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This paper presents logical foundations for an extension of Codd's relational model of data which aims at utilizing various aspects of similarity in data processing. A need for development of solid foundations for such extensions, sometimes called similarity-based relational databases, has repeatedly been emphasized by leading database experts. This paper argues that, contrary to what may be perceived from the literature, solid foundations for similarity-based databases can be developed in a conceptually sound way. In this paper, we outline the foundations and propose two query systems for similarity-based databases: a domain relational calculus (DRC) and a relational algebra (RA). We compare the expressive power of DRC and RA in our model and prove relational completeness of RA over DRC. A major implication of the paper is that similarity-based data querying can be made an integral part of an extended, similarity-based, relational model of data which is based on first-order predicate logics using residuated structures of truth values in much the same way as the querying in the Codd's model is based on the classic first-order predicate logic.