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Three-valued modal logic for theory construction with contradictory data

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Abstract

This article refers to theory construction on the basis of binary data, where a configuration of several yes/no-variables is used in order to explain a binary outcome. The methodology proposed for this purpose has been developed by the political scientist Charles Ragin and is known under the name qualitative comparative analysis. It is frequently confronted with situations, where the same configuration of explanatory variables has contradictory yes/no-values for the outcome-variable of different related cases. Since the standard-solutions to this problem are not always satisfactory, the author proposes the use of three-valued modal logic, which was originally developed by the Polish philosopher Jan Lukasiewicz. In this logic there is a third truth-value indeterminate, which serves to code contradictory or missing data. Moreover, by the proposed use of modal operators it becomes possible to differentiate between strict and possible triggers and inhibitors of a given outcome. Since indeterminate outcomes do not really contribute to knowledge accumulation and theory building, the article discusses strategies for promoting scientific progress by minimizing this indeterminacy.

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Fig. 1

Index I = (% of configurations with Y' = i due to missing instantiations) + (% of configurations with Y' = i due to contradictions)

Fig. 2

Index I = (% of configurations with Y' = i due to missing instantiations) + (% of configurations with Y' = i due to contradictions)

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Notes

  1. This contribution is a modified version of a working paper of the author, which appeared as Mueller (2014).

  2. For logical symbols in the next and all following formulas see the Glossary of logical symbols and expressions at the end of this contribution.

  3. fs/QCA is not only for fuzzy-set QCA but still contains modules for doing crisp-set QCA.

  4. (¬W AND C) OR (W AND¬C) OR (W AND C) = (¬W AND C) OR [W AND (¬C OR C)] = (¬W AND C) OR W = (¬W OR W) AND (C OR W) = (C OR W) = W OR C\(==\!>\)D.

  5. This principle does not hold for the three-valued implication \(==\!>\), which Lukasiewicz (1970) intentionally defined in such a way that i\(==\!>\)i is true. However, this irregularity is not relevant for the present article, since it does not make use of the implication i \(==\!>\) i.

  6. The details of the derivation are as follows: (¬W AND C) OR (W AND¬C) OR (W AND C) = [(¬W AND C) OR (W AND C)] OR [(W AND ¬C) OR (W AND C)] = [(¬W OR W) AND C)] OR [W AND (¬C OR C)] = C OR W = W OR C\(==\!>\)POS(D′).

  7. The details of the derivation are as follows: (¬W AND¬C) OR (¬W AND C) OR (W AND¬C) = [(¬W AND ¬C) OR (¬W AND C)] OR [(W AND ¬C) OR (¬W AND ¬C)] = [¬W AND (¬C OR C)] OR [(W OR ¬W) AND ¬C)] = ¬W OR¬C \(==\!>\)POS(¬D′).

  8. By the use of the original Y instead of the recoded Y′, we return to the language of theory building and leave the sphere of operationalizations, which is only needed at an intermediate stage of the analysis.

  9. Derivation of Eq. (10b) from (10a): (¬w AND c) OR (w AND ¬c) OR (w AND c) = (¬w AND c) OR (w AND (¬c OR c)) = (¬w AND c) OR (w) = w OR (¬w AND c) = (w OR ¬w) AND (w OR c) = (w OR c) \(==\!>\) POS(h′). Consequently, by definition: w OR ch.

  10. Derivation of Eq. (12): (¬W AND C AND B) OR (W AND C AND ¬B) OR (W AND C AND B) = [(¬W AND C AND B) OR (W AND C AND B)] OR [(W AND C AND ¬B) OR (W AND C AND B)] = (C AND B) OR (W AND C) = (W AND C) OR (C AND B) —> D.

  11. Derivation of Eq. (13): (¬W AND C AND B) OR (W AND ¬C AND ¬B) OR (W AND C AND ¬B) OR (W AND C AND B) = [(¬W AND C AND B) OR (W AND C AND B)] OR [(W AND ¬C AND ¬B) OR (W AND C AND ¬B)] = (C AND B) OR (W AND ¬B) = (W AND¬B) OR (C AND B)D.

