Deductive schemas with uncertain premises using qualitative probability expressions

ABSTRACT

The new paradigm in the psychology of reasoning redirects the investigation of deduction conceptually and methodologically because the premises and the conclusion of the inferences are assumed to be uncertain. A probabilistic counterpart of the concept of logical validity and a method to assess whether individuals comply with it must be defined. Conceptually, we used de Finetti's coherence as a normative framework to assess individuals' performance. Methodologically, we presented inference schemas whose premises had various levels of probability that contained non-numerical expressions (e.g., “the chances are high”) and, as a control, sure levels. Depending on the inference schemas, from 60% to 80% of the participants produced coherent conclusions when the premises were uncertain. The data also show that (1) except for schemas involving conjunction, performance was consistently lower with certain than uncertain premises, (2) the rate of conjunction fallacy was consistently low (not exceeding 20%, even with sure premises), and (3) participants' interpretation of the conditional agreed with de Finetti's “conditional event” but not with the material conditional.

Keywords:

• Uncertain reasoning
• deduction
• de Finetti's coherence
• conjunction fallacy

Acknowledgments

The authors thank Nicole Cruz and David Over for very helpful comments on a previous draft of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ The conditioning principle (Hacking, 1975) assumes that the revised probability upon learning the outcome D at time t₁ is equal to the probability of H conditioned on the (imagined or assumed) evidence D at a moment t₀ (that is, P(H|D) yielded by Bayes' rule): P_D(H) = P(H|D).

² There is another possible methodology based on the notion of p-validity. An argument is p-valid when the uncertainty of its conclusion cannot exceed the sum of the uncertainties of its premises (Adams, 1998). The experimenter examines the participant's assessment of the argument's conclusion with reference to its formal validity. The two notions are not equivalent as a p-valid assessment can be incoherent. Some paradoxes may arise with the p-validity methodology. For example, it suffices that the sum of the uncertainties of the premises reaches 1 for the conclusion to be p-valid. For a comparison between coherence and p-validity, see Baratgin and Politzer (in press).

³ Participants who selected the smaller option in response to the high or very high probability of the premises of AND-introduction could do so for correct reasons with respect to the upper bound (rejecting an increase in probability and avoiding the classic conjunction fallacy) but also because they had an evaluation below the lower bound and so be incoherent. However, the small frequencies observed compared with the very low, low and average cases where the lower bound reaches zero suggests that such violations were rare, if they existed at all.

⁴ We owe this suggestion to N. Cruz.

Additional information

Funding

This work was supported by a grant from the ANR Chorus 2011 (project BTAFDOC).