# An Algorithm for Producing Fuzzy Negations via Conical Sections
## 2 Preliminaries  
### Fuzzy Negation  
A fuzzy negation \(N\) is a generalization of the classical complement or negation \(\
eg\). The fuzzy negation truth table consists of the two conditions: \(\
eg 1\equiv 0\) and \(\
eg 0\equiv 1\). The fuzzy negation comes to cover the gap between \(0\) and \(1\), thus, maintaining the intuitive perspective of negation, as well as the concept of complementarity. The definitions and theorems laid out in this paragraph aim to present the insofar foundation of the fuzzy negation on axioms, by establishing, at the same time, the background of the tools used to support this study. The following definitions and theorems can be found in any introductory text book on Fuzzy logic (see [1; 4; 5; 6; 7]).  
**Definition 1**.: _A function \(N:[0,1]\rightarrow[0,1]\) is called a fuzzy negation if_  
\[N(0)=1,\;\;N(1)=0 \tag{11}\]  
\[N\;\text{is decreasing}. \tag{12}\]  
**Definition 2**.: _A fuzzy negation \(N\) is called strict if, in addition,_  
(1) \[N\;\text{is a strictly decreasing},\] (13) \[N\;\text{is a continuous}.\] (14)  
_(2)  A fuzzy negation N is called strong if the following property is met,_  
\[N(N(x))=x,x\in[0,1]. \tag{15}\]  
_(3)  The dual negation based on a fuzzy negation N is given by (see [2], p. 124)_  
\[N^{d}=1-N(1-x)\]  
**Theorem 1**.: ([1] Theorem 1.4.5). _If a function \(N:[0,1]\rightarrow[0,1]\) satisfies (12) and (13), then it also satisfies (11) and (13). Moreover, \(N\) is a bijection, i.e., it satisfies (13)._  
**Theorem 2**.: ([1] Lemma 1.4.9). _If \(N_{1}\),\(N_{2}\) are fuzzy negations such that \(N_{1}\circ N_{2}=\text{id}_{[0,1]}\), then_  
_(1) \(N_{1}\) is a continuous fuzzy negation,_  
_(2) \(N_{2}\) is a strictly decreasing fuzzy negation,__._
3. \(N_{2}\) _is a continuous fuzzy negation if and only if_ \(N_{1}\) _is a strictly decreasing fuzzy negation. In both cases_ \(N_{1}=N_{2}^{-1}\)_._  
**Remark 1**.: _Both the theorems mentioned above directly lead to the consequence that the N5 property is equivalent to the equation \(N(x)=N^{-1}(x)\). The latter equation leads to the fact that the graphics of the functions \(N\)\(\kappa\omega\)\(N^{-1}\) identify with each other, namely in other words, this means that the N graphic is symmetrical towards the straight line \(y=x\)._  
**Theorem 3**.: ([1] Theorem 1.4.13) or ([4] Theorem 5.2.1)_. For a function \(N:[0,\ 1]\rightarrow[0,1]\) the following statements are equivalent:_  
1. \(N\) _is a strong negation_ \(N\) _is_ \(\Phi-\)_conjugate with classical negation_ \(N_{C}\)_,_ \((N_{C}(x)=1-x)\) _there exists_ \(\varphi\in\Phi\)_, such that_ \(N(x)=(N_{C})_{\varphi}(x)=\varphi^{-1}(1-\varphi(x)),\ x\in[0,1]\)_._  
Following Theorem 3, the construction of a strong fuzzy negation is a simple case by using the \(\varphi\) function. Nevertheless, a different mode of construction is attempted in the context of the present study, which involved avoiding the use of the \(\varphi\) function. The main reasons for adopting the specific procedures were--first and foremost, the creation of negations from parts of the conical sections, the fact that the \(\varphi\) function might not easily ensure, and secondly the aim to ensure that the mode of constructing the negations shall be based on geometrical methods.