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Set-Theoretic Methods for the Social Sciences

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Set-Theoretic Methods for the Social Sciences
Cambridge University Press
9781107013520 - Set-Theoretic Methods for the Social Sciences - A Guide to Qualitative Comparative Analysis - By Carsten Q. Schneider and Claudius Wagemann
Frontmatter/Prelims

Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis

Qualitative Comparative Analysis (QCA) and other set-theoretic methods distinguish themselves from other approaches to the study of social phenomena by using sets and the search for set relations. In virtually all social science fields, statements about social phenomena can be framed in terms of set relations, and using set-theoretic methods to investigate these statements is therefore highly valuable. This book guides readers through the basic principles of set-theory and then on to the applied practices of QCA. It provides a thorough understanding of basic and advanced issues in set-theoretic methods together with tricks of the trade, software handling, and exercises. Most arguments are introduced using examples from existing research. The use of QCA is increasing rapidly and the application of set theory is both fruitful and still widely misunderstood in current empirical comparative social research. This book provides the comprehensive guide to these methods for researchers across the social sciences.

Carsten Q. Schneider is Associate Professor in the Department of Political Science and Founding Director of the Center for the Study of Imperfections in Democracies at Central European University, Hungary.

Claudius Wagemann is Professor of Qualitative Social Science Methods at Goethe University, Frankfurt.


Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis

Strategies for Social Inquiry

Editors

Colin Elman
Maxwell School of Syracuse University

John Gerring
Boston University

James Mahoney
Northwestern University

Editorial Board

Bear Braumoeller
David Collier
Francesco Guala
Peter Hedström
Theodore Hopf
Uskali Maki
Rose McDermott
Charles Ragin
Theda Skocpol
Peter Spiegler
David Waldner
Lisa Wedeen
Christopher Winship

This new book series presents texts on a wide range of issues bearing upon the practice of social inquiry. Strategies are construed broadly to embrace the full spectrum of approaches to analysis, as well as relevant issues in philosophy of social science.

Published Titles

John Gerring, Social Science Methodology: A Unified Framework, 2nd edition

Michael Coppedge, Democratization and Research Methods

Thad Dunning, Natural Experiments in the Social Sciences: A Design-Based Approach

Forthcoming Titles

Diana Kapiszewski, Lauren M. MacLean and Benjamin L. Read, Field Research in Political Science

Jason Seawright, Multi-Method Social Science: Combining Qualitative and Quantitative Tools


Set-Theoretic Methods for the Social Sciences

A Guide to Qualitative Comparative Analysis

Carsten Q. Schneider and Claudius Wagemann


CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City

Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org
Information on this title: www.cambridge.org/9781107601130

Translated and adapted from Qualitative Comparative Analysis (QCA) und Fuzzy Sets: Ein Lehrbuch für Anwender und alle, die es werden wollen published in German by Verlag Barbara Budrich 2007, © Verlag Barbara Budrich 2007.
First published in English by Cambridge University Press 2012 as Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis © Cambridge University Press 2012.

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

Printed and bound in the United Kingdom by the MPG Books Group

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Schneider, Carsten Q., 1972– author.
Set-theoretic methods for the social sciences : a guide to qualitative comparative analysis / Carsten Q. Schneider and Claudius Wagemann.
pages cm. – (Strategies for social inquiry)
Includes bibliographical references and index.
ISBN 978-1-107-01352-0 (hardback) – ISBN 978-1-107-60113-0 (paperback)
1. Social sciences – Comparative method. 2. Social sciences – Mathematical models. 3. Set theory. I. Wagemann, Claudius, author. II. Title.
H61.S379 2012
300.72–dc23 2012015930

ISBN 978-1-107-01352-0 Hardback
ISBN 978-1-107-60113-0 Paperback

Additional resources for this publication at www.cambridge.org/schneider-wagemann

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.


In honor of Philippe Schmitter, wise advisor, generous colleague, and good friend.

Dedicated to Sheila, Giulia, and Leo, without whom this book would have been finished much sooner.


