 2008). Euler’s explanation of
the failure to cross all bridges without crossing the same bridge twice is,
in this account, a highly abstract causal ex-planation: it is one that has ab-
stracted away from all elements in the causal web that are not difference-
makers. The only difference-makers are the structure of the bridges and
land masses (and the fact that walking around is equivalent to following
a curve on a graph). Mathematics helps us identify the true difference-
makers—it helps us understand why some elements are and why others
are not causal difference-makers—but it does not actually do the explaining.
For Huneman, by contrast, the bridges example constitutes a non-causal
explanation for the simple reason that the graph-theoretical property
coarse-grains over all the different bridge configurations that instantiate that
property. There are other bridge configurations that do not instantiate the
theorem conditions. Even if the laws of nature were radically different, the
counterfactual dependence of the outcome on the graph-theoretical property
would hold. In this way, for Huneman, the graph-theoretical property
coarse-grains the causal states; for Strevens, the same property identifies
the relevant causal difference-makers.
72 Granularity, Pragmatics, Non-Causal Explanation

The point here is not to confirm or criticize either stance, but rather
point to a curious difference in the set of possible explanantia each philos-
opher works with. For Strevens, the explanans must be drawn from the
causal web. The explanans that ultimately features in the causal expla-
nation must be a causal difference-maker, but the set of “candidate”expla-
nantia (see next section) is the causal web. By contrast, the set of candidate
explanantia for Huneman are alternative graph-theoretic theorems, only
one of which is true in our world. The set of candidate explanantia in
Strevens’sanalysisisthusmuchmorefine-grained than in Huneman’s
analysis. Strictly speaking then, they are not discussing the same expla-
nation, since one is operating with a set of candidate explanantia different
from the other’s.
3. Granularity of Analysis
In this section I will set out to make the preceding observations more
precise. Not just explananda, but also explanantia can be fine-grained, and
this seems to have at least something to do with diverging assessments of
non-causal explanation. Later in the paper I will propose an explanation of
this fact. For now, this and the next section will offer a more precise definition
of granularity, and a distinction between causal and non-causal explanation.
Intuitively, fine-graining implies some sense of “zooming in”on an
object. A zebra and a horse may seem identical at a distance, but on closer
inspection, we are able to perceive further properties, such as the patterns
in their hides, which distinguish the two animals. If we were to meta-
phorically “zoom in”even further to the level of genotypes, we may dis-
tinguish between them as separate species. As an initial characterization of
the concept, analyzing an object at a “finer grain”can be understood to
mean that the object is characterized by additional properties, allowing
a more powerful resolution by which the object can be distinguished from
other types of objects.
Similarly, we can “zoom in”on a phenomenon. Consider a population of
lynx and hare. As mentioned previously, an initial explanandum may be
the simple fact that the relative frequencies of hare and lynx ( p,q) change
periodically. This explanandum may be fine-grained to why the relative
frequencies ( p,q) follow one particular sequence of numbers rather than
another. Conversely, coarse-graining a phenomenon entails abstracting
away from some properties, so that the resulting explanandum is indistin-
guishable from some other explananda within a wider class. Thus, instead
of explaining why a lynx-hare population followed this particular path
rather than another towards an equilibrium ( p,q) (instead of equilibrium
(p
0
,q
0
)), we may want to know why the equilibrium ( p,q) (instead of
equilibrium ( p
0
,q
0
)) was attained, regardless of path.
73Perspectives on Science

In general, an explanandum can be thought of as some proposition p
1
concerning the actual state of the world (e.g., a fact, the occurrence of an
event, some state of affairs) that is contrasted with any number of other
propositions p
2
,p
3
,…corresponding to alternative (but non-actualized)
states of affairs. Let us assume for the sake of simplicity of exposition that
p
1
concerns the instantiation of some property P(“Xis P”), and that the
alternative proposition p
2
is the non-instantiation of that property (“X
is not-P”), so that the explanandum is “Xis Pinstead of not-P.”This
explanandum can be described at a finer grain by considering an additional
property Qwhich happens to be instantiated in X, and so the number of
alternative propositions is increased to three, where either only Pis instan-
tiated, or only Q, or neither of the two.
