Some Properties of Boolean-like Laws in Fuzzy Logic

Abstract

This article focuses on the relationships between fuzzy logic and classical logic properties using fuzzy t-norms, t-conorms, and fuzzy implications. It aims to contribute to fuzzy set theory by extending the Boolean laws in classical logic to fuzzy logic. We determine the necessary and sufficient conditions for validating the generalizations of the proposed properties from classical to fuzzy logic. Additionally, we provide examples demonstrating the practical applicability of this approach and its advantages over conventional methodologies, reinforcing its effectiveness.

1. Introduction

Fuzzy logic has garnered considerable interest as a framework for handling vague information. Over the past few decades, various non-classical logic systems have been proposed and explored. Among them, Lukasiewicz–Zadeh’s n-valued logic stands out as the most widely used, particularly in the fields of fuzzy sets and systems. In the 1930s, Lukasiewicz introduced n-valued logic, focusing predominantly on the logical operations of negation and implication. Zadeh later expanded on Lukasiewicz’s work by introducing the concept of membership functions, which are crucial for defining fuzzy sets. This combination of Lukasiewicz’s logical framework and Zadeh’s fuzzy set theory laid the groundwork for modern fuzzy logic applications across various domains, including artificial intelligence and control systems [1].

Given the significance of T-norms in various fields, including information science, fuzzy logic, and fuzzy sets, these norms play a crucial role in many applications [2]. A T-norm, also known as a t-norm or triangular norm, is a type of binary operation used in fuzzy logic and multi-valued logic frameworks. The concept of a t-norm was first introduced by Menger [3] in 1942. He established a unique class of two-place functions on the unit square, referred to as triangular norms, to properly formulate the triangle inequality. Although Menger did not impose the requirement of associativity, the boundary conditions he established were inherently weaker than those for modern t-norms. In 1958, Schweizer and Sklar [4] formulated the current axioms for t-norms, as stated in Definition 1. These axioms include commutativity, associativity, monotonicity, and the existence of a neutral element, which is 1. Their definition of associativity was based on a correct formulation of the polygonal inequality.

In 1965, Zadeh proposed alternative methods for fuzzy intersection (and union) that employed the minimum (and maximum) functions [5]. Additionally, Dubois [6] documented the use of t-norms within fuzzy sets and fuzzy logic frameworks for the first time in 1980. These alternative methods provided new insights into fuzzy logic and expanded the applications of t-norms beyond their initial scope. Dubois’s work in 1980 laid the foundation for the widespread use of t-norms in fuzzy sets and logic.

The dual operations known as T-conorms (also referred to as S-norms or triangular conorms) were introduced by Schweizer and Sklar in 1960 [7]. It is important to note that the term “s-norm” originates from the S in the original Schweizer–Sklar notation for t-conorms. Evidence suggests that both t-norms and t-conorms play a crucial role in the theory of fuzzy sets. Because they are dual to each other, any results derived for t-norms can be reformulated for t-conorms and vice versa. This duality allows for a deeper understanding of operations within fuzzy sets and provides flexibility in their application. However, it is possible to define t-conorms independently of t-norms using a similar set of axioms. The main distinction between the two lies in their neutral elements: 0 for t-conorms and 1 for t-norms. This distinction highlights the versatility of t-conorms within fuzzy set theory, enabling various interpretations and applications that differ from those of t-norms. Overall, the duality and independence of t-conorms contribute to the richness and complexity of operations in fuzzy set theory.

In fuzzy logic, fuzzy implications are essential for representing the relationship between input and output variables. By allowing for degrees of truth instead of relying solely on strict true or false values, fuzzy implications offer a more flexible and realistic framework for modeling uncertainty and imprecision across various domains. They generalize classical two-valued implications to a multi-valued context, proving significant in both theoretical and practical applications. This significance is evident in their use in multi-valued mathematical logic, fuzzy control systems, and data analysis. This discussion will focus on fuzzy implications from a theoretical perspective to highlight their fundamental principles.

Boolean laws, which are valid in classical logic, play a crucial role in fuzzy set theory and related Boolean-like principles through the use of t-norms, t-conorms, negation operators, and fuzzy implications. In the paper [8], the authors discuss certain Boolean-like laws that involve iterative variables in fuzzy logic. They explain that, in addition to the classical De Morgan triplets of connectives, which are defined by t-norms, t-conorms, and strong negations, there are fascinating models that can yield infinite solutions. Additionally, they highlight unexpected cases where no solutions exist. There has been considerable research on characterizing two types of fuzzy implication functions, specifically, R-implications and S-implications. However, there is limited knowledge about $QM$-implications. The papers [9,10] focus on studying certain characteristics of $QM$-implication operators. They examine the $QM$-implication operator both as an implication function and as a T-conditional function, providing useful tools for their characterization. In the paper [11], the author explores the well-known iterative Boolean-like law and discusses the property of approximation preservation related to the compositions of fuzzy implications. Finally, the necessary and sufficient conditions for the approximate equation of $(S,N)$-implications are provided.

The principle of distributivity, where disjunction distributes over conjunction, serves as a cornerstone in classical logic and forms the basis of algebraic structures defined by binary operators. In the context of fuzzy logic, this principle is extended through the use of t-norms and t-conorms, which provide a framework for generalizing classical properties. Several studies have focused on distributive equations for fuzzy implications, particularly about two aggregation functions, as outlined by Bala [12], Combs and Andrews [13], who pioneered the exploration of distributive equations by grounding their work in the logical validity within classical propositional logic. Their efforts aimed to address the complexity caused by the proliferation of combinatorial rules. Building on this, Trillas and Alsina [14] further investigated the general forms of these distributive equations. Moreover, other studies have identified solutions for fuzzy implications to these distributive equations for specific t-norms and t-conorms, as discussed by Pan [15]. This evolving discourse underscores the significance of distributivity in bridging classical and fuzzy logic, offering practical and theoretical insights into its application.

