On the hierarchical nature of partial preferences

In this work we generalize classical Decision Theory by considering that a preference relation might not be total. Incomplete preferences may be helpful to represent those situations where, due to lack of information, the decision maker would like to maintain different options alive and defer the final decision. In particular, we show that, when totality is pulled out, different formalizations of classical Decision Theory become not equivalent. We provide a hierarchical characterization of such formalizations and show that some derived properties of classical Decision Theory, such as justification, no longer hold. Consequently, whenever profitable, justification has to be reintroduced into the theory as an independent axiom.