New Fuzzy Implication Model Consisting Only of Basic Logical Fuzzy Connectives

Abstract

Fuzzy implication models play a crucial role in the field of fuzzy logic. The reason behind this reality is the fact that fuzzy implications are influenced by the properties of the model used for their creation. The importance of the mentioned models increases due to the fact that there is a need for new fuzzy implications for use in artificial intelligence and other applications. So, this paper aims to resolve this problem by creating a new model. This model, named (S,T,N) by the authors, is an evolution from previous models as it utilizes all of the basic logical fuzzy connectives in a new composition that emphasizes the use of as many connectives as practically possible. Moreover, a computer program has been developed to display various interpretations of the proposed model and allow the readers to form a deeper understanding of the paper’s research. The results provided by the research conducted are mainly due to the development of the new fuzzy implication model and, secondarily, the new tool for displaying the capabilities of the implication model. Finally, the conclusions drawn from the paper proved that the search for new fuzzy implications should not only be targeted at new research directions but also at more established ones. Furthermore, the program displayed the strong capabilities of computer-assisted computations since it allowed for rapid checking of multiple implications, thus easing the researcher’s task of practically verifying the new model’s validity.

1. Introduction

Fuzzy implications are a fundamental part of fuzzy logic. The importance of their role is mainly driven by the real-world impact of their applications (artificial intelligence, decision making, pattern recognition, etc.). As a result, in recent years, the need for new implications has become more apparent as the various fields that rely on fuzzy logic evolved and progressed.

Under these conditions, the motive for the current paper took shape. To be more specific, the main goals set by the authors when starting the research for this article were the following:

• To invent a new fuzzy implication model.
• To include as many fuzzy connectives in the composition of the said model in an effort to make it distinctly different in comparison to previous ones.
• To prove on an axiomatic basis all the research presented and validate the paper’s findings by displaying properties satisfied by the produced implications.
• To complement the theoretical part of the article with a computer program that allows the reader to validate the effectiveness of the proposed model not only mathematically (through the theorems of the paper) but also visually.
• To re-explore already established research directions of the field and prove that they can offer new findings.

However, before beginning research into the new implication model, it was necessary to conduct a literature review of the field in order to identify key publications and understand their impact on the subject.

The literature review of this paper will mainly focus on the presentation of existing fuzzy implication models that share the characteristic of exclusively using basic fuzzy connectives. This choice aims to not only highlight the novelty of the proposed model in comparison to previous ones but also emphasize the fact that even when dealing with a very restricted research window (the use of only basic logical fuzzy connectives) many valuable results can still be obtained.

The study of fuzzy implications is a field that spans multiple decades of research, with multiple books, articles and conferences contributing to its development. As a result, it can be understood that many implication models as well as single implications have been published over the years. In the following paragraphs, a list of fuzzy implication models has been crafted after extensive research on published material. However, as mentioned above, researchers did not only focus on the development of models but standalone implications, too. This reality led to the indirect discovery of new models through the publication of one of their standalone implications. As a result, the problem of documenting the known models became more complex since there are those that are only partly developed without a general formula, except for the general models that have been established.

This problem has been tackled by the authors by manually generating the general formulas of the single-published implications so that the literature review can be more straightforward and easy to understand even by readers who have no previous experience in fuzzy logic.

Moreover, in order to present the fuzzy implication models in an even more orderly fashion, they have been grouped into three categories according to which one of the three basic logical fuzzy connectives dominates their composition. As a result, three basic hyper-models were formed, each one dominated by a different connective.

Definition of the term “basic hyper-models”: The “basic” part refers to the fact that these models are composed of only basic logical fuzzy connectives, while the “hyper-model” characterization has been chosen because the three new categories are a composition of the models that came out of the literature review. After explaining the type of research that will be included in the literature review, as well as the way it will be presented to the reader, it is deemed important to proceed to discussing the structure.

To be more specific, a subsection has been dedicated to the detailed presentation of each one of the basic hyper-models, where the following information will be included:

• The standalone implications generated from its general formula (only in cases where the general formula has been generated by the authors);
• The citation of relevant publications either for the model or for the standalone implications.

The above mentioned subsections will be accompanied by a figure aimed at assisting the reader in achieving a more complete understanding of their contents. Figure 1, which has been composed using a free-to-use mind-mapping tool (yEd Graph Editor v.3,24), displays how the three basic logical hyper-models branch off into the various models presented in the literature review.