The ordering property $\left(OP\right)$ and the exchange principle $\left(EP\right)$ represent robust foundational principles in mathematics. Specifically, $\left(OP\right)$ ensures a well-defined ordering of elements within a set, commonly known as the degree ranking property, particularly in the interval $[0,1]$. Meanwhile, $\left(EP\right)$ permits the rearrangement of elements without altering the intrinsic properties of the set, generalizing the classical exchange principle expressed as a tautology. In the realm of fuzzy logic, $(S,N)$-implications are a type of fuzzy implication that adheres to both the left neutrality property $\left(NP\right)$ and the exchange principle $\left(EP\right)$. However, it is worth noting that not all $(S,N)$-implications satisfy the identity principle $\left(IP\right)$ or the ordering property $\left(OP\right)$ [16].

Classical logic’s application to fuzzy logic through equivalence showcases its practical utility, notably in preventing the explosion of combinatorial rules within fuzzy systems [17]. This approach effectively streamlines the complexity often encountered in such systems. Moreover, the research of Bala and Trillas [12,14] highlights the significance of the fuzzy logic translation of the distributivity functional equation. They successfully addressed this equation for a variety of implication functions derived from t-norms and t-conorms. These studies exemplify the seamless integration of classical logic principles into the fuzzy logic framework, offering both theoretical insights and practical benefits for handling complex logical structures.

In fuzzy logic, numerous laws of contraposition have been investigated, including the particular law of contraposition $\left(CP\right)$, the law of left contraposition to N$(L-CP)$, and the law of right contraposition to N$(R-CP)$ [17,18]. These laws have been examined for their relevance and fundamental role within the principles of classical logic. When N is defined as a strong negation, these three laws exhibit equivalence, as their properties align seamlessly. This demonstrates the simplicity and coherence of the contraposition laws under strong negation. However, when N is defined simply as a fuzzy negation rather than a strong negation, the equivalence between these contraposition principles fails. In this case, each law may exhibit different properties under fuzzy negation [17]. This distinction underscores the significance of strong negation in preserving the harmony among contraposition laws.

The rest of this paper is structured as follows:

In Section 2, we introduce some fundamental concepts and characteristics of algebraic structures and fuzzy set theory. We review several key definitions and results related to fuzzy negations, t-norms, t-conorms, and the properties that connect these fuzzy operators.

Section 3 examines the key themes and properties, focusing on fuzzy t-norms, t-conorms, and fuzzy implications, as well as the connections between classical logic properties and their fuzzy counterparts. We identify the necessary and sufficient conditions to validate the generalizations of classical logic properties within fuzzy logic. Furthermore, an example is provided to illustrate the practical applications of these results.

2. Preliminaries

We assume that the reader is already familiar with the fundamental concepts of classical logic connectives. However, this section will summarize several key ideas that will be utilized later in the text to ensure it is self-contained. Additionally, we will briefly revisit some of the ideas and findings that will be employed in the subsequent sections. This overview will serve as a refresher for those already well versed in classical logic and provide essential background information for those new to the topic of fuzzy logic.

The core concept of fuzzy logic can be better grasped by contrasting it with classical binary logic. Suppose there is a set of n logical variables, denoted as $x_1,x_2,\dots ,x_n$. These n variables can yield $2^n$ distinct combinations of truth values. As a result, the total count of logical functions that can define these variables is also $2^n$. Logical functions can be broadly classified into two main categories:

(a)

“negation”, “and function”, and “or function”;

(b)

“negation” and “the two implications”.

Through either set of primary logical functions, any other logical functions can be generated using suitable algebraic expressions, referred to as logical formulas. This approach facilitates the creation of complex logical operations by combining simpler ones, offering a powerful method for designing and analyzing digital circuits. The three main logical functions—negation (−), (∧), and (∨)—are combined in various types of logical formulas to create a logical function. It is important to specify the order of composition, for instance, by using parentheses to give a unique function. The logical expression

$$(x_1\vee x_3)\wedge ({\overline{x}}_1\vee x_2)\wedge ({\overline{x}}_2\vee {\overline{x}}_3)$$

represents a distinct logical function involving three variables. The expression $a\equiv b$ is commonly used to describe the equivalence of two logical formulas, represented by the symbols a and b. For example, the following equivalency is evident:

$$({\overline{x}}_1\wedge x_2)\vee (x_1\wedge x_3)\vee ({\overline{x}}_2\wedge {\overline{x}}_3)\equiv (x_1\wedge {\overline{x}}_2)\vee ({\overline{x}}_1\wedge {\overline{x}}_3)\vee (x_2\wedge x_3).$$

Boolean algebra may be used to verify that these expressions are equivalent with the variables $x_1$, $x_2$, and $x_3$ over all eight conceivable combinations of truth values. Boolean algebra is a binary logic structure with two values that bears the name of George Boole, a mathematician and logician in the 19th century. It provides a structured approach for manipulating logical expressions, facilitating the analysis and comparison of different formulations. By adhering to the principles of Boolean algebra, verifying the equivalence of logical expressions becomes a more streamlined and efficient process [19,20,21,22].

Boolean algebra consists of three fundamental logical operations: “negation ¬”, “and ∧”, and “or ∨”. By developing basic logical processes, Zadeh built on Lukasiewicz’s logic and introduced an infinite-valued logical framework. For ease of algebraic manipulation, these operations are often represented with the symbols −, ·, and +, respectively. A truth table can be used to clearly define a Boolean algebraic formula by listing all the variables as inputs and indicating the resulting output for the entire formula. Boolean algebraic formulas can be constructed for any truth table and can often be simplified to a more straightforward, equivalent form using various useful properties of Boolean algebra.

Note that several Boolean algebraic rules, including $a·0=0$, $a·1=a$, and $a+0=a$, are comparable to those of ordinary algebra. However, there are significant differences as well, for example, $a+1=1$. Therefore, great care should be taken to avoid mistakenly applying standard algebraic principles to logical formulations, as this can lead to errors. Furthermore, even if Boolean algebra works well for streamlining logical formulas, when numerous variables are involved, the procedure can become very tiresome in real-world applications. By examining the connections between Boolean laws and fuzzy logic properties, readers can gain insight into the versatility and applicability of these concepts across different mathematical frameworks. For example, De Morgan’s principles from classical logic can be applied to fuzzy logic by incorporating t-norms and t-conorms. This extension provides a deeper understanding of how fuzzy logic can more flexibly model uncertainty and vagueness compared to traditional Boolean logic.

We will review the main results regarding t-norms, t-conorms, fuzzy negation, and some concepts related to fuzzy implications that are referenced throughout this work [23,24].

