Zadeh (Zadeh, 1965) has also proved these laws.

Similarly, for any *x* ∈ *X*,

operation of Zadeh's theory.

equals either to zero or one, or is between them.

like postulates. The result is as follows.

This competes the proof.

*Ac* = *X A* where *A* ⊂ *X* and *Ac* is the complement of *A*.

*set theory.*

*Proof.*

Let *X* be a nonempty set. Consider a set of functions *μ* : *X* −→ **I**, i.e., the function set

$$\mathbb{I}^{X} = \{ \mu \mid \mu: X \longrightarrow \mathbb{I} \}\tag{3.6}$$

We extend the algebra of Theorem 3.1 into an *algebra of functions* (3.6)

$$L\_{\mathbf{I}^X} = \langle \mathbf{I}^X, \vee, \wedge, \neg, \mathbf{0}, \mathbf{1} \rangle \tag{3.7}$$

by pointwise calculation. Here **0** and **1** are constant functions, such that

$$\forall \mathbf{x} \in \mathbf{X}, \quad \mathbf{0}: \mathbf{x} \mapsto \mathbf{0}, \quad \mathbf{1}: \mathbf{x} \mapsto \mathbf{1} \tag{3.8}$$

The algebra (3.7) is a De Morgan algebra by its construction. This means that we calculate expressions *μ*(*x*) ∨ *ν*(*x*), *μ*(*x*) ∧ *ν*(*x*), ¬*μ*(*x*) etc. pointwise for any *x* ∈ *X*. Hence, the formulas (3.3) and (3.4) are applicable also in the function algebra (3.7).

As a special case, the algebra (3.7) has a subalgebra

$$L\_{\{0,1\}^X} = \langle \{0,1\}^X \rangle \text{max} \langle \text{min} \rangle \ \neg \text{,0} \ \mathbf{1} \rangle \tag{3.9}$$

being an algebra of characteristic functions of classical sets, *f* : *X* −→ {0, 1}. Sometimes we write **2** instead of {0, 1}, so, especially,

$$\mathbf{2}^{X} = \{ f\_{A} \mid f\_{A} : X \longrightarrow \{ 0, 1 \}, A \subset X \}$$

is the classical power set of a set *X* expressed by characteristic functions. The characteristic function of a given set *A* ⊂ *X*, *fA*, is the function

$$f\_A(\mathfrak{x}) = \begin{cases} 1 & \text{if} \quad \mathfrak{x} \in A\_{\sigma} \\ 0 & \text{if} \quad \mathfrak{x} \notin A\_{\sigma} \end{cases}$$

This function indicates by the value *fA*(*x*) = 1 that the element *x* ∈ *X* is an element of *A* and all the elements of *X* having the value *fA*(*x*) = 0 are elements of the complement of *A*. As a subalgebra of the algebra (3.7), the algebra (3.9) is a special De Morgan algebra, namely a *Boolean algebra*.

### **4. Algebra of standard fuzzy sets**

Consider the algebra (3.7). We may give a new label to it and use operation symbols max and min instead of ∨ and ∧, respectively, by the formulas (3.3) and (3.4). Hence, we have

$$\mathcal{Z}\_{\mathbb{N}\_1} = \langle \mathbb{I}^X, \mathbf{max}, \mathbf{min}, \neg, \mathbf{0}, \mathbf{1} \rangle \tag{4.1}$$

The subscript ℵ1 means the cardinality of continuum, so, **I***X* is a continuum because **I** is continuum, too. For short, we may refer to Zℵ1 by Z, without the subscript, if there is no possibility for confusion. The complementarity operation ¬ is a mapping

$$\neg : \mathbb{I}^X \longrightarrow \mathbb{I}^X \text{, } \mu \mapsto \mathbf{1} - \mu \tag{4.2}$$

Hence, the complement function of a function *μ* is **1** − *μ*, such that for all *x* ∈ *X*, (**1** − *μ*)(*x*) = **1**(*x*) − *μ*(*x*) = 1− *μ*(*x*). (The proof, that ¬ defined in this way is really a membership function, is given in the proof of Theorem 4.1.) This thing is analogous to the classical set complement expressed by subtraction a set *A* to be complemented from the universe of discourse *X*, i.e., *Ac* = *X A* where *A* ⊂ *X* and *Ac* is the complement of *A*.

The operations max and min are clearly commutative. Based on the fact that the algebra (3.7) is De Morgan algebra, the algebra (4.1) is De Morgan algebra, too. We call this algebra *Zadeh algebra* because it is an algebraic description of standard fuzzy set theory, similarly as in classical set theory, a certain Boolean algebra (set algebra or algebra of characteristic functions) is the algebraic description of the system of classical sets. Now, we have the following

**Theorem 4.1.** *Zadeh algebra* Z = �**I***X*, max, min, ¬, **0**, **1**� *is an algebraic approach to standard fuzzy set theory.*

*Proof.*

6 Will-be-set-by-IN-TECH

The algebra (3.7) is a De Morgan algebra by its construction. This means that we calculate expressions *μ*(*x*) ∨ *ν*(*x*), *μ*(*x*) ∧ *ν*(*x*), ¬*μ*(*x*) etc. pointwise for any *x* ∈ *X*. Hence, the formulas

being an algebra of characteristic functions of classical sets, *f* : *X* −→ {0, 1}. Sometimes we

**2***X* = { *fA* | *fA* : *X* −→ {0, 1}, *A* ⊂ *X*}

is the classical power set of a set *X* expressed by characteristic functions. The characteristic

This function indicates by the value *fA*(*x*) = 1 that the element *x* ∈ *X* is an element of *A* and all the elements of *X* having the value *fA*(*x*) = 0 are elements of the complement of *A*. As a subalgebra of the algebra (3.7), the algebra (3.9) is a special De Morgan algebra, namely a

Consider the algebra (3.7). We may give a new label to it and use operation symbols max and

The subscript ℵ1 means the cardinality of continuum, so, **I***X* is a continuum because **I** is continuum, too. For short, we may refer to Zℵ1 by Z, without the subscript, if there is no

Hence, the complement function of a function *μ* is **1** − *μ*, such that for all *x* ∈ *X*, (**1** − *μ*)(*x*) = **1**(*x*) − *μ*(*x*) = 1− *μ*(*x*). (The proof, that ¬ defined in this way is really a membership function,

min instead of ∨ and ∧, respectively, by the formulas (3.3) and (3.4). Hence, we have

possibility for confusion.  Consider a set of functions *μ* : *X* −→ **I**, i.e., the function set

We extend the algebra of Theorem 3.1 into an *algebra of functions* (3.6)

by pointwise calculation. Here **0** and **1** are constant functions, such that

(3.3) and (3.4) are applicable also in the function algebra (3.7).

