Given a set of entities S, an empirical relation R, the measurement procedure M and the corresponding numerical relation N, we have(S, R) --> (S, N, M)
where for a binary relation the representation condition means that
R(x, y) <==> N(M(x, M(y)).
Example: Let S be a set of persons and let Aimo and Niklas be two persons in S. Consider the binary relation R: taller_than. We know in the empirical world that Niklas is taller than Aimo, i.e., we have taller_than(Niklas, Aimo). Because taller_than is a binary relation we may also write this as "Niklas taller_than Aimo". Let M be the measurement length giving the length (in cm) as we normally understand this measure. For the numerical relation N we choose ">", i.e., grater_than.
By measuring we obtain length(Aimo)= 175 and length(Niklas)= 189. We now see that the statements "Niklas is taller than Aimo" and "length(Niklas) > length(Aimo)" both are true and that "Aimo is taller than Niklas" and "length(Aimo) > length(Niklas)" both are false.
It is easy to conclude that the representation condition is valid in this case. This means that the formal, relational world can be used to reason about facts in the empirical, relational world.
Introducing "length" also means that we may define new measures in the formal world that are not easily recognized in the empirical world. For instance we could calculate the average length of a set of persons which is a theoretical length possibly not possessed by any person in S.
In this way, introducing the measure length has made more precise what we mean by taller_than and facilitates further reasoning related to taller_than.