In Web resources and resource descriptions, many resources and descriptions of resources have different negations and fuzziness. In this paper, we investigate and analyze the cognition and characteristics of “negation” in philosophy, logic, linguistics and other knowledge fields, and based on the distinction between the concept of “contradiction” and the concept of “opposition”, and “clear concept” and “fuzzy concept”, we propose that there are three different kinds of negation in knowledge from the conceptual level, namely “contradictory negation”, “opposite negation” and “intermediary negation”. In order to establish a mathematical basis that can fully reflect the internal relations, properties and laws of these different negations and fuzziness, we propose a “Scoi: sets with contradictory negation, opposite negation and intermediary negation” and “Lcoi&PLcoi: logic with contradictory negation, opposite negation and intermediary negation”, and the operational properties of Scoi and the formal deduction properties of Lcoi have been proven. The soundness theorem, completeness theorem and compactness theorem for Lcoi are proved under the three-valued semantic interpretation of Lcoi, and the soundness and completeness for Lcoi&PLcoi are discussed under the given an infinite-valued semantics interpretation of Lcoi&PLcoi. Furthermore, we also demonstrated that descriptions of objective things such as “x is neither a positive nor a negative integer” and “he is neither rich nor poor” typle of statements are “intermediary negation” in Scoi and Lcoi&PLcoi. It shows that “intermediary negation” in Scoi and Lcoi&PLcoi has objectively.In order to enable RDF(Resource Description Framework) to describe Web resources with different negations and fuzziness, we extend RDF based on Scoi and Lcoi&PLcoi, and propose a resource description framework RDFcoi which can fully handle fuzzy attribute values and their different negations in attributes of resources. How to reflect the uncertainty of the relationship between Web resources and its attribute values, we believe that the relationship between a resource and fuzzy attribute value in one of its attributes should be a composite mapping from the resource domain to the fuzzy attribute value set and then to [0, 1], and a mathematical representation is given in this paper. Thus, RDFcoi differ from other RDF extensions. RDFcoi maintains the basic syntax and normative of RDF, and extends the basic syntax and semantics of RDF. Where, the concepts of RDFcoi triple with three kinds of negation, RDFcoi formula and RDFcoi graph are defined. RDFcoi vocabulary is determined and an interpretation is given. RDFcoi-entailment is defined and its properties are discussed. In addition, we also study the reasoning regimes of RDFcoi. By using the given logical translation method, a well-typed and interoperable RDFcoi graph can be transformed into a set of formulas in Lcoi&PLcoi, thus establishing “interoperability” between RDFcoi and Lcoi&PLcoi. From the perspective of reasoning mechanism of RDFcoi-entailment, RDFcoi reasoning can be carried out in Lcoi&PLcoi, and RDFcoi has the same reasoning mechanism as formal inference in Lcoi&PLcoi.