# Methods for Assessing Human-Machine Performance under Fuzzy Conditions
## 2 Fuzzy Sets and Generalizations  
The _\"Laws of Thought\"_[7], of [PERSON] (384-322 BC, Figure 1) that dominated for centuries the human reasoning are the following:  
* The principle of identity
* The law of the excluded middle
* The law of contradiction  
The law of the excluded middle (i.e., yes or no) was the basis for the traditional bi-valued logic and the precision of the classical mathematics owes undoubtedly a large part of its success to it. However, there were also strong objections to this law. The Buddha [PERSON], who lived in India a century earlier, had already argued that almost every notion contains elements from its opposite one, while [PERSON] (427-377 BC, Figure 1) discussed the existence of a third area beyond \"true\" and \"false\", where these two opposite notions can exist together. Modern philosophers including [PERSON], [PERSON], and [PERSON] adopted and further cultivated the above belief. The Polish philosopher [PERSON] (1878-1956) first proposed a systematic alternative of the bi-valued logic introducing in the early 1900s a three-valued logic by adding the term \"possible\" between \"true\" and \"false\" [8]. Eventually, he developed an entire notation and axiomatic system from which he hoped to derive modern mathematics. Later, he also proposed four- and five-valued logics and then finally arrived at the conclusion that, axiomatically, nothing could prevent the derivation of an infinite valued Logic. Similar ideas were also expressed by the famous Polish-American logician and mathematician [PERSON] (1901-1983) [9]. However, it was not until relatively recently that an infinite-valued logic was introduced, called _Fuzzy Logic (FL)_[10], because it is based on the notion of _Fuzzy Set (FS)_ initiated in 1965 by [PERSON] (1921-2017, Figure 2) [11], an electrical engineer of Iranian origin, born in Azerbaijan, USSR and Professor at the University of Berkeley, California. A _fuzzy subset_\(A\) of the set of the discourse \(U\) (or, for brevity, a FS in \(U\)) is defined by \(A=\{(x,m_{A}(x))\colon x\in U\}\), where \(m_{A}\colon U\to[0,1]\) is its _membership function_. The real number \(m_{A}(x)\) is called the _membership degree of \(x\) in \(A\)_. The greater is \(m_{A}(x)\), the more \(x\) satisfies the characteristic property of \(A\). Many authors, for reasons of simplicity, identify it with the function \(m_{A}\). Figure 1: [PERSON] (left) and [PERSON] in a [PERSON]’s fresco (1509). A crisp subset \(B\) of \(U\) can be considered as a special case of a FS in \(U\) with membership function \(\mathrm{m_{B}(x)}=\left\{ \begin{array}{l}1,\mathrm{x}\in\mathrm{B}\\ 0,\mathrm{x}\
otin\mathrm{B}\end{array} \right..\) Based on this, most notions and operations concerning the crisp sets, e.g., subset, complement, union, intersection, Cartesian product, binary and other relations, etc., can be extended to FS. For information on FS, we refer to the book [12]. Through FL, the fuzzy terminology is translated by algorithmic procedures into numerical values, operations are performed upon those values and the outcomes are returned into natural language statements in a reliable manner. As expected, this far-reaching theory aroused some objections in the scientific community. [PERSON] [13] argued that it can be shown that FL is unnecessary in all cases. [PERSON] [14] responded to her objections claiming that FL is useful for handling real-world situations that are inherently fuzzy, calculating the existence in such situations fuzzy data and describing the operation of the corresponding fuzzy systems. He indicated that traditional and FL theories need not be seen as competitive, but as complementary and that FL, despite the objections of classical logicians, has proved very successful in practical applications in almost all sectors of human activity (e.g., see [12] (Chapter 6), [15; 16; 17; 18; 19]).