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Front. Psychol. | doi: 10.3389/fpsyg.2018.02281

Is There a Conjunction Fallacy in Legal Probabilistic Decision Making?

  • 1University of Silesia of Katowice, Poland
  • 2City University of London, United Kingdom

Text Correction
In the original article, there was an error.
In the Discussion section, a small error was made in one of the quantum computations, which requires minor adjustment of the discussion. None of the empirical results, analyses, and other conclusions are affected.
Even though the error was small, it does affect some of the discussion and *must* be corrected. We cannot have a mathematical analysis published with a mistake, which affects the end results of that analysis.
As you can see from the suggested changes, this error affects a number of changes in the Discussion.
We apologize again for the error, but it does have to be corrected. We are happy to cover any type-setting charges.
A correction has been made to Discussion:

Corrections have been made to Discussion
(highlighted text shows the corrections)

For a double CF, we have: Prob(A), Prob(B)<Prob(A&B)
And then a bit later
For the participants with no legal background, we have a situation where Prob(A), Prob(B)<Prob(A&B)
And then a bit later
The observed results motivate the consideration of an initial representation for the mental space in a tensor product structure as above, but also a thought process which ‘mixes’ thoughts and beliefs between the two crimes (Broekaert et al., 2013; Pothos & Busemeyer, 2009). It is not our purpose presently to outline in detail a full cognitive model for the consideration of criminal cases and we focus on the technical elements of QPT potentially relevantfor coverage of the results (for more relevant details see e.g. Pothos & Busemeyer, 2009, or Trueblood & Busemeyer, 2011).
And then a bit later
…Given the above specification, we have:
Prob(〖Crime1〗_yes;ψ)=〖|P_Crime1yes⨂I∙U∙ψ|〗^2=〖|(■(■(1&0@0&1)&■(0&0@0&0)@■(0&0@0&0)&■(0&0@0&0)))∙(■(■(1&0@0&0)&■(0&0@-1&0)@■(0&-1@0&0)&■(0&0@0&1)))∙(■(■(a@b)@■(c@d)))|〗^2=〖|a|〗^2+〖|c|〗^2
Prob(〖Crime2〗_yes;ψ)=〖|I⨂P_Crime2yes∙U∙ψ|〗^2=〖|a|〗^2+〖|b|〗^2
Prob(〖Crime1〗_yes&〖Crime2〗_yes;ψ)=〖|a|〗^2
Recall that a mental state vector in QPT is normalized, therefore 〖|a|〗^2+〖|b|〗^2+〖|c|〗^2+〖|d|〗^2=1. But it should be clear that this scheme still cannot accommodate a CF, which illustrates that only certain space structures can produce a single CF (e.g., as in Pothos & Busemeyer, 2009) and it is unclear whether a double CF is possible at all. [The rest of this paragraph is not relevant.]
Overall, the present results revealed a double CF, for lay (regarding legal knowledge) individuals, but not for participants with more advanced levels of legal knowledge / experience with legal proceedings. As an empirical finding, this constitutes a salutary message regarding the ability of humans to embody rational decision making, in situations where there is a high expectation for such decision making. The double CF presents a challenge for decision models specifically developed to account for the CF and related fallacies. We focussed on one model, based on QPT. So far, QPT theory for the CF has been applied to the single CF, which is by far the most common finding. Modelling of the single CF with QPT involves incompatible questions, which lead to a psychological explanation based on how one question alters our perspective for the other. Regarding the double CF, we have outlined one possibility based on QPT, corresponding to compatible questions, and a ‘mixing’ thought process; our outline was intended to simply show indicative calculations, noting that for a single CF only particular space structures will work.
Psychologically this corresponds to a consideration of the two questions in a way that thoughts making each one individually more likely interfere with each other in the conjunctive case to produce probabilities inconsistent with CPT.

Keywords: Conjunction fallacy, Legal Decision Making, Quantum Cognition, quantum probability theory, legal psychology

Received: 25 Oct 2018; Accepted: 01 Nov 2018.

Edited by:

Eldad Yechiam, Technion – Israel Institute of Technology, Israel

Copyright: © 2018 Wojciechowski and Pothos. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Bartosz W. Wojciechowski, University of Silesia of Katowice, Katowice, Poland, bartosz.wojciechowski@us.edu.pl