## 1 Introduction
Let the set \(S=\{s_{1},s_{2},\ldots,s_{l},\ldots,s_{k}\}\) be the set of symptoms attributes and let \(F=\{f_{1},f_{2},\ldots,f_{l},\ldots,f_{n}\}\) be the set of faults attributes. Let the set \(X=O\times S\times F\) be called the _diagnostic range_ of objects \(o\) with the symptom \(s_{i}\) and faults \(f_{j}\) attributes. Some data about these objects are _fuzzy_, then the _diagnosis_ of the object \(x\in X\) with the attributes \(s_{i},f_{j}\), with some scaling, corresponds to the pair \((\mu_{i},v_{j})\) of fuzzy sets \(\mu_{i}:X\to[0,1],v_{j}:X\to[0,1]\). Observations of symptoms and faults are disjoint sets. The values of functions \(\mu_{i}\) and \(v_{j}\) satisfy the condition:  
\[\mu_{i}(o)+v_{j}(o)\leq 1. \tag{1}\]  
This means that the pair of fuzzy sets \((\mu_{i},v_{j})\) is the intuitionistic fuzzy set (IFSs) in the Atanassov sense [25]. In the [PERSON] study [25], the intuitionistic fuzzy sets (IFSs) consider that there is the value \(\mu\) of the membership function and additionally there is the value \(v\) of the non-membership function. In this research, the following constraint is taken into account: \(0\leq\mu+v\leq 1\). Moreover, in the [PERSON] study [26; 27], the Pythagorean fuzzy sets (PFSs) model is proposed in order to handle imprecise information. If we compare this model to the [PERSON] one, the IFSs condition is relaxed and has the form: \(0\leq\mu^{2}+v^{2}\leq 1\). In the literature, the research of [PERSON] and [PERSON] [28] can be found. They established the model of the Pythagorean fuzzy number (PFN). Furthermore, [PERSON]'s research [29; 30] shows the applications of PFSs in decision-making problems. The membership and non-membership values are usually measured for some **scale function**. Until now, they have been described on the scale 1:1 [31] or on the quadratic function scale of PFSs [32]. In [33], scales for the proven power \(n>2\) are used. In [24], the function scale is the power function \(Sc(x)=x^{a},a\in(1,+\inf))\). In addition, in the IFS articles [31; 34; 35; 36] and those related to PFSs [26; 27; 37; 32; 38; 39; 40], a number of mathematical and logical inaccuracies are corrected and presented in [33]. The paper [33] presented the elements of the deductive theory of n-PFS, which determines the membership degree \(\mu\) and the non-membership degree \(v\) not only for the square scale, but also for any power scale, i.e.,: \(0\leq\mu^{n}+v^{n}\leq 1\), for some \(\mu,v\in[0,1]\), if \(\mu+v>1\). This makes it possible to formulate any local deductions in terms of n-PFS. In this context, the motivation for this paper was the use of the described model to create analogous models, however based on any selected functional scales, e.g.,  
* The _map scale_\(f_{n}(x)=x/n:0\leq\mu/n+v/n\leq 1\),
* Or the sequence of _logarithmic scale_ functions \(f_{n}(x)=1-\log_{(1-1/n)}x\), for \(n>1,x\in(0,1]\), where \(0\leq\log_{(1-1/n)}\mu+\log_{(1-1/n)}v\leq 1\). However, two research problems arose:  
1. How to determine the sequences \(\{Sc_{n}\},\{Sc^{\prime}_{n}\}\) of scaling functions, such that from a certain number \(n\) there is \(0\leq Sc_{n}(\mu)+Sc^{\prime}_{n}(v)\leq 1\), for any results of diagnostic measurements \(\mu,v\in[0,1]\),
2. How to determine the number of \(n\) sequences \(\{Sc_{n}\},\{Sc^{\prime}_{n}\}\) satisfying the condition 1. In this paper, the solution to problems 1 and 2 is presented for the first time. Furthermore, the conceptual apparatus describing \(n\)-_PFSs_ has been generalized and clarified [33]. Furthermore, for the sequence \(\{Sc_{n}\}\) of some scaling functions, for any natural number \(n>0\), deductive theories of n-scaling intuitionistic fuzzy sets, with the condition \(0\leq Sc_{n}(\mu)+Sc_{n}(v)^{\prime}\leq 1\), are formulated. The constructions of these theories are presented below. Their formal description is called the _diagnostic granule system_.