Finally, it is important to declare that the functions $N\left(x\right),S(x,y)$ and $T(x,y)$ seen in Figure 1 are a symbolism for any fuzzy negation, disjunction and conjunction, respectively.

1.1. S-Dominated Basic Hyper-Model

This hyper-model is composed of three (3) sub-models, each with a different formula. The dominant characteristic displayed in this model is the main role of the T-conorm fuzzy connective (disjunction). In the following paragraphs, each of the sub-models will be presented in detail:

• $I(x,y)=S\left(N\right(x),y)$
  This model’s general formula is the simplest form of a fuzzy implication. A key implication derived from it is the following:
  • Kleene–Dienes (KD) Implication [1]: $I_{\mathrm{KD}}(x,y)=max(1-x,y)$
• $I(x,y)=S\left(N\right(x),T(x,y\left)\right)$
  This model’s general formula has been retrieved from [2]. Its key implementations, as well as the publication where they can be found, are the following:
  • Zadeh Implication [3]: $I_{\mathrm{Zadeh}}(x,y)=max(min(x,y),1-x)$
  • Reichenbach Implication [3]: $I_{\mathrm{Reichenbach}}(x,y)=1-x+x·y$
• $I(x,y)=S\left(T\right(N\left(x\right),N\left(y\right)),y)$
  This model’s general formula has been retrieved from [2].

1.2. T-Dominated Basic Hyper-Model

This hyper-model is composed of two (2) sub-models, each one with a different formula. The dominant characteristic displayed in this model is the main role of the T-norm fuzzy connective (conjunction). In the following paragraphs, each of the sub-models will be presented in detail:

• $I(x,y)=T\left(S\right(N\left(x\right),y),S(N\left(y\right),x\left)\right)$
  This model’s general formula has been generated by the authors, as it has not been directly mentioned in the literature. A key implication derived from it is the following:
  • Dienes Implication [4]: $I_{\mathrm{Dienes}}(x,y)=min\left(max(1-x,y),max(1-y,x)\right)$
• $I(x,y)=sup\{z\in [0,1]\mid T(x,z)\le y\},x,y\in [0,1]$
  This model’s general formula has been retrieved from [2].
• $I(x,y)=T(1,S(N\left(x\right),y\left)\right)$
  This model’s general formula has been generated by the authors, as it has not been directly mentioned in the literature. A key implication derived from it is the following:
  • Łukasiewicz Implication [3]: $I_{\mathsf{Ł}\mathrm{ukasiewicz}}=min(1,1-x+y)$

1.3. N-Dominated Basic Hyper-Model

This hyper-model is composed of one (1) sub-model. The dominant characteristic displayed in this model is the main role of the N-negation fuzzy connective (negation). In the following paragraphs, the sub-model will be presented in detail:

• $I(x,y)=N\left(T\right(x,N\left(y\right)\left)\right)$
  This model’s general formula has been retrieved from [5].

Since the literature review has been completed, it is deemed appropriate to discuss the practical use and role of the computer program and its products in the paper. Specifically, the reason the creation of the tool in question was pursued is that, due to the complex nature of fuzzy logic, it is challenging for researchers in other fields to come into contact with the content of the article. As a result, it would be difficult for the theoretical concepts introduced in the paper to be utilized in advancing the development of new practical applications, which would defeat one of the main purposes behind this work. So, the authors introduced the visual tool as a way to allow researchers to experimentally validate the research proposed without coming into direct contact with the complex mathematical formulas. This is possible due to the nature of the fuzzy implications, whose properties can easily be checked by simply looking at the graph of an implication. For illustration reasons, various examples of fuzzy implication graphs generated by the visualization tool are presented throughout the paper in an effort to not only complement the theoretical part of the paper with graphical aspects but also as an easy way for the reader to implement the theorem of the paper visually even without accessing the code. Before moving on from the introductory section of this article, it must be mentioned that the conclusions, as well as other aspects of this paper (e.g., results), will be properly presented in their appropriate sections later on. However, what can be said about the conclusions drawn from this paper is that they can be distilled into the following:

• Successfully achieving all of the research goals;
• Promoting innovation in the field by proposing, validating and presenting new research ideas.

2. Preliminaries

This section is dedicated to the display of basic principles and concepts relevant to the subject of the current paper. The author’s goal with the inclusion of the current section is to widen the range of readers who can understand the research presented. This target has been achieved by providing the necessary definitions for key concepts used throughout the paper. As a result, in the following pages, the definitions of fuzzy implications, triangular norms, triangular conorms and fuzzy negations will be presented in detail.