References

  • Beall, J.C., van Fraassen, B.C.: Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic, chap. 5. Oxford University Press, Oxford (2003)

    Google Scholar 

  • Bergmann, M.: An Introduction to Many-Valued and Fuzzy Logic. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  • Charmaz, K.: Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis. Sage, London (2006)

    Google Scholar 

  • Drass, K.A., Ragin, Ch.: fs/QCA Software, Release 2.0. http://www.u.arizona.edu/~cragin/fsQCA/software.shtml (2013). Accessed June 11, 2013

  • Kangas, O.: The politics of social security: on regressions, qualitative comparisons, and cluster analysis. In: Janoski, T., Hicks, A. (eds.) The Comparative Political Economy of the Welfare State, chap. 14. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  • Kuhn, Th: The Structure of Scientific Revolutions. University of Chicago Press, Chicago (1962)

    Google Scholar 

  • Lukasiewicz, J.: Selected Works (edited by L. Borkowski). North-Holland, Amsterdam (1970) [1920]

  • Mendelson, E.: Boolean Algebra and Switching Circuits. McGraw-Hill, New York (1970)

    Google Scholar 

  • Mueller, G.P.: Three-Valued Modal Logic for Reconstructing the Semantic Network Structure of a Corpus of Coded Texts. In: Friemel, Th. (ed.) Why Context Matters, pp. 37–54. VS Verlag, Wiesbaden (2008)

    Chapter  Google Scholar 

  • Mueller, G.P.: Three-Valued Modal Logic for Qualitative Comparative Policy Analysis with Crisp-Set QCA (= Working Papers SES 450). Faculty of Economics and Social Sciences at the University of Fribourg, Fribourg (2014)

  • Muzio, J.C., Wesselkamper, T.C.: Multiple-Valued Switching Theory. Adam Hilger, Bristol (1986)

    Google Scholar 

  • Popper, K.: The Logic of Scientific Discovery. Hutchinson, London (1959) [1934]

  • Ragin, Ch.: The Comparative Method: Moving Beyond Qualitative and Quantitative Strategies. University of California Press, Berkeley (1989)

    Google Scholar 

  • Ragin, Ch.: Using qualitative comparative analysis to study configurations. In: Kelle, U. (ed.) Computer-Aided Qualitative Data Analysis: Theory, Methods and Practice, chap. 13. Sage, London (1998)

    Google Scholar 

  • Rescher, N.: Many-valued Logic. McGraw-Hill, New York (1969)

    Google Scholar 

  • Rihoux, B., De Meur, G.: Crisp-set qualitative comparative analysis. In: Rihoux, B., Ragin, Ch. (eds.) Configurational Comparative Methods, chap. 3. Sage, Los Angeles (2009)

    Google Scholar 

  • Schneider, C., Wagemann, C.: Set-Theoretic Methods for the Social Sciences. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  • Strauss, A., Corbin, J.: Basics of Qualitative Research: Techniques and Procedures for Developing Grounded Theory. Sage, Thousand Oaks (1998)

    Google Scholar 

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  1. Faculty of Economics and Social Sciences, University of Fribourg, Blvd de Pérolles 90, 1700, Fribourg, Switzerland

    Georg P. Mueller

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  1. Georg P. Mueller
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Correspondence to Georg P. Mueller.

Glossary of logical symbols and expressions

f

False, also denoted by 0

t

True, also denoted by 1

i

Indeterminate truth in three-valued logic

X AND Y

Boolean conjunction of X and Y. For definition in three-valued logic see Table 5

X OR Y

Boolean disjunction of X and Y. For definition in three-valued logic see Table 5

¬X

Boolean negation of X. For definition in three-valued logic see Table 5

X \(==\!>\) Y

Boolean implication from X. For definition in three-valued logic see Table 5

NEC(X)

Necessity of X in modal logic. For definition see Table 6

POS(X)

Possibility of X in modal logic. For definition see Table 6

X Y

Possible triggering of Y by X

X Y

Possible inhibition of Y by X

X \({\hbox{---}}\!\!>\) Y

Strict triggering of Y by X

X Y

Strict inhibition of Y by X

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Mueller, G.P. Three-valued modal logic for theory construction with contradictory data. Qual Quant 53, 775–789 (2019). https://doi.org/10.1007/s11135-018-0788-4

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