As a special case, the algebra (3.7) has a subalgebra

function of a given set *A* ⊂ *X*, *fA*, is the function

write **2** instead of {0, 1}, so, especially,

**4. Algebra of standard fuzzy sets**

*Boolean algebra*.

max{*μ*, *ν*} = max{*ν*, *μ*} and max{*μ*, max{*ν*, *τ*}} = max{max{*μ*, *ν*}, *τ*}

for all *μ*, *ν*, *τ* ∈ **I***X* because these laws clearly hold for the elements of **I**, and these laws can be embedded to **I***X* by pointwice calculation of values of the functions *μ* ∈ **I***X* (*cf.* Lowen (Lowen, 1996)). The same properties hold for min, too.

$$(\max\{\mu, \mathbf{0}\})(\mathbf{x}) = \max\{\mu(\mathbf{x}), \mathbf{0}(\mathbf{x})\} = \max\{\mu(\mathbf{x}), \mathbf{0}\} = \mu(\mathbf{x})$$

Similarly, for any *x* ∈ *X*,

$$(\min\{\mu, \mathbf{0}\})(\mathbf{x}) = \mu(\mathbf{x})$$

(v) For any membership function *μ* ∈ **I***X*, there exists ¬*μ* ∈ **I***X*, such that for any *x* ∈ *X*,

$$(\neg \mu)(\mathbf{x}) = (\mathbf{1} - \mu)(\mathbf{x}) = \mathbf{1}(\mathbf{x}) - \mu(\mathbf{x}) = 1 - \mu(\mathbf{x})$$

taking values from the unit interval [0, 1]. Hence, ¬*μ* ∈ **I***X*, and ¬ is the complementarity operation of Zadeh's theory.

(vi) Clearly, Zadeh algebra Z satisfies the condition **0** �= **1**, by (iv). Hence, **2***X* ⊂ **I***X*. The constant functions **0** and **1** are the zero element and unit element of the algebra.

This competes the proof.

In classical set theory, an element either is or is not an element of a given set. In fuzzy set theory, we have three possibilities: a membership grade of an element in a given fuzzy set equals either to zero or one, or is between them.

For practical use, we may postulate Zadeh algebra by collecting the nevessary properties together. This means that we build Theor. 4.1 again using the main laws and properties like postulates. The result is as follows.

$$\mathbf{0}$$

Zadeh-algebra as a special case of De Morgan algebras give rise to closer analysis. Here we have done some part of it. The author thinks that Prof. Zadeh did not necessarily think about De Morgan algebras, when he created his crucial paper "Fuzzy Sets" (Zadeh, 1965). He thought the problem from another point of view, as can be seen in the construction of the paper. His leading idea was to model things in the eventful real world. In any way, it was a happy event that Prof. Zadeh's ideas met such a mathematical frame we have considered here. No others have been so successful to find such a *right interpretation* to some formal tools for modeling real world incidences. In the same time the *problem of interpretation* of many-valued logic got a solution. Many-valued logic began to give meaningful tools for analyzing and modeling things in real world. The role of many-valued logics were very nominal before Prof. Zadeh invented fuzzy set theory. After this, the study of many-valued logic met a new rise. Fuzzy set theory and fuzzy logic has helped the researchers to find new aspects from already

Standard Fuzzy Sets and some Many-Valued Logics 83

existing mathematical theories. This kind of work is now going on very strongly.

For example, imagine a set of beautiful women. Let us denote this set by *A*. There are women who do not belong to *A* with the highest grade 1. So, such a woman *does not* have some features which would make her beautiful. But she may have some of those features anyway. An intuitive hint about a possible answer to the question: "Where is the hiding-place of fuzziness?" can be found just on the second line above: "... *does not* ..." It seems that a partial

Let us compare Zadeh algebra with a general Boolean algebra with a supposition that the binary operations are associative because associativity holds in Zadeh algebra. The definition

**Definition 5.1.** Let ∧ (*meet*) and ∨ (*join*) be binary operations, and � (*complement*) a unary operation on a set *B*(�= ∅), and let **0** and **1** be the elements of *B*, such that the following

∀*x*, *y*, *z* ∈ *B*, *x* ∧ (*y* ∨ *z*)=(*x* ∧ *y*) ∨ (*x* ∧ *z*),

(BA4) ∀*x* ∈ *B*, *x* ∨ **0** = *x* and *x* ∧ **1** = *x*, i.e., **0** and **1** are the neutral elements (or identity

(BA5) For every element *x* ∈ *B* there exists an element *x*� ∈ *B*, such that *x* ∨ *x*� = **1** and

The only *structural difference* between these algebras is that between the axioms Z5 and BA 5. BA 5 is characteristic for complement operation, but Z5 does not satisfy the conditions of complement. So, fuzziness lies in the axiom Z5. The influence of this axiom is that also other

Hence, the set *B* together with these operations forms a *Boolean algebra* B = (*B*, ∨, ∧, �

*x* ∨ (*y* ∧ *z*)=(*x* ∨ *y*) ∧ (*x* ∨ *z*).

, **0**, **1**).

(BA1) ∧ and ∨ are commutative in *B*, i.e., ∀*x*, *y* ∈ *B*, *x* ∨ *y* = *y* ∨ *x* ja *x* ∧ *y* = *y* ∧ *x*;

**5. Where is the hiding-place of fuzziness?**

axioms hold:

*x* ∧ *x*� = **0**.

complementarity is somehow involved in this problem.

of this kind of Boolean algebra can be postulated as follows.

