Abstract—The concepts of sets and approximate reasoning within extended fuzzy logic (FLe) provide a systematic procedure for transforming unprecisiated knowledge into a nonlinear mapping over what we define here as f-sets. An f-set differs from a fuzzy set in that it is associated with the restriction of validity in addition to that of possibility. Therefore, by f-set, we can simultaneously deal with two different types of uncertainties: one that is related to ill-known objects represented by incomplete information—information with its one or more aspects being imprecise/vague/partial/nonspecific/undetermined—and another that is related to truth values considering gradualness. Here, we define new concepts of ϑ-cuts and αϑ-cuts, introduce the fextension principle, and consider arithmetic computations within FLe. We then address other aspects of the proposed FLe system such as fuzzification and validification operations in input processing stage, set-conversion and defuzzification in output processing stage, and inferencing. In fact, in this paper, we intend to develop FLe theoretically and practically from the stands of sets and systems to extend the concept of approximate reasoning. As a consequence of this development, we assert that considering the validity degree of methods and information can lead to more reasonable and trustworthy results through capturing more uncertainty.

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