2.1. Fuzzy Implications

This subsection is dedicated to providing the definition of fuzzy implications. It is important to note that the definition can be found in any of the following publications: (Baczyński M., p. 2, [6]), (Shi Y., p. 5, [7]) and (Fodor J., p. 299, [8])

Definition 1. 

A binary operator $I:{[0,1]}^2\to [0,1]$ is said to be an implication function, or an implication, if, for all $x,y,z\in [0,1]$, it satisfies the following:

$\left(I_1\right):\mathit{If}x\le y,\mathit{then}I(x,z)\ge I(y,z),i.e.,I(·,y)\mathit{is}\mathit{decreasing};$

$\left(I_2\right):\mathit{If}y\le z,\mathit{then}I(x,y)\le I(x,z),i.e.,I(x,·)\mathit{is}\mathit{increasing};$

$\left(I_3\right):I(0,0)=1,\mathit{boundary}\mathit{condition};$

$\left(I_4\right):I(1,1)=1,\mathit{boundary}\mathit{condition};$

$\left(I_5\right):I(1,0)=0,\mathit{boundary}\mathit{condition}.$

A function $I:{[0,1]}^2\to [0,1]$ is called a fuzzy implication only if it satisfies $\left(I1\right)$–$\left(I5\right)$. The set of all these fuzzy implications will be denoted by $FI$.

2.2. Fuzzy Negations

This subsection is dedicated to providing the definition of fuzzy negations. It is important to note that the definition can be found in any of the following publications: (Baczyński M., 1.4.1–1.4.2 Definitions, pp. 13–14, [6]), (Bedregal B.C., p. 1126, [9]), (Fodor J., 1.1–1.2 Definitions, p. 3, [10]), (Gottwald S., 5.2.1 Definition, p. 85, [11]), (Weber S., 3.1 Definition, p. 121, [12]) and (Trillas E., p. 49, [13]).

Definition 2. 

A function $N:(0,1)\to [0,1]$ is called a Fuzzy negation if

$\left(N1\right)N\left(0\right)=1,N\left(1\right)=0;$

$\left(N2\right)Nisdecreasing.$

A fuzzy negation N is called strict if, in addition to the former properties, the following apply:

$\left(N3\right)$ N is strictly decreasing;

$\left(N4\right)$ N is continuous.

A fuzzy negation N is called strong if the following property is satisfied:

$\left(N5\right)N\left(N\right(x\left)\right)=x,x\in [0,1].$

2.3. Triangular Norms (Conjunctions)

This subsection is dedicated to providing the definition of fuzzy conjunctions. It is important to note that the definition can be found in any of the following publications: (Klement E.P et al., 1.1 Definition, pp. 4–10, [14]), (Baczyński M., 2.1.1, 2.1.2 Definitions, pp. 41–42, [6]), (Weber S., 2.1 Definition, pp. 116–117, [12]) and (Yun s., p. 16, [7]).

Definition 3. 

A function $T:{[0,1]}^2\to [0,1]$ is called a triangular norm, or a t-norm, if it satisfies, for all $x,y\in [0,1]$, the following conditions:

$\left(T1\right)T(x,y)=T(y,x),\left(commutativity\right);$

$\left(T2\right)T(x,T(y,z\left)\right)=T\left(T\right(x,y),z),\left(associativity\right);$

$\left(T3\right)ify\le z,thenT(x,y)\le T(x,z),\left(monotonicity\right);$

$\left(T4\right)T(x,1)=x,\left(boundarycondition\right).$

2.4. Triangular Conorms (Disjunctions)

This subsection is dedicated to providing the definition of fuzzy disjunctions. It is important to note that the definition can be found in any of the following publications: (Klement E.P. et al., 1.13 Definition, p. 11, [14]), (Baczyński M., 2.2.1, 2.2.2 Definitions, pp. 45–46, [6]) and (Yun s., p. 22, [7]).

Definition 4. 