(BA2) The operations ∧ and ∨ are associative in *B*; (BA3) The operations ∧ and ∨ are distributive, i.e.,

elements) of the operations ∨ and ∧.

(BA6) For the elements **0** and **1** of *B* the condition **0** �= **1** holds.

**Proposition 4.1.** *Let* **I***X* = {*μ* | *μ* : *X* −→ **I**} *be the set of all functions from X to* **I***, where the operations* max *and* min *are pointwise defined between membership functions, and* ¬*μ* def = **1** − *μ. Then* Z = �**I***X*, max, min, ¬, **0**, **1**� *is Zadeh algebra if it satisfies the conditions*

$$(\mathcal{Z}6) \quad \mathbf{0} \neq \mathbf{1}.$$

**Definition 4.1** (Kleene algebra)**.** De Morgan algebra is *Kleene algebra* if it satisfies the additional condition

(K) *x* ∧ ¬*x* ≤ *y* ∨ ¬*y*.

**Theorem 4.2.** *Zadeh algebra (4.1) is a Kleene algebra.*

*Proof.* Zadeh algebra is De Morgan algebra. The condition (K) in Zadeh algebra has the form

$$\min\{\mu, \neg \mu\} \le \max\{\nu, \neg \nu\}$$

for all *μ*, *ν* ∈ **I***X*.

To prove this, we can easily show that always min{*μ*, ¬*μ*} ≤ 1 2 and 1 2 ≤ max{*ν*, ¬*ν*} for arbitrary *μ*, *ν* ∈ **I***X*, where the result follows immediately.

An alternative way is an easy task to check the four cases: (1◦) *μ* ≤ 1 2 , *ν* ≤ 1 2 , (2◦) *μ* ≤ 1 2 , *ν* > 1 2 , (3◦) *μ* > 1 2 , *ν* ≤ 1 2 , and (4◦) *μ* > 1 2 , *ν* > 1 2 , and find out that each of these cases satisfies the condition (K).

Zadeh algebra Zℵ1 has subalgebras which are Zadeh algebras, too. A range of membership functions can be a suitable subset of the unit interval [0, 1], such that the postulates of Prop. 4.1 are satisfied. Here the suitability means that the set is closed under the operations of the algebra.

**Example 4.1.** Consider a set *A* = {0, 1} which is a subset of [0, 1] consisting of the extreme cases of the unit interval. The algebra �*AX*, max, min, ¬, **0**, **1**� satisfies the conditions of Zadeh algebra. This algebra is really an extreme case, because it is the Boolean algebra of characteristic functions of strict (i.e., usual) sets. It is a subalgebra of Zℵ1 .

**Example 4.2.** Consider a set *A* = {0, 1 2 , 1} being a subset of [0, 1]. The set *A* is the range of functions *μ* : *X* −→ *A* where *X* �= ∅ is a set. These functions belong to the set **I***X*, by means of which *A* is a subset of **I***X*. The conditions of Prop. 4.1 are clearly satisfied. Hence, Z3 = �*AX*, max, min, ¬, **0**, **1**� is a 3-valued Zadeh algebra, and hence, a subalgebra of Zℵ1 .

**Example 4.3.** Consider a set *A* consisting of all the rationals from the unit interval [0, 1]. The number of the elements of *A* is countable, but infinite. Hence, the cardinality of *A* is ℵ0. Making similar considerations as in the previous example, we verify that Zℵ0 = �*AX*, max, min, ¬, **0**, **1**� is a subalgebra of Zℵ1 .

8 Will-be-set-by-IN-TECH

**Proposition 4.1.** *Let* **I***X* = {*μ* | *μ* : *X* −→ **I**} *be the set of all functions from X to* **I***, where the*

(Z4) *The neutral elements of the operations* max *and* min *are* **0** *and* **1***, respectively, i.e., for all μ* ∈ **I***X,*

**Definition 4.1** (Kleene algebra)**.** De Morgan algebra is *Kleene algebra* if it satisfies the additional

*Proof.* Zadeh algebra is De Morgan algebra. The condition (K) in Zadeh algebra has the form

min{*μ*, ¬*μ*} ≤ max{*ν*, ¬*ν*}

Zadeh algebra Zℵ1 has subalgebras which are Zadeh algebras, too. A range of membership functions can be a suitable subset of the unit interval [0, 1], such that the postulates of Prop. 4.1 are satisfied. Here the suitability means that the set is closed under the operations of the

**Example 4.1.** Consider a set *A* = {0, 1} which is a subset of [0, 1] consisting of the extreme cases of the unit interval. The algebra �*AX*, max, min, ¬, **0**, **1**� satisfies the conditions of Zadeh algebra. This algebra is really an extreme case, because it is the Boolean algebra of

of functions *μ* : *X* −→ *A* where *X* �= ∅ is a set. These functions belong to the set **I***X*, by means of which *A* is a subset of **I***X*. The conditions of Prop. 4.1 are clearly satisfied. Hence, Z3 = �*AX*, max, min, ¬, **0**, **1**� is a 3-valued Zadeh algebra, and hence, a subalgebra of Zℵ1 . **Example 4.3.** Consider a set *A* consisting of all the rationals from the unit interval [0, 1]. The number of the elements of *A* is countable, but infinite. Hence, the cardinality of *A* is ℵ0. Making similar considerations as in the previous example, we verify that Zℵ0 =

2 and 1

2 , and find out that each of these cases satisfies the

2 , 1} being a subset of [0, 1]. The set *A* is the range

2 , *ν* ≤ 1

2 ≤ max{*ν*, ¬*ν*} for

2 , (2◦) *μ* ≤ 1

2 , *ν* >

(Z5) *For any membership function μ* ∈ **I***X, there exists* ¬*μ* ∈ **I***X, such that* (¬*μ*) = 1 − *μ;*

def = **1** − *μ.*

*operations* max *and* min *are pointwise defined between membership functions, and* ¬*μ*

*Then* Z = �**I***X*, max, min, ¬, **0**, **1**� *is Zadeh algebra if it satisfies the conditions*

(Z1) *The operations* max *and* min *are commutative on* **I***X;* (Z2) *The operations* max *and* min *are associative on* **I***X;*

max{*μ*, **0**} = *μ and* min{*μ*, **1**} = *μ;*

**Theorem 4.2.** *Zadeh algebra (4.1) is a Kleene algebra.*

To prove this, we can easily show that always min{*μ*, ¬*μ*} ≤ 1

An alternative way is an easy task to check the four cases: (1◦) *μ* ≤ 1

characteristic functions of strict (i.e., usual) sets. It is a subalgebra of Zℵ1 .