Fine-graining the explanandum, as it will be understood in this paper,
thus involves introducing contrast classes in specifying an explanandum
(Van Fraassen 1980; Hitchcock 1996).
2
Fine-graining an explanandum in-
volves introducing an additional contrast class and monotonically increases
the size of the set of propositions that could have been true but are not.
If one is to scientifically explain a particular proposition about some part
of the world, one has to explain why all the alternative propositions are
not true.
A similar analysis can be done for fine- and coarse-graining explanantia,
but a distinction must be made between two ways of fine-graining. Consider
for instance the example of Euler’s bridges. One set of initial states can be
the set of possible starting points a walker can take in Königsberg. This
set of initial states can be fine-grained in two ways. The firstisthatwe
add further detail to the possible initial states; the walker starts out by
standing on one leg or on two legs, or by crossing her fingers. This fine-
graining allows factors to vary that had previously been ignored (without,
of course, necessarily resulting in a better explanation). The second way
of fine-graining allows factors to vary that had previously been assumed
to be constant; thus one can distinguish between initial states where graph
theory (as we know it) is true, and initial states where an alternative graph
theory is true (e.g., a graph theory, unimaginable to us but possible in some
vastly extended sense of possibility, where all seven bridges can be crossed
without crossing the same bridge twice).
To summarize this discussion, the level of granularity of an explanation
can refer both to the granularity of the explanandum (how many alter-
native propositions there are with which the actually true proposition is
to be contrasted) as well as to the granularity of the explanans (how many
2. The inverse operation, coarse-graining, corresponds to introducing an equivalence
class: See Desmond 2017.
74 Granularity, Pragmatics, Non-Causal Explanation

candidate propositions there are on which the explanandum depends). An
explanandum is fine-grained by the introduction of a novel contrast class;
an explanans is fine-grained by allowing a boundary condition to vary.
Note that I do not assume any time-asymmetry—for instance, in optimal-
ity explanations (discussed later more fully), possible explanantia can in-
clude different geometries or shapes that directly and without time-delay
affect the possible equilibrium states of the system.
4. A Pragmatic-Modal Account of the Causal Nature of the Explanation
4.1. Modal Strength
Since we want to account for how granularity of description affects the
boundary between causal and non-causal explanation, we need an initial
representation of this boundary. In the following I adopt Lange’suseof
the concept of modal strength.
3
Lange’s underlying intuition here is that
in non-causal explanations, the explanandum is entailed with a necessity
that is not possible in causal explanations. For instance, regardless of what
the laws of physics look like, it is impossible to cross all the bridges of
Königsberg without crossing the same bridge twice. No causal explanation
could have entailed the explanandum with the same “modal strength.”
In Lange’s framework, degrees of modal strength are tied up with a
metaphysics of necessity. The varieties of necessity form a pyramid-like
structure, including accidents, natural necessities (such as natural laws),
“broadly logical”necessities (e.g., mathematical truths), and “narrowly
logical”necessity (logical truths, such as the law of non-contradiction).
There is also a hierarchical relationship between these necessities. Natural
necessity is “lower”than mathematical necessity, in the sense that the
number of counterfactual situations under which the laws are preserved
is strictly less than the number of counterfactual situations under which
the logical truths are preserved (see Lange 2009, p. 77). Natural necessities
hold true in a greater number of possible worlds than accidents do; logical
necessities are true in an even greater number of possible worlds.
A distinctively mathematical explanation is then one that picks out one
of these broadly or narrowly logical necessities on which an explanandum
counterfactually depends. In contrast to Skow’s conception of causal and
non-causal explanation (cf. Lange 2016, p. 404n17), a distinctively math-
ematical explanation may implicitly involve causal dependencies, but
derives its explanatory force from the logical necessity it picks out. For
3. E.g., “If a fact has a distinctively mathematical explanation, then the modal
strength of the connection between causes and effects is insufficient to account for that fact’s
inevitability”(Lange 2016, p. 6).