A function $S:{[0,1]}^2\to [0,1]$ is called a triangular conorm (or t-conorm) if it satisfies, for all $x,y\in [0,1]$, the following conditions:

$\left(S1\right):S\left(x,y\right)=S\left(y,x\right)\left(commutativity\right);$

$\left(S2\right):S\left(x,S\left(y,z\right)\right)=S\left(S\left(x,y\right),z\right)\left(associativity\right);$

$\left(S3\right):\mathit{If}y\le z,\mathit{then}S\left(x,y\right)\le S\left(x,z\right)\left(monotonicity\right);$

$\left(S4\right):S\left(x,0\right)=x(\mathit{neutral}\mathit{element}0).$

3. Materials and Methods

This is the main section of the current paper with its primary focus being the presentation, and therefore validation, of the practices used by the authors during the research process.

3.1. Establishment of the Main Theorem

Having explained the scope of the current segment, it is deemed logical to proceed with the display of the central theorem of the article, where the new proposed model is mathematically proved.

Theorem 1. 

Let N, T and S be functions such that the following holds:

• $N:[0,1]\to [0,1]$, is a fuzzy negation;
• $T:{[0,1]}^2\to [0,1]$, is a triangular norm or a t-norm;
• $S:{[0,1]}^2\to [0,1]$, is a triangular conorm or a t-conorm.

Then there is a fuzzy implication $I^{S,T,N}:{[0,1]}^2\to [0,1]$, with the following formula:

$$I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)$$

(1)

Proof. 

In order to prove the validity of the fuzzy implication generated by Theorem 1, it must satisfy all of the properties of a fuzzy implication as defined in Definition 1.

• $I(·,y)\mathrm{is}\mathrm{decreasing}$: $\forall x\le y\iff I^{S,T,N}(x,z)\ge I^{S,T,N}(y,z)$

  $$\begin{array}{cc}\hfill & I^{S,T,N}(x,z)\ge I^{S,T,N}(y,z)\iff \hfill \\ \hfill & S\left(T\left(N\left(x\right),N\left(z\right)\right),S\left(T\left(N\left(x\right),z\right),T\left(x,z\right)\right)\right)\ge \hfill \\ \hfill & S\left(T\left(N\left(y\right),N\left(z\right)\right),S\left(T\left(N\left(y\right),z\right),T\left(y,z\right)\right)\right)\stackrel{S\uparrow }{\iff }\hfill \\ \hfill & T\left(N\left(x\right),N\left(z\right)\right)\ge T\left(N\left(y\right),N\left(z\right)\right)\stackrel{T\uparrow }{\iff }\hfill \\ \hfill & N\left(x\right)\ge N\left(y\right)\stackrel{N\downarrow }{\iff }\hfill \\ \hfill & x\le y\hfill \end{array}$$
• $I(x,·)\mathrm{is}\mathrm{increasing}$: $\forall y\le z\iff I^{S,T,N}(x,y)\le I^{S,T,N}(x,z)$

  $$\begin{array}{cc}\hfill & I^{S,T,N}(x,y)\le I^{S,T,N}(x,z)\iff \hfill \\ \hfill & S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\le \hfill \\ \hfill & S\left(T\left(N\left(x\right),N\left(z\right)\right),S\left(T\left(N\left(x\right),z\right),T\left(x,z\right)\right)\right)\stackrel{S\uparrow }{\iff }\hfill \\ \hfill & S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\le S\left(T\left(N\left(x\right),z\right),T\left(x,z\right)\right)\stackrel{S\uparrow }{\iff }\hfill \\ \hfill & T\left(x,y\right)\le T\left(x,z\right)\stackrel{T\uparrow }{\iff }\hfill \\ \hfill & y\le z\hfill \end{array}$$
• $\mathrm{First}\mathrm{Boundary}\mathrm{Condition}:I^{S,T,N}\left(0,0\right)=1$

  $$\begin{array}{cc}\hfill & I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{\begin{array}{c}x=0\\ y=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(0,0)=S\left(T\left(N\left(0\right),N\left(0\right)\right),S\left(T\left(N\left(0\right),0\right),T\left(0,0\right)\right)\right)\stackrel{\begin{array}{c}N\left(0\right)=1\\ T(0,0)=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(0,0)=S\left(T\left(1,1\right),S\left(T\left(1,0\right),0\right)\right)\stackrel{\begin{array}{c}T(1,1)=1\\ T(1,0)=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(0,0)=S\left(1,S\left(0,0\right)\right)\stackrel{S(0,0)=0}{\iff }\hfill \\ \hfill & I^{S,T,N}(0,0)=S\left(1,0\right)\stackrel{S(1,0)=1}{\iff }\hfill \\ \hfill & I^{S,T,N}(0,0)=1\hfill \end{array}$$
• $\mathrm{Sec}\mathrm{ond}\mathrm{Boundary}\mathrm{Condition}:I^{S,T,N}\left(1,1\right)=1$

