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The following paper presents deductive theories of n-Pythagorean fuzzy sets (n-PFS). N-PFS objects are a generalization of the intuitionistic fuzzy sets (IFSs) and the Yager Pythagorean fuzzy sets (PFSs). Until now, the values of membership and non-membership functions have been described on a one-to-one scale and a quadratic function scale. There is a symmetry between the values of this membership and non-membership functions. The scales of any power functions are used here in order to increase the scope of the decision-making problems. The theory of n-PFS introduces a conceptual apparatus analogous to the classic theory of Zadeh fuzzy sets, consistently striving to correctly define the n-PFS algebra.

The paper presents a certain class of the mathematical models of diagnostic information granules describing the fuzzy symptoms-faults relationship. A certain fuzzy diagnostic information retrieval system is described as an application of an expert diagnostic system. Symptoms and faults are fuzzy, and with some scaling of the symptom-fault concept pair values. These value pairs can be considered as intuitionistic fuzzy sets for the space of diagnosed objects. In this article, for scaling intuitionistic fuzzy sets (n-ScIFS), the deductive theory is formulated. There the intuitionistic fuzzy sets (IFSs) and the Pythagorean fuzzy sets (PFSs) are generalized to the n-ScIFS objects. The membership and non-membership values, as standard, can be described by the 1:1 scale or the quadratic function scale. However, any power scale n>2 can be used. In this paper, any n-Sc scaling functions retaining the IFSs properties are considered. The n-ScIFS theory introduces a conceptual apparatus analogous to the classical theory of Zadeh fuzzy sets and Yager PFSs, consistently striving, for the first time, to formulate the relational structure of n-ScIFSs as a model of a certain information granule system called here the diagnostic granule system. In addition, power- and linear-repeatable diagnostic granules are defined in the n-ScIFSs structure for serial or parallel diagnosis processes. The information granule base is determined and a diagnostic granule system model produced by this information granule base is shown. Certain algorithms have been given to establish the semantic language of diagnosis describing the system of diagnostic information granules.