Logical arithmetic is a logically correct technique for real arithmetic in Prolog which uses constraints over interval representations for its implementation. Four problems with the technique are considered: answers are conditional and uninformative; iterative computations may lead to unboundedly large constraint networks; it is difficult and ineffective to deal with negation; and computing extrema is often not effective. A solution to these problems is proposed in the form of "existential intervals" which record the existence of a solution to a set of constraints within an interval. It is shown how to operate on existential intervals and how they solve the four problems.