Definition 1. 

A t-norm is a function $T:[0,1]\times [0,1]\to [0,1]$, for all $a,b,c,d\in [0,1]$ that satisfies the following properties:

(T1) 

(Commutativity): $T(a,b)=T(b,a)$,

(T2) 

(Associativity): $T\left(T(a,b),c\right)=T\left(a,T(b,c)\right)$,

(T3) 

(Monotonicity): $T(a,b)\le T(c,d)$ if $a\le c$ and $b\le d$, (T is non-decreasing in each argument),

(T4) 

(1-identity): $T(a,1)=a$ (The number 1 acts as identity element).

Since a t-norm is a binary algebraic operation defined on the interval $[0,1]$, it can be represented using infix algebraic notation, where the t-norm is denoted by the symbol *. When examining the algebraic structure, the binary operation associated with the t-norm, represented in prefix notation as T, can also be expressed using the infix operator *. Consequently, the four axioms $\left(T1\right)$–$\left(T4\right)$ apply for every $a,b,c\in [0,1]$, as follows:

$$a\ast b=b\ast a$$

$$(a\ast b)\ast c=a\ast (b\ast c)$$

$$a\ast b\le a\ast c,\mathrm{if}b\le c$$

$$1\ast a=a$$

Example 1. 

The four most common t-norms are listed below:

(1) 

Minimum t-norm: $T_{min}(a,b)=min\{a,b\}$, also called the Gödel t-norm.

(2) 

Product t-norm: $T_{prod}(a,b)=a·b$ (the ordinary product of real numbers).

(3) 

Lukasiewicz t-norm: $T_{Luk}(a,b)=max\left\{0,a+b-1\right\}$.

(4) 

Drastic t-norm:

$$\begin{array}{c}\hfill {\top }_{\mathrm{D}}(a,b)=\left\{\begin{array}{cc}b,\hfill & ifa=1\hfill \\ a,\hfill & ifb=1\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

Remark 1. 

Considering the partial order on the family of all t-norms induced by the order on $[0,1]$, $T_{min}(a,b)$ is the greatest t-norm. Therefore, it is clear that for any t-norm T

$$T(a,b)\le T_{min}(a,b)\le b,for alla,b\in [0,1].$$

Remark 2. 

The minimum t-norm is the sole idempotent t-norm, i.e., $T_{min}(a,a)=a$, for all $a\in [0,1]$.

Definition 2. 

T-conorms (also known as S-norms) are dual to t-norms under the order-reversing operation that assigns $1-a$ to a on $[0,1]$. The complementary conorm S is defined for a t-norm T by

$$S(a,b)=1-T(1-a,1-b).$$

As a result, for all $a,b,c\in [0,1]$, a t-conorm meets the following criteria, which may be utilized for a similar axiomatic definition of t-conorms without regard to t-norms:

(S1) 

(Commutativity): $S(a,b)=S(b,a)$

(S2) 

(Associativity): $S\left(S(a,b),c\right)=S\left(a,S(b,c)\right)$

(S3) 

(Monotonicity): $S(a,b)\le S(c,d)$ if $a\le c$ and $b\le d$

(S4) 

(0-identity): $S(a,0)=a$ (The number 0 acts as identity element).

Example 2. 

The following four t-conorms are common:

(1) 

Maximum t-conorm: $S_{max}(a,b)=max\{a,b\}$ is dual to the minimum t-norm.

(2) 

Probabilistic sum: $S_{sum}(a,b)=a+b-a·b=1-(1-a)·(1-b)$ is dual to the product t-norm.

(3) 

Bounded sum: $S_{Luk}(a,b)=min\{a+b,1\}$ is dual to the Lukasiewicz t-norm.

(4) 

Drastic t-conorm:

$$S_D(a,b)=\left\{\begin{array}{cc}b,\hfill & ifa=0\hfill \\ a,\hfill & ifb=0\hfill \\ 1,\hfill & otherwise,\hfill \end{array}\right.$$

is dual to the drastic t-norm.

Remark 3. 

The least t-conorm is $S_{max}(a,b)$, taking into account the point-wise order on the family of all t-conorms induced from the order $[0,1]$. Consequently, given any t-conorm S,

$$b\le S_{max}(a,b)\le S(a,b)for alla,b\in [0,1].$$

Moreover, $S_{max}(a,1)=S_{max}(1,a)=1$ is evident from the Maximum t-conorm definition, which implies that for every t-conorm S, $S(a,1)=S(1,a)=1$, $a\in [0,1]$.

Definition 3. 

For every $a,b\in [0,1]$, a fuzzy negation (complement) is a function $N:[0,1]\to [0,1]$ that meets the following properties:

(N1) 

(Preservation of constants): $N\left(0\right)=1$, $N\left(1\right)=0.$

(N2) 

(Reversing of the order): $N\left(a\right)\le N\left(b\right)$ iff $b\le a$.

In addition, a fuzzy negation N is referred to as

(i) 

strict if it is strictly decreasing and continuous;

(ii) 

strong if it is an involution, i.e., $N\left(N\left(a\right)\right)=a$, for all $a\in [0,1]$.

Example 3. 

Though there are several functions that can be employed in negation, the classical negation function $N\left(a\right)=1-a$, $a\in [0,1]$, is typically utilized in applications. The classical negation is a strong negation, $N\left(a\right)=1-a^2$ is merely strict, and the least and greatest fuzzy negations, $N_{D1}$ and $N_{D2}$, are non-strict negations:

$$\begin{array}{cc}{N}_{D1}=\left\{\begin{array}{cc}1,\hfill & ifa=0,\hfill \\ 0,\hfill & ifa>0.\hfill \end{array}\right.,& N_{D2}=\left\{\begin{array}{cc}1,\hfill & ifa<1,\hfill \\ 0,\hfill & ifa=1.\hfill \end{array}\right.\hfill \end{array}$$

For every t-norm T, t-conorm S, and classical negation function N, it is easy to see that

$$S(a,b)=N\left(T\right(N\left(a\right),N\left(b\right)\left)\right).$$

As an extension of the traditional implication operation, fuzzy implications were presented and examined in the literature in accordance with the specified truth values:

$$1\Rightarrow 1\equiv 1,1\Rightarrow 0\equiv 0,0\Rightarrow 1\equiv 1,0\Rightarrow 0\equiv 1.$$

It is well known that the implication $p\Rightarrow q$ is equivalent to $\neg p\vee q$. This is the standard definition of implication. In fuzzy logic, there are many definitions of fuzzy implications found in the literature, especially in the earlier sections. In this study, we will use Definition 4, which aligns with the definition proposed by Fodor and Roubens [25].

