For a variety of reasons there has been considerable interest in upper and lower probabilities as a generalization of ordinary probability. Perhaps the most evident way to motivate this generalization is to think of the upper and lower probabilities of an event as expressing bounds on the probability of the event. The most interesting case conceptually is the assignment of a lower probability of zero and an upper probability of one to express maximum ignorance.

Simplification of standard probability spaces is given by random variables that map one space into another and usually simpler space. For example, if we flip a coin a hundred times, the sample space describing the possible outcome of each flip consists of 2100 points, but by using the random variable that simply counts the number of heads in each sequence of a hundred flips we can construct a new space that contains only 101 points. Moreover, the random variable generates in a direct fashion the appropriate probability measure on the new space.

What we set forth in this paper is a similar method for generating upper and lower probabilities by means of random relations. The generalization is a natural one; we simply pass from functions to relations, and the multivalued character of the relations leads in an obvious way to upper and lower probabilities.

The generalization from random variables to random relations also provides a method for introducing a distinction between indeterminacy and uncertainty that we believe is new in the literature.