75Perspectives on Science

instance, a mother may fail to distribute twenty-three strawberries evenly
among her three children, and the number of strawberries and the number
of children may be considered to be the causes of this failure: if the number
of strawberries or of children were different, then she might have suc-
ceeded (cf. Lange 2016, p. 19ff ). However, the best explanation of the
failure appeals to a mathematical fact, namely that twenty-three is
not divisible by three. The numbers of strawberries and children are only
explanatorily relevant insofar as they allow for this mathematical fact
to be applied. It is the mathematical fact that grounds the non-causal
nature of the explanation, and that allows the explanandum to be entailed
across all possible configurations of the causal web—even regardless of
whether a possible world is populated with a web of pseudo-processes
instead of causal processes (cf. Salmon 1984).
In the following I will bracket Lange’s metaphysics, and consider modal
strength purely from a pragmatic point of view. So, the approach is non-
ontic in the sense that the question of what parts of an ontology ground
different degrees of modal strength is bracketed. Instead, modal strength
is considered purely as a property of the internal structure of explanation,
and this is connected to how explanations are used and how their causal
character is judged. The following is a qualitative statement of the prag-
matic conception of modal strength:
An explanation is assessed as non-causal if the explanandum’s degree
of necessity is independent of explanans; otherwise it is deemed
causal.
At this point, this statement is simply stipulated; the motivation and
argument for its plausibility will be given in due course. For the moment
it is sufficient to note the pragmatic and non-ontic use of modal strength:
judging an explanation to be causal or not depends entirely on how the
explanandum and explanans are represented within an explanation, not
on whether the explanation is true or picks out genuine causal/non-causal
dependencies.
In the following subsection I use some generic features of the inter-
ventionist approach to explanation in order to develop this conception of
modal strength with some more quantitative precision. This helps make
sense of how precisely the choice of explanandum/explanans affects the
degree of modal strength.
4.2. A Quantitative Measure
A powerful way to analyze explanations quantitatively is through the
interventionist framework, where the explanandum and explanans are
76 Granularity, Pragmatics, Non-Causal Explanation

conceived as variables.
4
In this subsection I will adopt this approach, and
will formalize explanans and explanandum as variables, but I will divorce
this from conceiving of the relation between explanans and explanandum
in terms of physical interventions. Instead the relation will be conceived of
as a conditional probability, to allow for non-causal explanations to be rep-
resented within the framework as well.
In particular, let the triple (Y,X,R) represent the structure of an expla-
nation, where Yis the explanandum variable, Xis the explanans variable,
and Rthe set of conditional probabilities {P(y
k
|x
j
)|x
j
2X,y
k
2Y}. It is
useful to allow Xand Yto also denote the sets of possible values the
explanandum and explanans variables can take. The explanandum consists
of the set Y, consisting of a particular value {y}2Ytogether with its
contrast class Y\{y}. The explanans consists of Xand R. The set Xis
defined by the range of possible values, and thus implicitly reflects the
boundary conditions: certain conditions that all explanans values meet.
5
The relation Rrepresents how Xmaps onto Y.
This may not be the only way to quantitatively represent explanations;
however, it assumes only a very basic way of representing explanatory struc-
ture that most if not all accounts of explanation share, including traditional
accounts such as deductive-nomological explanation and statistical-relevance
explanation, as well as more recent ones such as the kairetic account (Strevens
2008) or the counter-factual theory of explanation (Reutlinger 2016).
Of course, given this level of generality, it should come as no surprise that
triples (Y,X,R) do not actually distinguish between explanatory and non-
explanatory relations between explanans and explanandum variables (ac-
counts of explanation typically provide extra conditions on the relation, such
as invariant generalizations, or law-like connection). However, the only pur-
pose here is to elucidate the relation between grain of analysis and causal nature.
