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A fuzzy inference is an extended form of formal inferences that enables symbolic and rigorous evaluation of the degree of a confidential level for a given causality on the basis of fuzzy expressions constructed with fuzzy sets and fuzzy logic operations. Fuzzy inferences are powerful denotational mathematical means for rigorously dealing with degrees of matters, uncertainties, and vague semantics of linguistic variables, as well as for precisely reasoning the semantics of fuzzy causalities. This paper presents a denotational mathematical framework of a set of mathematical structures of fuzzy inferences encompassing deductive, inductive, abductive, and analogical inferences. Each of the fuzzy inference processes is formally modeled and illustrated with real-world examples and cases of applications. The formalization of fuzzy inferences and methodologies enables machines to mimic complex human reasoning mechanisms in cognitive informatics, soft computing, and computational intelligence.