Logic, Sequences, and Explanation
James Mahoney
Northwestern University
Starting assumption: an individual
cause is normally one of five types:
(1) necessary but not sufficient;
(2) sufficient but not necessary;
(3) necessary and sufficient;
(4) INUS; or
(5) SUIN.
Other assumptions:
(1) Causation at the population level
exists because causes are operating
in the individual cases.
You can think of this as the
“methodological individualism” of
causation.
(2) Causes that raise the probability of an
outcome in a population may still
contribute to the absence of that
outcome in any one individual case.
(3) We first need to understand what
“cause” can mean at the level of the
individual case.
Then we need to explore whether and
how this understanding can be
extended to the population level.
Set-theoretic definition of necessary
cause:
X is a necessary cause of Y if Y is a
subset of X.
Set-Theoretic Conceptualization of a Necessary Cause
X
Y
Set-Theoretic Conceptualization of a Necessary Cause: Example
Set of all females
X
Y
Set of all
pregnant people
Set of
Females
Set of
pregnant
people
Set of
Females
Set of
pregnant
people
X
Not X
Y
Set of
not females
Set of
females
X
Not X
Y
Set of
pregnant
people
Set of
females
Set of
not females
Set of
pregnant
people
Set of not
pregnant
people
Method of Agreement (as conventionally
used): Eliminates necessary causes.
Case
Peasant Worker Social
Revolts Revolts Revolution
--------------------------------------------------------France Yes
Yes
Yes
Russia Yes
Yes
Yes
China
Yes
No
Yes
The Relative Importance of Necessary Causes: An Example
A
B
C
Y1
Y
A = Daytime
B = Sunshine
C = Sun shower
Y = Rainbow
Figure 2.2
Set of all females
Z
X
Y
Set of pregnant
individuals
Set of females
age 10-60
Framework for Assessing the Relative Importance of
Necessary Causes
X
Y
As X approaches
Y, it becomes less
trivial and more
important.
Set-theoretic definition of a sufficient
cause:
X is a sufficient cause of Y if X is a
subset of Y.
Set-Theoretic Conceptualization of a Sufficient Cause
Y
X
X
Y
Set of
males
Set of not
pregnant
people
Set of
males
Set of
Not
Males
Set of not
pregnant
people
A basic rule of logic:
If X is necessary for Y, then Not X is
sufficient for Not Y.
If X is sufficient for Y, then Not X is
necessary for Not Y.
Relationship between Sufficient/Necessary Causes
Females
Not pregnant
people
Not females
Not Y
X
Y
Not X
Pregnant people
Framework for Assessing the Relative Importance of
Sufficient Causes
Y
X
As X
approaches
Y, it becomes
less trivial
and more
important.
Children
Males
Not
pregnant
people
Method of Difference (as conventionally
used): Eliminates individually sufficient
causes.
Case
Relative
Social
Deprivation Revolution
--------------------------------------------------------France
Yes
Yes
Prussia
Yes
No
Set-Theoretic Conception of a Necessary and Sufficient Cause
X
Y
X
Not
X
Y
Not
Y
INUS Cause: A cause that it is one
part of a combination of causes that
are jointly sufficient for an outcome.
An INUS cause by itself is neither
necessary nor sufficient.
In Mackie’s words: An INUS cause is “an insufficient
but necessary part of a condition which is itself
unnecessary but sufficient for the result” (Mackie
1965: 246).
Logical AND (&/*): INUS causes are
connected together with the Logical
AND. Again, there are various
notations in vogue:
Y = A AND B AND C
Y=A*B*C
Y=A&B&C
Y = ABC
Here is what I like best:
A&B&C→Y
ABC → Y
Multiple Causation/Equifinality: Different
combinations that are sufficient for the same
outcome. Here the Logical OR (v/+) is used.
For example:
Y = (A AND B) OR (Not A AND C AND D)
Y = (A & B) v (~A & C & D)
Y = (A B) + (~A C D)
(A & B) v (~A & C & D) → Y
A B v ~A C D → Y
A version of Moore’s argument:
X & (A v B) → Y
where X = strong bourgeoisie; A =
alliance between bourgeoisie and
aristocracy; B = weak aristocracy;
and Y = democratic pathway.
Question: Which variables are INUS
causes?
Set-theoretic definition of an INUS
cause:
X is an INUS cause of Y if the
overlapping set created by X and one
or more other factors is a subset of Y.
Set-Theoretic Conception of INUS Causes
(X & Z) v . . . . → Y
Z
X
Y
X
Y
Z
We can also ask about the relative
importance of INUS causes. For
example, look at the two INUS
causes in the original diagram again.
Which one is more important?
Jim Mahoney’s Rule: Any cause becomes
more important as it becomes closer to
being a necessary and sufficient cause.
Necessary and sufficient cause = the gold
standard.
Thus, we can think of this rule as the
“golden rule” of causality.
SUIN Cause: A factor that is sufficient (but not
necessary) to constitute a necessary (but not
sufficient) cause.