2 , *ν* > 1

arbitrary *μ*, *ν* ∈ **I***X*, where the result follows immediately.

2 , and (4◦) *μ* > 1

(Z6) **0** �= **1***.*

(K) *x* ∧ ¬*x* ≤ *y* ∨ ¬*y*.

for all *μ*, *ν* ∈ **I***X*.

2 , (3◦) *μ* > 1

condition (K).

algebra.

2 , *ν* ≤ 1

**Example 4.2.** Consider a set *A* = {0, 1

�*AX*, max, min, ¬, **0**, **1**� is a subalgebra of Zℵ1 .

1

condition

(Z3) *The operations* max *and* min *are distributive to each other;*

Zadeh-algebra as a special case of De Morgan algebras give rise to closer analysis. Here we have done some part of it. The author thinks that Prof. Zadeh did not necessarily think about De Morgan algebras, when he created his crucial paper "Fuzzy Sets" (Zadeh, 1965). He thought the problem from another point of view, as can be seen in the construction of the paper. His leading idea was to model things in the eventful real world. In any way, it was a happy event that Prof. Zadeh's ideas met such a mathematical frame we have considered here. No others have been so successful to find such a *right interpretation* to some formal tools for modeling real world incidences. In the same time the *problem of interpretation* of many-valued logic got a solution. Many-valued logic began to give meaningful tools for analyzing and modeling things in real world. The role of many-valued logics were very nominal before Prof. Zadeh invented fuzzy set theory. After this, the study of many-valued logic met a new rise. Fuzzy set theory and fuzzy logic has helped the researchers to find new aspects from already existing mathematical theories. This kind of work is now going on very strongly.

### **5. Where is the hiding-place of fuzziness?**

For example, imagine a set of beautiful women. Let us denote this set by *A*. There are women who do not belong to *A* with the highest grade 1. So, such a woman *does not* have some features which would make her beautiful. But she may have some of those features anyway. An intuitive hint about a possible answer to the question: "Where is the hiding-place of fuzziness?" can be found just on the second line above: "... *does not* ..." It seems that a partial complementarity is somehow involved in this problem.

Let us compare Zadeh algebra with a general Boolean algebra with a supposition that the binary operations are associative because associativity holds in Zadeh algebra. The definition of this kind of Boolean algebra can be postulated as follows.

**Definition 5.1.** Let ∧ (*meet*) and ∨ (*join*) be binary operations, and � (*complement*) a unary operation on a set *B*(�= ∅), and let **0** and **1** be the elements of *B*, such that the following axioms hold:

$$\begin{aligned} \forall \mathbf{x}, \boldsymbol{y}, \boldsymbol{z} \in B, \quad & \mathbf{x} \wedge (\boldsymbol{y} \vee \boldsymbol{z}) = (\mathbf{x} \wedge \boldsymbol{y}) \vee (\mathbf{x} \wedge \boldsymbol{z}),\\ \mathbf{x} \vee (\boldsymbol{y} \wedge \boldsymbol{z}) = (\mathbf{x} \vee \boldsymbol{y}) \wedge (\mathbf{x} \vee \boldsymbol{z}).\end{aligned}$$

Hence, the set *B* together with these operations forms a *Boolean algebra* B = (*B*, ∨, ∧, � , **0**, **1**).

The only *structural difference* between these algebras is that between the axioms Z5 and BA 5. BA 5 is characteristic for complement operation, but Z5 does not satisfy the conditions of complement. So, fuzziness lies in the axiom Z5. The influence of this axiom is that also other

propositional wff's. Starting with propositional letters, we can combine them by connectives in the way shown by the production system (6.1). And finally, we can combine any formulas

Standard Fuzzy Sets and some Many-Valued Logics 85

We can refer to the propositional letters also by lower case letters in general, and to combined formulas by lower case Greek letters or by usual capital letters. These letters belong to the metalanguage we use when we discuss and describe object language. Here we use English

The definitions 6.1 and 6.2 above defines the language of propositional logic. Second, we consider some central semantical concepts being necessary for our consideration. This means that we will not present the whole machinery of formal semantics of standard many-valued

We have two important functions, *valuation* and *truth-function* we need in our consideration.

that associates truth values to propositional letters. **Prop** is a set of propositional variables.

where *n* is the number of propositional variables *p*, *q*, . . . in the formula defining a truth

In general, truth-functions are functions of several variables defined on the *n*-tuple of the

subindices *n* = 1, 2, 3, . . . are usually dropped. Actually, a propositional formula *ϕ* itself is a truth-function. Suppose that a formula *ϕ* consists of the propositional letters *p*, *q*, and *r*. Then we may write *ϕ* = *V*(*p*, *q*,*r*). The equality sign is used only between truth-functions and

In connected formulas, valuations of propositional variables give the values for the variables of the corresponding truth-function presented by the connected formula. Hence, a "valuation" of a connected formula is the value of the corresponding truth-function. Hence, to evaluate a truth value of the whole connected formula corresponding a given valuation for propositional variables, we calculate the value of the truth-function where the given valuation *v* first determines the values of the arguments of the truth-function. We may denote the truth value of a connected formula *ϕ* by *V*(*ϕ*), being like a valuation depending on a given valuation *v*

**Example 6.1.** Evaluate the truth value of a formula *p* ∧ (¬*q* ∨ *r*) with regard to a given valuation *v* for *p*, *q*, and *r*. Actually the formula is a truth-function *f*(*p*, *q*,*r*) = *p* ∧ (¬*q* ∨ *r*) where *p*, *q*, and *r* obtain their values from [0, 1]. These values are *v*(*p*), *v*(*q*), and *v*(*r*). Now,

*V*(*p* ∧ (¬*q* ∨ *r*)) = *v*(*p*) ∧ (¬*v*(*q*) ∨ *v*(*r*)

the truth value of the formula,given by this valuation *v*, is

*v* : **Prop** −→ **I** (6.2)

*Vn* : **I***n* −→ **I**, *n* = 1, 2, 3, . . . , (6.3)

*n* where the independent variables are proposition letters. The

equipped by some formal symbols (so-called meta-symbols) as a metalanguage.

according to the production system.

