At first glance it might seem that values might be measured in monetary terms, which are subject to arithmetic calculations. Suppose, for example, that thing X has a value of $40,000 to me, while Y has a value of $80,000. Then is the value of X not exactly twice the value of Y? The latter conclusion, however, rests on the premise that a second $40,000 in my monetary stock has the same value to me as my first $40,000—a premise which will be proven fallacious in Section 4. Thus money does not enable us to measure value in cardinal terms. (Of course, one can perform arithmetic calculations on the money amounts corresponding to values, but such calculations do not add to our understanding of an individual's value scale.)

Although ordinal measurements are not subject to arithmetic calculations, the orderings they measure (such as value scales) possess definite mathematical properties. In particular:

• Orderings are antisymmetric: if x and y are distinct units belonging to an ordering represented by the symbol >, then x > y or y > x, but not both.

• They are transitive: if x > y and y > z, then x > z.

Relationships that are antisymmetric but not transitive may be called partial orderings. One example is provided by the well-known child's game, "scissors-paper-rock":  scissors > paper > rock > scissors. A less frivolous example will be encountered in Section 5, in an analysis of voting systems. Frequently, it will be shown, majority preferences exhibit a similar "scissors-paper-rock" intransitivity.