Any set of confidence interval inferences on J - 1 linearly independent contrasts on J means, such as the two comparisons [mu][subscript 1] - [mu][subscript 2] and [mu][subscript 2] - [mu][subscript 3] on 3 means, provides a basis for the deduction of interval inferences on all other contrasts, such as the redundant comparison [mu][subscript 1] - [mu][subscript 3]. Deduced inference does not inflate the experimentwise error rate beyond the level associated with the set of direct statistical inferences from which it is deduced. This article shows that although deduced inference is not as precise as direct inference, it can be a useful alternative to the kind of post hoc analysis that often follows analyses of sets of linearly independent contrasts. The article also illustrates applications of deduced inference following analyses of factorial experiments, where the problems often associated with popular analysis strategies can be avoided if inferences on simple effect contrasts are deduced from confidence intervals on main and interaction effect contrasts. (Contains 5 tables and 4 footnotes.)