They are defined as follows.

*Truth-function* is a function

set of truth values, [0, 1]

for propositional letters.

**Definition 6.3.** *Valuation v* is a function

logics.

function.

truth values.

values can be considered as membership degrees than only 0 and 1. In Boolean algebras with the universe of discourse {0, 1} the postulate BA5 do not cause conflicts, like the intermediate values may do if these values are added to the universe.

Because complement operation satisfies the conditions of strong negation, a Boolean algebra B = ({0, 1}, ∨, ∧, � , **0**, **1**) is a special case of Z, i.e., the classical case is included in Z. Trivially, {0, 1} ⊂ [0, 1]. This means that crisp sets are special cases of fuzzy sets, as they should be also according to Zadeh's own theory. See also the proof of Theorem 2.1.

We may conclude that formally the core hiding-place of fuzziness is the statement Z5 in Proposition 4.1. In a concept being fuzzy there is always something that *does not* hold, i.e., some missing particle the concept does not have. Hence, the complementarity is somehow involved in a fuzzy concept.

### **6. Common features of many-valued logics based on Zadeh algebra**

We consider here some preliminary things being common for several many-valued logics. The main purpose is to find a connection between Zaheh algebra and the structures of some many-valued logics. We consider only some propositional logics, because the main concepts we consider here are basic to higher order many-valued logics, too. We restrict our considerations only to structural properties.

First, we need a formal language for our considerations. This language is that of propositional logic.

**Definition 6.1.** A propositional lanuage L consists of

These symbols are the *aplhabets* of the propositional language.

Usually only the so-called *primitive connectives* belong to the alphabet, but it is possible to choose some other connectives to the alphabet, too. Hence, we could drop either conjunction or disjunction from the alphabet if we like. Primitive connectives are connectives from which we can derive the other connectives.

In a standard nonclassical propositional language, the meanings of the connectives '∧' (*conjunction*), '∨' (*disjunction*), and '¬' (*negation*) can be given as follows: negation ¬ is a stong negation (i.e., it is a negation with involution property ¬¬*p* ≡ *p*), conjunction ∧ is *glb* (greatest lower bound), and disjunction ∨ is *lub* (least upper bound). The symbol '≡' and ⇐⇒ are used as a *meta-symbols* of equivalency, i.e., this symbol does not belong to the alphabet of the *object language* which is the language of a formal logic under consideration.

**Definition 6.2.** Well-formed formulas of L are given as follows:

$$\mathfrak{a} \implies \mathcal{p}\_{\mathbb{k}} \mid \neg \mathcal{q} \mid \mathfrak{q} \wedge \psi \mid \mathfrak{q} \vee \psi \,. \tag{6.1}$$

In this recursive production system of well formed formulas (wff's, for short) the symbol *pk* represents any propositional letter and lower case Greek letters are labels of any atomic or connected wff's. Hence, *α* is a label for any wff, and similarly *ϕ* and *ψ* represent any 10 Will-be-set-by-IN-TECH

values can be considered as membership degrees than only 0 and 1. In Boolean algebras with the universe of discourse {0, 1} the postulate BA5 do not cause conflicts, like the intermediate

Because complement operation satisfies the conditions of strong negation, a Boolean algebra

{0, 1} ⊂ [0, 1]. This means that crisp sets are special cases of fuzzy sets, as they should be also

We may conclude that formally the core hiding-place of fuzziness is the statement Z5 in Proposition 4.1. In a concept being fuzzy there is always something that *does not* hold, i.e., some missing particle the concept does not have. Hence, the complementarity is somehow

We consider here some preliminary things being common for several many-valued logics. The main purpose is to find a connection between Zaheh algebra and the structures of some many-valued logics. We consider only some propositional logics, because the main concepts we consider here are basic to higher order many-valued logics, too. We restrict our

First, we need a formal language for our considerations. This language is that of propositional

Usually only the so-called *primitive connectives* belong to the alphabet, but it is possible to choose some other connectives to the alphabet, too. Hence, we could drop either conjunction or disjunction from the alphabet if we like. Primitive connectives are connectives from which

In a standard nonclassical propositional language, the meanings of the connectives '∧' (*conjunction*), '∨' (*disjunction*), and '¬' (*negation*) can be given as follows: negation ¬ is a stong negation (i.e., it is a negation with involution property ¬¬*p* ≡ *p*), conjunction ∧ is *glb* (greatest lower bound), and disjunction ∨ is *lub* (least upper bound). The symbol '≡' and ⇐⇒ are used as a *meta-symbols* of equivalency, i.e., this symbol does not belong to the alphabet of the *object*

In this recursive production system of well formed formulas (wff's, for short) the symbol *pk* represents any propositional letter and lower case Greek letters are labels of any atomic or connected wff's. Hence, *α* is a label for any wff, and similarly *ϕ* and *ψ* represent any

*α* ::= *pk* | ¬*ϕ* | *ϕ* ∧ *ψ* | *ϕ* ∨ *ψ*. (6.1)

, **0**, **1**) is a special case of Z, i.e., the classical case is included in Z. Trivially,

values may do if these values are added to the universe.

according to Zadeh's own theory. See also the proof of Theorem 2.1.

**6. Common features of many-valued logics based on Zadeh algebra**

B = ({0, 1}, ∨, ∧, �

logic.

involved in a fuzzy concept.

considerations only to structural properties.

we can derive the other connectives.

**Definition 6.1.** A propositional lanuage L consists of 1. a set of propositional letters *p*0, *p*1,..., *pk*, . . . and 2. the truth-functional connectives '∧', '∨', and '¬'.

