Whether rapid search of short-term memory or a visual display can cease when a predesignated target is found or must proceed until all items are examined, is a major issue in elementary cognit_sn and information processing. This investigation summarizes past and recent theoretical results on the ability of self-terminating and exhaustive models to predict differences in slopes between positive (target-present) and negative (target- absent) set size functions. Position effect predictions are also covered and the empirical literature is reviewed with regard to the presence of slope ratios of 2/1 or greater, and what are called strong position effects. Self- terminating models can readily predict the results often associated with exhaustive processing, but exhaustive models seem to be incapable of predicting the major effects typically associated with self-termination.