Three valued logic for qualitative comprative analyses
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Transcript of Three valued logic for qualitative comprative analyses
Three-Valued Logic for Qualitative Comparative Analysis:
A New Application of the fs/QCA-Software of Charles Ragin
© Georg P. Mueller
Fac. of Economics and Social ScienceUniversity of Fribourg, SwitzerlandE-Mail: [email protected]
Transparenciespresented at the
3rd European Workshop onComputer-Aided Qualitative Research
Lisbon / PortugalOctober 7 - 8, 2010
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1. Qualitative comparative analysis QCA
Purpose of QCA:
Comparison of binary conditions X1, X2, ...with regard to a binary outcome Y
Tab. 1: An exemplary dataset: Deference to persons, by gender and status:
________________________________X1 X2 Y
Person Woman Chief Deference________________________________
1 1 1 12 0 1 13 1 0 04 0 0 0
________________________________Legend: Woman: 1 = yes, 0 = no. Chief: 1 = yes, 0 = no. Deference: 1 = deference to person, 0 = deference to others. Assumption: Y = X2.
The three steps of „classical“ QCA:
1) Translation of data with Y=1 into a Boolean expression in disjunctive normal form:
1 case = 1 set of conjunctionsE.g., from tab. 1 follows formula
Y = (X1 AND X2) OR (NOT X1 AND X2)
2) Simplification of this Boolean expression with the Quine-McCluskey algorithm.1)
E.g., from above follows: Y <==> X2
3) Exploration of the simplified Boolean formula E.g., from simplified formula follows:
Gender X1 has no influence on deference Y.
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Problems of „classical“ QCA:
1) Missing instantiations (cases) for certain sets of preconditions: E.g. case 1 in tab. 2.
2) Contradictory outcomes for certain sets of preconditions:
E.g. variable Y of cases 3a,b,c in tab. 2
Tab. 2: A modified exemplary dataset, based on tab. 1:________________________________________
X1 X2 Y Y*=Case Woman Chief Deference Rec. Y________________________________________
1 1 1 ? --2 0 1 1 1
3a 1 0 0 3b 1 0 0 3c 1 0 1 04 0 0 0 0
________________________________________Legend: Y*: Recoding of Y by „classical“ QCA-methodology. Other definitions: see tab. 1.
„Classical“ solutions to the problems of QCA:
1) Elimination of inconsistent cases 2) Missing values for missing cases 3) Quantification by fuzzy-set QCA
General critique of the „classical“ solutions:
a) Unwarranted simplifications/omissions of data b) Quantitative answers in qualitative research.
Alternative solutions to the problems of QCA:
Three-valued modal logic.
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2. An overview of three-valued modal logic
Basic feature 1:
Third truth-valuei = indeterminate whether true or false.
Examples with truth-value i:
Propositions about events in the future. Propositions with missing instantiations.
Boolean operators:
Extension from 2 to 3 truth-values possiblebut not needed for this article:
See Lukasiewicz (1970) and others.2)
Basic feature 2:
Two modal operators:POS = Possibility of a propositionNEC = Necessity of a proposition
Tab. 3: The definition of the modal operators:_________________________________________________________________
Y NOT Y POS Y NEC Y POS NOT Y NEC NOT Y_________________________________________________________________
0 1 0 0 1 1i i 1 0 1 01 0 1 1 0 0
_________________________________________________________________
Legend: NOT: Negation; POS: Possibility; NEC: Necessity; 0 = false; 1 = true; i = indeterminate.
Interpretation of modal operators:
X => NEC Y: „X is a strict trigger of Y“ X => POS Y: „X is a possible trigger of Y“
X => NEC NOT Y: „X is a strict inhibitor of Y“ X => POS NOT Y: „X is a possible inhibitor of Y“
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3. QCA with three-valued modal logic
Step 1:
Make missing knowledge more visible:a) Replace missing instantiations of Y by i b) Replace contradictory outcomes of Y by i
Results of step 1:
New three-valued dependent variable Y‘,difficult to treat with conventional QCA-software.
Step 2:
Creation of four new variables derived from Y‘:NEC Y‘, NEC NOT Y‘, POS Y‘, POS NOT Y‘ (see tab. 3)
Tab. 4: Results of the application of steps 1 and 2 to tab 2:_______________________________________________________________________
X1 X2 Y Y‘ = NEC NEC POS POSCase Woman Chief Deference Rec. Y Y‘ NOT Y‘ Y‘ NOT Y‘_______________________________________________________________________
1 1 1 -- i 0 0 1 12 0 1 1 1 1 0 1 0
3a 1 0 03b 1 0 03c 1 0 1 i 0 0 1 14 0 0 0 0 0 1 0 1
______________________________________________________________________Legend: Y ‘: Recoding of Y by methodology described in step 1. Other cols.: see previous tables.
Step 3:
Application of standard QCA to each of the variablesNEC Y‘, NEC NOT Y‘, POS Y‘, POS NOT Y‘,
which are all binary Boolean =>
Possibility of using fs/QCA-software.3)
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Results of step 3:
Four simplified Boolean expressions, which explainNEC Y‘, NEC NOT Y‘, POS Y‘, POS NOT Y‘
Example: X1 OR X2 => POS Y‘
Step 4:
Unification of the results of step 3by the use of four new Boolean operators:
(1) Strict implication X ––> Y‘ means X => NEC Y‘
(2) Strict inhibition X –//–> Y‘ means X => NEC NOT Y‘
(3) Possible implication X ----> Y‘ means X => POS Y‘
(4) Possible inhibition X --//--> Y‘ means X => POS NOT Y‘
Illustrative example of step 4:
X1 OR X2 => POS Y‘is replaced by
X1 OR X2 ----> Y‘
Step 5:
Exploration of the results of step 4 bydrawing logical inferences.
