Recently considerable attention has been given in the online information retrieval literature to techniques for producing a weighted output of documents in response to a request. One approach tries to maintain the form of and relationships among requests as they appear in current Boolean logic-based systems, while extending it to permit a weighted output. It is based on the mathematics of fuzzy-set theory, which assigns each potential member of a set a degree of membership between zero and one with intermediate values denoting partial membership in the set. Another approach is based on the mathematics of probability. It represents requests by sets of terms, and, by means of feedback information, assigns a weight to each term. Documents are ordered by the sum of weights of the terms in the request that match those in the documents. This paper provides an overview of both approaches and their advantages and disadvantages. It also outlines a basic theory of information retrieval, discusses the characteristics and problems of using Boolean logic in information retrieval, and lists areas for continuing research. An appendix describing mathematical constraints on fuzzy-complements and a 21-item bibliography conclude the paper. (Author/ESR)