Definition 4. 

A binary function $I:[0,1]\times [0,1]\to [0,1]$ is called a fuzzy implication if it satisfies the following conditions:

(I1) 

I is nonincreasing in the first place,

(I2) 

I is nondecreasing in the second one,

(I3) 

$I(0,0)=I(1,1)=1$ and $I(1,0)=0$.

Note that, according to Definition 4, certain potential properties are permissible for specific fuzzy implications:

• Boundary conditions: $I(0,0)=I(0,1)=I(1,1)=1$ and $I(1,0)=0$;
• Left antitonicity: if $a\le b$ then $I(b,c)\le I(a,c)$, for all $a,b,c\in [0,1]$;
• Right isotonicity: if $a\le b$ then $I(c,a)\le I(c,b)$, for all $a,b,c\in [0,1]$;
• Left boundary condition: $I(0,b)=1$, for all $b\in [0,1]$;
• Right boundary condition: $I(a,1)=1$, for all $a\in [0,1]$;
• Identity property: $I(a,a)=1$, for all $a\in [0,1]$.

Not all fuzzy implications possess the desirable qualities of classical implications defined on ${\{0,1\}}^2$ mapping to $\{0,1\}$. While it is widely accepted that the fuzzy implication concept generalizes classical implication operations, several of these positive attributes have been assumed in prior definitions of fuzzy implications. Therefore, it is essential and intriguing to explore how these attributes interact. Numerous studies [18,26] have proposed various properties of fuzzy implications. For example, the exchange principle $\left(EP\right)$ was included in the definition of a fuzzy implication in the paper [18] (see Definition 7 below). The most significant properties are listed below.

Fuzzy implications fall into three primary categories: $(S,N)-$, $R-$, and $QL-$ implications. From these, other classes can be derived. In the definition that follows, we will examine them.

Definition 5. 

Let T be a t-norm, S a t-conorm and N a fuzzy negation, then:

• A function $I:{[0,1]}^2\to [0,1]$ is called an $(S,N)$-implication (denoted by $I_{S,N}$) if

  $$I(a,b)=S\left(N\left(a\right),b\right).$$
• A function $I:{[0,1]}^2\to [0,1]$ is called an R-implication (denoted by $I_T$) if

  $$I(a,b)=sup\left\{t\in [0,1]:T(a,t)\le b\right\}.$$
• A function $I:{[0,1]}^2\to [0,1]$ is called a $QL$-implication (denoted by $I_{S,N,T}$) if

  $$I(a,b)=S\left(N\left(a\right),T(a,b)\right).$$

However, $QL$-implications in fuzzy logic have not been studied as extensively as $(S,N)$- and R-implications. One possible reason for this is that some members of this family do not satisfy the primary requirement for a fuzzy implication, which is left antitonicity. Additionally, earlier works were limited in terms of the class of operations from which $QL$-implications could be derived, as well as the properties that these implications satisfied. This was due to restrictions placed on the fuzzy logic operations used in the construction of $QL$-implications.

3. Proposed Properties and Boolean-like Laws

In fuzzy logic, traditional Boolean laws have been extended and investigated in the setting of functional equations or inequalities, which are known as Boolean-like laws. These laws are rarely found in standard structures such as $\left(\right[0,1],T,S,N,I)$, where T represents a t-norm, S a t-conorm, N a fuzzy negation, and I a fuzzy implication. Many features that hold in classical logic may not hold in fuzzy logic when using t-norms, t-conorms, and strong negations to perform conjunctions, disjunctions, and negations, respectively. A popular area of research in fuzzy logic involves investigating which of these operators preserve certain features of classical logic. This approach is often employed to solve functional equations using various types of operators. Researchers have raised questions about the conditions under which a Boolean-like law is valid in this context.

The Law of the Excluded Middle $\left(LEM\right)$ is a fundamental principle in classical logic, stating that for any proposition p, the expression $\neg p\vee p$ is always true. However, in the context of fuzzy logic, the Law of the Excluded Middle may not always apply due to the nature of fuzzy sets and the use of truth values that range between 0 and 1. Despite this, we can extend the principle of $LEM$ within the framework of fuzzy logic.

Definition 6. 

If S is a t-conorm and N is a fuzzy negation, the pair $(S,N)$ satisfies the $LEM$ iff

$$S\left(N\right(a),a)=1,\mathit{for}\mathit{all}a\in [0,1].$$

The distributivity of disjunction over conjunction is a fundamental principle in classical logic. Its application to fuzzy logic considers t-norms and t-conorms:

A t-conorm S is said to be distributive over a t-norm T if

$$S(a,T(b,c\left)\right)=T\left(S\right(a,b),S(a,c\left)\right).$$

The following is an important finding regarding the distributive property in laws similar to Boolean algebra:

Theorem 1 

([23]). If T is a t-norm and S is a t-conorm, then S is distributive over T iff $T=T_{min}$.

Certain studies have concentrated on the distributive equations of fuzzy implications in relation to two aggregation functions [12]. The distributivity of one binary operator over another forms the fundamental structure of the algebra defined by these operators in classical logic. Overall, there are four classical distributive equations that involve implications, which are as follows:

$$\begin{array}{c}\hfill (p\vee q)\Rightarrow r\equiv (p\Rightarrow r)\wedge (q\Rightarrow r)\\ \hfill (p\wedge q)\Rightarrow r\equiv (p\Rightarrow r)\vee (q\Rightarrow r)\\ \hfill p\Rightarrow (q\wedge r)\equiv (p\Rightarrow q)\wedge (p\Rightarrow r)\\ \hfill p\Rightarrow (q\vee r)\equiv (p\Rightarrow q)\vee (p\Rightarrow r)\end{array}$$

(1)

In classical logic, the equivalences discussed are considered tautologies. Recent research has focused on the distributivity of implication functions and aggregation functions, such as t-norms and t-conorms, which are essential in the framework of logical connectives. There is a significant body of work that examines the generalizations of these distributivity equations to fuzzy connectives, particularly those involving fuzzy implications.

