Causal necessitarianism and the monotonicity objection

Abstract

Do causes necessitate their effects? Causal necessitarianism (CN) is the view that they do. One major objection—the “monotonicity objection”—runs roughly as follows. For many particular causal relations, we can easily find a possible “blocker”—an additional causal factor that, had it also been there, would have prevented the cause from producing its effect. However—the objection goes on—, if the cause really necessitated its effect in the first place, it would have produced it anyway—despite the blocker. Thus, CN must be false. Though different from Hume’s famous attacks against CN, the monotonicity objection is no less important. In one form or another, it has actually been invoked by various opponents to CN, past and present. And indeed, its intuitive appeal is quite powerful. Yet, this paper argues that, once carefully analysed, the objection can be resisted—and should be. First, I show how its success depends on three implicit assumptions concerning, respectively, the notion of cause, the composition of causal factors, and the relation of necessitation. Second, I present general motivations for rejecting at least one of those assumptions: appropriate variants of them threaten views that even opponents to CN would want to preserve—in particular, the popular thesis of grounding necessitarianism. Finally, I argue that the assumption we should reject is the one concerning how causes should be understood: causes, I suggest, include an element of completeness that excludes blockers. In particular, I propose a way of understanding causal completeness that avoids common difficulties.

Full text

Synthese (2021) 199:2597–2627
https://doi.org/10.1007/s11229-020-02902-x
Causal necessitarianism and the monotonicity objection
Salim Hirèche1
Received: 1 October 2019 / Accepted: 5 October 2020 / Published online: 30 October 2020
© The Author(s) 2020
Abstract
Do causes necessitate their effects? Causal necessitarianism (CN) is the view that they
do. One major objection—the “monotonicity objection”—runs roughly as follows. For
many particular causal relations, we can easily find a possible “blocker”—an addi-
tional causal factor that, had it also been there, would have prevented the cause from
producing its effect. However—the objection goes on—, if the cause really necessi-
tated its effect in the first place, it would have produced it anyway—despite the blocker.
Thus, CN must be false. Though different from Hume’s famous attacks against CN,
the monotonicity objection is no less important. In one form or another, it has actually
been invoked by various opponents to CN, past and present. And indeed, its intuitive
appeal is quite powerful. Yet, this paper argues that, once carefully analysed, the objec-
tion can be resisted—and should be. First, I show how its success depends on three
implicit assumptions concerning, respectively, the notion of cause, the composition
of causal factors, and the relation of necessitation. Second, I present general moti-
vations for rejecting at least one of those assumptions: appropriate variants of them
threaten views that even opponents to CN would want to preserve—in particular, the
popular thesis of grounding necessitarianism. Finally, I argue that the assumption we
should reject is the one concerning how causes should be understood: causes, I suggest,
include an element of completeness that excludes blockers. In particular, I propose a
way of understanding causal completeness that avoids common difficulties.
Keywords Causation ·Necessitation ·Monotonicity ·Blockers ·Totality facts ·
Grounding
1 Introduction
An important philosophical question about causation concerns its modal status: Do
causes necessitate their effects? Causal necessitarianism (CN) is the view that they do.
BSalim Hirèche
salim.hireche@graduateinstitute.ch; salim.hireche@unige.ch
1Département de philosophie, Université de Genève, Rue de Candolle 2, 1205 Geneva, Switzerland
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2598 Synthese (2021) 199:2597–2627
Proponents of CN can be found throughout the history of philosophy—from Aristotle
to some contemporary “dispositional essentialists”.1CN has also been the target of
various attacks, most famously from Hume.2This paper focuses on one particular
objection: the “monotonicity objection”, which runs roughly as follows. For many
particular causal relations, it seems that we can find a possible “interferer”, or “block-
er”—an additional causal factor that, had it also been there, would have prevented the
cause from producing its effect. However—the objection goes on—, if the cause really
necessitated its effect in the first place, it would have produced it anyway—even with
the blocker. Therefore, CN is false.
Though different from Hume’s attacks,3the monotonicity objection is no less
important—in particular, its intuitive appeal is quite powerful. Indeed, it has been
invoked, in one form or another, by various opponents to CN, past and present.4Yet, it
has not, I think, received the attention that it deserves: more needs to be said about its
precise formulation, its background assumptions, and the possible strategies to answer
it. This is what I aim to do in this paper.
First, I offer an analysis of the objection. I propose a formulation of it as a valid
argument, consisting of three premises which may be briefly stated as follows: some
causes have possible blockers; necessitation is monotonic; necessitation is without
exceptions. And I argue that those premises, in turn, rely on three implicit assump-
tions: causal factors compose classically, namely non-holistically (when they obtain
together, each of them still individually obtains); causes do not include any complete-
ness component (e.g. a no-blocker or totality fact); necessitation is to be understood
classically, namely as a standard strict conditional. This analysis sheds light on the main
strategies to undermine the argument, each involving the rejection of one assumption.
Second, I present general motivations for rejecting at least one of those assumptions:
I argue that appropriate variants of them threaten views that even opponents to CN
would want to preserve: the modally weaker view that causes nomically necessitate
their effects; and the quite popular view that grounds necessitate what they ground.
Finally, I argue that the assumption that should be rejected, both as the best way
to undermine the argument and for independent reasons, is the one concerning the
content of causes: causes, I argue, do include a completeness component that excludes
blockers. In particular, I propose a way of understanding this completeness component
that avoids common drawbacks (e.g. making causal necessitation trivial, or vacuous).
1See Aristotle, Physics, II.7.198b5–9; Metaphysics VI.3.1027a32–b6, IX.5.1048a5–15; see also Stein
(2012). Other friends of CN (or some close variant) include Hobbes (1655, ch. 9); Spinoza (1677,PartI,
Axiom III); Mill (1843, III, V, §3, p. 241); Harré and Madden (1975, e.g. pp. 5, 90–91); Ellis (2001); Heil
(2012, ch. 6); Marmodoro (2015).
2SeeHume(1739, I, III, §IV–XIV).
3Indeed, a variant of the monotonicity objection also threatens Hume’s (1739 I, III, §XIV) own regularist
view of causation—see 3.1 below.
4See, in particular, Russell (1913,p.7),Schrenk(2010) and Mumford and Anjum (2011a,b,2014).
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2 Analysing the objection
2.1 Example and preliminaries
Before formulating the objection more precisely, let us consider an example used by
two of its main contemporary proponents, Anjum and Mumford (2011b, pp. 59–69).
I will not go through all the details, but focus on what I take to be most relevant to
our main purpose: understand how the reasoning behind the objection works, and put
some useful preliminaries in place. Suppose that, in a given room, the temperature
evolves in a certain way—say, it rises at a certain rate. We are interested in what
causes that particular rise in temperature. The natural focus will be on relevant causal
factors, namely things that are causally relevant to the evolution of temperature in that
room—e.g. a heater’s being on in a corner of the room, a window’s being open, some
people’s being present, and so on –, as opposed to things that are not—e.g. the fact
that the day is Tuesday.5
In general, we will assume that, for each particular effect, we can objectively identify
the relevant causal factors—e.g. the causal factors that are relevant to the temperature
in a room, when we are considering what causes a particular evolution of temperature
in that room; or relevant to the movement of a given object, when we are considering
what causes that object to move in a particular way. This should not be taken to imply
the epistemic claim that relevant causal factors are determinable in practise—that
we can establish an explicit list of them. In many actual cases, this would be very
complicated, and it would require an extensive knowledge of many different facts,
laws of nature, and so on. Rather, the intended assumption is a metaphysical one: for
each particular effect, as a matter of fact, certain things, but not others, are relevant
causal factors, so that it makes sense to refer to them. This only implies that they are
determinable in principle. This assumption seems reasonable—and, importantly, it is
equally acceptable to both sides in the debate.
Unless otherwise specified, “cause”, in what follows, will always mean full cause,
as opposed to partial cause; a cause will be assumed to include all the various relevant
causal factors, not only some of them. First, although we often talk, for convenience,
as if the relevant causal factors could be divided into “the cause” and “the background
or surrounding conditions”, I take it that the division is largely perspectival, rather
than properly metaphysical: it mainly amounts to giving a special status to relevant
causal factors that are particularly salient or interesting to us in a given context.6From
a metaphysical point of view, it is all the different relevant causal factors together that
bring about the effect. Second, if by “cause” we meant partial cause—only part of the
relevant causal factors –, a discussion of CN would be of little interest anyway—the
thesis would be trivially false. In our example, the effect is a particular rise in tempera-
ture in a room—call that effect “C”. The cause, accordingly, will include the plurality
A1,A
2, … of all the causal factors relevant for the evolution of temperature in that
5For Mumford and Anjum (2011a,b,2014), the main constituents of causes, namely what I refer to with the
generic term “relevant causal factors”, are activated causal powers, understood as irreducibly dispositional.
I remain neutral on that matter, as their monotonicity objection against CN does not depend on it.
6See Mill (1843, III, V, §3, p. 239), Lewis (1973b, pp. 558–559), Mumford and Anjum (2011a, p. 32) and
Heil (2012, pp. 126–130).
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2600 Synthese (2021) 199:2597–2627
room: the heater’s being on, the window’s being open, and so on. For simplicity, I will
call “A” the plurality A1,A
2, … of relevant causal factors.
Thus, we are assuming that A caused C. But did A necessitate C? Here is a reason to
think that it did not. It seems clear that there could have been a further relevant causal
factor B—say, a powerful cooler that would have been on at the same time—such
that A and B together would have caused the temperature in the room to decrease
instead. Thus, we would have had A and B, but not C. B provides an example of what
Mumford and Anjum call a possible “additive interference”—and that I will call a
possible “blocker”, as it would have prevented C from being caused. Now, how does
this show that A did not necessitate C in the first place? The idea is that, if A really did
necessitate C, then it would have done so in any situation, including one where B is also
there. Necessitation, in other words, seems to be subject to a principle of “antecedent
strengthening”, or “monotonicity”. As Mumford and Anjum (2011b, p. 68) put it:
What we have, therefore is an antecedent strengthening argument against the
necessitation of an effect by its cause. If C necessitated E, then C, plus anything
added, should also produce E.
Or, as Markus Schrenk (2010, p. 731) puts it,
The crucial point is that necessity is monotonic:ifCnecessarily leads to E, so
must C plus the antidote A.
According to this line of reasoning, the fact that a cause actually succeeded in causing
an effect is irrelevant: even if no blocker actually intervened, the point is that one
could have. Moreover, the fact that we are considering, not a single relevant causal
factor, but all the relevant causal factors—together constituting what we may call the
resultant causal factor—is also irrelevant:
[W]e can see that this antecedent strengthening consideration that counts against
the necessitation of its manifestation by a power applies just as much to any
resultant power as it does to the individual component powers. The resultant
too only disposes towards an outcome for it too is subject to the possibility of
additive interference. (Mumford and Anjum 2011b, p. 69; emphasis added.)