These symbols are the *aplhabets* of the propositional language.

*language* which is the language of a formal logic under consideration.

**Definition 6.2.** Well-formed formulas of L are given as follows:

propositional wff's. Starting with propositional letters, we can combine them by connectives in the way shown by the production system (6.1). And finally, we can combine any formulas according to the production system.

We can refer to the propositional letters also by lower case letters in general, and to combined formulas by lower case Greek letters or by usual capital letters. These letters belong to the metalanguage we use when we discuss and describe object language. Here we use English equipped by some formal symbols (so-called meta-symbols) as a metalanguage.

The definitions 6.1 and 6.2 above defines the language of propositional logic. Second, we consider some central semantical concepts being necessary for our consideration. This means that we will not present the whole machinery of formal semantics of standard many-valued logics.

We have two important functions, *valuation* and *truth-function* we need in our consideration. They are defined as follows.

**Definition 6.3.** *Valuation v* is a function

$$v: \mathbf{Prop} \longrightarrow \mathbb{I} \tag{6.2}$$

that associates truth values to propositional letters. **Prop** is a set of propositional variables.

*Truth-function* is a function

$$V\_{\mathfrak{n}} : \mathbb{I}^{\mathfrak{n}} \longrightarrow \mathbb{I}, \quad \mathfrak{n} = 1, 2, 3, \dots, \tag{6.3}$$

where *n* is the number of propositional variables *p*, *q*, . . . in the formula defining a truth function.

In general, truth-functions are functions of several variables defined on the *n*-tuple of the set of truth values, [0, 1] *n* where the independent variables are proposition letters. The subindices *n* = 1, 2, 3, . . . are usually dropped. Actually, a propositional formula *ϕ* itself is a truth-function. Suppose that a formula *ϕ* consists of the propositional letters *p*, *q*, and *r*. Then we may write *ϕ* = *V*(*p*, *q*,*r*). The equality sign is used only between truth-functions and truth values.

In connected formulas, valuations of propositional variables give the values for the variables of the corresponding truth-function presented by the connected formula. Hence, a "valuation" of a connected formula is the value of the corresponding truth-function. Hence, to evaluate a truth value of the whole connected formula corresponding a given valuation for propositional variables, we calculate the value of the truth-function where the given valuation *v* first determines the values of the arguments of the truth-function. We may denote the truth value of a connected formula *ϕ* by *V*(*ϕ*), being like a valuation depending on a given valuation *v* for propositional letters.

**Example 6.1.** Evaluate the truth value of a formula *p* ∧ (¬*q* ∨ *r*) with regard to a given valuation *v* for *p*, *q*, and *r*. Actually the formula is a truth-function *f*(*p*, *q*,*r*) = *p* ∧ (¬*q* ∨ *r*) where *p*, *q*, and *r* obtain their values from [0, 1]. These values are *v*(*p*), *v*(*q*), and *v*(*r*). Now, the truth value of the formula,given by this valuation *v*, is

$$V(p \land (\neg q \lor r)) = v(p) \land (\neg v(q) \lor v(r))$$

the evaluation rules (6.4), (6.5), and (6.6), the operations max, min, and ¬ exist at least in the logics having these evaluation rules. Let us compare the power set of fuzzy sets of the set *X*, i.e., the set **I***X*, to the set of all valuations *v* : **Prop** −→ **I**. Hence, the set of all valuations is **IProp**. Especially, **1** and **0** are constant valuations, such that **1** gives the truth value 1 to every propositional letters, and similarly, **0** gives the truth value 0. Hence, we also have the neutral elements corresponding those in Zadeh algebra. It seems that if we replace the set **I***X* by **IProp**

Standard Fuzzy Sets and some Many-Valued Logics 87

The values of valuations are truth values and those of membership functions are membership grades. Can these two interpretations for the elements [0, 1] be considered to be anyhow similar? According to formal consideration, we say *yes*. The both values are obtainable from the same set, namely from the unit interval [0, 1], and the construction of the both algebras are exactly the same. On the other hand, membership grades are in principle subjective opinions about the membership of an element in a given set. About truth values, a *degree of truth* of a given propositional letter in a given situation depends on the state of affairs associated to this situation. But there is a valuation for every state of affairs in any situation representing a suitable degree of truth expressed by a number obtained from [0, 1]. Hence, these degrees of truth correspond to suitable membership grades even so that for any valuation there exists a membership function that is identical with the valuation. Hence, these two apparently different interpretations can be considered to be the same. This means that we can interpret the values of the functions of the algebra (6.7) as truth values, or more accurately, degrees of

For historical reasons, we consider a piece of Kleene's 3-valued logic. S. C. Kleene was Zadeh's logic teacher, and it is natural that Zadeh compared his concept of fuzzy set with Kleene's

"Note that the notion of belonging", which plays a fundamental role in the case of ordinary sets, does not have the same role in the case of fuzzy sets. Thus, it is not meaningful to speak of a point *x* "belonging" to a fuzzy set *A* except in the trivial sense of *fA*(*x*) being positive. Less trivially, one can introduce two levels *α* and *β* (0 < *α* < 1, 0 < *β* < 1, *α* > *β*) and agree to say that (1) "*x* belongs to *A*" if *fA*(*x*) ≥ *α*; (2) "*x* does not belong to *A*" if *fA*(*x*) ≤ *β*; and (3) "*x* has an intermediate status relative to *A*" if *β* < *fA*(*x*) < *α*. This leads to a three-valued logic (Kleene, 1952) with three truth

The symbols of the truth values of Kleene's 3-valued logic are *T* (*true*), *U* (*unknown*), and *F* (*false*). In the literature, there are also some alternative symbols for the intermediate truth

Kleene introduced his 3-vaued logic in 1938. We denote it by **K**3. In order to describe Kleene's

"In Kleene's system, a proposition is to bear the third truth-value *I* not for fact-related, ontological reasons but for knowledge-related, epistemological ones: it is not to be excluded that the proposition may *in fact* be true or false, but it is merely *unknown*

3-valued logic. Zadeh ((Zadeh, 1965) p. 341-342) gives the following comment:

values *T* (*fA*(*x*) ≥ *α*), *F* (*fA*(*x*) ≤ *β*), and *U* (*β* < *fA*(*x*) < *α*).

value. For example, Rescher (Rescher, 1969) uses the symbol *I*.

logic, we refer to Rescher (Rescher, 1969), p. 34 - 36. He writes:

or undeterminable what its specific truth status may be.