  $$\begin{array}{cc}\hfill & I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{\begin{array}{c}x=1\\ y=1\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,1)=S\left(T\left(N\left(1\right),N\left(1\right)\right),S\left(T\left(N\left(1\right),1\right),T\left(1,1\right)\right)\right)\stackrel{\begin{array}{c}N\left(1\right)=0\\ T(1,1)=1\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,1)=S\left(T\left(0,0\right),S\left(T\left(0,1\right),1\right)\right)\stackrel{\begin{array}{c}T(0,0)=0\\ T(0,1)=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,1)=S\left(0,S\left(0,1\right)\right)\stackrel{S(0,1)=1}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,1)=S\left(0,1\right)\stackrel{S(0,1)=1}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,1)=1\hfill \end{array}$$
• $\mathrm{Third}\mathrm{Boundary}\mathrm{Condition}:I^{S,T,N}\left(1,0\right)=0$

  $$\begin{array}{cc}\hfill & I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{\begin{array}{c}x=1\\ y=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,0)=S\left(T\left(N\left(1\right),N\left(0\right)\right),S\left(T\left(N\left(1\right),0\right),T\left(1,0\right)\right)\right)\stackrel{\begin{array}{c}N\left(1\right)=0\\ N\left(0\right)=1\\ T(1,0)=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,0)=S\left(T\left(0,1\right),S\left(T\left(0,0\right),0\right)\right)\stackrel{\begin{array}{c}T(0,1)=0\\ T(0,0)=0\end{array}}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,0)=S\left(0,S\left(0,0\right)\right)\stackrel{S(0,0)=0}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,0)=S\left(0,0\right)\stackrel{S(0,0)=0}{\iff }\hfill \\ \hfill & I^{S,T,N}(1,0)=0\hfill \end{array}$$

□

Before presenting an in-depth analysis of the above theorem, it is important to explain shortly how the authors created its content. In the introductory segment of this paper, three fuzzy implication basic hyper-models were presented, each one composed of different sub-models. So, when it was time to start the development of a new model, the decision of which hyper-model it would belong to arose. The choice made was the S-dominated one since it includes the simplest form of a fuzzy implication, and the addition of the new model would contrast it, highlighting its different characteristics.

The reason behind this contrast is the increased complexity displayed by the new model, a byproduct of an intentional choice by the authors in order to differentiate the model from other ones by including more fuzzy connectives in its composition. This choice was rooted in the paper’s goals, which were not limited simply to the invention of a new model but rather the creation of one that displays new, unique characteristics in an effort to differentiate it from the other ones and to provide a valuable piece of research that can truly impact other fields through new research ideas.

The next logical step in the creation process of the model was the formation of its general formula. In order to make sure that the proposed model would satisfy the rule of including more fuzzy connectives in its composition, the $I(x,y)=S\left(T\right(N\left(x\right),N\left(y\right)),y)$ sub-model was chosen since it was the one that displayed the most connectives. Then, through manipulation of the before-mentioned formula, the authors managed to enrich it with even more fuzzy connectives, reaching the final equation seen in Theorem 1.

At this point, it is important to address any concerns arising from the increased complexity of the proposed formula. As it can be seen in Figure 1, the next-most complex sub-model of the S-dominated hyper-models (in which the new model also belongs) is the $I(x,y)=S\left(T\right(N\left(x\right),N\left(y\right)),y)$ model. So, it is reasonable to claim that since the two compared models have a similar complexity level between them, no complications will emerge from it, as is the case in the already established model. Moreover, the complexity displayed can be exploited in various applications where fuzzy implications with elevated characteristics and composition are needed.

Since the decisions that led to the creation of the model have been properly explained and justified, the rest of the Materials and Methods section will be dedicated to the following two directions:

• Providing a more complete picture of Theorem 1 by validating additional fuzzy implication properties;
• Presenting the computer tools that the authors created for the optimal visualization (see Figure 2) of the paper’s research.