Given these preliminaries, the two following notions of modal strength
can be considered:
(1) An explanandum value (Y¼y) is entailed with non-causal modal
strength within explanation (Y,X,R) if the probability P(Y¼y|
X¼x)isoneorzeroforeveryx.
4. From Woodward and Hitchcock 2003, p. 6: “Theexplanandumisatrue(or
approximately true) proposition to the effect that some variable (the ‘explanandum variable’)
takes on some particular value. The explanans is a set of propositions, some of which specify the
actual (approximate) values of variables (explanans variables), and others which specify relation-
ships between the explanans and explanandum variables.”See also Hitchcock 1996.
5. This is why mathematical facts can play a role in explanations even though they are not
explicitly represented by the explanans variable: mathematical facts can act as boundary con-
ditions (see below).
77Perspectives on Science

(2) An explanandum value (Y¼y) is entailed with non-causal modal
strength within explanation (Y,X,R) if the probability P(Y¼y|
X¼x)isequalforallx.
Definition (1) captures how some outcome is inevitable (or impossible)
within the context of a particular explanation (Y,X,R). Definition (2)
captures how the occurrence of some explanandum value is independent
of the explanans variable.
However, definition (1) is a flawed way of capturing the notion of
“modal strength.”For example, consider the explanation of the question,
“why did Paul hit the bull’s eye of a dart board given that Paul threw
the dart at a certain velocity v?”If Xis defined as the set of all possible
velocities, and Ythe set of all possible end points of the dart after it is
thrown, the probability of the outcome state “bull’s eye,”given a relatively
narrow range of velocities ex⊂Xis one (by the equations of kinetics, and
some boundary conditions like no sudden gusts of wind). Outside this
narrow range, the conditional probability of a bull’s eye is zero (P(y2ey
|x2X\ex)¼0). However, this is clearly a causal explanation: had Paul
thrown the dart with a different velocity, it might not have hit the bull’s
eye. The initial velocity of the dart—a causal factor that the explanation
assumes could have been different in different arrangements of the causal
network of the universe—is counterfactually relevant for the explanandum.
How would (Y,X,R) have to be redefined so that the explanation
would have a non-causal character? Consider how the explanation changes
when the question changes to “why did Paul hit the bull’seyewhen
throwing at some velocity (regardless of whether he used a catapult, his
arm, or a gun to give the dart the initial velocity)?”In this case, the set
of explanans values is constrained to ex—the initial velocity is now a bound-
ary condition in lieu of an explanans variable. Instead, explanans values
are distinguished by the cause of the initial velocity. The best explanation
of this new question will refer to the properties of kinematics and, more
specifically, to certain geometrical properties entailed by vector calculus.
Hence the resulting explanation is non-causal.
As a second example, consider a case where the explanandum is not
implied with probability one or zero. For instance, consider the explanation
of why Schrödinger’s cat died. Schrödinger put his cat in a box together
with a decaying uranium atom, a Geiger counter, and some poisonous gas
which would be released when the Geiger counter registered the decay of
the atom. Now, according to quantum theory, after some time has passed
(the half-life of uranium in this case), the two possible outcomes (cat is
alive, or cat is dead) have probability 1/2. Then I open the box and see that
the cat is alive. What explains this outcome? The explanation is causal if, in
78 Granularity, Pragmatics, Non-Causal Explanation

the definition of the set of initial states X, I allow the fundamental properties
of uranium to vary; the explanation is non-causal if I hold such proper-
ties to be fixed, and only allow other possible initial states to vary, such as
whether I open the box gently or abruptly, or whether the box is made
outofcardboardoriron.
What these two cases show is that the size of the probability of the
explanandum within the context of an explanation does not matter for
whether the explanation is causal or not. What matters is whether the
entailment relation is affected by the explanans values that are allowed
to vary. What matters is not whether the explanandum is shown to be
inevitable, but rather that the entailment of the explanandum is shown
to be independent of explanans values.
