Crisp Set Vs Fuzzy Set
Crisp Set Vs Fuzzy Set
Crisp Set Vs Fuzzy Set
.5 = neither .5 = cross-over:
fully in nor neither in nor out
fully out
0 = fully out 0 = fully out 0 = fully out 0 = fully out 0 = fully out
FUZZY MEMBERSHIP IN THE SET OF "RICH COUNTRIES"
.75 *
.625
*
R
I
C .5 *
H
*
.375 *
.25 *
.125 *
*
0 **********
.875
*
*
.75
.625 *
P *
O
O .5
R *
*
.375
.25 *
.125
*
*
0 ************************************************************
100 x
x
error x
O x
U x
T x
C 67 x
O x
M x
E x
x
Y x
x
33 x
x
x
x
x error
0 x
0 100
CAUSE (X)
• With fuzzy sets, cases in these regions of the plot have different
interpretations: Cases in the lower-right corner violate the argument that the
cause is a subset of the outcome; cases in the upper-left corner violate the
argument that the outcome is a subset of the cause.
THE CAUSE (X) IS A SUBSET OF THE OUTCOME (Y)
100 x x x x
x x xx xx
x x x xxx x
O x x x x x
U x x
T x x x
C 67 x xx x x x
O x xxx x
M x xx
E x xxxx x
x x x
Y x
x x x
33 xxx x
x x
x x
xxxx
x x
0 x
0 100
CAUSE (X)
• This plot illustrates the characteristic upper-triangular plot indicating the fuzzy
subset relation: X ≤ Y (cause is a subset of the outcome). This also can be
viewed as a plot supporting the contention that X is sufficient for Y.
• Cases in the upper-left region are not errors, as they would be in a
conventional quantitative analysis. Rather, these are cases with high
membership in the outcome due to the operation of other causes. After all, the
argument here is that X is a subset of Y (i.e., X is one of perhaps several ways
to generate or achieve Y). Therefore, cases of Y without X (i.e., high
membership in Y coupled with low membership in X) are to be expected.
• In this plot, cases in the lower-right region would be serious errors because
these would be instances of high membership in the cause coupled with low
membership in the outcome. Such cases would undermine the argument that
there is an explicit connection between X and Y such that X is a subset of Y.
THE OUTCOME (Y) IS A SUBSET OF THE CAUSE (x)
100 x
xx
x x
O x x x
U x x
T xx xx
C 67 x
O x x x x
E x x x
x x x
Y x x x
xx x x x
x x x x x
33 x xxxx x x
xx x x
x x x x x x
x x x x
xxx x x x xx x x
0 x xxx x x x xxx x x
0 100
CAUSE (X)
• This plot illustrates the characteristic lower-triangular plot indicating the fuzzy
superset relation: X ≥ Y (outcome is a subset of the cause). This also can be
viewed as a plot supporting the contention that X is necessary for Y.
• Cases in the lower-right region are not errors, as they would be in a
conventional quantitative analysis. Rather, these are cases with low
membership in the outcome, despite having high membership in the cause.
This pattern indicates that Y is a subset of X: condition X must be present for Y
to occur, but X is not capable of generating Y by itself. Other conditions may
be required as well. Therefore, cases of X without Y (i.e., high membership in
X coupled with low membership in Y) are to be expected.
• Cases in the upper-left region would be serious errors because these would be
instances of low membership in the cause coupled with high membership in the
outcome. In this plot, such cases would undermine the argument that there is
an explicit connection between X and Y such that Y is a subset of X.
REFINING THE FUZZY SUBSET RELATION:
1. CAUSE IS A SUBSET OF THE OUTCOME
• When the argument is that the cause (X) is a subset of the outcome (Y), cases
below the diagonal are "errors" because these X scores exceed the
corresponding outcome (Y) scores.
• As with crisp set analysis, logical and can be used to move scores to the
correct side of the diagonal. With logical and, conditions are compounded,
which in turn involves taking the minimum membership score of the
compounded sets as the membership of a case in the combinations.
Mathematically, A*B must be less than or equal to A.
• When the argument is that the outcome (Y) is a subset of the cause (X), cases
above the diagonal are "errors" because these X scores are less than the
corresponding outcome (Y) scores.
• As with crisp set analysis, logical or can be used to move scores to the correct
side of the diagonal. With logical or conditions are substitutable, which in turn
involves taking the maximum membership score of the substitutable sets. It
follows mathematically that A + B ≥ A.
