Probabilities of conditionals: The relevance effect might be confounded by the existence of boundary cases

The relevance between antecedent and consequent has recently been regarded as essential in modulating the probability assigned to a conditional in natural language. The empirical results are mixed. Positive evidence mainly comes from intensional probability studies about ordinary, unique events. Extensional probability studies about novel abstract events commonly fail to observe such an effect. In extensional probability studies, a set of events is typically provided to sustain participants' judgments. Depending on whether the antecedent and the consequent are true or false, the set can be divided into four subsets. When one or more subsets are empty, the set is called a boundary case. When the number of events becomes smaller, it becomes easier for boundary cases to occur. In previous extensional probability studies, however, boundary cases were normally not included in the test stimuli. In intensional probability studies, no explicit events are provided; participants have to mentally simulate a set of events from their own background knowledge to help them make judgments. The size of the mentally simulated sample is relatively small, especially when the judged statements are complex, like conditionals. It is then highly probable for the intensional probability studies to contain boundary cases, even though they cannot be directly observed. Based on the previous analyses, we suspect that the difference observed in previous studies might be confounded by the fact that boundary cases were included in the former case but not in the latter. To test this possibility, we introduced boundary cases into our experiment involving abstract multiple events and observed that (1) when boundary cases were included in the analyses, modulation effect was observed for three of the four parameters; (2) when boundary cases were excluded from analyses, no modulation effect was observed. Reanalyses of previous intensional studies corroborated our hypothesis. We also discussed the potential reason why relevance effect and boundary cases cooccur.