Conjunction fallacy

{{logic}}
The '''conjunction fallacy''' is a formal [[logical fallacy]] and a [[cognitive bias]], among other things associated with the sway of [[stereotype]]s over thought, and [[belief]] in the [[paranormal]] and in [[conspiracy theories]]. Depending on the context, it may also be referred to as the '''conjunction effect'''.

For two events ''A'' and ''B'' having independent [[Probability|probabilities]], the probability that both occur together is never larger than the probability that one of them (regardless of the other) does. This can be written as, <math>\Pr(A \land B) \leq \Pr(A)</math> and <math>\Pr(A \land B) \leq \Pr(B)</math>. Anything that contradicts this mathematical relationship is an example of the fallacy.

==Forms==
It can have several forms. The most well-known is the below.
{{syl
|1=P1|2='''A''' seems unlikely.
|3=P2|4='''B''' seems likely.
|5=C|6='''A''' and '''B''' seem more likely than '''A'''.
}}

The most well-known example of this is the "Linda problem" by {{wpl|Amos Tversky}} and {{wpl|Daniel Kahneman||,}} a question which they had people answer in their research – the majority choosing the second option.<ref name="tk81">{{cite report |last1=Tversky |first1=Amos |last2=Kahneman |first2=Daniel |year=1981 |title=Judgments of and by representativeness |  publisher  = Stanford University }}</ref><ref name="Tversky & Kahneman 1982">{{cite book |last1=Tversky |first1=A. |last2=Kahneman |first2=D. |year=1982 |chapter=Judgments of and by representativeness |editor-first=D. |editor-last=Kahneman |editor2-first=P. |editor2-last=Slovic |editor3-first=A. |editor3-last=Tversky |title=Judgment under uncertainty: Heuristics and biases |location=Cambridge, UK |publisher=Cambridge University Press |isbn=0-521-28414-7 }}</ref><ref name="tk83" />
{{quotebox|top=1|
''Linda is 31 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 [[anti-nuclear]] demonstrations.''

Which is more probable?
# Linda is a bank teller.
# Linda is a bank teller and is active in the [[feminist]] movement.}}

Variations in wording may affect how susceptible people are to this cognitive bias. Tversky and Kahneman found that changing the first alternative to ''"Linda is a bank teller whether or not she is active in the feminist movement"'' resulted in fewer making the error, though a majority of 57% still did. Studies have also varied the presentation in various ways to guard against other explanations for how people rank the options than the proposed cognitive bias.<ref name="tk83" />{{efn|Some people may read the first option as implicitly negating the conjunction added in the second option, in which case their choice of the second option is for a reason unrelated to the fallacy. To deal with this, for example, test subjects have been given lists with more options than the two of interest, including placing another option between the two, and then asked to rank them all as more or less probable.<ref name="tk83" />}}

Another fairly general form that ties the role of [[heuristic]]s or [[Intuition|intuitive]] judgement into it is the below. It captures more items from Tversky and Kahneman. For example, in 1980, respondents judged the scenario of the [[US]] breaking off diplomatic relations with the [[Soviet Union]] after the latter invades [[Poland]] more probable than merely a breaking off of US-Soviet diplomatic relations with nothing more specified. Tversky and Kahneman explain such skewed judgment in terms of a {{wpl|representativeness heuristic}} guiding the mind toward irrational conclusions.<ref name="tk81" /><ref name="tk83">{{Cite journal |last1=Tversky |first1=Amos |last2=Kahneman |first2=Daniel |date=October 1983 |title=Extension versus intuitive reasoning: The conjunction fallacy in probability judgment |journal=Psychological Review |volume=90 |issue=4 |pages=293–315 |url=http://content2.apa.org/journals/rev/90/4/293 |archive-url=https://archive.today/20130223193247/http://content2.apa.org/journals/rev/90/4/293 |url-status=dead |archive-date=2013-02-23 |doi=10.1037/0033-295X.90.4.293 }}</ref>
{{syl
|1=P1|2='''A''' seems so-and-so likely and gives some subjective impression.
|3=P2|4='''B''' seems so-and-so likely and gives some subjective impression.
|5=C|6='''A''' and '''B''' together give a balanced or more realistic impression and so seem most likely.
}}

The point here being that the subjective "realism" comes from the {{wpl|representativeness heuristic}} – which is swayed by stereotypes and by representative ideas and images which have formed through taking in all kinds of impressions, irrespective of [[rational]] evaluation.

