The “labeled interval calculus” is a formal system that performs quantitative inferences about sets of artifacts under sets of operating conditions. It refines and extends the idea of interval constraint propagation, and has been used as the basis of a program called a “mechanical design compiler,” which provides the user with a “high level language” in which design problems for systems to be built of cataloged components can be quickly and easily formulated. The compiler then selects optimal combinations of catalog numbers. Previous work has tested the calculus empirically, but only parts of the calculus have been proven mathematically. This paper presents a new version of the calculus and shows how to extend the earlier proofs to prove the entire system. It formalizes the effects of toleranced manufacturing processes through the concept of a “selectable subset” of the artifacts under consideration. It demonstrates the utility of distinguishing between statements which are true for all artifacts under consideration, and statements which are merely true for some artifact in each selectable subset.