Lℵ1 = �**IProp**, max, min, ¬, **0**, **1**�. (6.7)

then we have a special Zadeh algebra, namely, say, *propositional algebra*

truth.

**7. Description of Kleene's logic**

Suppose that *v* is a valuation where *v*(*p*) = 0.5, *v*(*q*) = 0.3, and *v*(*r*) = 1. Hence,

$$V(p \land (\neg q \lor r)) = 0.5 \land ((1 - 0.3) \lor 1) = 0.5 \land (0.7 \lor 1) = 0.5 \land 1 = 0.5$$

If we have two wff's representing the same state of affairs we use meta-equivalence sign '≡' or '⇐⇒' between them because the formulas are equivalent to each other, not identical. For example, the wff *ϕ* is the formula *p* ∧ (¬*q* ∨ *r*). So, we can write *ϕ* ≡ *p* ∧ (¬*q* ∨ *r*), or *ϕ* ⇐⇒ *p* ∧ (¬*q* ∨ *r*) to denote that we use an abbreviation *ϕ* for the formula *p* ∧ (¬*q* ∨ *r*). Another case is that we have two formulas being equivalent to each other, for example, ¬*p* ∨ ¬*q* ≡ ¬(*p* ∧ *q*). This equivalency describes one of De Morgan's laws. However, the expression

$$\forall \mathbf{x}, y \in [0, 1], \quad \neg \mathbf{x} \lor \neg y = \neg (\mathbf{x} \land y)$$

emphasizes that two truth-functions ¬*x* ∨ ¬*y* and ¬(*x* ∧ *y*) are identical.

Instead of propositional letters, we prefer to use "usual" variable symbols as variables of a truth-function, because of possible confusions.

We are interested in the logics where the *evaluation rules* for these connectives are

$$V(p \lor q) = \max\{v(p), v(q)\}\tag{6.4}$$

$$V(p \wedge q) = \min\{v(p), v(q)\}\tag{6.5}$$

$$V(\neg p) = 1 - v(p) \tag{6.6}$$

where *v*(*p*), *v*(*q*) ∈ [0, 1], or *v*(*p*), *v*(*q*) ∈ *A* where *A* is a suitable subset of [0, 1].

Evaluation rules are rules for evaluating truth values to connected logical formulas in a given logic.

We must remember that all the logics are not truth-functional. For example, modal logics are non-truth-functional.

In practice, we need some other connectives, too. Two of them are the connectives *implication* and *equivalency*.

The way to choose implication separates the logics based on Zadeh algebra or any De Morgan algebra. Hence, implication must be presented by means of disjunction and negation, or by means of conjunction and negation. There are several ways to define different implications from other connectives, depending on the logic in question. We consider these things in the case of each logic to be considered below.

The formulas (6.4), (6.5), and (6.6) somehow emphasize the relationship to algebraic construction. We have two binary operations and one unary operation defined on a nonempty set just as in usual algebraic system. Additionally, the binary operations are combined together, for example, being distributive. The formulas (6.4), (6.5), and (6.6) are the same as Zadeh's operations defined on the set of fuzzy sets. The bridge between standard fuzzy sets and some many-valued logics seems to be obvious. Having got this kind of motivation, we continue our construction of the bridge between standard fuzzy sets and some many-valued logics.

Consider Zadeh algebra Zℵ1 = �**I***X*, max, min, ¬, **0**, **1**� (*cf.* formula (4.1)). Now the question is wether there is a counterpart to this algebra in the scope of many-valued logic. According to 12 Will-be-set-by-IN-TECH

*V*(*p* ∧ (¬*q* ∨ *r*)) = 0.5 ∧ ((1 − 0.3) ∨ 1) = 0.5 ∧ (0.7 ∨ 1) = 0.5 ∧ 1 = 0.5

If we have two wff's representing the same state of affairs we use meta-equivalence sign '≡' or '⇐⇒' between them because the formulas are equivalent to each other, not identical. For example, the wff *ϕ* is the formula *p* ∧ (¬*q* ∨ *r*). So, we can write *ϕ* ≡ *p* ∧ (¬*q* ∨ *r*), or *ϕ* ⇐⇒ *p* ∧ (¬*q* ∨ *r*) to denote that we use an abbreviation *ϕ* for the formula *p* ∧ (¬*q* ∨ *r*). Another case is that we have two formulas being equivalent to each other, for example, ¬*p* ∨ ¬*q* ≡ ¬(*p* ∧ *q*).

∀*x*, *y* ∈ [0, 1], ¬*x* ∨ ¬*y* = ¬(*x* ∧ *y*)

Instead of propositional letters, we prefer to use "usual" variable symbols as variables of a

Evaluation rules are rules for evaluating truth values to connected logical formulas in a given

We must remember that all the logics are not truth-functional. For example, modal logics are

In practice, we need some other connectives, too. Two of them are the connectives *implication*

The way to choose implication separates the logics based on Zadeh algebra or any De Morgan algebra. Hence, implication must be presented by means of disjunction and negation, or by means of conjunction and negation. There are several ways to define different implications from other connectives, depending on the logic in question. We consider these things in the

The formulas (6.4), (6.5), and (6.6) somehow emphasize the relationship to algebraic construction. We have two binary operations and one unary operation defined on a nonempty set just as in usual algebraic system. Additionally, the binary operations are combined together, for example, being distributive. The formulas (6.4), (6.5), and (6.6) are the same as Zadeh's operations defined on the set of fuzzy sets. The bridge between standard fuzzy sets and some many-valued logics seems to be obvious. Having got this kind of motivation, we continue our construction of the bridge between standard fuzzy sets and some many-valued