===Without a choice of conjunction===
A conjunction effect can be seen even when a person only considers one option, rather than a choice involving having or not having the logical conjunction. It's then typically not called a fallacy but rather the "conjunction effect". In other words, there's not always any formal argument structure involved in errors of this kind, rather the intuitive impression when considering some thing is just biased. Tversky and Kahneman actually studied this first.

For example, in a variation on the "Linda problem" study, each individual test subject is given one of two lists of options, each list lacking either the main option with a conjunction or the main option without it. When ranking how probable things seem, each person only considers one of those two options of interest, and yet, the effect is shown in the relative rankings of how probable the two options are judged along with more options across the whole study.<ref name="tk83" />

===Class-inclusion problems===
A related form of faulty reasoning, shown to be common in the thought processes of small children, is the group of class-inclusion problem, going back to the child psychologist Jean Piaget. If, for instance, children are shown a picture of 7 cows and 3 horses and asked whether there are more [[cow]]s or [[animal]]s, they may reply, "more cows", although they are aware of the fact that both horses and cows are animals (this example of the error was shown to occur until around age 10). This error shows a failure to apply knowledge of hierarchies or to reason in a logical fashion about hierarchies, instead only considering one level (cow) without considering the others. It is also connected with riddles based on semantic ambiguity, such as "What barks but is not a dog?" "A bitch" (where "dog" refers ambiguously to the male of the animal or all canines); children are able to appreciate such riddles from around age 8, although even adults may not always get the answer.<ref>Politzer, Guy. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4951384/ "The class inclusion question: a case study in applying pragmatics to the experimental study of cognition"]. Springerplus. 2016; 5(1): 1133. Published online 2016 Jul 19. doi: 10.1186/s40064-016-2467-z</ref><ref>C.J. Brainerd, V.F. Reyna. [https://www.sciencedirect.com/topics/psychology/conjunction-fallacy "Advances in Child Development and Behavior"], in ''Advances in Child Development and Behavior'', 2002, included in "Conjunction Fallacy", Science Direct, accessed October 2, 2023</ref>

==In assorted woo==
There's research suggesting that people who believe in paranormal ideas are particularly susceptible to the conjunction fallacy,<ref>Rogers, P., Davis, T., & Fisk, J. (2009). [https://psycnet.apa.org/record/2009-04949-005 Paranormal belief and susceptibility to the conjunction fallacy.] ''Applied Cognitive Psychology'', 23(4), 524–542. https://doi.org/10.1002/acp.1472</ref> and that the same also goes for conspiracy theorists.<ref>Brotherton, R., & [[Chris French|French, C. C.]] (2014). [https://psycnet.apa.org/record/2014-08687-010 Belief in conspiracy theories and susceptibility to the conjunction fallacy.] ''Applied Cognitive Psychology'', 28(2), 238–248. https://doi.org/10.1002/acp.2995</ref> Intuitively, the conjunction fallacy would seem not only to make it easier to accept bogus ideas, but also to possibly promote [[crank magnetism]] by making it easier to accept more elaborate combinations of all kinds of [[crank]] ideas. We can only speculate on such matters here at [[RationalWiki]], though.

To speculate on how the conjunction fallacy may work in relation to [[woo]] teachings, it may be useful to consider which stereotypes and cliché ideas and images are popular – more generally what the {{wpl|representativeness heuristic}} may latch on to. For example, to a person who thinks the ''[[X-Files]]'' are close to [[reality]] and really wants to believe, what kinds of things may seem representative of reality? Or, to someone in a [[cult]] which teaches that suppressed knowledge is all-important for [[salvation]]? What kinds of additions to "this and …" may seem to boost how likely it all is to such people? There could possible exist interesting examples to be found in the world of [[conspirituality]].

From looking at examples of crank magnetism in action, a tongue-in-cheek suggestion for an aspiring cult leader could be, in order to suck in those predisposed to be believers in wild combinations of claims added to claims: claim a lot, at once and over time. If you assert some things to your would-be believers without [[evidence]], and they don't care much for that message, then try piling on more, not less. Conjure up emotional images, draw upon stereotypes and cognitively accessible clichés and fictional tropes, the kind of stuff people tend to intuitively view as fitting together as part of what they want to be real, and those predisposed to this fallacy may be able to swallow a lot of claims in conjunction all in one go. Of course, it can't necessarily all be chalked up to this fallacy when people do swallow such teachings, though – the [[psychology]] of [[belief]] is complicated.

==See also==
*[[Cognitive bias]]
*[[Crank magnetism]]
*[[Logical fallacy]]

==Notes==
{{notelist}}

==References==
{{reflist|2|80%}}

{{logfal}}
[[Category:Formal fallacies]]
[[Category:Cognitive biases]]

Categories: Formal fallacies, Cognitive biases