3.2. Satisfaction of Additional Fuzzy Implication Properties

In the preliminaries section, the definitions for the main concepts mentioned in the paper were given. One of those was dedicated to fuzzy implications, which were presented as having five basic properties that determine which functions constitute a fuzzy implication. Those five properties have been successfully proved to be satisfied by the proposed model, by Theorem 1; however, they are not the only ones. To clarify, there are six additional fuzzy implication properties that, in contrast to the first five, their satisfaction is not necessary in order for a function to be a fuzzy implication. In the following Proposition 1, the additional properties are mentioned as well as the proof that the proposed model satisfies them. At this point it is important to note that not all of the additional properties are satisfied $\forall x,y\in [0,1]$, but rather for limited values of x and y. In the properties where this is the case, it will be mentioned clearly during the Proof segment of the Proposition 1.

Proposition 1. 

Additional Fuzzy Implication Properties

• Left Boundary: $I^{S,T,N}(0,y)=1,\forall y\in [0,1]applyifandonlyify=0ory=1$
• Right Boundary: $I^{S,T,N}(x,1)=1,\forall x\in [0,1]applyifandonlyifx=0orx=1$
• Identity Principle: $I^{S,T,N}(x,x)=1,\forall x\in [0,1]applyifandonlyifx=0orx=1$
• Ordering Property: $If{I}^{S,T,N}(x,y)=1\iff x\le y,\forall x,y\in [0,1]$
• Neutrality Property: $I^{S,T,N}(1,y)=y,\forall y\in [0,1]$
• Exchange Principle: $I^{S,T,N}(x,I^{S,T,N}(y,z))=I^{S,T,N}(y,I^{S,T,N}(x,z))applyifandonlyifx=y,\forall x,y,z\in [0,1]$

Proof. 