Example of step 5:
X1 OR X2 ----> Y‘implies
X1 ----> Y‘ and X2 ----> Y‘
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4. On the use of fs/QCA software
Purpose of fs/QCA software:
Qualitative comparative analyses:a) Fuzzy set method of Ch. Raginb) Crisp set method of Ch. Ragin
Source a free software copy:
http://www.u.arizona.edu/~cragin/fsQCA/software.shtmlSoftware runs under Windows XP
Exemplary data:
See tab. 4: Deference to persons,by gender and status.
Fig. 1a: Definition of data and variables as step 1 inthe use of fs/QCA:
Legend: For definitions of the variables Women, Chief, and POS_Defsee tab. 4, cols. Women, Chief, and POS Y‘.
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Fig. 1b: Data gathering as step 2 in the use of fs/QCA:
Legend: Data from tab. 4, cols. Women, Chief, and POS Y‘.
Step 3 in the use of fs/QCA:
Choice of method of analysis from software menu:Crisp Sets with Truth Table Algorithm.
Fig. 1c: Model specification as step 4 in the use of fs/QCA:
Legend: Outcome and Causal Conditions from the data-pool Variables.
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Fig. 1d: Data cleaning of outcome variable as step 5 in the use of fs/QCA:
Legend: Dependent variable POS_Def has same value as before, because of100% consistency consist of the original data-table.
Fig. 1e: Specification of meaning of values as step 6 in the use of fs/QCA:
Legend: No Don‘t Care Cases, no Contradictions, no Remainders due to contradiction-free, complete data.
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Fig. 1f: Extraction of Boolean expression asstep 7 in the use of fs/QCA:
Legend: Fourth line from below: (OR) added by the author.
Interpretation of figure 1f:
Woman OR Chief =>POS_Def = Possibility of Deference
In other words:Woman OR Chief ----> Deference
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5. An exemplary application to ethno-political conflictResearch question:
What determines the ethno-political mobilization Eof a region in a situation of wealth:
Size S, Linguistic ability L, Economic growth G?
Reference study:
„Classical“ QCA by Ch. Ragin (1989), pp. 133-149,4)
based on data about 36 ethnic regions in Europe.
Tab. 5: Ethno-political conflict in wealthy regions:_________________________________________________________________Config. Size Ling. Abil. Growth Ethn. Mob. Ethn. Mob.
Nr. S L G E E*_________________________________________________________________
1 0 0 0 0 02 0 0 1 i -3 0 1 0 1 14 0 1 1 i 15 1 0 0 i -6 1 0 1 i 17 1 1 0 1 18 1 1 1 i 1
_________________________________________________________________Legend: S = Size. L = Linguistic ability. G = Economic growth. E* = Ethno-political mobilization, coded by Ragin, 1989, Tab. 13. E = Ethno-political mobilization, coded by the author: E=1, ifRagin, 1989, Tab. 12, reports for all cases conflict level 2; E=0, if Ragin, 1989, Tab. 12, reports forall cases conflict level 0 or 1; E=i, for all other cases. Sample: Wealthy subnations with W=1 (see Ragin, 1989, Tab. 13). Source: Ch. Ragin. 1989. The Comparative Method. Berkeley: Universityof California Press.
Four Boolean expressions representingthe empirical results of QCA with 3-valued logic:
NOT G AND L ––> E S OR G OR L ---> E G OR NOT L --//--> E
NOT S AND NOT G AND NOT L –//–> E
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Fig. 2: The effects of different Boolean terms on the conflict E:
LS
G
NOT L
NOT G AND L
NOT L ANDNOT G AND NOT S
--------> E
–––––> E
E <----//----
E <––//––
Legend: E = Ethno-political mobilization; G = Growth; L = Linguistic ability; S = Size.
Interpretation of fig. 2:
The presence of G or S or Lmay trigger a conflict E.
L must trigger a conflict E if inaddition there is no growth G.
Linguistic ability L as a prerequisiteof ethnic identity.
The presence of G or the absence of Lmay inhibit a conflict E.
The absence of L must inhibit a conflict Eif in addition G and S are both absent.
The presence of G mayboth trigger or inhibit a conflict E:
Growth G makes a region more important but threatens its ethnic identity.
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6. Three-valued QCA: What is different?Fig. 3a,b: 2- versus 3-valued QCA of tab. 5:
Differences and similarities:
L AND NOT G
L AND G S AND G––––> E
E <––––
Mueller with 3-valued QCA
Ragin with 2-valued QCA
NOT L AND NOT G
––//––> E
E <––//––
Mueller with 3-valued QCA
NOT L AND NOT S AND NOT G
NOT L ANDNOT S AND G
Ragin with 2-valued QCA
Comments on fig. 3a,b:
The strict triggers of E in 3-valued QCA are a subset of the strict triggers in 2-valued QCA.
The strict inhibitors of E in 3-valued QCA are asubset of the strict inhibitors in 2-valued QCA.
General summary:
3-valued QCA is a „prudent“ methodology:It points to the limits of our theories,
which may be hidden by the use 2-valued QCA.
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Notes:
1: For Quine-McCluskey algorithm see: Mendelson, Elliot (1970): Boolean Algebra and Switching Circuits: chap. 4. New York: McGraw-Hill.
2: Lukasiewicz, Jan (1970 [1920]): Selected Works. Ed. by L. Borkowski. Amsterdam: North-Holland.
3: For fs/QCA-software see: http://www.u.arizona.edu/~cragin/fsQCA/ 4: Ragin, Charles (1989): The Comparative Method. Berkeley: University of California
Press.
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