Hi Roel,

Apologies for this late reply, I rarely check my RG messages and Gmail did not let me know, for some reason.

Not sure I correctly follow the notation. Y

Hi Adrian,

Many thanks for your reply. I’m sorry I confused you by suggesting I’m referring to necessity. My question is much simpeler. The PRI as we know it, checks for X being simultaneously a subset of Y and ~Y. However, it is also possible in fuzzy sets that both X and ~X are a subset of Y. As far as I know, we don’t have a metric to check for this. But, following the PRI formula, the metric to check for both X and ~X being a subset of Y may be:

[min(XY)-min( X~XY)] / [X-min(X~XY)]

Does that make sense?

I see now.

But in this case, I don'treally see what the problem ia because it is not surprising for both X and not X to be a subset of Y.

Imagine a small set X completely included in a veey large Y. There is plenty of space outside X (that is, not X) which is included in Y. That is only natural, so what exactly are we going to measure/capture with this new PRI...?

Dear Roel,

first, X -> Y being true and X -> ~Y being true is not a logical contradiction. A logical contradiction is a proposition for which it is impossible to assign truth values to its constituent variables that make the proposition true. That you can look up in any intro textbook on logic (e.g., Bergman 2008, p.15). A contradiction is the opposite of a tautology, for the latter of which it is impossible to assign truth values to its constituent variables that make the proposition false. Since X -> Y and X -> ~Y are true for X = 0, (X -> Y)*(X -> ~Y) is a proposition that is true for X = 0, and thus cannot be a logical contradiction.

Second, that paradoxical subset relations in fsQCA can exist for both sufficiency and necessity relations has already been shown ten years ago in Thiem and Dusa (2013, p.64), Figure 4.3. See, in particular, Equations 4.10 and 4.13 on p. 64 and p.69, respectively, on dedicated PRI measures.

Third, if you are worried about simultaneous subset relations, which is not a bad idea when using a problematic method such as fsQCA, why don't you straightforwardly compute inclusion / consistency scores for X -> Y and X -> ~Y, X -> Y and ~X -> Y, respectively, instead of trying to invent formulae that may even miss what you want to capture (as an aggregate measure, PRI breaks down in some cases!)? The fuzzy package for Stata by Longest and Vaisey (2008), which is unfortunately not maintained any longer I believe, does it just like that by computing both consistency scores simultaneously. In particular, see p.95 in Longest and Vaisey (2008).

Best wishes

Alrik

References

• Bergmann, Merrie. 2008. An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems. Cambridge: Cambridge University Press.
• Longest, Kyle C., and Stephen Vaisey. 2008. "fuzzy: A program for Performing Qualitative Comparative Analyses (QCA) in Stata." Stata Journal 8 (1):79-104.
• Thiem, Alrik, and Adrian Duşa. 2013. Qualitative Comparative Analysis with R: A User's Guide. New York: Springer.

In Qualitative Comparative Analysis (QCA), a subset relationship occurs when the presence of one set of conditions (subset) is always associated with the presence of another set of conditions (superset).

Simultaneous subset relationships occur when two or more subsets of conditions are always associated with the presence of a superset of conditions. In other words, the presence of both subsets is necessary to produce the outcome associated with the superset.

For example, suppose we are studying the conditions that lead to successful organizational innovation. We find that the presence of a supportive organizational culture (subset A) and access to financial resources (subset B) are both necessary to achieve successful innovation (superset C). If we find that the presence of subset A alone or subset B alone is not always associated with successful innovation, then we have a simultaneous subset relationship between subsets A and B and superset C.

Simultaneous subset relationships are

A logical contradiction is a situation where two or more statements, when taken together, are mutually exclusive and cannot both be true at the same time. In other words, a logical contradiction occurs when the premises of an argument lead to a conclusion that contradicts one of the premises.