Given this motivation for definition (2), one can propose the following
information-theoretic measure as an operationalization of modal strength:
Within an explanation (Y,X,R), the explanandum value y(instead
of its contrast class Y\{y}) is entailed with non-causal modal
strength if and only if the mutual information I(X;Y) equals zero.
Mutual information I(X;Y)isdefined as H(Y)−H(Y|X), the difference
between the unconditional entropy over outcome states and the condi-
tional entropy over outcome states. The entropy Hof a probability distri-
bution (conditional or unconditional) is a measure of the uncertainty
involved in predicting what the precise outcome will be. If the probability
distribution is uniform, the entropy is maximal; if one outcome has prob-
ability one, the entropy is zero. Mutual information, by contrast, does not
measure uncertainty, but rather “uncertainty reduction”given knowledge
of an explanans value x. If knowing xaffects the predictability of a possible
outcome y, then this knowledge reduces the uncertainty of the outcome.
Thinking of mutual information in terms of uncertainty reduction is suf-
ficient for our purposes here, but for precise definitions, see Cover and
Thomas (2006) or Desmond (2017).
This definition of the degree of modal strength captures key properties
of modal strength as defined in (2). First, mutual information is zero when
knowing that one explanans value rather than another occurred and does
not affect the probability of the explanandum variable taking a particular
value. The unconditional probability of the outcome—which assumes that
any initial state could occur—is the same as the conditional probability of
the outcome, where there is knowledge that only one initial state occurs.
Definition (2) of modal strength is the core of the pragmatic-modal
account of the causal nature of explanation. It is not an account of causal
explanation, because it makes no attempt to pinpoint how structures such
as (Y,X,R) are explanatory in the first place. Thus, it is not a competitor
79Perspectives on Science

with the ontic view on causal explanation. The target of the pragmatic-
modal account is rather to identify the conditions under which we will
judge a given explanation to be causal, or to be non-causal.
In this sense, it can also be contrasted with the “modal view of expla-
nation”(e.g., Mellor 1976; see discussion in Salmon 1989, p. 118ff),
which does aim to be a full competitor of the ontic view. According to
the modal view, explaining some event or phenomenon means showing
that it happened with necessity given the explanans (universal laws and
particular conditions). Mellor’s version of the modal view is slightly
weaker, requiring only that explanation show that the explanandum was
necessitated to a certain degree (“partial entailment”). Degree of necessi-
tation is identified with the size of the conditional probability of the ex-
planandum given the explanans. By contrast, in the pragmatic-modal
account, no account of explanatoriness is given, and the size of the prob-
ability does not matter for judging an explanation to be causal or not.
5. Application
Until now the reader has been motivated to go along with the pragmatic
(non-ontic) analysis of modal strength, as well as (to a lesser extent) with
the quantitative operationalization of modal strength. We will now con-
sider some harder arguments for the virtues of the pragmatic-modal
account. I will do so by outlining three areas where the account is philo-
sophically fruitful: accounting for the fundamental connection between
granularity and the causal nature of explanation; dissolving disagreements
concerning the causal nature of specific examples; and dissolving (at least
some) disagreements concerning the definition of causal explanation.
5.1. Modal Strength and Granularity
Recall that in the ontic view of causal (as well as of non-causal) explanation,
it is somewhat of a mystery why fine-graining an explanans should not
reveal anything but causal explanations, or anything but non-causal expla-
nations. If a vaguely described explanation is like a grey-scale photograph,
why should zooming in reveal darker and brighter shades, and not just the
same shade of grey? By contrast, within the pragmatic-modal framework,
a natural connection can be made between granularity and modal strength.
