List of figures List of tables Acknowledgements

Transcription

1 List of figures List of tables Acknowledgements page xii xiv xvi Introduction 1 Set-theoretic approaches in the social sciences 1 Qualitative as a set-theoretic approach and technique 8 Variants of QCA 13 Plan of the book 16 How to use this book 19 Part I Set-theoretic methods: the basics 21 1 Sets, set membership, and calibration The notion of sets Sets and concepts The pros and cons of crisp sets Properties of fuzzy sets What fuzzy sets are not The calibration of set membership Principles of calibration The use of quantitative scales for calibration The direct and indirect methods of calibration Does the choice of calibration strategy matter much? Assessing calibration 40 2 Notions and operations in set theory Conjunctions, Boolean and fuzzy multiplication, intersection, logical AND 42 vii in this web service

2 viii 2.2 Disjunctions, Boolean and fuzzy addition, union, logical OR Negations, complements, logical NOT Operations on complex expressions Rules for combining logical operators Negation, intersection, and union of complex sets Calculating membership in complex sets Relations between sets Notational systems in set-theoretic methods 54 3 Set relations Sufficient conditions Crisp sets Fuzzy sets Necessary conditions Crisp sets Fuzzy sets Causal complexity in set-theoretic methods Defining causal complexity INUS and SUIN conditions The notion of asymmetry Set-theoretic methods and standard quantitative approaches 83 4 Truth tables What is a truth table? How to get from a data matrix to a truth table Crisp sets Fuzzy sets Analyzing truth tables Matching similar conjunctions Logically redundant prime implicants Issues related to the analysis of the non-occurrence of the outcome 112 Part II Neat formal logic meets noisy social science data Parameters of fit Defining and dealing with contradictory truth table rows Consistency of sufficient conditions Coverage of sufficient conditions 129 in this web service

3 ix 5.4 Consistency of necessary conditions Coverage of necessary conditions Issues related to consistency and coverage Limited diversity and logical remainders Limited diversity in set-theoretic methods: how to see it when it is there Sources of limited diversity Arithmetic remainders Clustered remainders Impossible remainders What limited diversity is not The Standard Analysis procedure: identifying logical remainders for crafting plausible solution terms The dimension of set relations The dimension of complexity The dimension of types of counterfactuals The Standard Analysis procedure in a nutshell The Truth Table Algorithm From the data matrix to truth table Attributing an outcome value to each truth table row Logically minimizing the truth table Implications of the Truth Table Algorithm 190 Part III Potential pitfalls and suggestions for solutions Potential pitfalls in the Standard Analysis procedure and suggestions for improvement Beyond the Standard Analysis: expanding the types of counterfactuals The Enhanced Standard Analysis: forms of untenable assumptions and how to avoid them Incoherent counterfactuals I: contradicting the statement of necessity Incoherent counterfactuals II: contradictory assumptions Implausible counterfactuals: contradicting common sense Putting the Enhanced Standard Analysis procedure into practice 209 in this web service

4 x 8.3 Theory-Guided Enhanced Standard Analysis: complementary strategies for dealing with logical remainders Choosing entire truth table rows as good counterfactuals Formulating conjunctural directional expectations Comparing the different strategies for the treatment of logical remainders Potential pitfalls in the analysis of necessity and sufficiency and suggestions for avoiding them Pitfalls in inferring necessity from sufficiency solution terms Hidden necessary conditions The appearance of false necessary conditions The analytic consequences of skewed set-membership scores The coverage of necessary conditions and the problem of trivialness The consistency of sufficient conditions and the problem of simultaneous subset relations A general treatment of skewed set membership in fuzzy-set analyses 244 Part IV Variants of QCA as a technique meet QCA as an approach Variants of QCA The two-step approach Multi-value QCA Principles of mvqca: notation and logical minimization An assessment of mvqca Set-theoretic methods and time Forms of causally relevant notions of time Informal ways of integrating notions of time into set-theoretic methods Sequence elaboration Temporal QCA Data analysis technique meets set-theoretic approach Recipe for a good QCA The appropriateness of set-theoretic methods The choice of the conditions and the outcome 276 in this web service

5 xi The choice of the QCA variant Calibration of set-membership scores Analysis of necessary conditions Analysis of sufficient conditions Presentation of results Interpretation of results Reiteration of the research cycle The use of software Robustness and uncertainty in QCA How do we see robustness in set-theoretic methods when it is there? The effects of changing calibration The effects of changing consistency levels The effect of dropping or adding cases The evaluation of theories in set-theoretic methods Why standard hypothesis testing does not fit into set-theoretic methods The basics of theory evaluation in set-theoretic methods Extending theory evaluation by integrating consistency and coverage Summarizing set-theoretic theory evaluation Set-theoretic methods and case selection Types of cases after a QCA Forms and aims of (comparative) within-case studies after a QCA Post-QCA case selection principles Looking back, looking ahead Looking back: the main topics of this book Myths and misunderstandings Looking ahead: tasks and developments in the coming years 318 Glossary 322 Bibliography 336 Index 346 in this web service