I would encourage readers to simply ignore self-referencing advice and instead point to the fundamentals, one being "Introduction to Logic" by Irving M. Copi, Carl Cohen, and Kenneth McMahon. According to this book, a logical contradiction occurs when "two statements have opposite truth values; that is, when one is true, the other is false, and vice versa" (Copi et al., 2017, p. 34).

Reference

Copi, I. M., Cohen, C., & McMahon, K. (2017). Introduction to logic (15th ed.). Routledge.

Hi Roel

The situation you refer to can occur when Y is (almost) a constant and as such it can be a consistent superset of both X and ~X. If this is what you mean, I believe this is addressed by the Relevance of Necessity (RoN) score.

Thanks all for your replies. I’m still concerned that: if studying is sufficient for passing the exam - as in substantively sufficient, providing a plausible explanation for passing the exam. Then ~studying cannot also be sufficient for (i.e., substantively explain) passing the exam. Even though this is mathematically possible in fsQCA. I believe that violates the (Aristotle’s) principle of non-contradiction. In any case, it makes no sense at all. Since this concerns sufficiency, the RoN to me does not seem an obvious metric to check for X and ~X both being sufficient for Y. That’s why I still think the metric I suggested may be useful. Any further comments, thoughts are appreciated. I’ll also ask our colleagues at the AQCA conference this week.

Cheers,

Roel.

Dear Roel, if interested in logic, I would absolutely recommend asking an analytical philosopher trained in logic, and not some social scientist at some QCA conference. You wouldn't call a tiler for fixing your TV, either. When in 2013 I asked the audience at the QCA conference in Zurich about the definition of the Boolean conditional (->) to check whether we were on the same page before continuing the discussion, only one doctoral student knew under which constellation a Boolean conditional is false (when the antecedent is true and the consequent false). That was as revealing as it was shocking to me. Ten years later, nothing much seems to have changed, though.

Above, Adrian Dusa says "a logical contradiction occurs when the premises of an argument lead to a conclusion that contradicts one of the premises." That is a vacuous statement because it is circular. You can't define contradiction by saying that it occurs when something contradicts something else. Equally, you wouldn't find it useful to say that a second is defined as the time it takes for the sixtieth part of a minute to pass. Just as Bergman (2008, p.15), whom I've already referenced above, Copi et al. (2007, p.215) provide a proper definition: "A contradiction is a statement that is false in virtue of its form. There is no assignment of truth values to the simple component propositions of a contradiction which will make the statement true."

Thus, again, a contradiction simply is a statement that can never be true, such as, for instance, p * ~p (read: p and not-p), which is always false irrespective of whether p is true or false. Thus, I believe that what you actually want to say is not that (X -> Y) * (X -> ~Y) is a contradiction, which it clearly is not, but that

(X -> Y) * ~(X -> Y)

is a contradiction. As you can easily check for yourself when setting up a small truth table, every possible assignment of truth values to X and Y will make the proposition false. This turns your statement about studying and passing exams into a slightly different form: if studying is sufficient for passing the exam, and if it is not the case that studying is sufficient for passing the exam, then we have a contradiction.

Small addition: note that (X -> Y) * (~X -> Y) is no contradiction, either, because the proposition is true if both X and Y are true or if X is false and Y is true.

Best wishes

Alrik

References

• Bergmann, Merrie. 2008. An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems. Cambridge: Cambridge University Press.
• Copi, Irving M., Carl Cohen, and Daniel E. Flage. 2007. Essentials of Logic. 2nd ed. Upper Saddle River: Pearson/Prentice Hall.

Dear Aurik,

You hit the target but miss the point. QCA is a qualitative method. Hence there is an important difference between mathematical and substantive sufficiency. Published QCA papers are often flawed not because authors got the “technicalities” wrong but because they failed to interpret them into plausible explanations. And that’s the problem with X and ~X both being mathematically sufficient for Y. I cannot think of an example in social reality where X is a plausible explanation of Y, but also ~X. Let me know if you have one. QCA being a qualitative method, formal logic, counterfactual analysis and regularity always come second to knowledge of cases, context and concepts.

