Is The Notion of Conjunctive Fallacy ("Conjunction Fallacy") Based on Fallacy?
Some Notes on Supposed Irrationalities in Human Reasoning

©2011 Edward G. Rozycki, Ed. D.

RETURN
edited 10/31/19
with addendum*

Our problem is to retain what is useful and valid in intuitive judgment while correcting the errors and biases to which it is prone. -- Kahnemann, et al, (1982) p. 98 [1]

Introduction

The existence of a "conjunctive fallacy", or "conjunction fallacy," as it is also called, is widely taken to establish that human beings are less than rational. Using set theory as a gold standard, Kahnemann and Tversky report certain inconsistencies in the ways people understand likelihoods in practical situations with what the probability calculus is interpreted to indicate should be the correct judgments.

In this paper I argue

a. that their argument from the set theory axiom, p(A) ≥ p(A∩B) is a non-sequitur. It remains to be proven whether conjunctive "problems" are anything more than a Procrustean distortion of intuition.

b. That, in natural discourse, understanding two categories as contrastive permits a different set theoretic interpretation that violates no axioms.

c. A fortiori, when natural language categories, e.g. "bank teller" and "feminist," are elaborated as in a manner that makes them distinct from set-theoretic entities, it is unlikely that the probability calculus can be unproblematically applied to them.

Who Has The Superior Rationality?

We meet our friend, Amos, for lunch, and, having rendered homage to our growling stomachs, we pose him flat out, without any lead in or preparation, the following question: Which of these two statements is the most probable:

a. Linda is a bank teller.

b. Linda is a bank teller and a feminist?

He replies that statement a is the more probable since there is a basic theorem of set theory, easily proven, that holds that the probability of a set A is greater than or equal to the probability of A's intersection with B, a second set: p(A) ≥ p(A∩B). So Linda's being a bank teller, alone, is more probable than her being both a bank teller and a feminist.

But has Amos been posed a trick question? Are these all the alternatives? Has he fallen into a trap because he in an important sense "knows too much?" Why does he think set theory is relevant here? Suppose we had asked, not "Which is more probable?" but "Which is more likely?"

That evening at dinner we meet with our friend, Daniel. We want to ask him the question we posed to Amos at lunch, but this time we preface it with a short narrative, as follows:

Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations. [2]

We ask Daniel, "Which of these two statements is the most probable:

a. Linda is a bank teller.

b. Linda is a bank teller and a feminist?

Daniel replies, "Why that's remarkable! You described my own niece, Linda, almost to a T. She majored in philosophy at a small, religious college, was very active publicly in social issues and actually boarded an atomic submarine to break a bottle of blood on it. After she graduated she considered entering a religious order, but fell in love with an art student. She took the job at the bank because it enables her and her husband to make ends meet and, although boring, working in a bank is not too taxing so she has the energy to spend a couple of evenings a week participating in groups that interest her. Statement b, she is a bank teller and a feminist. Not only is that more probable, it is almost 100% certain."

But what might Amos or Daniel have responded, had we asked them which of the following was the correct response to the question Which alternative is more probable?:

a. It is more probable that Linda is a bank teller.

b. It is more probable that Linda is both a bank teller and a feminist.

c. We cannot tell which of a or b is more probable, since there is no reason to rule out that the set of bank tellers is not the same as the set of feminists, except on evidential grounds pertinent to very specific situations. "Logic" does not demand it.

Who Is The More Logical?

Daniel is on reasonably firmer ground. Amos' argument lacks a crucial premise; one that is empirical in nature, rather than conceptual. Daniel's reasons have empirical grounding to connect Linda, his niece's avatar in the argument, to Daniel's own conclusion.

Of the choices offered immediately above, i.e. it is false both that it is more probable that a. Linda is a bank teller and b. Linda is both a bank teller and a feminist. c. is correct, so long as we are looking for purely logical connection with the probability axiom.

Amos' argument invokes a fact about set theory, p(A) ≥ p(A ∩ B). But this theorem is a conjunction: it states not only that p(A) is greater than p(A ∩ B) but also that p(A) may be equal to p(A ∩ B). In order to logically establish that p(A) is the greater, a secondary premise is required: one that establishes that A ≠A∩B. But such a premise would need to be extraneously introduced. It does not follow automatically from any facts about set theory. It could merely be an empirical fact specific to a given situation.

