Hence, x, y, and z are equal and can be substituted for each other. Orderings come about mathematically, when one or more of these statements about equivalent relational properties on a set are false. And we now define a preference relation on a set U. if it is true that the relation R is irreflexive, asymmetric, and transitive. We will arbitrarily eliminate irreflexivity from further consideration by assuming that all empirical preference orderings possess this property. Likewise we will not discuss asymmetry.
Asymmetric consistency is closely akin to reliability or reproducibility.
The same, of course, would be true of a group of judgements by several respondents on one or more occasions. Instead, we will focus on transitivity, which has received far less attention. These and other intriguing problems can be engendered by a consideration of the logic of preference relations. However, in non-literate populations, with little or no Western schooling, there may be special problems in making measurements. Complex judgemental tasks are frequently plagued with errors due to failures to adequately communicate instructions and translate concepts.