What is required for the truth of a general conditional?

General conditionals, if p then q, can be used to make assertions about sets of objects. Previous studies have generally found that people judge the probability of one these conditionals to be the conditional probability of q given p, P(q|p). Two experiments investigated the qualitative relation between the exhaustive possibilities, p & q, p & ¬q, ¬p & q, and ¬p & ¬q, and truth and possibility judgements about general conditionals. In Experiment 1, for truth judgements, people evaluated a general conditional as "true" in sets containing p & q cases but no p & ¬q, and "true" judgements depended only on P(q|p). In Experiment 2, for possibility judgements, people's responses implied that only p & q cases have to be possible in a set for a general conditional to be true of the set. Our results add to earlier findings against representing a general conditional as the material conditional of extensional logic, and they provide novel disconfirmation of two recent proposals: the modal semantics of revised mental model theory and certain inferentialist accounts of conditionals. They supply new support for suppositional theories of conditionals.