2.2 The monotonicity argument against causal necessitarianism
The above gave us an illustration of how the reasoning works: starting from poten-
tial blockers, we get a failure of monotonicity, which in turn leads to a failure of
necessitation. This may be formulated as a general argument against CN:
(Monotonicity Argument Against CN)
(POSSIBLE BLOCKERS) Some causes have possible blockers:ForsomeA
and C, A causes C, but it is possible that there is a further relevant causal factor
B, A and B obtain together, and C does not obtain.
(MONOTONIC NECESSITATION) Necessitation is monotonic: For any A
and C, if A necessitates C, then, for any B, A and B together necessitate C.
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Synthese (2021) 199:2597–2627 2601
(EXCEPTIONLESS NECESSITATION) Necessitation is exceptionless:For
any A and C, if A necessitates C, then it is not possible that A obtains but C does
not.
Therefore,
CN is false: It is not the case that, for any A and C, if A causes C, then A
necessitates C.
Note, first, that I take A, B and C to be broad pluralities of facts—i.e. one or more
facts. Typically, the effect C is a single fact, whereas the cause A is a strict plurality of
facts—in which case “A obtains” is short for “A1,A
2, … obtain together”. However,
nothing crucial, in the following discussion, should rely on understanding causation
specifically in terms of facts (as opposed to other fact-like entities, such as events).7
This choice is mainly practical: in particular, at some points, I will establish parallels
between causation and grounding, and a common way of understanding the latter is
precisely as a relation between facts.8Following usual conventions, I will sometimes
use “[p]” as short for “the fact that p”, and “<p>” for “the proposition that p”. Second,
for the main proponents of the monotonicity objection, blockers are possible for all
causes.9Thus, in a more accurate reconstruction, premise (POSSIBLE BLOCKERS)
would be strengthened—as (POSSIBLE BLOCKERS FOR ALL CAUSES) –, and the
resulting argument would threaten, not only CN, but partial CN—the view that some
causes necessitate their effects. For the time being, I will consider the weaker argument
formulated above, as it suffices as an argument against CN.
Third, the above argument is clearly valid. By (POSSIBLE BLOCKERS), for some
A and C, A causes C, but there could have been a blocker, namely a further relevant
causal factor B such that A and B obtain together, and C does not. In particular, given
(EXCEPTIONLESS NECESSITATION), there is a B such that A and B together do
not necessitate C. This implies, given (MONOTONIC NECESSITATION), that A
does not necessitate C. As we assumed that A causes C, CN must be false. More-
over, all three premises are at least prima facie plausible. As the room temperature
example illustrated, (POSSIBLE BLOCKERS) is based on the apparently obvious fact
that, at least for many particular causal relations, there could have been a blocker. As
regards (MONOTONIC NECESSITATION), it is quite plausible as well: if A necessi-
tates C, it seems that it should have done so in whatever alternative circumstances—in
particular, together with any other B. Finally, as regards (EXCEPTIONLESS NECES-
SITATION), A necessitating C sounds trivially incompatible with the possibility that
A obtains without C also obtaining.
7Theviewthattherelata of causation are events is common [e.g. Davidson (1967), Lewis (1973b), Kim
(1973)], but not unanimous: in particular, some indeed take them to be facts (Bennett 1988; Mellor 1995;
see also Wilson 2018)—and there are still other candidates, e.g. states of affairs (Armstrong 1997).
8See e.g. Rosen (2010), Audi (2012a); others prefer to formulate grounding as a sentential connective
(because)—seee.g.Fine(2001,2012b) and Correia (2010).
9See e.g. Mumford and Anjum (2011b,p.68;2014, p. 105).
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2.3 Three implicit assumptions
It may indeed be tempting to conclude that (POSSIBLE BLOCKERS), (MONO-
TONIC NECESSITATION) and (EXCEPTIONLESS NECESSITATION) are, not
only prima facie plausible, but quite clearly true. And the main reason, I suggest,
is that they implicitly rely on three further assumptions that are usually just taken for
granted:
(Classical Composition) The composition of relevant causal factors is clas-
sical: when a plurality of relevant causal factors A A1,A
2, … obtain together,
each of them still fully obtains—i.e. “A1,A
2, … obtain together”, or for short “A
obtains”, is correctly formalised as a classical conjunction: <A1obtains> ∧<A2
obtains> ∧….
(No Completeness) A cause (i.e. a full cause) includes the relevant causal
factors, but no completeness fact (e.g. a “totality” fact, a “ceteris paribus” fact,
a “no-blocker” fact).
(Classical Necessitation) A necessitates C iff (<A obtains> →<C
obtains>), where →is the material conditional and is the necessity opera-
tor with the usual possible-worlds semantics (PWS).
These assumptions are quite natural and may seem to need no justification. As regards
(Classical Composition), in the room temperature example, there is a plurality A 
A1,A
2, … of different causal factors (the fact that the heater is on, the fact that the
window is open, and so on). Intuitively, the composition of A1,A
2, … is classical: when
they obtain together, each of them still fully obtains; accordingly, although this may
simply be formalised, for simplicity, as one proposition—<A obtains> , or <A1,A
2,…
obtain together> –, it may also be correctly formalised as a classical conjunction: <A1
obtains> ∧<A2obtains> ∧…. Likewise, to describe the counterfactual situation where
a further relevant causal factor B (the powerful cooler) obtains, we may simply add
it as a further conjunct: <A1obtains> ∧<A2obtains> ∧…∧<B obtains> —or, for
short, <A obtains> ∧<B obtains> . And similar remarks would apply if, for instance,
the effect were the acceleration of an object o, and the relevant causal factors, A 
A1,A
2, …, were the forces being exerted on it: A1[Force f1is exerted on o], A2
[Force f2is exerted on o], …
As regards (No Completeness), completeness facts (ceteris paribus facts, totality
facts, and the like) are often regarded as unwanted for various reasons. For one thing,
including a completeness fact in the cause simply seems superfluous—even if “cause”,
as it is assumed here, means full cause, all you need to get a cause is all the relevant
causal factors, taken together. Moreover, it may seem to make the claim that the cause
is necessitating circular, or vacuous—something roughly amounting to the claim that
the cause, excluding possible cases where it would have failed to produce its effect,
would have produced it in all possible cases. A third kind of reason concerns the
formulation of the completeness fact itself; one may think that it is doomed to be too
complex or insufficiently clear. Finally, as regards (Classical Necessitation), using the
strict conditional with the usual PWS is a very common and simple way of formalising
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Synthese (2021) 199:2597–2627 2603
the relation of necessitation—indeed, one that yields, in at least most cases, results
that fit our intuitions.
Later on, I will argue that each assumption (or something close to it) is plausi-
bly needed for the monotonicity argument to work—that if we drop one of them,
there is a clear way to undermine the argument. Now, let us see more precisely how
they jointly make it work. Consider first premise (POSSIBLE BLOCKERS). As we
have seen, for at least many causal relations, it seems that we can find a possible
blocker. The most natural way to avoid that possibility would be to say that the cause,
in addition to relevant causal factors, includes some appropriate completeness fact
excluding the blocker; but this is precisely what (No Completeness) does not allow.
Thus, given (No Completeness), it seems that (POSSIBLE BLOCKERS) should be
accepted. Premise (MONOTONIC NECESSITATION) should also be accepted, given
(Classical Composition) and (Classical Necessitation). By (Classical Necessitation),
A necessitates C just in case (<A obtains> →<C obtains>) holds. In particular,
if (<A obtains> →<C obtains>) holds, then ((<A obtains> ∧<B obtains>) →<C
obtains>) holds for any B: for any possible world W, if both A obtains and B obtains in
W, then A obtains in W; and as by hypothesis C obtains in any world where A obtains,
C also obtains in W. This clearly establishes the monotonicity of necessitation for
all cases where A and B compose classically. Are there cases where they do not? In
the present context, relevant cases would be ones where A and B are, more precisely,
relevant causal factors composing non-classically; but this is precisely what (Classi-
cal Composition) excludes. Hence, we have no reason, given (Classical Composition)
and (Classical Necessitation), to reject (MONOTONIC NECESSITATION). Finally,
given (Classical Necessitation), A necessitates C just in case (<A obtains> →<C
obtains>) holds, namely just in case it is not possible that A obtains without C also
obtaining (there is no possible world in which A obtains and C does not). Hence,
(EXCEPTIONLESS NECESSITATION) should also be accepted.
3 Addressing the objection
3.1 General motivations: causal regularity lost
Apparently, there are good reasons to keep (Classical Composition), (No Complete-
ness) and (Classical Necessitation); but if we do, the monotonicity argument against
CN quite clearly works. Some may not see this as an important cost: after all, unless
you hold some necessitarian view about causal laws in the first place,10 it is unlikely
that you will even consider CN as plausible. However, the costs of keeping those three
assumptions are not limited to the loss of CN. First, it most plausibly leads to the
falsity of causal nomic necessitarianism (CNN): the weaker view that, if A causes C,
then A nomically necessitates C—where nomic necessitation is necessitation given
the laws of nature (in particular, the causal ones). As I noted earlier, some (e.g. dis-
positional essentialists) understand laws of nature as universal generalizations that
10 E.g. “dispositional essentialist” views, on which causal laws derive from the essential causal properties
of the relevant natural kinds—see Shoemaker (1980), Swoyer (1982), Ellis and Lierse (1994), Ellis (2001),
Bird (2007) and Heil (2012).
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2604 Synthese (2021) 199:2597–2627
are metaphysically necessary; on their view, CNN is just as strong as CN. However,
many rather take laws to have a primitive modal force, strictly weaker than metaphys-
ical necessity.11 Indeed, on broadly Humean accounts, laws are contingent: they are
simply true universal generalizations, perhaps meeting some additional (epistemic or
practical) criteria.12 On such non-necessitarian views of laws, CNN is clearly weaker
than CN. Yet, CNN, whatever modal status we attribute to laws, faces an analogous
monotonicity argument:
(Monotonicity Argument against Causal Nomic Necessitarianism)
(NOMICALLY POSSIBLE BLOCKERS) Some causes have nomically pos-
sible blockers: For some A and C, A causes C, but it is nomically possible that
there is a further relevant causal factor B, A and B obtain together, and C does
not obtain.
(MONOTONIC NOMIC NECESSITATION) Nomic necessitation is mono-
tonic: For any A and C, if A nomically necessitates C, then, for any B, A and B
together nomically necessitate C.
(EXCEPTIONLESS NOMIC NECESSITATION) Nomic necessitation is
exceptionless: For any A and C, if A nomically necessitates C, it is nomically
impossible that A obtains, and C does not.
Therefore,
CNN is false.
