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: Georg.Mueller@Unifr.ch

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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