However, if all the bank tellers in, say, the single bank in this small town are feminists, because the bank president, being a feminist biases his hiring in favor of them, then, indeed, A = B which makes A = A ∩ B. What Amos needs is some fact about the world that would disestablish this possibility. He mentions none.

(He might have said something like, "There are far more bank tellers than feminists in the world, so it is reasonable to suppose that the probability encountering a bank teller is higher than that of encountering a feminist." However, a disagreement with him on this empirical assertion does not reflect on anyone's rationality, but on each person's information and the restrictions he or she places on it, e.g. where, in particular, is one counting the numbers of bank tellers and feminists.)[3]

This is the problem. Amos was posed a "problem" which is all too comfortably interpreted as a logic problem. Thus it becomes an issue of what is a bank teller vs. what a feminist is, in general. Outside a context of specific purposes to be serve, it is far from clear how this might, or whether it can, be settled. By playing the logic game, Amos has no empirical recourse to developing connecting premises to reject A =(A∩B)

What about the meanings of the words "bank teller" and "feminist?" They have criteria of identification that seem to be logically independent of one another. That is, you can identify bank tellers without knowing they are feminists, and vice versa.

Yes, but on the basis of those criteria alone, you can't rule out the possibility that there are subsets of A that are equal to subsets of B. Besides there is sufficient controversy as to what "feminist" means to rely too heavily on some kind of standardized meaning. In some American communities, any woman who steps out of the arenas of Kitchen, Church and Children is considered to be a feminist (even, say, Phyllis Schlafly). Concepts in natural languages seldom, if ever, are clearly defined simple entitities with clear-cut borders. That is why judges, and lawyers and interpreters and scholars of all kinds have a function to perform.

The critical claim, repeated in innumerable places, has been this: not to move easily from p(A) ≥ p(A∩B) to rejecting the possibility that linda could be both teller and feminist, is to make a logical error -- to evidence irrationality, even. That claim, despite insistent repetition, has not been established by valid logical argument.

To reiterate: it does not follow because p(A) ≥ p(A∩B), that p(A) > p(A∩B)

What About Daniel?

Has Daniel "compromised his rationality?" He may have; but that is a bit strong of a characterization, if what we mean is only that he has made a mistake. We might decide that his story provides supporting but not quite sufficient grounds for accepting his judgment. But then, he is probably not making a judgment about the relationship the concept of "bank teller" to that of "feminist." He is making a judgment about Linda. And if we think evidence is missing we can asking him for the missing information. He can build a reasonable, if inelegant, argument to support his specific claims.

It's important here to recall the distinction between a sound argument, which is a valid argument with true premises, and a valid argument, which has any of certain standardly recognized structures, e.g. modus ponens for syllogisms, etc. The claim being argued in this essay is that Amos' argument is invalid; Daniel's is likely valid (or can approach validity) even though it may be unsound, i.e. contain some false premises, or needs additional ones to make connections.

A person is not necessarily exhibiting irrationality solely because he or she is uninformed or misinformed or has trouble understanding the problem.[4] As the distinction between soundness and validity indicates, truth, logic and rationality may be only haphazardly related.

Another Way of Understanding Linda's Status

Natural language categories almost always represent intersections of "simpler" categories, especially if the identification of their members involves multiple criteria. To be a bank teller is to meet multiple criteria; similarly, to be a feminist is to meet multiple criteria. So being nothing more than a bank teller is not, in natural language, logically the same as being a bank teller.

Let's reconsider:

a. Linda is a bank teller.
b. Linda is a bank teller and is active in the feminist movement.

Because b. offers a contrast not explicitly articulated in a., we can , following natural usage, expand a. and get a better sense of comparison:

a. Linda is only bank teller (and nothing else important, i.e. not a feminist, nor a Presbyterian, nor a city council member, nor a heiress, nor a champion runner, nor a … )

b. Linda is a bank teller and is active in the feminist movement.

But the p(a) can now be interpreted as a more highly restricted

p(a) ≡ p{a∩x'∩y'∩z'...}

(where Z' is read "not-Z", where A = bank teller, X = feminist, Y= city council member, Z = Ph.D., ... etc.) which, depending upon the numbers in each category, is very much more likely to be less than

so that: p(b) might be greater than (or equal to) p(a). That is: it is possibly more likely (probable) that Linda is both a bank teller and active in the feminist movement than that she is a bank teller and nothing else of importance.