6 Figures xii 0.1 Venn diagram for relation of sufficiency page Set-theoretic approaches in the social sciences Membership in fuzzy set of Länder with underdeveloped all-day schools plotted against percentage of pupils enrolled in all-day schools Two-by-two table sufficiency Venn diagram sufficiency XY plots in crisp-set analysis distribution of cases for sufficient conditions XY plot distribution of cases for sufficient condition X XY plot fully consistent sufficiency solution Two-by-two table necessity Venn diagram necessity XY plot distribution of cases for necessary condition X XY plot non-consistent necessary condition Two-by-two table necessity and sufficiency XY plot contrasting perfect set relation with perfect correlations Venn diagram with three conditions Three-dimensional property space Logical minimization of primitive expressions to prime implicants Venn diagram with logically redundant prime implicant Venn diagrams consistent and inconsistent sufficient conditions XY plot consistent and inconsistent sufficient conditions Venn diagrams different levels of coverage sufficiency XY plot different levels of coverage sufficiency Venn diagram equifinal solution term and types of coverage XY plot condition STOCK, outcome EXPORT Venn diagrams trivial and non-trivial necessary conditions XY plot condition MA+STOCK, outcome EXPORT XY plot the tension between consistency and coverage of sufficient conditions Conservative, intermediate, and most parsimonious solution terms Venn diagram types of counterfactuals in Standard Analysis procedure 176 in this web service

7 xiii List of figures 7.1 XY plot for path C~P Steps in the Truth Table Algorithm XY plot combined with two-by-two table Venn diagram types of counterfactuals, extended list Venn diagram different sources of trivialness necessity XY plot trivial necessary condition XY plots for condition PSR and outcomes U and ~U XY plot four areas and eight potential subset relations Logical minimization of sequence of events XY plot with two-by-two table and types of cases 308 in this web service

8 Tables xiv 1.1 Verbal description of fuzzy-set membership scores page Calibration of condition many institutional veto points QUALITATIVE versus direct method of calibration for set many institutional veto points Important operations in set-theoretic methods Determining membership in complex sets Basic operations and notations in set-theoretic approaches Sufficiency: stylized data matrix Hypothetical data matrix with ten cases and set-membership scores in three conditions and the outcome Hypothetical data matrix with complements of three conditions Hypothetical data matrix with some conjunctions Hypothetical data matrix with fuzzy-set membership scores Data matrix necessity Hypothetical data matrix with all complements of single conditions and conjunction ~A+C Data matrix with ten cases, three conditions, and outcome Hypothetical truth table with three conditions Hypothetical data matrix with fuzzy-set membership scores Fuzzy-set data matrix with two cases Fuzzy-set membership in ideal types for hypothetical data matrix Fuzzy-set ideal types for hypothetical data matrix Fuzzy-set membership in rows and outcome Truth table derived from hypothetical fuzzy-set data Example of hypothetical truth table Prime implicant chart Two-by-two tables consistent and inconsistent sufficient conditions Two-by-two tables different levels of coverage sufficiency Fuzzy-set membership in solution and outcome (Vis 2009) Fuzzy-set membership in path PS and outcome (Vis 2009) Two-by-two tables consistent and inconsistent necessary conditions Analysis necessity, single conditions (Schneider et al. 2010: 255) 142 in this web service

9 xv List of tables 5.7 Analysis necessity, functional equivalents (Schneider et al. 2010: 255) Truth table with three conditions and limited diversity Truth tables with all logically possible combinations of simulated values for logical remainders Hypothetical truth table with five conditions and limited diversity Fuzzy values data matrix, 44 cases Distribution of cases to ideal types Fuzzy-set membership scores of cases in ideal type ~C~P~NR Consistency values of ideal types Truth table based on fuzzy-set data matrix Truth table for outcome ~U (Vis 2009) Truth table Lipset data (Ragin 2009) Truth table, outcome CA (Ragin et al. 2003) Truth table (Koenig-Archibugi 2004) Types of assumptions included in Standard Analysis vis-à-vis additional strategies Truth table (Stokke 2004) Truth table with logical contradictions and hidden necessary condition Test of necessity, outcome Y Crisp-set membership scores (Vis 2009) Truth table, outcome U (Vis 2009) Simultaneous consistent subset relation of X with both Y and ~Y Simultaneous inconsistent subset relation of X with both Y and ~Y Consistency of truth table rows for outcome and its complement Consistency, PRI, and PRODUCT for simultaneous subset relation Synopsis of software packages for performing set-theoretic analyses Intersections of theory (T) and solution term (S) with types of cases Post-QCA case selection principles 311 in this web service