Probability in reasoning: A developmental test on conditionals

Abstract

a b s t r a c tProbabilistic theories have been claimed to constitute a new paradigm for the psychology of reasoning. A key assumption of these theories is captured by what they call the Equation, the hypothesis that the meaning of the conditional is probabilistic in nature and that the probability of If p then q is the conditional probability, in such a way that P(if p then q) = P(q|p). Using the probabilistic truth-table task in which participants are required to evaluate the probability of If p then q sentences, the present study explored the pervasiveness of the Equation through ages (from early adolescence to adulthood), types of conditionals (basic, causal, and inducements) and contents. The results reveal that the Equation is a late developmental achievement only endorsed by a narrow majority of educated adults for certain types of conditionals depending on the content they involve. Agerelated changes in evaluating the probability of all the conditionals studied closely mirror the development of truth-value judgements observed in previous studies with traditional truth-table tasks. We argue that our modified mental model theory can account for this development, and hence for the findings related with the probability task, which do not consequently support the probabilistic approach of human reasoning over alternative theories.

Logic, probability, and human reasoning

“…Some studies support the Equation, although to varying degrees [54,[75][76][77][78][79][80], and others do not [81,82]. It may fail because of ignorance of the probability calculus: a recent study reports that the judgments of only a minority of well-educated individuals corroborated it and only for some sorts of conditional [83].…”

Arguments from Developmental Order

“…The three most notable groups of theories are Mental Models accounts ( Johnson-Laird and Byrne, 2002 ; Thompson and Byrne, 2002 ; Byrne and Johnson-Laird, 2009 ; Johnson-Laird, 2010 ), Suppositional accounts ( Edgington, 1995 ; Bennett, 2003 ; Evans and Over, 2004 ; Oaksford and Chater, 2007 ), and Mental Logic accounts ( Rips, 1994 ; Braine and O’Brien, 1998 ). In several recent studies, Gauffroy and Barrouillet (2014) , and Barrouillet and Gauffroy (2015) have proposed a developmental trajectory for conditional reasoning. They advocate a modified Mental Models account as fundamental for understanding conditionals (cf.…”

“… Barrouillet and Gauffroy, 2015 , p. 34). Barrouillet and Gauffroy (2015) propose that the interpretation of conditionals, specifically probabilistic conditionals, evolves from an initial incomplete mental model (which treats a conditional as equivalent to a conjunction) to a more fleshed-out mental model that corresponds to a defective biconditional interpretation to a model that corresponds with the Suppositional Theory’s strategy for evaluating conditionals (i.e., some variant of applying the Ramsey Test, cf. Edgington, 1995 ; Willer, 2010 ).…”

“…Thereby, one would focus on the developmental stability – some adult participants in their experiments still use the ‘conjunctive’ conditional rather than the ‘suppositional’ conditional – and on the continuities in reasoning – one would be able to justify a re-interpretation of the ‘suppositional’ responses as more refined mental-model-responses (cf. Barrouillet and Gauffroy, 2015 , p. 34). The late developmental achievements found in adult reasoners are taken to speak against the Suppositional account, as that explanation is supposedly incompatible with the earlier developmental data (the reliance on a conjunctive understanding of the conditional).…”

Missing-link conditionals: pragmatically infelicitous or semantically defective?

“…To begin with, one of the most robust findings in psychology of reasoning is that around 40 percent (in some studies even close to 50 percent) of adult participants seem to interpret conditionals as conjunctions; among children as grown as 9 years old this even appears to be the predominant interpretation (Barrouillet and Lecas 1999;Barrouillet et al 2000;Evans et al 2003; Caroline and Barrouillet 2014; Barrouillet and Gauffroy 2015). Assuming a similar percentage of conjunctive responders among our participants, something like 0.4 × (877 + 31) ≈ 363 of the combinations of TTT + TTF responses given to normal conditionals will have come from such responders, and something like 0.4 × (629 + 407) ≈ 414 of the combinations of TTT + TTF responses given to missing-link conditionals will have come from such responders as well.…”