Fuzzy systems are well known for their capability to approximate any continuous function with a high degree of precision. However, achieving this level of accuracy typically requires a large rule base. Many researchers have investigated distributive equations to mitigate the excessive expansion of combinatorial rules. Combs and Andrews [13] were the first to propose this concept, based on the logical validity of the second equation of (1) in classical propositional logic. In 2002, Trillas and Alsina [14] explored the general form

$$I\left(T\right(a,b),c)=S\left(I\right(a,c),I(b,c\left)\right)$$

of the second equation of (1). Additionally, some studies have identified fuzzy implication solutions to the equations in (1) for specific t-norms and t-conorms [15].

Definition 7. 

$\left(i\right)$ Fuzzy implication I is considered to fulfill the left neutrality property if,

$$\begin{array}{c}\hfill I(1,b)=b,b\in [0,1]\left(NP\right)\end{array}$$

$\left(ii\right)$ Fuzzy implication I is said to satisfy the exchange principle, if

$$\begin{array}{c}\hfill I\left(a,I(b,c)\right)=I(b,I(a,c)),a,b,c\in [0,1]\left(EP\right)\end{array}$$

$\left(iii\right)$ Fuzzy implication I is said to satisfy the ordering property, if

$$\begin{array}{c}\hfill I(a,b)=1\iff a\le b,a,b\in [0,1]\left(OP\right)\end{array}$$

The ordering property $\left(OP\right)$ and the exchange principle $\left(EP\right)$ are both strong conditions. The $\left(OP\right)$ ensures that the elements in a set are well ordered, while the EP allows for the rearrangement of elements without changing the set’s properties. These principles are fundamental in various mathematical concepts and proofs. The property $\left(EP\right)$ generalizes the classical exchange principle, which is a tautology expressed as:

$$p\Rightarrow (q\Rightarrow r)\equiv q\Rightarrow (p\Rightarrow r).$$

The ordering property (OP), also known as the degree ranking property, establishes an ordering for the underlying set $[0,1]$.

Theorem 2 

([25]). If a function $I:{[0,1]}^2\to [0,1]$ satisfies $\left(EP\right)$ and $\left(OP\right)$, then I satisfies $\left(I1\right)$ and $\left(I3\right)$.

The above conclusion demonstrates that both the conditions $\left(EP\right)$ and $\left(OP\right)$ render any function $I:{[0,1]}^2\to [0,1]$ essentially a fuzzy implication. The only property lacking from an I that fulfills $\left(EP\right)$ and $\left(OP\right)$ is $\left(I2\right)$.

All $(S,N)$-implications are fuzzy implications that meet the criteria $\left(NP\right)$ and $\left(EP\right)$. However, not every $(S,N)$-implication satisfies the identity principle $\left(IP\right)$ or the ordering property $\left(OP\right)$ [16]. The following solution presents an equivalence condition under which $(S,N)$-implications fulfill these properties.

Theorem 3 

([16]). The following claims are equivalent for a fuzzy negation N and a t-conorm S:

(i) 

$\left(IP\right)$ is fulfilled by the $(S,N)$-implication $I_{S,N}$.

(ii) 

$\left(LEM\right)$ is satisfied by the pair $(S,N)$.

It is clear that the $QL$-implication derived from the triple $(S,N,T)$ does not satisfy $\left(OP\right)$ if S is a positive t-conorm. A necessary condition for a $QL$-implication to fulfill $\left(OP\right)$ is provided by the following result:

Theorem 4. 

The negation N is strictly decreasing if a $QL$-implication $I_{S,N,T}$ derived from a non-positive t-conorm S satisfies $\left(OP\right)$.

Proof. 

Assume that the property $\left(OP\right)$ is satisfied by a $QL$-implication $I_{S,N,T}$ derived from a non-positive t-conorm S. Furthermore, let there exist $a,b\in [0,1]$ such that $N\left(a\right)=N\left(b\right)$ despite $a<b$. We can infer this using the ordering property $\left(OP\right)$.

$$\begin{array}{ccc}{I}_{S,N,T}(a,b)=1\hfill & \Rightarrow & S\left(N\left(a\right),T(a,b)\right)=1\left(\text{By def. of (QL)-implication}\right)\hfill \\ & \Rightarrow & S\left(N\left(a\right),T(b,a)\right)=1\left(\text{T1-Commutativity}\right)\hfill \\ & \Rightarrow & S\left(N\left(b\right),T(b,a)\right)=1\left(\text{By Assumption}\right)\hfill \\ & \Rightarrow & I_{S,N,T}(b,a)=1\left(\text{By def. of (QL)-implication}\right)\hfill \\ & \Rightarrow & b\le a\left(\text{By the ordering property (OP)}\right).\hfill \end{array}$$

However, since it contradicts itself, the negation N is strictly decreasing. □

Examples are given by the equivalence

$$\begin{array}{c}\hfill (p\wedge q)\Rightarrow r\equiv (p\Rightarrow r)\vee (q\Rightarrow r)\end{array}$$

(2)

which prevented combinatorial rule explosion in fuzzy systems [17]. Fuzzy logic translation of Equation (2) yields

$$\begin{array}{c}\hfill I\left(T\right(a,b),c)=S\left(I\right(a,c),I(b,c\left)\right)a,b,c\in [0,1]\end{array}$$

(3)

where I is an implication function, T is a t-norm, and S is a t-conorm. In [12,14], this distributivity functional equation was solved for several types of implication functions that were obtained from t-norms and t-conorms.

The law of contraposition is one of the most important tautologies in classical two-valued logic, expressed as:

$$p\Rightarrow q\equiv \neg q\Rightarrow \neg p$$

This law is essential for proving many results by contradiction. It is meaningful to generalize this concept to fuzzy logic, utilizing fuzzy negations and fuzzy implications. When working with fuzzy implications, the contrapositive symmetry of a fuzzy implication I concerning a fuzzy negation N is crucial in fuzzy logic.

Additionally, in classical logic, the following rules are also tautologies, as the classical negation adheres to the law of double negation:

$$\neg p\Rightarrow q\equiv \neg q\Rightarrow p$$

$$p\Rightarrow \neg q\equiv q\Rightarrow \neg p$$

As a result, we can investigate several laws of contraposition in the context of fuzzy logic.