Contents

List of figures
xii
List of tables
xiv
Acknowledgements
xvi
Introduction
1
Set-theoretic approaches in the social sciences
1
Qualitative Comparative Analysis 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
23
1.1       The notion of sets
24
1.1.1     Sets and concepts
24
1.1.2     The pros and cons of crisp sets
24
1.1.3     Properties of fuzzy sets
27
1.1.4     What fuzzy sets are not
30
1.2       The calibration of set membership
32
1.2.1     Principles of calibration
32
1.2.2     The use of quantitative scales for calibration
33
1.2.3     The “direct” and “indirect” methods of calibration
35
1.2.4     Does the choice of calibration strategy matter much?
38
1.2.5     Assessing calibration
40
2         Notions and operations in set theory
42
2.1       Conjunctions, Boolean and fuzzy multiplication, intersection, logical AND
42
2.2       Disjunctions, Boolean and fuzzy addition, union, logical OR
45
2.3       Negations, complements, logical NOT
47
2.4       Operations on complex expressions
47
2.4.1     Rules for combining logical operators
48
2.4.2     Negation, intersection, and union of complex sets
49
2.4.3     Calculating membership in complex sets
51
2.5       Relations between sets
52
2.6       Notational systems in set-theoretic methods
54
3         Set relations
56
3.1       Sufficient conditions
57
3.1.1     Crisp sets
57
3.1.2     Fuzzy sets
65
3.2       Necessary conditions
69
3.2.1     Crisp sets
69
3.2.2     Fuzzy sets
75
3.3       Causal complexity in set-theoretic methods
76
3.3.1     Defining causal complexity
78
3.3.2     INUS and SUIN conditions
79
3.3.3     The notion of asymmetry
81
3.3.4     Set-theoretic methods and standard quantitative approaches
83
4         Truth tables
91
4.1       What is a truth table?
92
4.2       How to get from a data matrix to a truth table
93
4.2.1     Crisp sets
93
4.2.2     Fuzzy sets
96
4.3       Analyzing truth tables
104
4.3.1     Matching similar conjunctions
105
4.3.2     Logically redundant prime implicants
108
4.3.3     Issues related to the analysis of the non-occurrence of the outcome
112
Part II   Neat formal logic meets noisy social science data
117
5         Parameters of fit
119
5.1       Defining and dealing with contradictory truth table rows
120
5.2       Consistency of sufficient conditions
123
5.3       Coverage of sufficient conditions
129
5.4       Consistency of necessary conditions
139
5.5       Coverage of necessary conditions
144
5.6       Issues related to consistency and coverage
148
6         Limited diversity and logical remainders
151
6.1       Limited diversity in set-theoretic methods: how to see it when it is there
152
6.2       Sources of limited diversity
153
6.2.1     Arithmetic remainders
154
6.2.2     Clustered remainders
154
6.2.3     Impossible remainders
155
6.3       What limited diversity is not
157
6.4       The Standard Analysis procedure: identifying logical remainders for crafting plausible solution terms
160
6.4.1     The dimension of set relations
161
6.4.2     The dimension of complexity
165
6.4.3     The dimension of types of counterfactuals
167
6.4.4     The Standard Analysis procedure in a nutshell
175
7         The Truth Table Algorithm
178
7.1       From the data matrix to truth table
179
7.2       Attributing an outcome value to each truth table row
182
7.3       Logically minimizing the truth table
186
7.4       Implications of the Truth Table Algorithm
190
Part III  Potential pitfalls and suggestions for solutions
195
8         Potential pitfalls in the Standard Analysis procedure and suggestions for improvement
197
8.1       Beyond the Standard Analysis: expanding the types of counterfactuals
198
8.2       The Enhanced Standard Analysis: forms of untenable assumptions and how to avoid them
200
8.2.1     Incoherent counterfactuals I: contradicting the statement of necessity
201
8.2.2     Incoherent counterfactuals II: contradictory assumptions
203
8.2.3     Implausible counterfactuals: contradicting common sense
206
8.2.4     Putting the Enhanced Standard Analysis procedure into practice
209
8.3       Theory-Guided Enhanced Standard Analysis: complementary strategies for dealing with logical remainders
211
8.3.1     Choosing entire truth table rows as good counterfactuals
212
8.3.2     Formulating conjunctural directional expectations
215
8.4       Comparing the different strategies for the treatment of logical remainders
217
9         Potential pitfalls in the analysis of necessity and sufficiency and suggestions for avoiding them
220
9.1       Pitfalls in inferring necessity from sufficiency solution terms
221
9.1.1     Hidden necessary conditions
221
9.1.2     The appearance of false necessary conditions
227
9.2       The analytic consequences of skewed set-membership scores
232
9.2.1     The coverage of necessary conditions and the problem of trivialness
233
9.2.2     The consistency of sufficient conditions and the problem of simultaneous subset relations
237
9.2.3     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
251
10        Variants of QCA
253
10.1      The two-step approach
253
10.2      Multi-value QCA
255
10.2.1    Principles of mvQCA: notation and logical minimization
256
10.2.2    An assessment of mvQCA
258
10.3      Set-theoretic methods and time
263
10.3.1    Forms of causally relevant notions of time
264
10.3.2    Informal ways of integrating notions of time into set-theoretic methods
265
10.3.3    Sequence elaboration
266
10.3.4    Temporal QCA
269
11        Data analysis technique meets set-theoretic approach
275
11.1      Recipe for a good QCA
275
11.1.1    The appropriateness of set-theoretic methods
276
11.1.2    The choice of the conditions and the outcome
276
11.1.3    The choice of the QCA variant
277
11.1.4    Calibration of set-membership scores
277
11.1.5    Analysis of necessary conditions
278
11.1.6    Analysis of sufficient conditions
278
11.1.7    Presentation of results
280
11.1.8    Interpretation of results
280
11.1.9    Reiteration of the research cycle
281
11.1.10   The use of software
282
11.2      Robustness and uncertainty in QCA
284
11.2.1    How do we see robustness in set-theoretic methods when it is there?
285
11.2.2    The effects of changing calibration
287
11.2.3    The effects of changing consistency levels
291
11.2.4    The effect of dropping or adding cases
293
11.3      The evaluation of theories in set-theoretic methods
295
11.3.1    Why standard hypothesis testing does not fit into set-theoretic methods
296
11.3.2    The basics of theory evaluation in set-theoretic methods
297
11.3.3    Extending theory evaluation by integrating consistency and coverage
300
11.3.4    Summarizing set-theoretic theory evaluation
304
11.4      Set-theoretic methods and case selection
305
11.4.1    Types of cases after a QCA
306
11.4.2    Forms and aims of (comparative) within-case studies after a QCA
308
11.4.3    Post-QCA case selection principles
310
12        Looking back, looking ahead
313
12.1      Looking back: the main topics of this book
313
12.2      Myths and misunderstandings
316
12.3      Looking ahead: tasks and developments in the coming years
318
Glossary
322
Bibliography
336
Index
346



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