Additional Fuzzy Implication Properties

• $\begin{array}{cc}& I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{x=0}{\iff }\\ & I^{S,T,N}(0,y)=S\left(T\left(N\left(0\right),N\left(y\right)\right),S\left(T\left(N\left(0\right),y\right),T\left(0,y\right)\right)\right)\stackrel{\begin{array}{c}N\left(0\right)=1\\ T(0,y)=0\end{array}}{\iff }\\ & I^{S,T,N}(0,y)=S\left(T\left(1,N\left(y\right)\right),S\left(T\left(1,y\right),0\right)\right)\stackrel{\begin{array}{c}T(1,N(y\left)\right)=N\left(y\right)\\ T(1,y)=y\end{array}}{\iff }\\ & I^{S,T,N}(0,y)=S\left(N\left(y\right),S\left(y,0\right)\right)\stackrel{S(y,0)=y}{\iff }\\ & I^{S,T,N}(0,y)=S\left(N\left(y\right),y\right)\left\{\begin{array}{cc}& \stackrel{\begin{array}{c}S\left(N\right(y),y)=y\\ Ify>N\left(y\right)\end{array}}{\Rightarrow }{I}^{S,T,N}(0,y)=y\\ & \stackrel{\begin{array}{c}S\left(N\right(y),y)=N\left(y\right)\\ Ify<N\left(y\right)\end{array}}{\Rightarrow }{I}^{S,T,N}(0,y)=N\left(y\right)\end{array}\right.\end{array}$
• $\begin{array}{cc}& I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{y=1}{\iff }\\ & I^{S,T,N}(x,1)=S\left(T\left(N\left(x\right),N\left(1\right)\right),S\left(T\left(N\left(x\right),1\right),T\left(x,1\right)\right)\right)\stackrel{\begin{array}{c}N\left(1\right)=0\\ T(x,1)=x\end{array}}{\iff }\\ & I^{S,T,N}(x,1)=S\left(T\left(N\left(x\right),0\right),S\left(T\left(N\left(x\right),1\right),x\right)\right)\stackrel{\begin{array}{c}T\left(N\right(x),0)=0\\ T\left(N\right(x),1)=N\left(x\right)\end{array}}{\iff }\\ & I^{S,T,N}(x,1)=S\left(0,S\left(N\left(x\right),x\right)\right)\\ & \left\{\begin{array}{cc}& \stackrel{\begin{array}{c}S\left(N\right(x),x)=x\\ Ifx>N\left(x\right)\end{array}}{\Rightarrow }{I}^{S,T,N}(0,x)=S\left(0,x\right)\stackrel{S(0,x)=x}{\iff }{I}^{S,T,N}(x,1)=x\\ & \stackrel{\begin{array}{c}S\left(N\right(x),x)=N\left(x\right)\\ Ifx<N\left(x\right)\end{array}}{\Rightarrow }{I}^{S,T,N}(0,y)=S\left(0,N\left(x\right)\right)\stackrel{S(0,N(x\left)\right)=N\left(x\right)}{\iff }{I}^{S,T,N}(x,1)=N\left(x\right)\end{array}\right.\end{array}$
• $\begin{array}{cc}& I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{y=x}{\iff }\\ & I^{S,T,N}(x,x)=S\left(T\left(N\left(x\right),N\left(x\right)\right),S\left(T\left(N\left(x\right),x\right),T\left(x,x\right)\right)\right)\stackrel{\begin{array}{c}T\left(N\right(x),N(x\left)\right)=N\left(x\right)\\ T(x,x)=x\end{array}}{\iff }\\ & I^{S,T,N}(x,x)\left\{\begin{array}{cc}& \stackrel{\begin{array}{c}T\left(N\right(x),x)=x\\ Ifx<N\left(x\right)\end{array}}{=}S\left(N\left(x\right),S\left(x,x\right)\right)\stackrel{S(x,x)=x}{=}S\left(N\left(x\right),x\right)=N\left(x\right)\\ & \stackrel{\begin{array}{c}T\left(N\right(x),x)=N\left(x\right)\\ Ifx>N\left(x\right)\end{array}}{=}S\left(N\left(x\right),S\left(N\left(x\right),x\right)\right)\stackrel{S\left(N\right(x),x)=x}{=}S\left(N\left(x\right),x\right)=x\end{array}\right.\end{array}$
• $\begin{array}{cc}& I^{S,T,N}(x,y)=1\iff \\ & S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)=1\iff \\ & \left\{\begin{array}{cc}& T\left(N\left(x\right),N\left(y\right)\right)=1or\\ & S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)=1\end{array}\right.\iff \\ & \left\{\begin{array}{cc}& N\left(x\right)=1andN\left(y\right)=1or\\ & T\left(N\left(x\right),y\right)=1orT\left(x,y\right)=1orT\left(N\left(x\right),y\right)=1andT\left(x,y\right)=1\end{array}\right.\iff \\ & \left\{\begin{array}{cc}& x=0andy=0or\\ & (N\left(x\right)=1andy=1)or\left(x=1andy=1\right)or\\ & \left[\left(N\left(x\right)=1andy=1\right)and\left(x=1andy=1\right)\right]\end{array}\right.\iff \\ & \left\{\begin{array}{cc}& x=0andy=0or\\ & (x=0andy=1)or\left(x=1andy=1\right)or\\ & \left[\left(x=0andy=1\right)and\left(x=1andy=1\right)\right]\end{array}\right.\iff \\ & \left\{\begin{array}{cc}& x=0andy=0or\\ & x=0andy=1or\\ & x=1andy=1\end{array}\right.\iff x\le y\end{array}$
• $\begin{array}{cc}& I^{S,T,N}(x,y)=S\left(T\left(N\left(x\right),N\left(y\right)\right),S\left(T\left(N\left(x\right),y\right),T\left(x,y\right)\right)\right)\stackrel{x=1}{\iff }\\ & I^{S,T,N}(1,y)=S\left(T\left(N\left(1\right),N\left(y\right)\right),S\left(T\left(N\left(1\right),y\right),T\left(1,y\right)\right)\right)\stackrel{\begin{array}{c}N\left(1\right)=0\\ T\left(1,y\right)=y\end{array}}{\iff }\\ & I^{S,T,N}(1,y)=S\left(T\left(0,N\left(y\right)\right),S\left(T\left(0,y\right),y\right)\right)\stackrel{\begin{array}{c}T\left(0,N\left(y\right)\right)=0\\ T(0,y)=0\end{array}}{\iff }\\ & I^{S,T,N}(1,y)=S\left(0,S\left(0,y\right)\right)\stackrel{S\left(0,y\right)=y}{\iff }\\ & I^{S,T,N}(1,y)=S\left(0,y\right)\stackrel{S\left(0,y\right)=y}{\iff }\\ & I^{S,T,N}(1,y)=y\end{array}$
• $I^{S,T,N}(x,I^{S,T,N}(y,z))=I^{S,T,N}(y,I^{S,T,N}(x,z))applyifandonlyifx=y,\forall x,y,z\in [0,1]$

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3.3. Computer Programs Aimed at Research Visualization

In the previous segments of the current paper, the new proposed fuzzy implication model was presented. However, the authors believe that the mathematical part of the paper should be paired with a visual representation of the paper’s research. The reason behind this choice is the fact that the addition of a visual element to the paper would not only assist readers who are not familiar with the field but also ones who do have a deep comprehension of matters related to fuzzy logic.