Dear Roel,

permanent insistence that QCA is absolutely special and doesn't have to respect any formal logic or follow any established mathematical (Boolean algebra) or philosophical (INUS causation) tradition may provide the small QCA community with a means to immunize itself against attempts of serious methodological evaluation, but that will not help in the long run.

The big mistake you and many QCA people make is that you equate logical sufficiency with causation, thus your gut feeling that something is not quite right with logical sufficiency, and hence the ad hoc introduction of QCA-exclusive ideas such as "substantive" or "robust" sufficiency.

However, only some forms of logical sufficiency reflect evidence for causal relations, namely irredundant forms, as Mackie (1965) had already shown. Now, neither (X -> Y) * (~X -> Y), nor X + ~X -> Y for that matter, are logical contradictions, no matter how far you bend Boolean logic. Yet, X + ~X -> Y is not an irredundant relation of sufficiency because X + ~X is a tautology. Boolean optimization proper thus never outputs X + ~X -> Y.

The obvious mismatch between this insight and the fact that something like

X + ~X -> Y is still mathematically possible in fsQCA only stems from the fact that fuzzy sets are not compatible with Boolean optimization. Yet, fsQCA still operates with Boolean truth tables and (Boolean) Quine-McCluskey optimization. Mathematically and substantively, you should thus ask yourself what is wrong with fsQCA, and not with the statement "studying and not studying are sufficient for passing the exam".

Have a good day

Alrik

References

• Mackie, John L. 1965. "Causes and Conditions." American Philosophical Quarterly 2 (4):245-64.

QCA is perfectly fine. Wrong is Alrik Thiem's understanding and his constant misinterpretation of QCA. That form of logic called material implication is just not compatible with QCA, and I have already shown why in my SMR reply.

QCA applies Boolean minimization, so it is obvious fuzzy sets have no play here because the word "Boolean" is tightly linked with the word "crisp". I recommend Mr. Thiem to properly study Charles Ragin's "From fuzzy sets to crisp truth tables" and maybe, just maybe he will understand what QCA is.

Adrian, ResearchGate is meant for scientific exchange, not bouts of emotions or insults. As long as you respect that, we can continue discussing here. If you can't handle critical questions on QCA in that way, you should not use this forum.

It would first help the debate if you could properly cite what you are referring to. If you mean your article "Critical Tension" (Dusa 2022), that one has been thoroughly debunked by Baumgartner (2022). I'm also not sure how that exactly relates to the content here.

In any case, if I sum up what you and Roel have said, I take away two main points:

Is that correct? If yes, it perhaps makes no sense to continue the debate here. You should first clarify at your next QCA meeting what it actually is that QCA is or does. For instance, Ragin (1987, chapter 6) and Schneider and Wagemann (2012, chapter 2) claim that QCA uses Boolean algebra, and Ragin couldn't be clearer in saying that QCA is a method for detecting INUS causation (Ragin and Strand 2008, 431-432). 39 years after QCA's introduction in Ragin et al. (1984), it would thus be high time to come up with some agreement amongst yourself that would help avoid having to always come back to such basic questions. As you notice, it makes it very difficult, if not impossible, to move forward in any sensible way.

Have a good day

Alrik

References

• Baumgartner, Michael. 2022. "Qualitative Comparative Analysis and robust sufficiency." Quality & Quantity 56 (4):1939-63.
• Dușa, Adrian. 2022. "Critical Tension: Sufficiency and Parsimony in QCA." Sociological Methods & Research 51 (2):541-65.
• Ragin, Charles C. 1987. The Comparative Method: Moving beyond Qualitative and Quantitative Strategies. Berkeley: University of California Press.
• Ragin, Charles C., and Sarah Ilene Strand. 2008. "Using Qualitative Comparative Analysis to Study Causal Order: Comment on Caren and Panofsky (2005)." Sociological Methods & Research 36 (4):431-41.
• Ragin, Charles C., Susan E. Mayer, and Kriss A. Drass. 1984. "Assessing Discrimination: A Boolean Approach." American Sociological Review 49 (2):221-34.
• Schneider, Carsten Q., and Claudius Wagemann. 2012. Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis (QCA). Cambridge: Cambridge University Press.