Definition 8. 

Let I be a fuzzy implication and N be a fuzzy negation.

(i) 

We say that I satisfies the law of contraposition (also known as the contrapositive symmetry) with respect to N, if

$$I(a,b)=I\left(N\right(b),N(a\left)\right),a,b\in [0,1].\left(CP\right)$$

(ii) 

We say that I satisfies the law of left contraposition with respect to N, if

$$I\left(N\right(a),b)=I\left(N\right(b),a),a,b\in [0,1].(L-CP)$$

(iii) 

We say that I satisfies the law of right contraposition with respect to N, if

$$I(a,N(b\left)\right)=I(b,N(a\left)\right),a,b\in [0,1].(R-CP)$$

If I satisfies the (left, right) contrapositive symmetry with respect to N, then we also denote this by $CP\left(N\right)$ (respectively, by $L-CP\left(N\right)$, $R-CP\left(N\right)$).

When N is a strong negation, it is clear that all three qualities are equal. Many authors have explored the classic law of contraposition (CP) [17,18]. Generally speaking, N must be a strong negation, which means that it is unnecessary to consider three separate laws of contraposition. However, if N is merely a fuzzy negation without any additional assumptions, the various principles of contraposition may not be equivalent [17].

The well-known Boolean-law in Boolean logics

$$\begin{array}{c}\hfill p\Rightarrow (q\Rightarrow p)\equiv 1\end{array}$$

(4)

is referred to as “Weakening”, since it is said to be the weakening of “$p\Rightarrow p$”. Similarly, assuming “⇒” as the material implication, we see that (4) is also a weakening formalization of Aristotle’s Law of the Excluded Middle “$\neg p\vee p$” since, “$p\Rightarrow (q\Rightarrow p)$” would be equal to “$\neg p\vee (\neg q\vee p)$”. It is possible to generalize that law (4) to the fuzzy context as

$$\begin{array}{c}\hfill I(a,I(b,a\left)\right)=1\end{array}$$

(5)

where I is a fuzzy implication. This law can be viewed as a fuzzy modification of the classic Weakening axiom from the perspective of formal logic [27,28].

One theoretical reason to examine the Equation (5) is that various definitions of fuzzy implications aim to capture the correct (common sense) understanding of what constitutes a logical implication. Consequently, exploring the framework of fuzzy implication classes can shed light on the meaning of implications in a fuzzy context. Additionally, the concept of implication across different logical systems provides a broad motivation for defining these classes. For instance, the implications found in classical and quantum logic are generalized by $(S,N)$-implication and $QL$-implication, respectively. Thus, characterizing approximate reasoning through each type of implication also benefits from analyzing the conditions under which logical laws are valid within each of these classes.

Next, we will present the necessary and sufficient conditions for the $(S,N)$-, R-, and $QL$-implications of the Boolean-like law (5) to hold. The following theorems will illustrate these fundamental properties.

Theorem 5. 

Let $T,N$ and $I_{S,N}$ be a t-norm, a fuzzy negation, and $(S,N)$-implication, respectively. An $(S,N)$-implication $I_{S,N}$ fulfills Equation (5), which extends the law (4) into the fuzzy context iff the pair $(S,N)$ satisfies $\left(LEM\right)$.

Proof. 

Firstly, we assume that an $(S,N)$-implication $I_{S,N}$ satisfies (5). It is evident from the definition of $I_{S,N}$ that

$$I(1,a)=S\left(N\left(1\right),a\right)=S(0,a)=a$$

and from the Identity property

$$I\left(a,I(1,a)\right)=I(a,a)=1.$$

Then, we obtain that

$$S\left(N\left(a\right),a\right)=S\left(N\left(a\right),I(1,a)\right)=I\left(a,I(1,a)\right)=1\text{for all}a\in [0,1].$$

Thus, the pair $(S,N)$ satisfies $\left(LEM\right)$.

We now suppose that the pair $(S,N)$ satisfies $\left(LEM\right)$, i.e., for all $a\in [0,1]$, $S\left(N\right(a),a)=1$. Then, we derive that

$$\begin{array}{ccc}I\left(a,I(b,a)\right)\hfill & =& I\left(a,S\left(N\left(b\right),a\right)\right)=S\left(N\left(a\right),S\left(N\left(b\right),a\right)\right)\left(\text{By def. of (S,N)-implication}\right)\hfill \\ & =& S\left(N\left(a\right),S\left(a,N\left(b\right)\right)\right)\left(\text{S1-Commutativity}\right)\hfill \\ & =& S\left(S\left(N\left(a\right),a\right),N\left(b\right)\right)\left(\text{S2-Associativity}\right)\hfill \\ & =& S\left(1,N\left(b\right)\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& 1\left(\text{Remark 3}\right)\hfill \end{array}$$

Thus, it has been established that $\left(I_{S,N}\right)$ satisfies Equation (5). □

The result outlined in Theorem 5 demonstrates and highlights the adaptability of classical logic principles when integrated into the framework of fuzzy logic, thereby enriching its theoretical depth. Moreover, the law (4) has been rewritten as follows:

$$\begin{array}{c}\hfill p^{\prime }\vee (q^{\prime }\vee p)\equiv p^{\prime }\vee (p\vee q^{\prime })\equiv (p^{\prime }\vee p)\vee q^{\prime }\equiv 1\vee q^{\prime }\equiv 1\end{array}$$

(6)

By extending the law (6) to the fuzzy logic domain, we obtain results that align with the principles and properties inherent to fuzzy systems:

$$\begin{array}{c}\hfill S\left(N\left(a\right),S(a,N(b\left)\right)\right)=1\end{array}$$

(7)

Corollary 1. 

Let $T,S,N$ be a t-norm, a t-conorm, and a fuzzy negation, respectively. If the pair $(S,N)$ satisfies the $\left(LEM\right)$ property, then Equation (7) is valid.

Proof. 