The argument is valid, and its premises quite clearly follow from assumptions (Clas-
sical Composition) and (No Completeness), and an adapted version of (Classical
Necessitation):
(Classical Nomic Necessitation) A nomically necessitates C iff N(<A
obtains> →<C obtains>), where Nis the nomic necessity operator with the
usual PWS (with possible worlds restricted to nomically possible worlds).
Just like (POSSIBLE BLOCKERS) in the original argument, (NOMICALLY POS-
SIBLE BLOCKERS) should be accepted, given (No Completeness). The key point is
that, usually, when considering a particular causal relation, and looking for a possible
blocker, we do not even need to think about a situation obtaining in a distant meta-
physically possible world—one with different causal laws, or alien kinds. Clearly,
for at least many causes, we will find a nomically possible world where A obtains
together with a further relevant causal factor B, and C does not obtain—for instance,
the powerful cooler, in the above temperature case, or the additional force, in the
above acceleration case, were not only possible, but nomically possible blockers. As
regards premises (MONOTONIC NOMIC NECESSITATION) and (EXCEPTION-
LESS NOMIC NECESSITATION), they follow, by the same reasoning as before,
from (Classical Composition) and (Classical Nomic Necessitation).
11 See e.g. Armstrong (1983) and Fine (2005).
12 See e.g. Hume (1739)andLewis(1973a, §3.3).
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I take it that anyone accepting (Classical Necessitation) would also accept (Classical
Nomic Necessitation): the reasons one may have for understanding necessitation tout
court or nomic necessitation in the classical way should essentially be the same. If
so, then what seems to be the best way to make the monotonicity argument against
CN work—i.e. to assume (Classical Composition), (No Completeness) and (Classical
Necessitation)—quite naturally leads to the loss of CNN. To put it otherwise, what is
ultimately at stake with the monotonicity objection is not the idea that causal relations
involve metaphysically necessary connections, but the more general idea that they
involve corresponding lawful regularities—non-accidental “constant conjunctions”.
And this idea may be equally appealing to necessitarians and non-necessitarians about
laws.
3.2 General motivations: grounding necessitarianism threatened
Some may remain unimpressed by the above considerations, thinking that causation is
a particularly “fuzzy” matter anyway and that, although casual talk of causal regularity
(“same cause, same effect”) is quite common, there is no such thing, strictly speaking.13
However, the consequences of keeping (Classical Composition), (No Completeness)
and (Classical Necessitation) may not be limited to the realm of causation: it may
ultimately affect another sort of explanatory relation, which has become central in
recent philosophical debates—metaphysical grounding.14 Even among opponents to
CN, many would probably be keen on preserving grounding necessitarianism (GN):
the view that, if A grounds (i.e. fully grounds) C, then A necessitates C. This posi-
tion might seem natural: grounding, often considered as a “stronger” explanatory
relation—metaphysical explanation as opposed to “mere” causal or nomic or natural
explanation –, should yield necessitation, even though causation does not. Indeed, it
seems fair to say that, in the grounding literature, GN is majority.15 And sometimes
this modal feature is precisely presented as one of the main ways in which grounding
differs from causation.16 Yet, GN is threatened by an analogous argument:
(Monotonicity Argument against Grounding Necessitarianism)
(POSSIBLE BLOCKERS FOR GROUNDING) Some grounds have possi-
ble blockers: For some A and C, A grounds C, but it is possible that there is
a further relevant grounding factor B, A and B obtain together, and C does not
obtain.
(MONOTONIC NECESSITATION) Necessitation is monotonic: For any A
and C, if A necessitates C, then, for any B, A and B together necessitate C.
13 See e.g. Russell (1913) and Mumford and Anjum (2011a, ch. 3, 2014).
14 See e.g. Fine (2001,2012b), Correia (2005), Schaffer (2009); Rosen (2010), and Correia and Schnieder
(2012); for some more sceptical views, see Sider (2012), Wilson (2014), Koslicki (2015) and Hofweber
(2016).
15 See Correia (2005), deRosset (2010,2013a,b), Rosen (2010), Audi (2012a,b), Fine (2012b,2015),
Trogdon (2013), Dasgupta (2014), Wilsch (2015). But there are also grounding contingentists—e.g. Dancy
(2004, ch. 3), Schnieder (2006), Schaffer (2010), Chudnoff (2011, ms), Leuenberger (2014a) and Skiles
(2015).
16 See e.g. Rosen (2010, p. 118), Fine (2012b, §1) and Wilson (2018, p. 731).
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2606 Synthese (2021) 199:2597–2627
(EXCEPTIONLESS NECESSITATION) Necessitation is exceptionless:For
any A and C, if A necessitates C, then it is not possible that A obtains but C does
not.
Therefore,
GN is false.
The premises of this valid argument quite clearly follow from assumption (Classical
Necessitation), together with grounding versions of (Classical Composition) and (No
Completeness):
(Classical Composition for grounding) The composition of relevant grounding
factors is classical: when a plurality of relevant grounding factors A A1,A
2,…
obtain together, each of them still fully obtains—i.e. “A1,A
2, … obtain together”,
or for short “A obtains”, is correctly formalised as a classical conjunction: <A1
obtains> ∧<A2obtains> ∧….
(No Completeness for grounding) A ground (i.e. a full ground) includes the
relevant grounding factors, but no completeness fact (e.g. a “totality” fact, a
“ceteris paribus” fact, a “no-blocker” fact).
Premise (EXCEPTIONLESS NECESSITATION) follows from (Classical Necessi-
tation), as before. Assuming also (Classical Composition for grounding), it seems that
we have no reason, in this context, to deny (MONOTONIC NECESSITATION). Now
consider premise (POSSIBLE BLOCKERS FOR GROUNDING). First, whereas it
seems plausible that, assuming (No Completeness), all causes have possible blockers,
it seems much less plausible that all grounds do, even assuming (No Completeness
for grounding). For instance, let A be [Chris is an unmarried man], and let C be [Chris
is a bachelor]. Arguably, A grounds C—with A including no sort of completeness
fact. However, we would not be able to find any fact B such that A and B could have
obtained together, without C obtaining. It seems that this ground (and you will easily
find other examples) has no possible blocker.
Yet, this should not lead us to the extreme conclusion that blockers are impossible
in all grounding cases. Indeed, some are very similar to causal cases in this respect.
Let C be the following restricted accidental generalization: [∀x(Mx →Hx)], where M
and H are the property of being a mammal in Sarah’s garden, and the property of being
a hedgehog, respectively. And suppose that C obtains: the mammals in Sarah’s garden
are indeed three hedgehogs—a,band c. For some philosophers, there are good reasons
to think that the right ground for C is simply the plurality of relevant grounding factors
constituted by C’s instances—i.e. A [Ha], [Hb], [Hc].17 The main other candidate
would be A+, namely A plus a totality fact T, with T [a,band care the only M’s].18
However, as a sort of completeness fact, T cannot be part of the ground if we assume
(No Completeness for grounding). This leaves us with A grounding C. Clearly, our
ground has possible blockers—e.g. B, the fact that there is a fox in Sarah’s garden.
17 See Skiles (2015, §4.1).
18 On totality facts (in truthmaking), see Russell (1918, p. 207), Armstrong (1997, ch. 13), Heil (2003,
pp. 70–71); on their use in grounding, see Rosen (2010, §8) and Fine (2015, §7). (See also §5 below.).
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Synthese (2021) 199:2597–2627 2607
As a different illustration, suppose that, in my room, there is a collection of different
objects put together, with certain shapes, arranged in a certain way, such that the whole
thing forms a complex object, o, that has the property of being a chair—suppose that
there is a reasonable functional definition of a chair, including the condition that
someone may actually be sitting on it. Arguably, A, [Object ois in my room], grounds
C, [There is a chair in my room]. But now consider a further relevant grounding factor
that could have obtained: B, the fact that there is an additional leg attached to the
original object, o, and that this leg is much longer than the other four ones, making
olose its stability. Call o’ the resulting object—o’ is just oplusthe additional leg,
so that owould still be in my room if o’ were. Suppose that o’ would not have the
property of being a chair—one could not be sitting on it anymore. So A grounds C,
but A could have obtained together with B, without C obtaining. One may be tempted
to say that A should include some sort of completeness fact excluding B; but this is
not an available option if we assume (No completeness for grounding).
As a last example, suppose that I like the taste of chocolate very much: when I
ate some yesterday, the taste was pleasant—as it always is. We may say my tasting
chocolate (perhaps with further facts about my tasting preferences)—fact A—was the
ground for my having a pleasant tasting experience—fact C. Now, instead of simply
tasting chocolate, I could have tasted both chocolate and fish at the same time. B, the
fact that I taste fish, would have been a further relevant grounding factor for my tasting
experience, and it would have obtained together with A. In this alternative situation,
C might well have failed to obtain—my tasting experience might not have been so
pleasant.
Given such grounding examples,19 it seems that (POSSIBLE BLOCKERS FOR
GROUNDING) is true if we assume (No Completeness for grounding); and assuming
also (Classical Composition for grounding) and (Classical Necessitation), (MONO-
TONIC NECESSITATION) and (EXCEPTIONLESS NECESSITATION) are true as
well—which leads to the falsity of GN. Earlier, I assumed that anyone accepting (Clas-
sical Necessitation) would most plausibly accept (Classical Nomic Necessitation) as
well. Similarly, I suggest that (Classical Composition for grounding) should probably
be accepted if (Classical Composition) is: if the composition of relevant causal factors
is always classical, the same should be true of relevant grounding factors. It is far
from clear why one would think that the composition of A and B is classical when,
for instance, A and B correspond to different forces being exerted on a given object,
or to different causal factors relevant for the temperature in a room, but not when, for
instance, A and B correspond to different mammals being in Sarah’s garden. Likewise,
I suggest that (No Completeness for grounding) should be accepted if (No Complete-
ness) is: if no cause includes any completeness fact, then presumably the same goes
for grounds. The idea is that what is at stake is the more general question whether we
should accept completeness facts in explanatory facts, be they causal or metaphysical,
and that the reasons for accepting them in one case or the other should essentially
be the same (more on this in Sect. 5.1). The above examples already provide some
motivation for this claim: it is unclear why one would accept a completeness fact in
19 Leuenberger (2014a, pp. 157–161) offers a different, though analogous, example—roughly, A is some
physical facts (without a totality fact), C is some phenomenal fact, A fully grounds C, but there is a possible
blocker.
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2608 Synthese (2021) 199:2597–2627
the ground when what is to be grounded is the fact that all mammals in Sarah’s garden
are hedgehogs, or that there is a chair in my room, but exclude any completeness fact
from the cause when, for instance, what is to be caused is the acceleration of an object
o. Indeed, the analogy between those cases seems quite clear.
