The aim of this paper is to evaluate the capabilities and th~ limitations of purely ordinal approaches to decision under uncertainty (DMU) and Multi-Criteria Decision Making (MCDM). Results obtained in this paper are valid for the two decision problems, due to their strong mathematical similarity. In the axiomatic approach followed by this paper, ordinality is captured by an axiom requiring that two ordinally equivalent pairs of alternatives should be ranked in the same way. We prove that this requirement is basically incompatible widi the axiomatic framework that underlies most numerical approaches. Ordinality is especially incompatible with the transitivity of prefèrence. When the requirement of transitivity is relaxed to quasi-transitivity, the resulting ordinal decision rules are more elaborate than what impossibility theorems in social choice suggest. Yet they prove to hâve a rather limited expressive power. Some ways out of this difficulty are Suggested at the end of the paper.