This paper presents a monotonic core logic for default conditionals and \u000Ainvestigates possible nonmonotonic extensions. First, we introduce a possible \u000Aworlds semantics characterizing HRC0, a strong logic for non‐nested normal \u000Aimplication. Next, we formulate several basic regularity postulates for \u000Adefeasible entailment relations extending our monotonic inference relation and \u000Aclaiming to provide an appropriate interpretation of our default conditional. \u000AWe then construct a preferential semantic framework describing a class of \u000Aconsequence relations verifying our standard requirements. In particular, we \u000Adefine and analyze elementary hyperentailment, a new semantic‐based plausible \u000Ainference relation for boolean combinations of facts and defaults.