The discussion in later sections will bring some further motivation for the claims
that (Classical Composition for grounding) should be accepted if (Classical Com-
position) is, and that (No Completeness for grounding) should be accepted if (No
Completeness) is. Assuming them for now, we can conclude that what seems to be
the best way to make the monotonicity argument against CN work—namely holding
(Classical Composition), (No Completeness) and (Classical Necessitation)—would
lead, in addition, to the loss of GN. Of course, opponents to GN may remain unim-
pressed by such considerations. They may even take them as further support for their
own view—in general, it is not uncommon for opponents to GN to assume the falsity
of CN, and to rely on such analogies between causation and grounding to motivate
their position.20 On the other hand, many philosophers endorse GN. For them, if the
above considerations are sound, abandoning at least one assumption among (Clas-
sical Composition), (No Completeness) and (Classical Necessitation) should be an
attractive option.21
3.3 Specific strategies: non-exceptionless necessitation, non-monotonic
necessitation, holistic composition, completeness
I have presented general reasons why one may want to abandon at least one assumption
among (Classical Composition), (No Completeness) and (Classical Necessitation). But
which one(s), if any, should be abandoned, and on what independent grounds? There
are four basic strategies to resist the monotonicity argument against CN, each one
involving the rejection of one assumption. Those strategies are independent from each
other—pursuing one of them successfully is sufficient to undermine the argument, but
it is compatible with pursuing any other strategy simultaneously. First, one may reject
(Classical Necessitation), arguing that necessitation is not exceptionless—or perhaps
that the notion of necessitation which is relevant in the context of causation is not.
This would clearly undermine premise (EXCEPTIONLESS NECESSITATION). In
what follows, I will not consider this strategy as an option: CN, as I understand it here,
involves the unqualified notion of necessitation, namely necessitation tout court.22
20 See e.g. Leuenberger (2014a, p. 152) and Skiles (2015, §4.1, §5.4, §6).
21 Indeed, if (Classical Composition), (No Completeness) and (Classical Necessitation) ultimately threaten
GN, it seems that they should also threaten the weaker view that grounding involves non-accidental reg-
ularity. I argued for an analogous claim in the causal case (§3.1), and it seems that the argument may
be adapted to the grounding case. If so, then keeping (Classical Composition), (No Completeness) and
(Classical Necessitation) would be quite costly, not only for the defenders of GN, but indeed for all those
taking grounding relations to support at least non-accidental generalizations [e.g. Rosen (2010, p. 131),
Audi (2012b, p. 693), Schaffer (2016, p. 57) and Wilson (2018,p.729)].
22 As opposed to e.g. necessitation other things being absent,orinterferers being absent—such qualified
notions have been suggested to capture, for instance, the modal relation between physical facts and other
actual facts (Leuenberger 2008), or between a triggered disposition and its manifestation (Bird 2005).
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Synthese (2021) 199:2597–2627 2609
And I will assume that this notion would not deserve the name if it were not strong
enough to exclude exceptions.
Second, one may argue that (Classical Necessitation) should be replaced, not
(necessarily) with a non-exceptionless formalisation, but with a non-monotonic
one—thereby undermining (MONOTONIC NECESSITATION). This strategy (or
something similar) has actually been defended,23 but I will not pursue it here. There
may indeed be independent reasons to go for an alternative, non-monotonic formali-
sation of necessitation that is stronger than (Classical Necessitation)—say, (Stronger
Non-Monotonic Necessitation), a formalisation that is non-monotonic because it
requires, in addition to what (Classical Necessitation) requires, that the antecedent
be relevant to the consequent.24 However, although (Stronger Non-Monotonic Neces-
sitation) would indeed undermine (MONOTONIC NECESSITATION) as it stands, it
seems that the proponent of the monotonicity argument would have an easy response.
She could replace (MONOTONIC NECESSITATION) by the following premise:
(ALTERNATIVE MONOTONICITY) For any A and C, if A necessitates C,
then, for any B, it is not possible (i.e. classically possible) that A and B obtain
together, and C does not obtain.
Still assuming (Classical Composition), (ALTERNATIVE MONOTONICITY) is
as plausible with (Stronger Non-Monotonic Necessitation) as it is with (Classical
Necessitation). And now (POSSIBLE BLOCKERS) and (ALTERNATIVE MONO-
TONICITY) suffice as the two premises of a new valid argument against CN—indeed
a successful one, given (No Completeness). More generally, as long as we assume
(Classical Composition), (ALTERNATIVE MONOTONICITY) would hold, not only
assuming (Stronger Non-Monotonic Necessitation), but any formalisation that is
stronger than (Classical Necessitation). Thus, if we wanted to save CN just by replac-
ing (Classical Necessitation) with an alternative formalisation—while keeping (No
Completeness) and (Classical Composition) –, the alternative formalisation would
have to be weaker—such that, even assuming (Classical Composition), (ALTERNA-
TIVE MONOTONICITY) would be false. However, it seems that such a formalisation
would then be too weak to yield an intuitive notion of necessitation. The strategy under
consideration, then, seems to face a sort of dilemma. Be that as it may—for the rest
of this paper, I will assume that necessitation is correctly formalised with (Classical
Necessitation).
23 Lowe (2012) aims at undermining Mumford and Anjum’s (2011a) monotonicity objection against CN;
however, he does not reformulate the objection exactly as I do here. In particular, he takes them to hold, not
exactly (MONOTONIC NECESSITATION), but the following claim: (P necessitates Q) entails that, for any
R, (If PandR,then Q). To undermine this claim, he argues, not that necessitation is not monotonic—indeed
he does not deny (Classical Necessitation) –, but that there are plausible formalisations of conditionals (“If…,
then…”) that are non-monotonic, such as the following: If P, then Q iff (P→Q)∧(♦P∨Q). In
the present context, however, one may indeed reject (Classical Necessitation) on the grounds that Lowe’s
stronger, non-monotonic conditional is better at grasping the notion of necessitation, and then undermine
(MONOTONIC NECESSITATION) on that basis (although I will argue shortly that such a strategy is not
very promising).
24 For instance, one may argue that necessitation is better captured by the non-monotonic relation that Fine
(2012a, pp. 235–236) calls “exact entailment”, or “exact consequence”—A exactly entails C iff any state
that exactly verifies A (i.e., roughly, classically verifies A and is wholly relevant to A) also exactly verifies
C.
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2610 Synthese (2021) 199:2597–2627
The third strategy consists in undermining (MONOTONIC NECESSITATION) by
rejecting, not (Classical Necessitation), but (Classical Composition), claiming that
the way relevant causal factors compose is not (always) classical. Something like this
strategy has also been at the centre of a debate between critics and proponents of the
monotonicity objection.25 As it may indeed look more promising than the first two
ones, I will briefly discuss it in the next part of this paper (Sect. 4). Finally, the fourth
strategy is to reject (No Completeness), arguing that causes do include some sort of
completeness fact excluding blockers. This would be a way to undermine (POSSI-
BLE BLOCKERS)—causes would not have possible blockers, strictly speaking. This
strategy will also be discussed, indeed defended, in some more detail (Sect. 5).
It is worth noting that those two strategies, though distinct, are similar in that both
aim at arguing that, in the alleged blocker situations described, the cause would not
obtain. Yet, as we will see in more detail, the two strategies differ as to why exactly
it is so. On the former strategy, the obtaining of the cause together with the blocker,
though possible, would amount to the obtaining of a holistic fact, in which the cause
would somehow be lost—it would no longer obtain (in the way it actually does). By
contrast, on the latter strategy, the obtaining of the cause together with the blocker is
simply impossible, as they are incompatible.
4 Against the holistic composition strategy
4.1 Holistic composition, monotonicity, and blockers
Is it so clear that causal factors compose classically? Perhaps the various relevant causal
factors A1,A
2, … for the evolution of temperature in a given room form a complex
web of interdependent factors, whose obtaining together is best understood as the
obtaining of one holistic fact, H—not reducible to the obtaining of A1, the obtaining
of A2, etc. Likewise, if a further relevant causal factor B (a powerful cooler’s being on)
had obtained together with H, the latter would no longer have obtained (as it would
have without B): the obtaining of H together with B, though possible, would have
amounted to the obtaining of a new holistic fact, H*—in which neither H nor B, the
two “ingredients” that would have holistically composed into H*, would have been
preserved.26
Such considerations may motivate the rejection of (Classical Composition). Indeed,
simply considering that (Classical Composition) might be false should naturally lead us
to reject (MONOTONIC NECESSITATION) in its current form. When A necessitates
C, it may be intuitive that A and B together should also necessitate C, but only if the
obtaining of A and B together still implies the obtaining of A—not if it amounts to
the obtaining of a holistic fact instead. (MONOTONIC NECESSITATION) may then
be replaced with a weaker, more plausible premise:
25 The relevant debate opposed John Heil and Olivier Massin to Rani Anjum and Stephen Mumford (see
Mumford and Anjum 2014, pp. 113–115).
26 The point that relevant causal factors may compose holistically was made by John Heil and Olivier
Massin, as an answer to Mumford and Anjum’s monotonicity objection (Mumford and Anjum 2014,
pp. 113–115).
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Synthese (2021) 199:2597–2627 2611
(CLASSICALLY MONOTONIC NECESSITATION) Necessitation is classi-
cally monotonic: For any A and C, if A necessitates C, then, for any B such that
A and B compose classically, A and B together necessitate C.
Accordingly, (POSSIBLE BLOCKERS) should be replaced with a premise requiring
that the possible blockers be such that they would have composed classically with the
cause. The resulting new valid argument is the following:
(Classical Monotonicity Argument against CN)
(POSSIBLE CLASSICAL BLOCKERS) Some causes have possible classi-
cal blockers: For some A and C, A causes C, but it is possible that there is a
further relevant causal factor B, A and B compose classically, A and B obtain
together, and C does not obtain.
(CLASSICALLY MONOTONIC NECESSITATION) Necessitation is clas-
sically monotonic: For any A and C, if A necessitates C, then, for any B such
that A and B compose classically, A and B together necessitate C.
(EXCEPTIONLESS NECESSITATION) Necessitation is exceptionless:For
any A and C, if A necessitates C, then it is not possible that A obtains but C does
not.
Therefore,
CN is false.27
With this modified argument, the dialectics of the discussion changes. Unlike (MONO-
TONIC NECESSITATION), (CLASSICALLY MONOTONIC NECESSITATION)
does not depend on (Classical Composition) anymore: it simply follows from (Classi-
cal Necessitation), just as (EXCEPTIONLESS NECESSITATION) does. What does
rely on (Classical Composition), as well as on (No Completeness), is now (POSSI-
BLE CLASSICAL BLOCKERS). For whether causes have possible classical blockers
depends on two things. First, it depends on whether causes have any sort of possi-
ble blockers in the first place. As we have seen, assuming (No Completeness), many
causes, plausibly all of them, quite clearly do. The second question, then, is whether
those possible blockers are classical; and this crucially depends on (Classical Com-
position), and more generally on whether all, some or no pluralities of causal factors
compose classically.