Given that the pair $(S,N)$ adheres to the conditions set forth by $\left(LEM\right)$, it can be inferred that

$$\begin{array}{ccc}S\left(N\left(a\right),S(a,N(b\left)\right)\right)\hfill & =& S\left(S\left(N\left(a\right),a\right),N\left(b\right)\right)\left(\text{S2-Associativity}\right)\hfill \\ & =& S\left(1,N\left(b\right)\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& 1\left(\text{Remark 3}\right).\hfill \end{array}$$

□

As a result, extending the law (6) into the fuzzy logic domain, we obtain results that align with the principles and properties inherent to fuzzy systems. This adaptation provides a theoretical framework that bridges classical logic with the nuances of fuzziness, ensuring consistency and expanding the applicability of the established law.

The following theorem demonstrates that Equation (5) is satisfied by every R-implication.

Theorem 6. 

The Equation (5) holds for every R-implication.

Proof. 

Let T be any t-norm and $I_T$ an R-implication generated from it. By Remark 1, we know that fixing arbitrarily $a,b\in [0,1]$, $T(b,a)\le min\{b,a\}\le a$. By definition of R-implication, the formula

$$I(b,a)=sup\{t\in [0,1]:T(b,t)\le a\}$$

holds that $a\le I(b,a)$. Then, taking into account I-Right isotonicity, we have

$$\begin{array}{c}I(c,a)\le I\left(c,I(b,a)\right)\text{for all}c\in [0,1]\end{array}$$

or particularly, we can write that

$$I(a,a)\le I\left(a,I(b,a)\right).$$

Since $I(a,a)=1$, we obtain $I\left(a,I(b,a)\right)=1$. It means that R-implication satisfies (5). □

Theorem 7. 

If $(S,N)$ satisfies $\left(LEM\right)$ and $T=T_{min}$, then $QL$-implication $I_{S,N,T}$ satisfies (5).

Proof. 

Assume that $(S,N)$ satisfies $\left(LEM\right)$ and $T=T_{min}$. By Lemma 1, S is distributive over T iff $T=T_{min}$. Therefore, for a $QL$-implication $I_{S,N,T}$, we write that

$$\begin{array}{ccc}I\left(a,I(b,a)\right)\hfill & =& S\left(N\left(a\right),T\left(a,I(b,a)\right)\right)\hfill \\ & =& S\left(N\left(a\right),T\left(a,S\left(N\left(b\right),T(b,a)\right)\right)\right)\left(\text{By def. of QL-implication}\right)\hfill \\ & =& S\left(N\left(a\right),T\left(a,T\left(S\left(N\left(b\right),b\right),S\left(N\left(b\right),a\right)\right)\right)\right)\left(\text{S is distributive over T}\right)\hfill \\ & =& S\left(N\left(a\right),T\left(a,T\left(1,S\left(N\left(b\right),a\right)\right)\right)\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& S\left(N\left(a\right),T\left(a,S\left(N\left(b\right),a\right)\right)\right)\left(\text{T4-identity}\right)\hfill \\ & =& T\left(S\left(N\left(a\right),a\right),S\left(N\left(a\right),S\left(N\left(b\right),a\right)\right)\right)\left(\text{S is distributive over T}\right)\hfill \\ & =& T\left(1,S\left(N\left(a\right),S\left(N\left(b\right),a\right)\right)\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& S\left(N\left(a\right),S\left(N\left(b\right),a\right)\right)\left(\text{T4-identity}\right)\hfill \\ & =& S\left(N\left(a\right),S\left(a,N\left(b\right)\right)\right)\left(\text{S1-Commutativity}\right)\hfill \\ & =& S\left(S\left(N\left(a\right),a\right),N\left(b\right)\right)\left(\text{S2-Associativity}\right)\hfill \\ & =& S\left(1,N\left(b\right)\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& 1\left(\text{Remark 3}\right).\hfill \end{array}$$

Thus, we obtain the desired result. □

Theorem 8. 

If the $QL$-implication $I_{S,N,T}$ is constructed using a strictly increasing t-conorm S on $[0,1]$, a t-norm T, and a fuzzy negation N, and it satisfies the condition (5), then it follows that the pair $(S,N)$ fulfills $\left(LEM\right)$, and $T=T_{min}$.

Proof. 

Suppose that t-conorm S strictly increasing in $[0,1)$, a t-norm T, a fuzzy negation N and $QL$-implication $I_{S,N,T}$ satisfy the Equation (5). Therefore, for any $b\in [0,1]$, we can write that

$$\begin{array}{ccc}\hfill 1=I\left(1,I(b,1)\right)& =& S\left(N\left(1\right),T\left(1,I(b,1)\right)\right)\hfill \\ & =& S\left(N\left(1\right),T\left(1,S\left(N\left(b\right),T(b,1)\right)\right)\right)\left(\text{By def. of QL-implication}\right)\hfill \\ & =& S\left(0,S\left(N\left(b\right),b\right)\right)\left(\text{N1-Preservation of constants and T4-identity}\right)\hfill \\ & =& S\left(N\left(b\right),b\right)\left(\text{S4-identity}\right).\hfill \end{array}$$

Thus, $(S,N)$ satisfies $\left(LEM\right)$. From assumption, since $QL$-implication $I_{S,N,T}$ satisfies the Equation (5), we have $I\left(a,I(1,a)\right)=1$. Therefore, we deduce that

$$\begin{array}{ccc}\hfill 1=I\left(a,I(1,a)\right)& =& S\left(N\left(a\right),T\left(a,I(1,a)\right)\right)\hfill \\ & =& S\left(N\left(a\right),T\left(a,S\left(N\left(1\right),T(1,a)\right)\right)\right)\left(\text{By def. of QL-implication}\right)\hfill \\ & =& S\left(N\left(a\right),T\left(a,S\left(0,a\right)\right)\right)\left(\text{N1 and T4-identity}\right)\hfill \\ & =& S\left(N\left(a\right),T(a,a)\right)\left(\text{S4-identity}\right).\hfill \end{array}$$

Moreover, $(S,N)$ satisfies $\left(LEM\right)$, for any $a\in [0,1]$, we have

$$\begin{array}{c}\hfill S\left(N\left(a\right),T(a,a)\right)=1=S\left(N\left(a\right),a\right).\end{array}$$

(8)

Now, for case $a=1$ yields $T(a,a)=a$. And for case $a\in [0,1)$, since S is strictly increasing t-conorm in $[0,1)$

$$\begin{array}{c}\hfill S\left(N\left(a\right),T(a,a)\right)=S\left(N\left(a\right),a\right)\end{array}$$

implies $T(a,a)=a$. Consequently, we have proved for any $a\in [0,1]$ that $T(a,a)=a$. It means that T is an idempotent t-norm. We know that $T_{min}$ is the only idempotent t-norm. Thus, we obtain $T=T_{min}$. □

Theorem 9. 