Natural Categories vs Set-Theoretic Categories

To reiterate: Natural language categories almost always represent intersections of "simpler" categories, especially if the identification of their members involves multiple criteria. These criteria may be shared among categories; they would be represented by overlaps in the sets represented in Venn or Euler diagrams.

The criteria are of different subtly interacting types, observable, contextual, conventional, defeasible, etc. which complicate analysis substantially. In addition, there are many "assumed" criteria, logical substrates, that only reveal themselves in a forensic context, being "taken for granted" in normal discourse. This again complicates matters, especially when it comes to reasoning on the fly.[5]

Elementary set theory, most commonly used, is based on a binary axiom "a or not-a" so that a single criterion, call it, say, x, bifurcates the universe of discourse into two sets, A, whose members meet x, and not-A, whose members do not meet x. If necessary conditions for being an A are also sufficient for being an A, then when a necessary condition is not met, we have a case of a not-A.

Natural language categories are more "porous," so to speak. A grandfather clock that has lost its pendulum is not a not-clock. It is a special "kind" of clock, a damaged clock. By converse example, a road-killed racoon is not a not-racoon. Natural languages have all sorts of transformational and historical indicators that justify our attributing or retaining an item's class membership despite substantial deviation from the ideal, or paradigm examplar: for example, aged, broken, copy, imposter, counterfeit, unripe, under renovation. Items can be quite amazingly deviant from paradigm yet still retain their identity, e.g. "My grandfather's old rocker is now just a pile of ashes."

If the categories we wish to probabilize in any formal sense do not constitute a partitioned set, a collection of mutually exclusive and exhaustive categories, then probabilities, which are assignments of real numbers greater than or equal to zero or less than or equal to one, cannot be coherently done. One cannot say that it is impossible to create such a partition, but it will likely take more than a casual attitude towards natural language to bring it about.[6]

ENDNOTES

[1] Daniel Kahneman, Paul Slovic, Amos Tversky (eds) Judgment under uncertainty: Heuristics and biases. (Cambridge: Cambridge U. Press, 1982.)

[2] See A. Tversky and D. Kahnemann, "On the Study of Statistical Intuitions," p. 496 in Kahnemann, (1982). I don't think I have created a straw man here. I have never, in many years, read this problem explained in a way that justifies a connection between the set theory theorem to the sole conclusion that p(A) is greater. Basically, invoking the theorem is an ignoratio elenchi.

[3] Some researchers report being unable to find evidence of the conjunction fallacy. See Michael D. Lee, Emily Grothe, Mark Steyvers. (2009) Conjunction and Disjunction Fallacies in Prediction Markets. 

Also, see G Charness, E Karni & D Levin (2009) On the Conjunction Fallacy in Probability Judgment: New Experimental Evidence Regarding Linda.

See, also, A Sides, D Osherson, N Bonini, & R Viale, "On the Reality of the Conjunction Fallacy," (2002) Memory & Cognition, 30 (2) 191-198.

[4] Before we attack rationality, there are many other factors to consider in evaluating responses. See Allyson L. Holbrook, Jon A. Krosnick, David Moore, Roger Tourangeau. (2007) Response Order Effects in Dichotomous Categorical Questions Presented Orally: the Impact of Question and Respondent Attributes. Public Opinion Quarterly, Vol. 71, No. 3. pp. 1-25. 

[5] For an analysis of deviance-classes as recognition-equivalence classes generated by such transformational or historical indicators, see E. G. Rozycki, "The Things We Recognize."

[6] Clearly, "bank teller" and "feminist" are not mutually exclusive. On what basis do we assign probabilities to these categories? See E. G. Rozycki, "The Functional Analysis of Behavior: theoretical and ethical limits" for expanded parallel argument.

* I recently (2/25/12) found a strong argument supporting some of the major contentions of this paper. I had missed that article early on because I had encountered the "fallacy" by reading in economics and psychology journals where it was not mentioned. The article is by Jaacko Hintikka and is called, "A Fallacious Fallacy," in pages 25 - 35 in Synthese 140, Issue 1-2, May 2004. For download see http://www.jstor.org/pss/20118439

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