This article presents a family of paraconsistent inference relations which aim at reasoning with inconsistent knowledge bases. These inference relations are defined from inconsistency measures. We check under which conditions these inference relations satisfy a series of properties for nonmonotonic reasoning. We also show that they are paraconsistent when the corresponding inconsistency measures satisfy some rationality postulates, and even strictly paraconsistent in most cases. In addition, some dependencies and incompatibilities among some rationality postulates are shown.