# An Algorithm for Producing Fuzzy Implications via Conical Sections
## 2 Preliminaries  
In this section, we present the necessary mathematical propositions that constitute the basis of the developed algorithm. These fundamental building blocks pertain to fuzzy implications, fuzzy disjunctions, as well as fuzzy negations. Their definitions and some of their well-known properties are examined below. These are included in many well-known textbooks on fuzzy logic, e.g., [10; 13; 14]. A fuzzy implication expands the notion of the classical implication operator \(\rightarrow\) to the domain of fuzzy logic. In classical logic, the implication operator is expressed via the tautology \"IF \(P\) THEN \(Q^{\prime}\)\" = \"NOT \(P\) OR \(Q^{\prime\prime}\). Fuzzy implications generalize the truth table of the above relation. Similarly, a fuzzy negation generalizes the truth table of the classical negation \(\
eg\) and a fuzzy disjunction generalizes the notion of the classical \(\vee\) (or) operator. We begin by presenting the definition and some additional properties of fuzzy implications. **Definition 1**. : _A bivariate function \(J:[0,1]\times[0,1]\rightarrow[0,1]\) is called a fuzzy implication if \(\forall\ u,u_{1},u_{2},v,v_{1},v_{2}\in[0,1]\) we have the following conditions:_  
\[u_{1}\leq u_{2} \Rightarrow J(u_{1},v)\geq J(u_{2},v), \tag{1}\] \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\] (2) \[J(0,0) =1,\] (3) \[J(1,1) =1,\] (4) \[J(1,0) =0. \tag{5}\]  
Equations (1-5) correspond to a set of functions denoted by \(\mathcal{FJ}\). In addition to the above, there have been many complementary properties in different publications, e.g., [3; 4; 5]. Each of these properties defines a subset of \(F\), i.e., a subclass of the encompassing set of fuzzy implications. Some common ones that we will be using in this paper are provided in the following definition. **Definition 2**. : _Given a function \(J\in F\), it is said to satisfy_  
1. _the left neutrality property if_ \[J(1,v)=v,\quad v\in[0,1];\] (NePr)
2. _the exchange principle if_ \[J(u,J(v,w))=J(v,J(u,w)),\quad u,v,w\in[0,1];\] (ExPr)
3. _the identity principle if_ \[J(u,u)=1\quad u\in[0,1];\] (IdPr)
4. _the ordering property if_ \[J(u,v)=1\Leftrightarrow u\leq v,\quad u,v\in[0,1].\] (OrdPr)  
**Definition 3**. : _A univariate function \(N:[0,1]\to[0,1]\) is called a fuzzy negation if_  
\[N(0)=1,\ N(1)=0,\] (N1) \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]0 (N2)  
The set of functions that satisfy the above conditions can be denoted as \(FN\). These conditions are generic and can be true for a wide array of functions. In addition to the above, several properties pertaining to fuzzy negations have been investigated, which define distinct subclasses of \(FN\). Specifically, as we have mentioned, the fact that the negations that form the basis of the proposed methodology are \"strong\" leads to many favorable properties for the generated parametric family of implications. To this end, we present below the pertinent definitions. **Definition 4**. : _A fuzzy negation is called_  
1. _strict if_ \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]1 (N3) \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]2 (N4)
2. _strong if_ \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]3 _i.e., the function_ \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]4 (N5)  
The law of contraposition is another well-known tautology from the realm of classical logic. Formally, for two propositions \(P\) and \(Q\), this can be expressed via the formula \"IF \(P\) THEN \(Q\)\" \(\equiv\) \"NOT \(Q\) THEN NOT \(P\)\". This can be extended in fuzzy logic as follows [10] (p. 20):  
**Definition 5**. : _Let \(J\in\mathcal{FJ}\). Furthermore, let \(N\in FN\) be a fuzzy negation. Then, \(J\) satisfies the following:_  
1. _law of contraposition if_ \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]5 (CP)
2. _left law of contraposition if_ \[v_{1}\leq v_{2} \Rightarrow J(u,v_{1})\leq J(u,v_{2}),\]6 (L-CP)
3.