Let $I_{S,N,T}$ be a $QL$-implication generated by a strictly increasing t-conorm S in $[0,1)$, a fuzzy negation N and a t-norm T. $QL$-implication $I_{S,N,T}$ satisfies (5) iff $(S,N)$ satisfies $\left(LEM\right)$ and $T=T_{min}$.

Proof. 

Straightforward from Theorems 7 and 8. □

These results indicate that the generalization of a classical implication to a fuzzy context—a $(S,N)$-implication—satisfies Equation (5) if, and only if, $(S,N)$ adheres to $\left(LEM\right)$.

The following example illustrates that the conditions of the established theorems align with the well-known results of classical logic.

Example 4. 

In classical logic, stating that for any proposition p and q, the following expression is a tautology

$$\begin{array}{c}\hfill \left[(p\Rightarrow q)\wedge q^{\prime }\right]\Rightarrow p^{\prime }\end{array}$$

(9)

Indeed, according to the definition of implication, by employing the law of classical negation, the principle of distributivity, and Aristotle’s Law of the Excluded Middle, this equivalence can be established:

$$\begin{array}{ccc}\left[(p\Rightarrow q)\wedge q^{\prime }\right]\Rightarrow p^{\prime }\hfill & \equiv & {\left[(p^{\prime }\vee q)\wedge q^{\prime }\right]}^{\prime }\vee p^{\prime }\equiv \left[(p\wedge q^{\prime })\vee q\right]\vee p^{\prime }\hfill \\ & \equiv & \left[(p\vee q)\wedge (q^{\prime }\vee q)\right]\vee p^{\prime }\equiv \left[(p\vee q)\wedge 1\right]\vee p^{\prime }\hfill \\ & \equiv & (p\vee q)\vee p^{\prime }\equiv (p\vee p^{\prime })\vee q\hfill \\ & \equiv & 1\vee q\equiv 1\hfill \end{array}$$

Now, the conditions for translating Equation (9) to Formula (10) within the fuzzy logic framework can be established. These conditions will outline the necessary relationships and properties required for the translation to hold, ensuring consistency with the underlying principles of fuzzy systems.

Theorem 10. 

Let $T,S,N$ and $I_{S,N}$ be a t-norm, t-conorm, a fuzzy negation, and $(S,N)$-implication, respectively. If $(S,N)$ fulfills $\left(LEM\right)$ and $T=T_{min}$, then, the subsequent translation is valid:

$$\begin{array}{c}\hfill I\left(T\left(I\left(a,b\right),N\left(b\right)\right),N\left(a\right)\right)=1,a,b\in [0,1]\end{array}$$

(10)

Proof. 

To prove that, by using definition of $(S,N)$-implication, we conclude that

$$\begin{array}{ccc}I\left(T\left(I\left(a,b\right),N\left(b\right)\right),N\left(a\right)\right)\hfill & =& I\left(T\left(S\left(N\left(a\right),b\right),N\left(b\right)\right),N\left(a\right)\right)\hfill \\ & =& S\left(N\left(T\left(S\left(N\left(a\right),b\right),N\left(b\right)\right)\right),N\left(a\right)\right)\hfill \\ & =& S\left(S\left(T\left(a,N\left(b\right)\right),b\right),N\left(a\right)\right)\left(\text{By def. of classical negation}\right)\hfill \\ & =& S(T\left(S\left(a,b),S\left(N\left(b\right),b\right)\right),N\left(a\right)\right)\left(\text{the distributive property}\right)\hfill \\ & =& S(T\left(S\left(a,b),1\right),N\left(a\right)\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& S\left(S(a,b),N\left(a\right)\right)\left(\text{T4-identity}\right)\hfill \\ & =& S\left(a,S\left(b,N\left(a\right)\right)\right)\left(\text{S2-Associativity}\right)\hfill \\ & =& S\left(a,S\left(N\left(a\right),b\right)\right)\left(\text{S1-Commutativity}\right)\hfill \\ & =& S\left(S\left(a,N\left(a\right)\right),b\right)\left(\text{S2-Associativity}\right)\hfill \\ & =& S\left(1,b\right)\left(\text{(S,N) satisfies (LEM)}\right)\hfill \\ & =& 1\left(\text{Remark 3}\right)\hfill \end{array}$$

□

4. Discussion

This paper delves into the connections between classical and fuzzy logic properties, offering a comprehensive framework for analysis. It demonstrates their application through t-norms, t-conorms, fuzzy negations, and implications. Key principles from classical logic—such as the Law of the Excluded Middle, the distributive property of disjunction over conjunction, and the law of contraposition—can be generalized by adapting the combinatorial rules of classical equations to a fuzzy logic context. We examine the well-known Boolean law (4) by converting it into a fuzzy context using Equation (5), addressing how specific classical properties may not hold equivalently when N is treated solely as a fuzzy negation without additional assumptions. We outline the conditions for various properties of this generalization to be valid within fuzzy logic. It allows for the extension of logical rules by reinterpreting classical combinatorial equations within fuzzy frameworks. Moreover, the provided examples underscore the practical advantages of this methodology and its impact across diverse applications. The study also highlights the broader potential of fuzzy logic to address complexities beyond the scope of traditional Boolean logic by providing a more in-depth explanation of the subject.

5. Conclusions

The study focuses on the critical role that Boolean laws play in connecting classical logic and fuzzy set theory. We establish the conditions under which these generalizations are valid within fuzzy logic by utilizing t-norms, t-conorms, negation operators, and fuzzy implications. By exploring Boolean laws alongside fuzzy logic properties, the study provides readers with valuable insights into the adaptability and scope of these concepts across various mathematical structures. We give examples to demonstrate the methodology’s strengths and practical advantages over established techniques. For further research, it can be investigated whether some properties that are valid in classical logic can also be applied to fuzzy logic. If certain principles are incompatible with fuzzy logic, necessary and sufficient conditions can be developed to ensure the reliability of these generalizations.