Example: X & Z → Y; A v B → Y.
where Y = democratic pathway; X = strong
bourgeoisie; Z = politically subordinate
aristocracy; A = alliance between bourgeoisie
and aristocracy; and B = weak aristocracy.
Here A and B are SUIN causes of a democratic
path.
Set theoretical definition of SUIN
cause:
X is a SUIN cause of Y if Y is a subset
of the joint space created by X when
combined with one or more other
causal factors.
Set-Theoretic Conception of SUIN Causes
A = Weak landed elites
Y = Democratic regime
Y
A
Set-Theoretic Conception of SUIN Causes
X
X
A
Z
A = Weak Landed Elites
X = Small Landed Class
Z = Politically Marginal
Landed Class
Set-Theoretic Conception of SUIN Causes
X
Z
Y
X = Small Landed Class
Z = Politically Marginal
Landed Class
Y = Democratic Regime
It is worth asking which SUIN cause in
the diagram is the more important
one.
Remember the golden rule.
Why do I believe that this list of five causes
is exhaustive from a logic and set theory
perspective?
Because, from a set theory perspective, a
cause or a combination of causes must be
necessary and/or sufficient for an
outcome.
And these five seem to express the range of
possibilities.
What is the relationship between logical
types of causes and the understanding of
causation used in statistical analysis (e.g.,
possible outcomes model)?
Possible answer: Variables that show up
as exerting effects in correctly specified
statistical models have those effects
because they picking up important INUS
causes (i.e., INUS causes that come close
to being necessary and/or sufficient).
Z
Y
Y
V
x
x
xx
xxx
Y
x xx
x xx
x
x x xx
x
Z
Y
It is possible to
imagine the
Venn diagram
that corresponds
to different
scatter plots.
x
x
x
x xx
x x
x
x
x x x x
x x
V
x
x
x
x
x
The main point here is that we can
think about correlations in terms of
logic. I think we can, in a sense,
“reduce” them to their logical status.
We learn something important when we ask
about the extent to which a causal
variable approximates a necessary and/or
sufficient cause.
Example: economic development is
associated with democracy is because it
approximates a sufficient cause.
If a cause is close to being a necessary, then
its effect will be more like an “enabler” –
i.e., it is a variable that allows some
outcome to happen.
You can think about it this way: the cause
exists in most of the various different
combinations of factors that are each
sufficient for the outcome.
If a cause is close to being sufficient, then it
is more like a factor that does not need
much help in order to generate the
outcome.
It does not need to combine with many
other factors to produce the outcome. It
is almost enough by itself.
Now I want to discuss a method for assessing
the relative importance of causes in case
study research.
I want to do so by building on the notions of
causal importance introduced earlier,
including especially the golden rule.
Normally, we evaluate causal importance
using populations of cases:
X
Z
Y
Y
X = Female, Z = Female age 10-60, Y = pregnant individual
In the example, the size of the circles
corresponds to the number of cases that
are members of the categories
represented by the circles.
But how could we assess causal importance
when N is small, or when N = 1?
Let me make three points to start . . .
(1) Small-N scholars routinely debate the
importance of causes associated with
different time periods.
For example:
Yashar and Mahoney on Central America;
Precolonial, colonial, and postcolonial
arguments about the NICs; and
“Critical juncture” arguments more
generally.
(2) Scholars have trouble deciding if and
when an intervening cause should be
treated as more important than the
original cause.
For example:
Colonialism => Developmental State =>
Sustained High Growth in Korea and
Taiwan
(3) Are proximate or historical causes more
important in general?
Elster: The best explanation is the one that
identifies the causes most proximate to
the outcome.
Diamond: An ultimate explanation is the
one that identifies the original historical
causes.
The Method Sequence Elaboration
(Mahoney, Koivu, and Kimball 2009)
Sequence Elaboration: Like Lazarsfeld’s
“elaboration model” but applies to the
kinds of causes used in historical
explanation.
Basic idea: Start with an initial two factor
relationship (e.g., X =n=> Y) and
elaborate it by introducing new
antecedent and/or intervening causes.
With correlations, we know that controlling
for a third variable can influence how we
think about the original relationship. The
timing of that third variable is crucial.
For example: imagine that strong positive
correlation drops to zero when we control
for a third variable. It makes a big
difference if that control variable is
antecedent or intervening.
What happens when we start with a two
variable relationship that is expressed in
terms of the logical types of causes (e.g.,
X is necessary for Y), and then introduce
a third variable (either antecedent or
intervening) that is also a logical type of
cause?
For example: Start with X =n=> Y. Then
add an antecedent cause:
Z –?–> X –n–> Y
║--------?----------↑
One possibility is:
Z –n–> X –n–> Y
║--------n----------↑
Z –n–> X –n–> Y
║--------n----------↑
Downing: Z = Roman Empire, X =
medieval constitutionalism, and Y = early
democracy.
Question: How do the new relationships
affect our understanding of the original
relationship? What is more important, X
or Z? The Roman Empire or medieval
constitutionalism?
Answer: X is the more important cause
(from a logical perspective).