4.2 Can holistic composition save necessitarianism?
In the new dialectical situation just described, one may simply reject (Classical Com-
position), arguing for holistic composition—while keeping (Classical Necessitation)
27 When recognizing that composition may be holistic in some cases, Mumford and Anjum (2014, p. 114)
stress that, in such cases, even if their “antecedent strengthening test” for causal necessitation is not failed,
it is not passed either—it simply does not apply. Thus, what they seem to recognize is that their original
(2011a) test—corresponding to the monotonicity argument formulated in §2.2—should be replaced with
a modified test, corresponding to the classical monotonicity argument—on the new test, if a cause has
possible classical blockers, then it is not necessitating.
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2612 Synthese (2021) 199:2597–2627
and (No Completeness). In principle, pursuing this strategy successfully would be suf-
ficient to undermine (POSSIBLE CLASSICAL BLOCKERS). But can it be pursued
successfully? It is difficult to deny that at least some causes have possible classical
blockers. Suppose that A includes the forces exerted on an object o, and that A causes
C, the fact that oaccelerates accordingly. B, the fact that a further force is exerted
on o, is quite clearly a possible blocker. And there seems to be no reason to think
that A and B would not have composed classically: intuitively, there is nothing in the
additional force’s being exerted on othat would have prevented the initial set of forces
from being fully exerted on o—and vice versa.28
Some have argued that composition may be classical in this and similar cases, but
holistic in other cases: roughly, composition is classical when it can be described
as “linear”, as in the case of Newtonian forces, whose composition can simply be
formalised as an addition of vectors; and it is holistic when composition is more
complex than simple addition.29 However, even assuming that we can draw such a
distinction between “linear” and “non-linear” composition, it is not even clear that
composition should be holistic in the latter case. For instance, in the room temperature
example, a precise mathematical description of how the various relevant causal factors
combine to bring about the effect may involve quite complex, non-linear equations;
and their composition may, in that sense, be considered as “non-linear”. Yet, there
may still be a legitimate impression that the actual relevant causal factors (the heater’s
being on, the window’s being open, etc.) compose classically with each other, and
would have composed classically with a further relevant causal factor (the powerful
cooler).
Be that as it may, even if composition were holistic for some types of relevant
causal factors, it would not suffice to falsify (POSSIBLE CLASSICAL BLOCKERS).
Indeed, it may not even suffice to falsify the stronger premise, (POSSIBLE CLASSI-
CAL BLOCKERS FOR ALL CAUSES), and undermine the corresponding classical
monotonicity argument against partial CN, based on (POSSIBLE CLASSICAL
BLOCKERS FOR ALL CAUSES), (CLASSICALLY MONOTONIC NECESSITA-
TION) and (EXCEPTIONLESS NECESSITATION). For unless some causes have no
possible blocker whatsoever—which is quite implausible, given (No Completeness) –,
undermining (POSSIBLE CLASSICAL BLOCKERS FOR ALL CAUSES) requires
showing that composition would have been holistic, not simply for some causes and
some of their possible blockers, but any of their possible blockers—which looks like a
significantly more difficult task. In sum, it is unclear whether the holistic composition
28 Taking the composition of forces as a case of classical composition implies some form of realism about
mere component forces—a view that is controversial (Cartwright 1983, p. 59; Wilson 2009), but plausible
(Mumford and Anjum 2011a, pp. 38–44; Massin 2017). In any case, we may take a similar example that
does not directly appeal to forces. Let A be the fact that two objects oand p, with respective masses m1
and m2, are at a distance dfrom each other; and suppose that A causes C, the corresponding acceleration
of oand ptowards each other. Clearly, there is a possible situation where A obtains together with B, the
fact that a further object q(with mass m3and a given position relative to oand p) is also around, and C
does not obtain. And intuitively, A and B would have composed classically.
29 The linearity criterion for classical composition was suggested by Mumford and Anjum (2011a,
pp. 86–87; 2014, p. 114)—more precisely, they suggest that non-linearity is one important reason why
composition may be holistic.
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Synthese (2021) 199:2597–2627 2613
strategy can ultimately save partial CN; and it seems indeed very unlikely that it can
save CN itself.
Note that, in this respect, the grounding case is analogous. Based on some putative
examples of holistic composition of grounding factors (e.g. my tasting chocolate and
my tasting fish), one may reject (Classical composition for grounding) and replace
the initial monotonicity argument against GN with a classical monotonicity argument.
However, it is very unlikely that there are enough cases of holistic composition to
undermine this argument: composition is clearly classical for at least many types of
grounding factors—e.g. the mammals being in Sarah’s garden. Thus, it seems that the
causal case and the grounding case are on par with respect to the holistic composition
strategy: it is plausibly doomed to fail in both cases.
5 In defence of the completeness strategy
5.1 Motivating completeness
The fourth strategy is to reject (No Completeness), arguing that causes do include
some appropriate completeness fact excluding possible blockers—and saying more
about what form this completeness fact would take, a question that I will address
shortly (Sects. 5.2–5.3). Thus, the cause is not simply A, namely the relevant causal
factors, but A+, namely A plus the completeness fact. One motivation for pursuing
this strategy concerns the specific purpose of saving, not just partial CN, but CN itself,
from the monotonicity objection. For this requires a strategy that is applicable to all
causation—to any cause (that has at least a putative possible blocker) and any putative
possible blocker. The strategy discussed above was not general in that sense. The idea
was not to say that the obtaining of the cause together with the alleged blocker was
impossible, but that it amounted to the obtaining of a holistic fact, in which the cause
was not preserved. Thus, the strategy was only applicable to cases involving a specific
sort of causal factors, and a resulting specific mode of composition. By contrast, the
completeness strategy, if applicable at all, is applicable to all causes, however the
causal factors involved compose: the obtaining of the cause together with a blocker is
simply impossible, because any putative blocker, classical or holistic, is incompatible
with the completeness fact. Thus, it offers a more promising way to undermine, not
just (POSSIBLE CLASSICAL BLOCKERS FOR ALL CAUSES), but (POSSIBLE
CLASSICAL BLOCKERS), and indeed (POSSIBLE BLOCKERS). (And of course,
if (POSSIBLE BLOCKERS) is false, so is, a fortiori, (NOMICALLY POSSIBLE
BLOCKERS) in the argument against CNN.)
Beyond such “strategic” motivations for completeness, what about independent
motivations? One is that it seems to allow for a better account of what causes are and,
in particular, what would have happened to them if a further relevant causal factor
had come into play. For assuming (No Completeness), a cause is simply a plurality
of relevant causal factors, A. Thus, in particular, whenever we consider a given cause
and what would have happened if a further relevant causal factor B had obtained
together with A, the crucial question (Cause?) “Would the cause still have obtained?”
reduces to a question about the composition of A and B: (Classical?) “Would Ahave
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2614 Synthese (2021) 199:2597–2627
composed classically with B?” When discussing the holistic composition strategy, I
was assuming that this picture was correct—I only argued that, on this picture, saving
CN is hopeless. Now, I suggest that this very picture is incorrect. For it seems that, in
the examples considered, the answer to (Cause?) is always negative: the cause would
not have obtained if B had been there. This impression is incompatible with the picture
on which the cause is A, and (Cause?) reduces to (Classical?): in at least some cases,
the answer to (Classical?) is quite clearly positive—A and B would have composed
classically. Yet, it perfectly fits the alternative picture on which the cause is A+—B
is incompatible with the completeness fact in A+, so that the answer to (Cause?) is
always negative, whatever the answer to (Classical?).
Consider the acceleration case: A is the relevant causal factors for the acceleration
of an object o—say, the forces being exerted on o—and C is the fact that oaccelerates
accordingly. What would have happened if a further relevant causal factor B—a further
force exerted on o—had obtained? Presumably, C would not have obtained. But it also
seems that that the initial cause would no longer have obtained: the answer to (Cause?)
is negative. For it does not sound right to say that the cause of C would still have been
there, just as in the actual case, but that it would somehow have “failed” to cause C, or
been “prevented” to do so; it sounds more natural to say that a new cause, including both
A and B, would have been there instead—also succeeding in causing its own effect.
Although the fact that an additional force is there seems to be irrelevant to whether
the initial set of forces is still there—hence the positive answer to (Classical?) –, it
is relevant to whether this set of forces still constitutes the resultant force, which in
turn seems relevant to whether the actual cause still fully obtains—hence the negative
answer to (Cause?). Those intuitions, which seem at least reasonable, are incompatible
with the picture on which A is the cause and (Cause?) reduces to (Classical?). But
they perfectly fit the alternative picture on which the cause is A+. What caused the
actual acceleration of owas not simply the fact that there were certain forces exerted
on it, but also the fact that there were no further forces.
Now take the room temperature example. How should we answer (Cause?)—would
the cause still have obtained if B (the powerful cooler) had been there? Just as in the
acceleration case, there seems to be something wrong in saying that the actual cause
of C (the rise in temperature) would still have obtained, but would somehow have
“failed” to cause C. Rather, there would have been a new cause, including A and B,
which would have succeeded in causing its effect (a decrease in temperature). Thus,
the answer to (Cause?) seems negative. On the old picture, the only way to account
for this impression is to say that the answer to (Classical?) is also negative: A and
B would have composed holistically. However, even assuming that they would have
(which is in itself disputable), it cannot be the main reason for the impression that the
initial cause would not have obtained anymore. For if it were, we would expect not to
have that same impression when considering a clear(er) case of classical composition.
Yet, in the acceleration case, there was the same impression. Whether composition
would have been classical or holistic in the temperature case, what can clearly explain
that same impression is that the actual cause is A+—namely A “alone”, in particular
without B. The absence of B in the actual situation may or may not be relevant to the
full obtaining of the relevant causal factors, A; but the absence of B definitely seems
relevant to the full obtaining of the cause of C.
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Synthese (2021) 199:2597–2627 2615
A second motivation for completeness is indirect: it concerns the relation between
CN and the modally weaker thesis of causal nomic necessitarianism (CNN). As I
have argued (Sects. 3.3–4), the other strategies to undermine the monocity argument,
based on the rejection of (Classical Necessitation) or (Classical Composition), are
hardly promising, so that the completeness strategy remains the only available one.
As a result, (No Completeness) very plausibly stands or falls with the monotonicity
argument against CN. But given the close links between CN and CNN (Sect. 3.1), it is
easy to see that the same goes for the analogous monotonicity argument against CNN.
Thus, no less than defenders of CN, defenders of the more general idea that causation
supports (non-accidental) regularities have a good reason to go for completeness.
