This paper discusses the issue of categorical acceptability of indicative and concessive conditionals. It presents experimental results in favour of two claims concerning the role of the evidential support relation for acceptability (or otherwise) of conditionals of both types. In particular, the results show that, contrary to fairly standard philosophical theorising, high probability of a conditional's consequent given its antecedent is necessary but not sufficient for the acceptability of that conditional, and that the antecedent being evidence for the consequent is a further acceptability condition. The results further show that the evidential support relation is crucial in differentiating between the acceptability of an indicative condi...