Z –n–> X –n–> Y
║--------n----------↑
Z
Z
X
Z
X
Y
Y
Z –n–> X
X –n–>Y
Z –n–> Y
X
Y
Z –n–> X –n–> Y
║--------n---------↑
Z
X
Y
We call this “contextualizing” the original
X-Y finding. We learn something about
the causes of our original cause.
As a Rule: In a sequence of linked
necessary causes, the importance a cause
increases as it becomes more temporally
proximate to the outcome.
“For want of a nail, a horseshoe was lost. For want of a
horseshoe, a horse was lost. For want of a horse, a
rider was lost. For want of a rider, the battle was lost.
For want of a battle, the war was lost.”
This approach identifies logical or
empirical importance. It does not tell us
about theoretical or normative
importance.
Some factors (e.g., slavery; US
intervention; ideas) may be important as
causes because they raise critical
normative issues, or because they are
associated with important theoretical
traditions.
To what does the size of a circle correspond
in this example?
Possible answer:
(1) The real case/s in which the necessary
causes and the outcome are present (this
determines the size of the outcome circle);
(2) Hypothetical/counterfactual cases in which
the necessary cause is present but the
outcome is not. The number of these cases
can be inferred from the logical structure
of the argument.
Question: What would have happened to
Downing’s argument if the Roman
Empire was sufficient of medieval
constitutionalism (and still necessary for
the final outcome of democracy)?
That is:
Z –s–> X –n–> Y
║---------n---------↑
Z = Roman Empire, X = medieval constitutionalism, and Y
= early democracy.
Answer: the Roman Empire (Z) would be
the more important cause.
Z –s–> X –n–> Y
║--------n---------↑
X1
X1
Z1
Z1 –s–> X1
Z1
X1
Z1
Y1
Y1
X1 –n–>Y1
Y1
Z1 –n–> Y1
Z –s–> X –n–> Y
║--------n---------↑
Z
X
Z = Roman Empire
X = medieval constitutionalism
Y = early democracy.
Y
We call this: Diminishment. The
importance of the original causal
factor is diminished.
Great merit of sequence elaboration:
gives precise answers!
This structure is common. One tries to
“diminish” an initial necessary cause
by:
(a) Finding a new necessary cause;
(b) Showing that the new necessary cause
comes before the original one;
(c) Showing that the new necessary cause is
sufficient for the original one.
Illogical Relationships: Not all
combinations are logically possible.
Sequence elaboration tells us when a set
of causal claims is impossible.
For example:
Z –n–> X –n–> Y
║----------s---------↑
Z –n–> X –n–> Y
║----------s---------↑
Z1
Y1
Z1
X1
Y1
X1
Y1
Z1
Z1 –s–> X1
X1 –n–>Y1
Z1 –s–> Y1
X1
Z1
Z –n–> X –n–> Y
║----------s---------↑
This doesn’t work;
can’t be done.
Several other interesting types exist:
(1) When we introduce an intervening
cause, the original cause may be less
important. We call this discovering
the mechanism through which the
original cause exerts its effect.
Example from Waldner (1999)
X–s–>Y
X–s–>Z–s–>Y
║-------s-------↑
Z is a more important cause
than X. We call this identifying
the mechanism through which
X does its causal work.
Y
Z
X = high elite conflict
Z = nondevelopmental state
Y = failed development
X
X = high elite conflict
Y = failed development
X
Y
X = high elite conflict
Z = nondevelopmental state
Y = failed development
X
Z
Y
(2) Of course, logically speaking, it is also
possible for the intervening cause to be
less important than the original cause.
We call this identifying a partial
mechanism.
For example:
X–n–>Y
X–s–>Z–n–>Y
║------n--------↑
Z
X
Y
For example:
X–n–>Y
X–s–>Z–n–>Y
║------n--------↑
X
Z
Y
We also distinguish between:
(3) Background Contextualization; and
(4) Pathway Contextualization.
So let’s do it.
Background Contextualization
(Example from Mahoney 2001)
X–s–>Y
Z–n–>X–s–>Y
║------n--------↑
Z
Y
Y
X
Z provides a
background
within which X
does its causal
work.
Z = partially centralized state/incipient export economy
X = radical liberalism (critical juncture)
Y = military authoritarianism
Background
Contextualization
Z–n–>X–s–>Y
║------n--------↑
Z
X
Y
Pathway Contextualization
Luebbert:
~LL =n=>SD
~LL –n–>RG –s–>SD
║--------n----------↑
~LL
SD
RG
The intervening
cause identifies
one pathway through
which ~LL exerts its
necessary causal
effect.
LL = liberal-labor alliance; RG = red-green
alliance; SD = social democracy.
~LL –n–>RG –s–>SD
║--------n----------↑
~LL
RG
Pathway
Contextualization
SD
(1) There is a whole world of methodology
out there waiting to be developed. It is
based in the philosophy of logic, not
statistical/probability theory. But most
graduate students learn next to nothing
about formal logic. 
(2) This world of methodology could have
profound implications for statistical
analysis. A unified theory of causality
might resolve major methodological
problems in the social sciences. 