There has been a good deal of work on information retrieval systems that have continuous weights assigned to the index terms that describe the records in the database, and/or to the query terms that describe the user queries. Recent articles have analyzed retrieval systems with continuous weights of either type and/or with a Boolean structure for the queries. They have also suggested criteria which such systems ought to satisfy and record evaluation mechanisms which partially satisfy these criteria. We offer a more careful analysis, based on a generalization of the discrete weights. We also look at the weights from an entirely different approach involving thresholds, and we generate an improved evaluation mechanism which seems to fulfill a larger subset of the desired criteria than previous mechanisms. This new mechanism allows the user to attach a “threshold” to the query term.