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Classical measurement theory authors have used both versions. The advantage of the second version is that partial orders in the empirical system can be mapped to integers or reals that are both totally ordered.
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EXAMPLE 5.1 HEIGHT OF PEOPLE
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The classic example of mapping an empirical system to a numerical system is the height of people. In the empirical system, there is a well-understood height relationship among people. Given two people who are standing next to each other, everyone would agree about who is taller. This is the empirical system: people are the entities and the well-understood relation is shorter or the same height. The numerical system is the real number system (either metric or imperial units) with the standard relation of less than or equal. The mapping is just the standard measured height of people. This is usually measured barefoot, standing straight against a wall. The representation condition (either version) is satisfied, since if Fred is shorter than or equal to Bill, then Fred s measured height is less than or equal to Bill s measured height.
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Develop a measure, BIG, for people that combines both weight and height. Empirically, if two people are the same height, the heavier is bigger, and if two people are the same weight, the taller is bigger. If we use this notion, we can have a partial order that most people would agree with. The only pair of persons that we would not order by this would be if one was heavier and the other was taller. Numerically, we can use a tuple, < height, weight >. Each part of the tuple would be a real number. Two tuples would be related if both parts were related in the same direction. That is, if x ; y are tuples, than x is less than or equal to y in bigness if x height < y height and x weight < y weight. This is also a partial order, and both versions of the representation condition are satisfied.
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An important characteristic of a measure is monotonicity. It means that the value of the measure of an attribute does not change direction as the attribute increases in the object. For example, the count of lines of code will not decrease as more code is added.
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EXAMPLE 5.3
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