There is no point in continuing an argument with someone who is constantly tweaking the very basic fundamentals of logics. If "QCA does not use Boolean algebra", I would invite you to write a formal paper to demonstrate that (if you can). When you've done it, I will grant you my attention, otherwise throwing unfounded mumble-jumble "logical gymnastics" type of arguments on RG is not a proof of that happening.

Not to let that pass without a response, you might imagine I am writing a formal debunk to Michael's debunk. It is soon to be submitted, I am just not feeling rushed in any ways.

I never said that QCA does not use Boolean algebra. It does. I never said that QCA does not try to identify INUS-condition. It does. I never said or suggested anything like it and it pisses me off that you claim I did, because it means you’re not engaging seriously with my arguments.

Michael never debunked anything. All he did is to deliberately misunderstand QCA for years. The distinction between mathematical sufficiency and robust or substantive consistency is been part and parcel of QCA from the day Charles Ragin developed it.

Hi Roel, what I wanted to say if you exchange X and y in the RoN formula, it might capture just what you mean. Your proposed formula might well do a better job, but it might be interesting to compare. Good luck with this interesting endeavour, it's good to see such active and productive QCA scholars.

Hi

This is addressed in Oana et al 2019: 66. PRI deals with simultaneous subset relations for sufficient condition. There is no such relations for necessity.

Dear Roel,

let's stay with scientific arguments. This is what this forum is for.

Ok, so you say QCA uses Boolean algebra, but you reject the Boolean-algebraic implication / conditional / sufficiency operator (->). However, since this operator is a non-fundamental operator in Boolean algebra, which is defined in terms of "and" (*), "or" (+) and "not" (~) (see any intro book to Boolean algebra, such as Hohn, 1966, chapter 3), you should also reject *, +, and ~. Is that correct? If not, please clarify how you reconcile rejecting "->", but accepting "*", "+" and "~". I'm losing you here, and every analytical philosopher properly trained in Boolean logic would as well.

Then you say QCA searches for INUS structures. Ok, fine, I'm with you here as you know. But why do many others in the QCA community then deny this? When I browse through QCA books and articles (except those written by myself, Michael Baumgartner and our colleagues), virtually all say that QCA looks for necessary and sufficient conditions. No mention of INUS causation - let alone an explanation of what INUS conditions actually are - at all.

Would you be so kind as to reply only to those two issues? Many thanks.

Best wishes

Alrik

References

Hohn, Franz E. 1966. Applied Boolean Algebra: An Elementary Introduction. 2nd ed. New York: Macmillan.

Dear both,

I find it fascinating to observe the interplay between qualitative and quantitative sensibilities. However, I would like to pose a question to Roel: If QCA is an interpretive method, why do data patterns matter to you? FYI some of your questions might already find a solution here: Veri, F. (2019). Aggregation bias and ambivalent cases: A new parameter of consistency to understand the significance of set-theoretic sufficiency in fsQCA. Comparative Sociology, 18(2), 229-255.

Hi Francesco, I’m interested in patterns for the same reason as Charles Ragin. Conditions describe cases and configurations describe cases in complex ways. From configurations you interpret causal explanations. You know Critical Realism and you know my paper on “Uncertainty, possibility and causal power in QCA”. Patterns are epistemically relativist statements about our knowledge of the world. We cannot mistake them for ontological statements about the nature of the world. The step from patterns (epistemology) to causal claims (ontology) goes via interpretation: What does the pattern mean in the light of our knowledge of cases, context and concepts.

This article might provide some insights into the issue discussed hereArticle Two-sample test for ambivalent subset relationship in fuzzy ...