A different indirect motivation was already mentioned earlier: including a com-
pleteness fact in certain grounds is often considered as a legitimate move; and as those
grounding cases look analogous to causal cases, it is unclear why including a com-
pleteness fact in causes should be less legitimate. Yet, this indirect motivation might
be disputed. For it might be that the specific reasons why completeness is often con-
sidered as legitimate in the grounding case are not (clearly) applicable to the causal
case, or anyway not relevant to the present defence of CN against the monotonicity
objection. Indeed, one sort of argument for completeness in the grounding case simply
relies on the very claim that GN is true, whether this claim is just assumed or inde-
pendently motivated: GN is true; certain grounds (such as those considered earlier)
are necessitating only if they include a completeness fact; therefore, they must include
it. Such an argument will not be of much help here. First, although GN may just be
assumed in certain contexts, as it is quite widely accepted, the same cannot be said
about CN. And even when necessitation in the grounding case is not just assumed,
but based on independent arguments, such arguments may not be as convincing once
adapted to the causal case.30 Second, in the present context, we are anyway trying to
undermine a particular objection, the monotonicity objection, against necessitarian-
ism; thus, whether in the grounding or the causal case, it would be question begging to
invoke necessitarianism itself—whether it is just assumed or independently motivated.
Rather, what we need here is an argument for completeness that is both independent
from necessitation and clearly applicable to the causal case as well.
One such argument concerns explanation. It is widely agreed that a (partial) ground
provides a metaphysical explanation for what it grounds, with explanation understood
in the objective, mind-independent sense—being a correct or appropriate metaphysical
explanation does not depend on our interests or contingent capacities for understand-
ing.31 And if partial grounds provide metaphysical explanations, it seems natural to
30 One such argument for GN involves essence: roughly, when A grounds C, it is essential to A (or C, or
both) that, if A obtains, then A grounds C (see Fine 2012b, p. 74). Assuming that grounding is factive and
that essentiality entails necessity, GN follows. Another argument runs roughly as follows: if A grounds C,
then A provides a good explanation for C; if A provides a good explanation for C, then A does not have
“confounding cases”, namely possible cases where A obtains and C does not; thus, GN is true (see DeRosset
2010).
31 My use of “providing a metaphysical explanation” is meant to cover both being a metaphysical
explanation (e.g. Fine 2001, p. 15; Litland 2013; Dasgupta 2014)andbacking such an explanation
(e.g. Rodriguez-Pereyra 2005; Audi 2012b, pp. 687–688; Schaffer 2012,2016, pp. 58–59; Wilson 2018,
pp. 728–729). For a more sceptical view on the systematic relation between grounding and explanation, see
e.g. Wilson 2014.
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2616 Synthese (2021) 199:2597–2627
think that grounds (i.e. full grounds) should provide full, or complete, metaphysical
explanations. On that basis, one may argue that, in the relevant grounding cases, the
ground should include a completeness fact. For instance, the mere fact that a,band c,
three mammals in Sarah’s garden, are hedgehogs—fact A—does not seem to provide a
full metaphysical explanation for the fact that all the mammals in her garden are hedge-
hogs—fact C. By contrast, A plus the completeness fact that a,band care the only
mammals in her garden—which excludes, for instance, that a fox is also there—seem
to provide a full explanation for C. This reason for completeness is independent from
necessitation, and arguably quite convincing.32 Importantly, it is also applicable to
the causal case. For it is also widely agreed that causation is an explanatory relation,
(partial) causes providing objective causal explanations. In particular, it is natural to
think that causes (i.e. full causes) should provide full causal explanations. And this
intuitively counts as a reason for completeness, just as in the grounding case. For
instance, the fact that a number of forces are exerted on object odoes not seem to
provide a full causal explanation for its accelerating precisely as it does. Rather, it
seems that a full explanation would also involve some completeness fact excluding,
in particular, that other forces be exerted on it.
Note that a similar argument could be run, based on a different core feature that
causation shares with grounding: both are relations of generation, or building.33 The
argument would be distinct, but perfectly analogous to the one based on explanation: in
short, just like the analogous sort of full grounds, full causes should be fully generative
(i.e. they should fully generate, or bring about, their effects), which intuitively they
would not be without including the relevant completeness fact—e.g. the fact that no
further force is exerted on o.
More would be needed for a proper defence of completeness, but the above already
gives us good motivations. There is, however, another issue to be addressed. For even
granting that including some appropriate completeness fact in causes is both effective
as a way to undermine the monotonicity objection, and legitimate on independent
grounds, more needs to be said about what this completeness fact would look like.
Indeed, this is one respect in which the causal and the grounding case may not seem
to be on par: in the causal case, it may seem more difficult to find an acceptable
formulation of the relevant completeness fact—one that is clearly able to exclude all
alleged possible blockers while avoiding the sort of common drawbacks mentioned
earlier (e.g. rendering necessitation vacuous). This is the problem to which I now turn.
5.2 Formulating completeness: common difficulties
The problem of dealing with alleged blockers is not specific to the debate on CN—or
CNN, or GN. It concerns various claims that a given relation “supports” some sort of
modally loaded conditional. One is the claim that the relation between the triggering
conditions and the manifestation of a disposition supports a counterfactual conditional:
32 It may be disputed, however: one may argue that the completeness fact is not explanatory relevant
(Chudnoff ms); or that, even if it were, it would not automatically follow that it is part of the ground (Skiles
2015, §5.4).
33 See e.g. Skiles 2015, §1; Schaffer 2016, p. 50.
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Synthese (2021) 199:2597–2627 2617
if x has disposition D (e.g. fragility), then, if D’s triggering conditions (x’s being struck)
were to obtain, D’s manifestation (x’s breaking) would also obtain.34 Such claims, too,
have been attacked on the basis of alleged potential blockers35 (e.g. a fragile object’s
being protected by some packaging). In the case of causation, as in the other cases,
a natural way to deal with blockers is to introduce some completeness fact or clause,
T. And this may be done in two main ways. One is to include T in the cause itself (or
e.g. in the disposition’s triggering conditions). The other is to add T as an external
condition: if A causes C, then, given T , A necessitates C.36 In the context of our
discussion, the former case amounts to rejecting (No Completeness) and including T
in the cause itself, which is what I am suggesting here (the latter case would amount
to renouncing necessitation tout court, as expressed e.g. by (Classical Necessitation),
and defending only a weaker, conditional modal claim).
In the causal case, the main natural ways to formulate T face important difficulties
(the case of dispositions is similar in this respect, while the grounding case is arguably
less problematic). Those common ways may be divided into three categories, which
I will now briefly present before considering them in turn. First, T may be an explicit
no-blocker fact; the main problem, as I will argue, is that a cause’s necessitating its
effect then becomes an entirely trivial matter. Second, T may be a totality fact that
excludes everything but the relevant causal factors, A; the problem with the result-
ing “all exclusive” understanding of causes is that it is only applicable to “idealized”
causes, not actual ones. Third, the cause may be taken to include everything in a
certain space–time area (the “reverse light cone” of the effect), with T only exclud-
ing everything outside this area; the problem is that, on the resulting “all inclusive”
understanding of causes, they include many facts that are intuitively irrelevant. Let us
consider those three options in some more detail.
First, if T is meant to be an effective no-blocker fact, it should exclude, not only
some explicit list of particular possible blockers, but all of them. The problem is that
T will then run the risk of either being too vague or imprecise—“ceteris paribus”,
“under normal conditions”, “excluding non-standard cases”—or being more precise
but making the relevant conditional trivial, or vacuous.37 For instance, one may suggest
that a cause, A+, has the form A plus T, where A is the relevant causal factors, and
T is the (second-order) fact that, for any fact B, if it is possible that A and B obtain
together without the effect C also obtaining, then B does not obtain. The explicit
no-blocker fact, T, which directly refers to C and excludes the possibility of C’s non-
obtaining if A obtains (together with any fact B), would indeed succeed in excluding all
possible blockers; and it would be more precise than a general “in standard conditions”
fact. However, it would also render the causal necessitation claim—A+necessitates
C—trivial. And applying this strategy to all causation, causes would become trivially
necessitating. Indeed, on some historical accounts, it seems to be part of the very
34 This is the left-to-right direction of the “simple conditional analysis” of dispositions (see e.g. Quine
1960, ch. 46).
35 Known as finks (Martin 1994), and maskers, or antidotes (Bird 1998).
36 For instance, Choi (2006) pursues the former strategy for dispositions; Bird (2005) pursues the latter
strategy for dispositions and related laws of nature; Leuenberger (2008) pursues the latter strategy for
physicalism as the claim that physical facts ground all actual facts.
37 See e.g. Martin (1994, pp. 5–6), Fara (2005) and Yli-Vakkuri (2010).
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2618 Synthese (2021) 199:2597–2627
definition of a cause that it is necessitating.38 However, such moves would now appear
to most of us as ad hoc attempts to save necessitarianism: intuitively, it is unclear why
the very notion of cause should include necessitation.
And the point could be made even stronger. With causes of the form A plus T,
it is not simply that causes are trivially necessitating: there is a sense in which they
logically entail their effects.39 Here is briefly how the argument goes (I assume that
“p” and “[p] obtains” are equivalent). Let us suppose that the cause obtains. Thus, A
obtains, and so does T, namely the universally quantified fact [For any fact B, if it is
possible that A and B obtain and the effect C does not, then B does not obtain], as well
as any instance TB[If it is possible that A and B obtain and C does not, then B does
not obtain], for any arbitrary fact B. In particular, there is a B that is a logical fact: B 
[t], where t is some tautology. The corresponding instance, TB, is logically equivalent
to [If B obtains, then necessarily, if A and B obtain, then C obtains]. And as B obtains
as a matter of logic, by modus ponens, we get that [Necessarily, if A and B obtain,
then C obtains] obtains; so does [Necessarily, if A obtains, then C obtains], since B
is a logical fact; and so does [If A obtains, then C obtains], assuming that necessity is
factive. As we were assuming that A obtains, by modus ponens, we get that C obtains.
Thus, starting from the obtaining of the cause, we have logically derived the obtaining
of the effect. Yet, even philosophers inclined to think that causes are necessitating,
perhaps trivially so in some sense, would have been reluctant to claim that causation
is indeed a logical relation.
