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Keren G (1997). On the calibration of probability judgments: Some critical comments and alternative perspectives. Journal of Behavioral Decision Making, 10, 269–278. doi: 10.1002/(SICI)1099-0771(199709)10:3<269::AID-BDM281>3.0.CO;2-L [ CrossRef] [ Google Scholar] Lichtenstein S, & Fischhoff B (1977). Do those who know more also know more about how much they know? Organizational Behavior and Human Performance, 20, 159–183. doi: 10.1016/0030-5073(77)90001-0 [ CrossRef] [ Google Scholar] Aczel B, Szollosi A, & Bago B (2016). Lax monitoring versus logical intuition: The determinants of confidence in conjunction fallacy. Thinking & Reasoning, 22, 99–117. doi: 10.1080/13546783.2015.1062801 [ CrossRef] [ Google Scholar] Mata A, Ferreira MB, Voss A, & Kollei T (2017). Seeing the conflict: An attentional account of reasoning errors. Psychonomic Bulletin & Review, 24, 1980–1986. doi: 10.3758/s13423-017-1234-7 [ PubMed] [ CrossRef] [ Google Scholar]

For recognition, all incorrect reasoners’ responses were included in the analyses because their answers came in the form of a multiple-choice forced response. A mixed effects logistic regression (with subject as the random variable) was again conducted due to the dichotomous dependent variable (with or without “more than”). Once again, incorrect reasoners usually recognized the standard problem, but not the control, as containing “more than” (see Table 1), with this effect of condition significant, b = 3.00, odds ratio ( OR) = 20.07, χ 2 = 50.34, p< .001, 95% confidence interval ( CI) [8.76, 45.94]. De Neys W, & Bonnefon J-F (2013). The ‘whys’ and ‘whens’ of individual differences in thinking biases. Trends in Cognitive Sciences, 17, 172–178. doi: 10.1016/j.tics.2013.02.001 [ PubMed] [ CrossRef] [ Google Scholar] For the recall response, a mixed effects logistic regression (with subject as the random variable) was conducted due to the dichotomous dependent variable (with or without “more than”), considering only those participants who wrote down an answer that could be coded as with or without the “more than” phrase (e.g., “don’t know” responses were excluded). Incorrect reasoners usually recalled the standard problem, but not the control, as containing “more than” (see Table 1), with this effect of condition significant, b = 3.02, odds ratio ( OR) = 20.44, χ 2 = 21.69, p< .001, 95% confidence interval ( CI) [5.74, 72.75]. The analysis of variance was conducted on proportions to enable comparison with the findings of De Neys et al. (2013), who used an analysis of variance on percentages to compare standard and control problems. However, because the dependent variable is dichotomous (0 or 1), a mixed effects logistic regression (with subject as the random variable) is more appropriate and yielded comparable results. Specifically, participants were significantly more likely to respond accurately to isomorphic control questions than to standard variants, b = −3.73, odds ratio ( OR) = 41.57, χ 2 = 153.54, p< .001, 95% confidence interval ( CI) [0.01, 0.04]. That is, participants’ odds of answering the isomorphic control questions correctly was about 42 times more likely than answering the standard variants correctly. Frey D, Johnson ED, & De Neys W (2017). Individual differences in conflict detection during reasoning. Quarterly Journal of Experimental Psychology, 71, 1188–1208. doi: 10.1080/17470218.2017.1313283 [ PubMed] [ CrossRef] [ Google Scholar]System 2 thinking, on the other hand, is slow, rational, reflective, and effortful. It gets into the driver’s seat when you focus and concentrate on a complicated problem. Kelley CM, & Lindsay DS (1993). Remembering mistaken for knowing: Ease of retrieval as a basis for confidence in answers to general knowledge questions. Journal of Memory and Language, 32, 1–24. doi: 10.1006/jmla.1993.1001 [ CrossRef] [ Google Scholar]

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Response times were log transformed for the analyses although the means and standard deviations reported here are untransformed values (in seconds). Again, a 3 × 2 mixed factorial analysis of variance was conducted. In this case, as mentioned earlier, the analysis was limited to the response times from incorrect reasoners, defined as those who were incorrect on the standard problem (ignoring the correctness of their response on the control problem). There was a main effect of condition such that incorrect reasoners responded faster to the control question than to the standard question (see Table 1), F(1, 224) = 38.833, p< .001, η p 2 = .15. As for the response confidence scores, the opinion judgments are reported here as proportions rather than percentages. There was a positive relationship between standard question confidence and standard question opinion judgments for incorrect reasoners, r(225) = .480, p< .001, such that incorrect reasoners who were less confident in their response were also less likely to think other reasoners could answer the standard question correctly. This strong relationship lends support to the notion that opinion judgments and response confidence scores are reflecting similar cognitive processes. Bourgeois-Gironde S, & Vanderhenst J-B (2009). How to open the door to System 2: Debiasing the Bat and Ball problem. In Watanabe S, Bloisdell AP, Huber L, & Young A (Eds.), Rational animals, irrational humans (pp. 235–252). Tokyo: Keio University Press. [ Google Scholar]