The aim of this paper is to analyse from various points of view the relationships between abduction and deduction. In particular, we consider a meta-level definition of abduction in terms of deduction, similar to various definitions proposed in the literature, and an object-level definition in which abductive conclusions are expressed as a logicalconsequence of the observations and of a simple transformation of the domain theory based on predicate completion. The equivalence between the two definitions is proved for domain theories of considerable expressive power. The object-level characterization we propose uses very simple forms of reasoning and the equivalence result allows us to make explicit some of the assumptions underlying meta-level definitions of abduction. The use of predicate completion in characterizing abductive explanations shows a relation between abduction and foundations of logic programming.