First let’s consider an intuitive description of this connection. To reca-
pitulate from the previous section, an explanandum variable is explained
non-causally when the probabilities of the occurrence of the explanandum
values are not affected by the explanans variable. Hence, if the explanan-
dum is fine-grained, there are more explanandum values where there could
be a dependence on the explanans variable. Thus, it will be “more diffi-
cult”(as it were) to show that the various “explanans values”no longer
80 Granularity, Pragmatics, Non-Causal Explanation

matter. As an example, consider the use of the Lotka Volterra model to
explain why a particular limit cycle Λwas reached (within the context
of a Lotka-Volterra explanation). The set of initial states is the set of initial
relative frequencies; the set of possible outcomes is the set of possible limit
cycles. Knowing the precise initial state is not relevant for what limit cycle
will occur; the limit cycle Λcan be deduced as a mathematical con-
sequence of certain structural features of the equation—the explanation
is thus non-causal. However, if the set of outcome states is fine-grained
to consist not of possible limit cycles, but of possible sequences of fre-
quencies followed by a population over time t,(p
t
,q
t
), there will be many
possible paths corresponding to a single limit cycle (but not vice versa).
Here knowing the precise initial frequencies ( p
0
,q
0
) will be relevant for
the resulting outcome: the explanation is causal.
However, not every fine-graining of the explanandum of a non-causal
explanation yields a causal explanation. The failure to cross the seven brid-
ges of Königsberg in the right way is not affected by the starting position,
but a fine-graining of this explanandum—for instance, the failure to cross
the bridges in a particular order—is not affected by the starting position
either. This makes sense: the outcome of crossing all seven bridges in the
right way is impossible given the set of possible initial states, and this means
that every particular way of crossing the seven bridges is impossible too.
This intuition is encapsulated by the following theorem:
Fine-graining the set of explanandum values (or explanans values)
monotonically increases the causal character of the explanation.
A proof can be given as follows. Let Y¼{y
1
,…,y
n
} be the set of outcome
states, and consider the fine- graining Y
0
where y
n
is replaced by {w
1
,…,
w
m
}, such that Y
0
¼{y
1
,y
2
,…,y
n-1
,w
1
,…,w
m
}. Then, relying on the log
sum inequality (cf. Cover and Thomas 2006, p. 31):
IY
0;XðÞ¼
X
y0X
x
Py
0;xðÞlog Py
0;xðÞ
Py
0
ðÞPxðÞ
¼X
wm
y0¼w1X
x
Py
0;xðÞlog Py
0;xðÞ
Py
0
ðÞPxðÞ−Py
n;xðÞlog Py
n;xðÞ
Py
n
ðÞPxðÞ
"#
þIY;XðÞ
≥IY;XðÞ
Because mutual information is symmetric (I(X;Y)¼I(Y;X)), the same
proof is applicable to fine-graining the explanans values. Furthermore, a
sufficient condition to reach equality is when P(y
n
,x)¼0 for all x, i.e.,
81Perspectives on Science

when the explanandum value y
n
is impossible, regardless of the value of
the explanans variable.
Note that this theorem only represents a limited result. What it does
not say is that the associated explanation of a fine-grained explanandum
has a more causal character. Rather, this result only holds if the explanans
is held constant throughout the fine-graining operation. If the fine-grained
explanandum necessitates a different explanans variable, then the asso-
ciated explanation may have a very different modal structure, anywhere
between the causal or non-causal end of the spectrum.
Nonetheless, even this limited result can help give deeper insight into
why fine-graining an explanandum should affect the causal nature of the
associated explanation. It also identifies two previously overlooked proper-
ties. The first is the importance of fine-graining an explanans variable: this
is discussed in more detail in the next section. The second is that some
non-causal explanations retain their non-causal character no matter how
much the explanandum is fine-grained (when the equality condition in
the proof above is satisfied: e.g., the bridges of Königsberg example). Like-
wise, the causal character of an explanation can reach a maximum when
knowledge of the explanans value allows the explanandum value to be
specified with absolute certainty (i.e., the case of determinism). In this
case, fine-graining the explanandum will not further alter the causal char-
acter of the explanation. To return to the metaphor of the photograph:
some grains remain black and some grains remain white, no matter how
much you zoom in on them.
