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Keren G (1991). Calibration and probability judgments: Conceptual and methodological issues. Acta Psychologica, 77, 217–273. doi: 10.1016/0001-6918(91)90036-Y [ CrossRef] [ Google Scholar] De Neys W, Rossi S, & Houdé O (2013). Bats, balls, and substitution sensitivity: Cognitive misers are no happy fools. Psychonomic Bulletin & Review, 20, 269–273. doi: 10.3758/s13423-013-0384-5 [ PubMed] [ CrossRef] [ Google Scholar] Lichtenstein S, & Fischhoff B (1980). Training for calibration. Organizational Behavior and Human Performance, 26, 149–171. doi: 10.1016/0030-5073(80)90052-5 [ CrossRef] [ Google Scholar]

The body of work on conflict detection in decision-making provides the theoretical framework for explaining the findings outlined above (e.g., Aczel, Szollosi, & Bago, 2016; De Neys, 2012; Pennycook, Fugelsang, & Koehler, 2012). In this line of theorizing, reasoning errors arise from an inability to inhibit prepotent intuitive responses that, importantly, cannot be explained solely due to miserly cognition because incorrect reasoners demonstrate that they are unsure their reasoning was accurate. Indeed, there is a rich body of research in support of error sensitivity in particular and logical intuitions more generally (e.g., Bago & De Neys, 2017; De Neys, 2012, 2014; De Neys & Bonnefon, 2013; De Neys & Glumicic, 2008; De Neys et al., 2013; Gangemi, Bourgeois-Gironde, & Mancini, 2015; Mata, Schubert, & Ferreira, 2014; but see Singmann, Klauer, & Kellen, 2014, for some caveats regarding this literature). Furthermore, sensitivity to error processing has been demonstrated though converging evidence in latency response investigations ( De Neys & Glumicic, 2008; Frey, Johnson, & De Neys, 2017; Johnson, Tubau, & De Neys, 2016) neuroimaging ( De Neys, Vartanian, & Goel, 2008), and alternative measurements of confidence ( De Neys, Cromheeke, & Osman, 2011). However, sensitivity effects were not obtained by indexing reasoner’s mouse-movements ( Travers, Rolison, & Feeney, 2016) nor their eye-movements ( Mata, Ferreira, Voss, & Kollei, 2017; see the subsequent debate concerning the studies by Mata et al. and Frey et al. discussed by Mata & Ferreira, 2018). On a more general level, unconscious processes have not been found to have much (if any) explanatory power (for a review, see Newell & Shanks, 2014). De Neys W (2012). Bias and conflict: A case for logical intuitions. Perspectives on Psychological Science, 7, 28–38. doi: 10.1177/1745691611429354 [ PubMed] [ CrossRef] [ Google Scholar]B. T., & Stanovich KE (2013). Dual-process theories of higher cognition: Advancing the debate. Perspectives on Psychological Science, 8, 223–241. doi: 10.1177/1745691612460685 [ PubMed] [ 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] Arkes HR, Christensen C, Lai C, & Blumer C (1987). Two methods of reducing overconfidence. Organizational Behavior and Human Decision Processes, 39, 133–144. doi: 10.1016/0749-5978(87)90049-5 [ CrossRef] [ Google Scholar] Gangemi A, Bourgeois-Gironde S, & Mancini F (2015). Feelings of error in reasoning—in search of a phenomenon. Thinking & Reasoning, 21, 383–396. doi: 10.1080/13546783.2014.980755 [ CrossRef] [ Google Scholar]

Keren G (1988). On the ability of monitoring non-veridical perceptions and uncertain knowledge: Some calibration studies. Acta Psychologica, 67, 95–119. doi: 10.1016/0001-6918(88)90007-8 [ PubMed] [ CrossRef] [ Google Scholar] The question really belongs more to the science of the mind than it does to mathematics and logic— it is about the assumptions we make, rather than whether or not we have the ability to solve the question. Travers E, Rolison JJ, & Feeney A (2016). The time course of conflict on the Cognitive Reflection Test. Cognition, 150, 109–118. doi: 10.1016/j.cognition.2016.01.015 [ PubMed] [ CrossRef] [ Google Scholar]Johnson ED, Tubau E, & De Neys W (2016). The Doubting System 1: Evidence for automatic substitution sensitivity. Acta Psychologica, 164, 56–64. doi: 10.1016/j.actpsy.2015.12.008 [ PubMed] [ CrossRef] [ Google Scholar] Psychology of learning and motivation (Vol. 26, pp. 125–173). Academic Press. doi: 10.1016/S0079-7421(08)60053-5. [ CrossRef] [ Google Scholar] 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. Finally, latency responses to both the standard and control versions were collected. Correlations between response time and response confidence were expected, in line with previous work (e.g., Johnson et al., 2016; Kelley & Lindsay, 1993; Thompson, Prowse Turner, & Pennycook, 2011; Thompson et al., 2013).