The second main way to exclude blockers is with a totality fact making the cause
“all exclusive”: roughly, the cause is A, namely the relevant causal factors, plus the
fact that that’s all. The intuitive idea is to consider A as obtaining in isolation—just
as some causal factors are put in quasi-isolation in scientific experiments, except that
here isolation is total. The totality fact—and that’s all—may be expressed with T
understood as a totality operator on facts.40 Roughly, T(A) is the (second-order) fact
that no fact other than A obtains. Or, perhaps more simply, we may take T to be a
factive operator, so that Abecomes redundant in A and T(A), and the cause is just
T(A), amounting to A plus the fact that no other fact obtains—i.e. A plus [For any B,
if B obtains, then B is identical to (included in) A]. The suggested formulation may
of course be refined. In particular, T(A) excludes more than needed. For instance, as
the idea is to have A in causal isolation, there may be no need to exclude “abstract”
facts (facts involving only abstract entities), like [2 + 2 4]; we may not even need to
exclude “concrete” facts outside the “reverse light cone” of s, where s is the spacetime
location at which C obtains. We may then restrict totality to facts that have the property
of being first-order, concrete facts in the reverse light cone of s—call that complex
property “RLC(s)” (for “reverse light cone of s”). On the resulting view, the cause
is the factive, restricted totality fact, TRLC(s)(A)—i.e. A plus [For any B, if B is an
obtaining RLC(s), then B is (included in) A]. This formulation may still need further
38 E.g. Hobbes (1655, ch. 9) and Mill (1843, III.V.3, p. 332).
39 I owe this further point to an anonymous referee for this journal.
40 On totality operators on facts, see Chalmers and Jackson (2001) and Leuenberger (2014b).
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Synthese (2021) 199:2597–2627 2619
improvements.41 But it will do for our present purpose, which is to consider how
the type of cause considered, consisting in A as being in isolation, is able to meet
the challenges considered above. First, TRLC(s)(A) succeeds in excluding all alleged
possible blockers, and the totality fact involved is arguably more precise than a general
“in standard conditions” fact: it is a ceteris absentibus fact—all (concrete) things other
than A being absent. Moreover, unlike the sort of formulation considered earlier, it does
not render the necessitation claim vacuous or trivial: there is nothing in TRLC(s)(A)
that already contains, or logically implies, C.42 However, this strategy has a major
drawback: a fact like TRLC(s)(A) cannot be an actual cause, only an “idealization” of
it—in actual cases, the relevant causal factors, A, do not obtain in isolation. Thus, we
can only say that, if an ideal cause of form TRLC(s)(A) had obtained (in another possible
world), then it would have necessitated its effect—at least, necessitation would not
have been threatened by possible blockers. Yet, our main problem here is whether
actual causes have possible blockers.
One natural way out is what I called the “all inclusive” option: the cause is not simply
A in isolation, but all the first-order, concrete facts actually obtaining in the reverse
light cone of s—all the actual RLC(s)’s—in isolation. Noting “I” (for “irrelevant”)
the plurality of all the facts other than A that are also actual RLC(s)’s, the cause
maybeformulatedasT
RLC(s)(A, I)—i.e. A plus [For any B, if B is an obtaining
RLC(s), then B is identical to (included in) A or I]. Unlike TRLC(s)(A), TRLC(s) (A, I)
is now applicable to actual causes. And it still excludes all possible blockers, without
making the necessitation claim trivial. However, first, it seems that a cause including
both A and I, namely a whole reverse light cone of concrete facts, can only be the
one instantiation of the corresponding repeatable type of cause—i.e. the only actual
TRLC(s)(A, I)-like fact. Some may find it problematic, as we usually see particular
causes as instantiations of repeatable, indeed often repeated, types.43 It is not obvious
to me why this should count as an important problem: I can understand why, from a
practical point of view, we may want types of causes to be relatively few, and often
41 In particular, I am assuming that, in the definition of RLC(s), the reverse light cone of s is understood
strictly—i.e. excluding s itself. This is appropriate assuming that all causation is non-simultaneous. In this
case, as any possible blocker would havebeen an RLC(s), any possible blocker is excluded by TRLC(s) (A), as
intended. However, at least some causes (or, more generally, relevant causal factors) might be simultaneous
with their effects (Taylor 1966;Brand1980; Mumford and Anjum 2011a, ch. 5). If so, then there may be
possible simultaneous blockers, namely blockers that would have obtained at s; and those blockers, failing
to be RLC(s)’s, are not excluded by TRLC(s)(A). Thus, to allow for simultaneous causation, we may rather
understand the reverse light cone of s broadly (i.e. including s). The problem, then, is that TRLC(s)(A)
even excludes C; and we certainly would like C to be compatible with the cause—without being already
contained in, or logically implied by, it. Perhaps we may assume that a blocker B (e.g. a further force
exerted on o) is never in itself incompatible with C (a given acceleration of o), whereas an effect other than
C(adistinct acceleration of o)is incompatible with C. If this is correct, then a way to make TRLC(s)(A)
compatible with both the effect obtaining and its failing to obtain, while still excluding all blockers, may be
the following: TRLC(s)(A) is A plus [For any B, if B is an obtaining RLC(s), then B is identical to (included
in) A or B is identical to (included in) C or B is incompatible with C].
42 In any possible world where TRLC(s)(A) obtains, it is excluded, in particular, that there is a blocker and C
does not obtain; but nothing a priori excludes that there is no blocker and C fails to obtain for another reason:
for instance, one may argue that the causal laws applying to the case in question are not deterministic, or
not metaphysically necessary, in which case there clearly is a possible world where TRLC(s)(A) obtains but
C does not.
43 See Russell (1913, pp. 8–9) and Mumford and Anjum (2011a, ch. 3).
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2620 Synthese (2021) 199:2597–2627
repeated; but it is unclear to me why its not being the case should be a problem in itself
if what we are after is a correct metaphysical account of causation. Be that as it may,
I think that the view under consideration faces a more important issue: what makes it
difficult to accept a cause of the form TRLC(s)(A, I) as intuitive has less to do with I
containing a huge number of facts than with its containing facts that, by hypothesis,
are not relevant causal factors, and that seem, more generally, mostly irrelevant to C’s
being caused.
Finally, there are general worries about causes including completeness facts, which
apply to all types of formulation considered above: the “all inclusive” one, just con-
sidered; the “all exclusive” or “isolation” one, only applicable to idealized causes; and
the no-blocker one, making causes trivially necessitating. For in all cases, the cause
includes, or entails, a fact of the following form: [For any fact B, if B meets such-and-
such conditions, then B does not obtain.] To take the simplest example, an unrestricted
totality fact like T(A) includes [For any fact B, if B is not (included in) A, then B does
not obtain]—and the same goes for the other formulations considered, except that the
conditions on which B does not obtain are more complex. Such second-order facts,
as part of the cause, may be objected on two grounds. First, they include negative
facts, namely the non-obtaining of certain facts—indeed, an infinity of them. And one
may argue that a cause should not involve anything negative: causation is a matter of
generating, or bringing about, effects; but absences, it seems, are unable to bring about
anything.44 Second, one may worry that the cause will involve an infinity of facts that
seem intuitively irrelevant. As we have seen, TRLC(s) (A, I) already faces such a diffi-
culty—I seems mostly irrelevant to C’s being caused. But the problem considered here
is another, more general relevance problem, which also affects T(A) or TRLC(s)(A),
for instance. For all those formulations involve a second-order fact that, for any fact
B, if B meets certain conditions, then B does not obtain. And even forgetting about the
“negativity” issue, the worry is anyway that all those B’s—including e.g. [2 + 2 4],
[Today is Tuesday], [Socrates is a philosopher]–, and whether they meet the relevant
conditions, may seem irrelevant to C’s being caused.45
5.3 Formulating completeness: a suggested solution
The above briefly illustrated common (families of) ways of understanding causes
as excluding alleged blockers. They all come with drawbacks, some of which are
particularly important. Is there a better way? A natural idea, in the light of the previous
general discussion about completeness (Sect. 5.1), is the following. It is assumed by
both parties in the debate about the monotonicity objection that the cause includes A,
namely the plurality A1,A
2,A
3,… of relevant causal factors (e.g. the causal factors
relevant for the evolution of temperature in a given room). It is also agreed that these
causal factors are the only (actual) relevant causal factors, so that together they form
the resultant causal factor. Indeed, this latter fact is useful to explain why the effect
obtains (e.g. why the temperature increases exactly as it actually does). However, the
proponents of the monotonicity objection claim that it is not part of the cause; I now
44 See e.g. Mumford and Anjum (2011a, p. 66; and ch. 6).
45 For a similar worry about totality facts in grounds, see Skiles (2015, §4.1).
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Synthese (2021) 199:2597–2627 2621
suggest that it is—the cause includes the completeness fact that A exhausts the relevant
causal factors.
It should be stressed that what the relevant causal factors are, in each particular
case, does not directly depend on C itself. As I already pointed out, what relevant
causal factors are relevant to is not simply C, but a larger category or domain of
potential effects, that I will call “D(C)”: when the effect C is a particular evolution
of temperature in a given room, for instance, D(C) is the evolution of temperature in
that room in general; when C is a particular acceleration of an object, D(C) is the
movement or acceleration of that object in general. To put it otherwise, D(C) is related
to C as is a determinable to one of its determinates, or a variable to one of its values.
Thus, the relevant causal factors are, more explicitly, the D(C)-relevant causal factors.
Let me briefly address a potential worry.46 The above presupposes that there is a
(non-arbitrary) way to associate any determinate effect C with a unique determinable
D(C), which one might dispute—in particular, a determinate may in principle be a
determinate of several determinables (e.g. scarlet is a determinate of both red and
colourful). A helpful way to think of determinable-determinate relations is in terms
of “determination spaces” (Funkhouser 2006,2014, pp. 25–36). For instance, colour-
fulness has a three-dimensional determination space: being coloured is having three
particular values falling within the ranges of possible values for the variables hue, satu-
ration and brightness, respectively. Redness is a determinate of colourfulness because
its determination space is a proper subset of the determination space for colourfulness:
the space has the same three dimensions, but with more restricted ranges of possible
values.
Mutatis mutandis, we may establish analogous determination relations between
certain facts involving the instantiation of properties—relying on the determination
relations between those properties. For instance, [Object ois red] is a determinate of the
determinable [Object ois coloured]: the former’s determination space is a proper subset
of the latter’s, which is constituted of all “point-like” facts of the form [Object ohas
values hfor hue, sfor saturation, bfor brightness], for some triplet (h,s,b) in the three-
dimensional determination space of colourfulness. In particular, it seems reasonable
to think of any given effect C as involving the instantiation of some properties, and
accordingly as being the determinate of some determinable D(C). For example, C
[ohas acceleration (x”, y”, z”) (1,0,-1)], where 1, 0 and -1 are the particular
values for the x-, y- and z-components of o’s acceleration in m/s2, is a determinate
of determinable D(C) [ohas some acceleration], the latter’s determination space
being constituted of all the facts [ohas acceleration (x”, y”, z”) (ax,ay,az)], for any
(ax,ay,az) in the three-dimensional determination space of all possible values for o’s
acceleration.
Now, why think that D(C) is unique for any effect C? In the above example, another
determinable for C would be D(C)* [ohas acceleration (x”, y”, z”) ∈[(0,0,-2),(2,
0, 0)]], itself a determinate of D(C) [ohas some acceleration]. Yet, keeping in mind
that here C is not any determinate fact but an effect, and that D(C) or D(C)* is not
any determinable but what the relevant causal factors (e.g. the forces exerted on o)are
supposed to be (objectively) causally relevant to, the two candidates are intuitively not
46 Thanks to an anonymous referee for pressing me on this point.
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2622 Synthese (2021) 199:2597–2627
on par. The property of othat the forces exerted on it are causally relevant to, the one
that they play a causal role in determining within the relevant determination space, is
o’s acceleration tout court, namely its property of having an acceleration—rather than
its property of having an acceleration-within-this-specific-range. The latter property
of ojust does not seem sufficiently natural or joint-carving; neither does, in turn, the
forces’s property of being causally relevant to that property of o.