5.2. Analyzing Diverging Assessments I
5.2.1. Constraint Explanations. We can now revisit the contrast be-
tween Strevens’and Huneman’s diverging assessments of the role of math-
ematics in scientific explanation. For Strevens, an explanans variable can
possibly range over any aspect of the causal web. The explanatory role of
mathematics is to help us abstract away from those aspects of the causal
web that do not make a difference for the explanandum variable, and thus
to help us identify the true causal difference-makers. The mathematical
proposition thus is a condition for what aspects of the causal web can
qualify as a true causal difference-maker, but the mathematical proposi-
tion does not itself represent a value the explanans variable can assume. In
any case, for Strevens, in a given constraint explanation the explanans
variable ranges over any difference-making aspect of the causal web, so
that knowing what value the explanans variable takes affects the proba-
bility of what value the explanandum variable will take. In other words,
the explanation is causal.
82 Granularity, Pragmatics, Non-Causal Explanation

By contrast, for Huneman—at least according a basic construal of his
account—the mathematical property or proposition acts as a boundary
condition on the explanans variable. The explanans variable ranges only
over configurations or states of affairs that instantiate a certain mathemat-
ical property, or that allow a certain mathematical theorem to be applied.
(So Huneman’s explanans variable ranges over fine-grained states of affairs
that instantiate a relevant topology, whereas Strevens’explanans variable
ranges over coarse-grained states of affairs that both do and do not instan-
tiate the relevant topology.) In other words, the explanans variable thus
makes no difference for the probability of the explanandum variable.
The explanation is non-causal.
Something similar is implicitly going on in Lange’sanalysisofthe
strawberry example. While Lange acknowledges that both the number
of strawberries and the number of children are causal difference makers
for the explanandum value (i.e., failure to distribute strawberries evenly),
and thus can be considered different values of the explanans variable, the
set of explanans values is coarse-grained according to whether a mathe-
matical property is instantiated or not (i.e., one number being indivisible
by another). When emphasizing the “distinctively mathematical”aspect of
this explanation, Lange is implicitly speaking of the mathematical prop-
erty as a boundary condition.
It is possible to identify a third construal that combines parts of the
previous two. Allow the explanans variable to range over all possible con-
figurations or states of the physical system under consideration. Only some
of these configurations instantiate the mathematical property (topological,
structural, etc.) that is being invoked as explanatory. So here the explanans
variable ranges over fine-grained states of affairs that both do and do not
instantiate the relevant topology. Strevens’construal is obtained by coarse-
graining the explanans variable: the mathematical property induces an
equivalence class on the values of the explanans variable, such that two
explanans values are considered equivalent if they correspond to instanti-
ations of the same mathematical property. In this explanation, knowing
the value of the explanans variable matters only insofar as one knows
which equivalence class the value is a part of.
Note that the third construal is a fine-graining of (my reading of)
Strevens’construal, but that this fine-graining does not increase the causal
character. By contrast, the third construal is also a fine-graining of (my
reading of ) Huneman’s construal but does increase the causal character.
This can be seen as follows: in the third construal the relevant mathe-
matical property is allowed to be variably instantiated, but in Huneman’s
construal the mathematical property is taken as a boundary condition.
(This is a coarse-graining because the explanans values are characterized
83Perspectives on Science

by fewer properties.) This is an illustration of how not all fine-grainings
have the same effect on the causal character of the associated explanation.
5.2.2. Optimization Explanations. Consider optimization phenomena
as a second example of how mutual information may help analyze contro-
versial examples of scientific explanation. The canonical example here is
the marble finding its way to the bottom of the bowl, regardless of where
on the bowl’s rim it is initially placed (see Sober 1983). Elliott Sober
argued (a view echoed more recently in Rice 2015) that optimization ex-
planations are not causal because they do not cite the precise path followed
by the marble—instead, the outcome state is explained as an attractor state
given certain structural features, such as the geometry of the bowl. This
would be a construal of the explanation where the explanans variable
ranges over the possible paths a marble may take, and structural features
function as a boundary condition.