A a I i M m U u M+U i*(M+U)
Australia 0.8 0.2 0.6 0.4 0.4 0.6 0.6 0.4 0.6 0.4
Belgium 0.6 0.4 0.2 0.8 0.2 0.8 0.8 0.2 0.8 0.8
Denmark 0.6 0.4 0.4 0.6 0.2 0.8 0.8 0.2 0.8 0.6
France 0.6 0.4 0.8 0.2 0.2 0.8 0.2 0.8 0.2 0.2
Germany 0.6 0.4 0.8 0.2 0.4 0.6 0.4 0.6 0.4 0.2
Ireland 0.2 0.8 0.6 0.4 0.8 0.2 0.6 0.4 0.8 0.4
Italy 0.4 0.6 0.8 0.2 0.2 0.8 0.6 0.4 0.6 0.2
Netherlands 0.6 0.4 0.4 0.6 0.2 0.8 0.4 0.6 0.4 0.4
Norway 0.6 0.4 0.4 0.6 0.6 0.4 0.8 0.2 0.8 0.6
Sweden 0.8 0.2 0.4 0.6 0.8 0.2 1.0 0.0 1.0 0.6
UKingdom 0.6 0.4 0.6 0.4 0.8 0.2 0.6 0.4 0.8 0.4
UStates 1.0 0.0 0.8 0.2 0.4 0.6 0.2 0.8 0.4 0.2
FUZZY SETS AND CONFIGURATIONS
Country Income Inequality Manufacturing Strong Unions
I i M m U u i*m*u i*m*U i*M*u i*M*U I*m*u I*m*U I*M*u I*M*U
Australia 0.6 0.4 0.4 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.4 0.4
Belgium 0.2 0.8 0.2 0.8 0.8 0.2 0.2 0.8 0.2 0.2 0.2 0.2 0.2 0.2
Denmark 0.4 0.6 0.2 0.8 0.8 0.2 0.2 0.6 0.2 0.2 0.2 0.4 0.2 0.2
France 0.8 0.2 0.2 0.8 0.2 0.8 0.2 0.2 0.2 0.2 0.8 0.2 0.2 0.2
Germany 0.8 0.2 0.4 0.6 0.4 0.6 0.2 0.2 0.2 0.2 0.6 0.4 0.4 0.4
Ireland 0.6 0.4 0.8 0.2 0.6 0.4 0.2 0.2 0.4 0.4 0.2 0.2 0.4 0.6
Italy 0.8 0.2 0.2 0.8 0.6 0.4 0.2 0.2 0.2 0.2 0.4 0.6 0.2 0.2
Netherlands 0.4 0.6 0.2 0.8 0.4 0.6 0.6 0.4 0.2 0.2 0.4 0.4 0.2 0.2
Norway 0.4 0.6 0.6 0.4 0.8 0.2 0.2 0.4 0.2 0.6 0.2 0.4 0.2 0.4
Sweden 0.4 0.6 0.8 0.2 1.0 0.0 0.0 0.2 0.0 0.6 0.0 0.2 0.0 0.4
UKingdom 0.6 0.4 0.8 0.2 0.6 0.4 0.2 0.2 0.4 0.4 0.2 0.2 0.4 0.6
UStates 0.8 0.2 0.4 0.6 0.2 0.8 0.2 0.2 0.2 0.2 0.6 0.2 0.4 0.2
With three fuzzy sets, the vector space has eight corners. It is possible to calculate the
membership of each case in each corner.
The corners can be viewed as ideal typic cases; the membership of a case is a corner is
the degree to which it conforms to the ideal type represented by the corner.
SE
NO
DN
.75
BL
FI
OS
NT
GR
IT UK
IR
.50 FR NZ
JA SZ
.25
CA AL
US
.00
1. The researcher calibrates fuzzy membership scores for all relevant causal conditions and the
outcome. It is OK to mix crisp and fuzzy causal conditions.
2. fsQCA calculates the membership of each case in all the logically possible combinations of
membership. In each calculation the minimum membership score is used.
4. The results of these 2k analyses are recorded in a truth table. In effect, each row of the
(crisp) truth table represents a corner of the multidimensional space defined by the causal
conditions.
5. fsQCA also calculates the number of cases with greater than .5 membership in each
combination of conditions. This information is used to assess patterns of limited diversity.
6. The researcher uses the information in the truth table (from #4 and #5) to code the outcome
in the truth table. This involves decisions about frequency and consistency thresholds.
7. fsQCA analyzes the truth table. Usually, two solutions are derived, one with remainders set
to false (0), the other with remainders set to don’t care (-). These two solutions establish the
range of plausible solutions (the complexity/parsimony continuum).
NSF Tends to Fund Research That:
⋅ has a research design that is specified at the outset of the research, usually in
the proposal itself; and
⋅ culminates in the creation of a public good--a large-N data set that is, in effect,
purchased by the NSF for the social scientific community.
⋅ develops the research design in the course of the research, as the investigator
learns more about his/her cases; and
In short, the typical qualitative research project completely contradicts the NSF
funding template.
What All “Good” Principle Investigators Should Do:
• Write clearly and engagingly for a broad audience of social scientists. For example,
define and explain disciplinary or project specific jargon.
• Situate the research in relation to existing theory whether the research goal is to
challenge conventional views of some phenomenon or to develop new theory or chart
new terrain.
• Locate the research in the literature citing existing studies of related phenomena,
specifying comparable cases, building on findings of other researchers, and bringing
this research into dialogue with the work of others.
• Outline clearly the research procedures including details about where, when, who,
what, and how the research will be conducted.
• Discuss a plan for data analysis including a discussion of different strategies for
managing the various types of data to be gathered, how data will be stored and
accessed, and the procedures for making sense of the information obtained.
• Describe a strategy to refine the concepts and construct theory as more is learned
about the case(s) under investigation.
• Provide information about replicability, in particular try to consider and suggest ways
in which this research might be reproduced by others.
• Describe the data archive that will be left behind for others to use and the plan for
maintaining confidentiality.