The claim that any effect C comes with just one natural D(C) may require further
argument, but the above suggests that it is at least reasonable. Let us now turn to a
more precise formulation of the view of causes suggested above. For each particular
effect C, the relevant causal factors are, more precisely, the D(C)-relevant causal
factors—for short, the D(C)-RCF’s. The suggested cause, then, may be formulated as
the following factive, restricted totality fact: TD(C)-RCF(A)—i.e. A plus [For any B, if
B is an obtaining D(C)-RCF, then B is (included in) A]. Put simply, the cause amounts
to A and that’s all as regards relevant causal factors.47
Let us now see how this way of understanding completeness compares to the main
alternatives considered above (Sect. 5.2). First, the suggested completeness fact is
arguably more precise than a mere “in standard conditions” fact, while still being gen-
eral enough to exclude all possible blockers: if the cause is TD(C)-RCF(A), its obtaining
is incompatible with the obtaining of any additional relevant causal factor—and thus,
of any alleged blocker. Second, while excluding all possible blockers, it does not
make the cause trivially necessitating, or CN vacuous: just like TRLC(s)(A, I)—the “all
inclusive” cause—or TRLC(s)(A)—the “all exclusive” cause –, TD(C)-RCF(A) does not
contain, or logically imply, C.48 Third, unlike the “all exclusive” view– i.e. TRLC(s)(A)
as the cause –, the suggested view can apply to actual causes: unlike TRLC(s)(A),
TD(C)-RCF(A) does not exclude all other facts (in the reverse light cone), but only fur-
ther relevant causal factors—which by hypothesis do not actually obtain. Fourth, the
proposed view is applicable to actual causes without their having to include a huge
collection of irrelevant facts, I (all the concrete facts obtaining with A in the whole
reverse light cone): unlike TRLC(s)(A, I)—the “all inclusive” cause –, TD(C)-RCF(A) is
compatible with, but does not include, I.
Let us now briefly consider the more general worries about causes involving a fact
of the form [For any fact B, if B meets such-and-such conditions, then B does not
obtain]—in the case of TD(C)-RCF(A), the relevant fact would be [For any B, if B is a
D(C)-RFC that is not (included in) A, then B does not obtain]. One worry was that
47 The formulation of TD(C)-RCF(A) may require some refinements. In particular, more than one plurality
of facts may intuitively count as forming the resultant causal factor. In the temperature case, the relevant
causal factors may be correctly identified with some macro-physical facts A1,A
2, …(the fact that the
heater is on, etc.), but they may also be correctly identified with micro-physical facts that ground them,
respectively, A’1,A’
2, … Likewise, in the acceleration case, one may say that the relevant causal factors
are A1,A
2, …, corresponding to the forces exerted on o(the fact that force f1is exerted on o, etc.), but one
may instead invoke facts that arguably ground A1,A
2, …, namely facts A’1,A’
2, … about the masses of
oand other objects, the distances between them, etc. Thus, various pluralities of facts, at various “levels of
fundamentality”, may correctly identify the relevant causal factors; and none of them should be excluded
by the cause, TD(C)-RCF(A), where A is one such plurality. We may then say that TD(C)-RCF(A) is, more
precisely, A plus [For any B, if B is a D(C)-RCF, then B is (included in) A or B grounds A or B is grounded
in A]. Put simply, the cause amounts to A and that’s all as regards relevant causal factors that neither ground
nor are grounded in A.
48 Seefn42above.
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Synthese (2021) 199:2597–2627 2623
such a fact includes negative facts—the non-obtaining of certain B’s; the other worry
was that all those B’s (and whether they meet the given condition) are intuitively not
relevant to C’s being caused. First, as those worries apply to all types of completeness
facts, they are irrelevant to a comparison between the formulation suggested and the
main alternatives considered. Second, it is unclear how much weight should be given
to such worries. In particular, it is not obvious that a cause’s involving negative facts
should in itself be problematic.49 Third, even assuming that those worries are serious,
the general reasons for completeness presented in Sect. 5.1 may be considered as more
important—so that TD(C)-RCF(A), as a candidate cause, may still be preferred to A,
overall.
Fourth, it is not even clear that TD(C)-RCF(A) has to face the worries considered in
the first place. For TD(C)-RCF(A), on current interpretation, amounts to A plus [The
obtaining D(C)-RCF’s are at most A]. But we may instead understand the cause,
TD(C)-RCF(A), simply as [The obtaining D(C)-RCF’s are exactly A]. If this fact is then
taken to reduce to the conjunction of (a) [The obtaining D(C)-RCF’s are at least A] and
(b) [The obtaining D(C)-RCF’s are at most A], then the worries under consideration
may apply—at least if we take (b) to amount to [For any B, if B is a D(C)-RCF that
is not (included in) A, then B does not obtain]. However, one may precisely resist
the claim that the intended “exact fact”, TD(C)-RCF(A), reduces to an “at least fact”,
(a), plus an “at most fact”, (b). Instead, TD(C)-RCF(A)—though necessarily obtaining
just in case (a) and (b) obtain—may be considered as distinct from, and not reducible
to, (a) plus (b).50 If so, then TD(C)-RCF(A) avoids the worries considered—it may still
imply (b), but it does not include it. In general, it is common to take what I call “at
least” and “at most” facts as the more basic building blocks in terms of which we may
build “exact” facts. This may have various practical and other advantages. But I fail
to see any clear metaphysical reason why an exact fact, such as [The people in the
room are (exactly) Mary and Simon], could not also be considered as basic, at least
not reducible to the relevant “at least” and “at most” facts.51
In sum, I suggest that a cause includes the fact that the actual relevant causal factors,
A, exhaust the relevant causal factors. Beyond its precise formulation, which would
probably need further refinements, the proposed way of understanding causal com-
pleteness avoids important difficulties faced by the main alternative ways considered
earlier. More generally, the suggested completeness fact seems to be needed as part of
49 See e.g. Mellor (1995, pp. 131–135).
50 This is an adapted version of Fine’s (2012b, §7) suggestion that we may take an unrestricted totality fact
of the form T(a, b, c, …)—i.e. the fact that a, b, c, … are exactly the objects that there are—to be a basic
fact that necessarily co-obtains, but is not identical, with the fact that a, b, c, … exist plus the universal
generalisation that, for any x, x aorxb or … [For worries about such basic facts, see Skiles (2015,
§4.2).]
51 A general worry about the proposed view may be that, once reformulated in terms of events (as many
views of causation are), it would be committed to “totality events”—which are, in particular, a sort of
second-order events. A detailed answer to that worry is beyond the scope of this paper, but here are two
brief remarks. First, totality events, and second-order events in general, may not be very common, and I can
see how they might be considered as somewhat unnatural, prima facie. Yet, it is still not obvious why they
should be so problematic—in particular, substantially more so than the analogous sorts of facts. Second, in
any case, there may be no need to reformulate the proposed view in terms of events. After all, thinking of
causation in terms of facts, as I did in this paper, may be intuitive enough. Indeed, the view that the relata
of causation are events is not unanimous, and some actually take them to be facts instead (see fn 7).
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2624 Synthese (2021) 199:2597–2627
the cause—not to make it necessitating, but to make it an intuitive full cause in the first
place. If more or less relevant causal factors had obtained, not only would the effect
have been different and incompatible with the actual effect, but the cause itself would
have been different and incompatible with the actual cause. In order to get a precise
cause, you need to specify what the relevant causal factors are exactly—not at least,
or at most. In that respect, subtracting and adding causal factors are on par: just as the
initial cause is lost if you subtract some of the relevant causal factors (which anyone
would grant), it is also lost, and for analogous reasons, if you add some relevant causal
factors. This seems to be a central feature of how causation works. It is the reason
why, for instance, it is usually accepted that causal laws need to be formulated with
some completeness (or ceteris paribus) clause, so as to determine what causal fac-
tors there are exactly. The proponents of the monotonicity objection would probably
not deny that causation works in that way: that feature of causation is precisely what
their objection mainly relies on, and what makes the possibility of blockers seem so
plausible at first sight. All I am suggesting here is that we take that feature on board
by acknowledging that completeness is a component of causes—not some external
condition. What causes an effect is not simply the obtaining of certain relevant causal
factors, but the fact that exactly those relevant causal factors obtain.
6 Conclusion
Though quite common, and prima facie powerful, the monotonicity objection against
CN can, and should, be resisted. Once carefully analysed, it relies on assumptions
that ultimately threaten the weaker view of CNN, as well as GN—views that many,
beyond friends of CN, may find plausible. To put it otherwise, the ultimate scope of
the objection is, in two respects, wider than usually thought: it is not simply an objec-
tion against causal necessitation, but against (non-accidental) regularity, whatever the
precise modal status; and it is not simply about causation, but plausibly a larger range
of explanatory relations, including grounding. Beyond such general motivations to
address the objection, each of the assumptions underlying it may be disputed on inde-
pendent grounds. For instance, one may reasonably think that (Classical Necessitation)
should be replaced by a stronger, non-monotonic notion of necessitation; and (Classi-
cal Composition) may reasonably be rejected on the grounds that some pluralities of
causal factors may compose holistically.
The corresponding two strategies to defend CN, however, seem hopeless. The
assumption which is crucial to reject, both to undermine the objection and for the
general purpose of providing a satisfactory account of causes, is (No Completeness).
My defence of the corresponding strategy included a proposed formulation of causal
completeness that avoids common difficulties, as well as independent motivations for
causal completeness in general—some of which relied on the suggested strong anal-
ogy between causation and grounding (indeed, given this analogy, some independent
elements of my overall defence of CN may, in turn, bring further support to GN against
the relevant monotonicity objection). Of course, even if this defence is convincing,
it does not show that there are no good reasons to reject CN, or indeed the weaker,
more widely acceptable thesis of CNN. But it does show that, if there are any such
123
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Synthese (2021) 199:2597–2627 2625
reasons, they are not to be found in common considerations regarding monotonicity
and alleged blockers.
Acknowledgements I am grateful to Fabrice Correia, Stephan Leuenberger, Anna Marmodoro, audiences
at the Universities of Geneva (eidos seminar, 2015) and Glasgow (Philosophy Senior Seminar, 2018), as
well as three anonymous referees, for helpful comments and discussion on earlier drafts. I also thank the
Swiss National Science Foundation (Projects 168387 and 169313) for supporting my research on this topic.
Funding Open access funding provided by University of Geneva.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included
in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If
material is not included in the article’s Creative Commons licence and your intended use is not permitted
by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the
copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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