In order to achieve the above mentioned goal of offering a tool for the visual representation of the paper’s research, the authors created a computer program. The programming environment chosen for the development of the code is “MATLAB v.2021b” made by the company “MathWorks”. When deciding the characteristics of the program, the main concern of the authors was to keep the balance between a potent research tool that properly displays the paper’s contents and a tool able to be used by readers of all backgrounds.

The way this balance was achieved is through a very user friendly user interface (see Figure 3). To be more specific, the program was coded as a MATLAB application as a way to bypass the interaction between the user and the command line, which may confuse readers not accustomed to programming environments. Moreover, the inputs necessary for the functionality of the app were specifically designed as sliders or buttons so that the user could use them without the need for specialized knowledge.

At this point, since the development choices behind the app’s development have been explained, the next paragraphs will be dedicated to the detailed presentation of the technical part of the code and its inner workings.

The general structure of the app is the following: for each category of fuzzy connectives (negations, disjunctions, and conjunctions), three basic examples were chosen. It is important to state that any other appropriate connective could be used in each of these positions; however, due to the nature of the tool, only three in each category are displayed, whose number and identity can easily modified by the reader through the code (see [15]). Then, after the user has inputted his/her fuzzy connectives of choice, the program integrates them automatically into Equation (1) and generates a 3-D figure of the configured implication (see Figure 4). Certain parametric connectives require an extra parameter value. However, because the parameters used behave in different ways mathematically, they have been grouped into two categories, the $\lambda$ and the $\omega$ parameter, for which two sliders were coded as input. Furthermore, the program recognizes internally when the implication generated does not require the use of a parameter, and any value assigned to the parameters through the sliders is ignored. After the graph has been composed (see Figure 5), the user can utilize the figure-manipulation tools that the MATLAB environment offers so that they can influence the figure in any way they see fit and even save it to their computer for later reference.

4. Results

This section is dedicated to the presentation of the results produced by the research conducted in the current paper. However, a simple enumeration of the article’s outcomes would defeat its purpose since the scientific importance of the practices proposed can be realized only when the results are compared with the goals set by the authors at the start of the paper since many of them were about the expected results.

So, while keeping in mind the targets set in the introductory section, the research output of the article can be as summed up as follows:

• A new fuzzy implication model;
• The successful implementation of the idea of including more fuzzy connectives in the new model’s composition than previous ones;
• The successful design and creation of a programming tool aimed at displaying the paper’s research;
• Proof that innovation can still provide new research findings, even when dealing with established fields.

Taking the above into consideration, the current paper has managed to provide practical results with real-world applications and also serve as an example of how new ideas and concepts can be extracted from well-established fields.

5. Discussion

As mentioned in the section above, the current paper has produced a number of novel results that can be utilized both as theoretical and practical advances in the field of fuzzy logic. Specifically, the introduction of a new fuzzy implication model consisting purely of basic logical fuzzy connectives is a research direction that has not been explored in recent years. This, combined with the fact that the list of already established models is pretty short, elevates the paper’s importance and highlights its novelty. Moreover, the composition of the model can be viewed as another of the differentiating points since its unique approach to fuzzy connective implementation provides it with a range of useful characteristics. At this point, it is important to emphasize the importance of the computer program that serves as the display tool, which, other than its practical use, serves as a successful example of how different scientific fields can share practices and assist in the development and promotion of new research directions. Finally, the findings of this article introduce multiple new research directions, with the most prominent ones being the exploration of the various applications of the newly introduced model, the possible creation of a (S,T,N) fuzzy implication family, or even evolution of the display tool into a more general application that provides more capabilities to the researcher.

6. Conclusions

Taking everything into consideration, the current paper has achieved the goals that motivated the writing. Specifically, it has established its findings both on a theoretical and practical level, while at the same time trying to render the research presented accessible and easy to understand by other scientists. Finally, even though the current article presents a mix of both theoretical and practical elements, another important aspect of this work that should be mentioned is how, as mentioned above, the authors tried to uphold the scientific quality of the paper at every step of the research process through various conscious decisions (like the inclusion of a more reader-oriented literature review, the mathematical validation of the proposed formula and the creation of a computer program for visualization purposes), which aimed to present to the reader a piece of work as representative of reality as possible.