However, it is possible to give optimization explanations a causal con-
strual, in analogy with constraint explanations. Strevens has argued that
such explanations are causal because they pick out difference-makers:
not just the geometry of the bowl, but also the fact that the marble is
placed somewhere on the bowl’s rim (Strevens 2008, p. 271). The expla-
nans variables range over all difference-making aspects of the causal web,
including possible geometries of the bowl, the smoothness of the bowl
surface, as well as all the possible starting positions of the marble (includ-
ing a starting point outside the bowl). Only some of these values will lead
to the outcome where the marble finds its way to the middle (and bottom)
of the bowl. This is a causal explanation.
5.3. Analyzing Diverging Assessments II
In light of the preceding discussion, a tentative suggestion can be
made: what seem like disagreements about the definition of causal expla-
nation are to some extent really disagreements about what explanans var-
iable to use. Ostensibly, there seems to be a divergence in the definition of
causal explanation between, say, Lange and Skow, and at first sight this
divergencemayseemtoexplainwhydifferent assessments are reached
about particular explanations. However, upon closer analysis, it could be
argued that Lange and Skow operate with different explanans variables.
Skow’s preferred definition of causal explanation is, in his own formula-
tion, an explanation that cites “what causes, if any, [the explanandum]
had; or if it [cites] what it would have taken for some specific alternative
or range of alternatives to [the explanandum] to have occurred instead”
(Skow 2014, p. 449). So, an explanation is causal if either the explanans
variable ranges over the causal difference-makers; or, if there seems to be no
84 Granularity, Pragmatics, Non-Causal Explanation

identifiable cause of the phenomenon, and the explanans variable ranges
over causal difference-makers of the other possible (but non-actualized)
values of the explanandum variable. In either case, knowing what value
the explanans variable assumes reduces the uncertainty over what value
the explanandum variable will obtain.
By contrast, and as previously analyzed, for Lange the explanans variable
is either coarse-grained according to whether a mathematical property is
instantiated or not, or the mathematical property simply acts as a boundary
condition on the range of values of the explanans variable. The latter case
describes “distinctively mathematical”explanations, and the explanans var-
iable is allowed to roam over all possible worlds where the laws of nature
might look very different—just as long as the mathematical property is
instantiated.
The contrast between Strevens’and Huneman’sdefinition of causal ex-
planation is perhaps less dramatic, but nonetheless Strevens operates with
his own (kairetic) account of difference-making where difference-makers
are abstracted from the causal web, whereas Huneman seems to operate
with a mechanistic conception of causation (at least in Huneman 2018).
Here too, the divergence is less deep that it would seem: as previously
argued, each gives a different construal of constraint explanations, with
different explanans variables.
6. Conclusion
The ontic view of the causal nature of explanation is: causal explanations
pick out causal dependencies between explanans and explanandum, and
non-causal explanations pick out non-causal dependencies. Confusion as
to the causal nature of instances of explanation arises when the explanation
is not precisely described. The pragmatic-modal account is not a full com-
petitor of the ontic view—it does not attempt to give an account of expla-
nation as such in the way that e.g., Reutlinger (2016) does—but it does
compete with it in giving an account of why fine-graining the explanan-
dum and explanans seems to affect our assessment of the causal nature of
the resulting explanation. I have argued that the pragmatic-modal account
is superior because, unlike the ontic view, it does not write off the con-
nection between granularity and causal nature as a coincidence. Rather,
the connection reflects basic and generic features of the structure of expla-
nation. Furthermore, I have suggested how the pragmatic-modal account
can help dissolve not only diverging assessments of explanations, but even
diverging accounts of causal ex-planation. Disagreements about causal and
non-causal explanation perhaps do not go quite as deep as sometimes as-
sumed, and can be avoided by greater precision in defining not just expla-
nandum variables but also explanans variables.
85Perspectives on Science

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