Consider Zadeh algebra Zℵ1 = �**I***X*, max, min, ¬, **0**, **1**� (*cf.* formula (4.1)). Now the question is wether there is a counterpart to this algebra in the scope of many-valued logic. According to

*V*(*p* ∨ *q*) = max{*v*(*p*), *v*(*q*)} (6.4) *V*(*p* ∧ *q*) = min{*v*(*p*), *v*(*q*)} (6.5) *V*(¬*p*) = 1 − *v*(*p*) (6.6)

Suppose that *v* is a valuation where *v*(*p*) = 0.5, *v*(*q*) = 0.3, and *v*(*r*) = 1. Hence,

This equivalency describes one of De Morgan's laws. However, the expression

We are interested in the logics where the *evaluation rules* for these connectives are

where *v*(*p*), *v*(*q*) ∈ [0, 1], or *v*(*p*), *v*(*q*) ∈ *A* where *A* is a suitable subset of [0, 1].

emphasizes that two truth-functions ¬*x* ∨ ¬*y* and ¬(*x* ∧ *y*) are identical.

truth-function, because of possible confusions.

case of each logic to be considered below.

logic.

logics.

non-truth-functional.

and *equivalency*.

the evaluation rules (6.4), (6.5), and (6.6), the operations max, min, and ¬ exist at least in the logics having these evaluation rules. Let us compare the power set of fuzzy sets of the set *X*, i.e., the set **I***X*, to the set of all valuations *v* : **Prop** −→ **I**. Hence, the set of all valuations is **IProp**. Especially, **1** and **0** are constant valuations, such that **1** gives the truth value 1 to every propositional letters, and similarly, **0** gives the truth value 0. Hence, we also have the neutral elements corresponding those in Zadeh algebra. It seems that if we replace the set **I***X* by **IProp** then we have a special Zadeh algebra, namely, say, *propositional algebra*

$$\mathcal{L}\_{\aleph\_1} = \langle \mathbf{I}^{\mathbf{Prop}}, \mathbf{max}, \min, \neg, \mathbf{0}, \mathbf{1} \rangle. \tag{6.7}$$

The values of valuations are truth values and those of membership functions are membership grades. Can these two interpretations for the elements [0, 1] be considered to be anyhow similar? According to formal consideration, we say *yes*. The both values are obtainable from the same set, namely from the unit interval [0, 1], and the construction of the both algebras are exactly the same. On the other hand, membership grades are in principle subjective opinions about the membership of an element in a given set. About truth values, a *degree of truth* of a given propositional letter in a given situation depends on the state of affairs associated to this situation. But there is a valuation for every state of affairs in any situation representing a suitable degree of truth expressed by a number obtained from [0, 1]. Hence, these degrees of truth correspond to suitable membership grades even so that for any valuation there exists a membership function that is identical with the valuation. Hence, these two apparently different interpretations can be considered to be the same. This means that we can interpret the values of the functions of the algebra (6.7) as truth values, or more accurately, degrees of truth.

### **7. Description of Kleene's logic**

For historical reasons, we consider a piece of Kleene's 3-valued logic. S. C. Kleene was Zadeh's logic teacher, and it is natural that Zadeh compared his concept of fuzzy set with Kleene's 3-valued logic. Zadeh ((Zadeh, 1965) p. 341-342) gives the following comment:

"Note that the notion of belonging", which plays a fundamental role in the case of ordinary sets, does not have the same role in the case of fuzzy sets. Thus, it is not meaningful to speak of a point *x* "belonging" to a fuzzy set *A* except in the trivial sense of *fA*(*x*) being positive. Less trivially, one can introduce two levels *α* and *β* (0 < *α* < 1, 0 < *β* < 1, *α* > *β*) and agree to say that (1) "*x* belongs to *A*" if *fA*(*x*) ≥ *α*; (2) "*x* does not belong to *A*" if *fA*(*x*) ≤ *β*; and (3) "*x* has an intermediate status relative to *A*" if *β* < *fA*(*x*) < *α*. This leads to a three-valued logic (Kleene, 1952) with three truth values *T* (*fA*(*x*) ≥ *α*), *F* (*fA*(*x*) ≤ *β*), and *U* (*β* < *fA*(*x*) < *α*).

The symbols of the truth values of Kleene's 3-valued logic are *T* (*true*), *U* (*unknown*), and *F* (*false*). In the literature, there are also some alternative symbols for the intermediate truth value. For example, Rescher (Rescher, 1969) uses the symbol *I*.

Kleene introduced his 3-vaued logic in 1938. We denote it by **K**3. In order to describe Kleene's logic, we refer to Rescher (Rescher, 1969), p. 34 - 36. He writes:

"In Kleene's system, a proposition is to bear the third truth-value *I* not for fact-related, ontological reasons but for knowledge-related, epistemological ones: it is not to be excluded that the proposition may *in fact* be true or false, but it is merely *unknown* or undeterminable what its specific truth status may be.

Clearly, the algebraic approach to **K**3 is Kleene algebra, i.e., a 3-valued Zadeh algebra with the

Standard Fuzzy Sets and some Many-Valued Logics 89

*Kleene-Dienes many-valued logic* is an extension of **K**3 into **K**ℵ1 having the set of truth values [0, 1]. The evaluation rules for conjunction, disjunction, and negation are the same as in the

def

being in accordance with the implication of **K**3. This means that the evaluation rule for

The equations (6.4), (6.5), (6.6), (7.7), and (7.9) are the truth value evaluation rules for disjunction, conjunction, negation, implication, and equivalence, respectively, of

The implication operation of Kleene-Dienes many-valued logic is a typical example about a so-called *S-implication*. Another example is the implication operation of classical logic. The

We begin with Łukasiewicz' many-valued logic Łℵ1 having the closed unit interval [0, 1] as the

As we know, Łukasiewicz chose the connectives of *negation* and *implication* as primitives. This is a remarkable difference, for example, between Kleene's logic and Łukasiewicz logic. Hence, the connection between standard fuzzy set theory and Łℵ1 cannot be seen immediately.

Let *v* be any valuation of Łℵ, then the truth value evaluation rules for negation and implication

⇐⇒ ¬*p* ∨ *q* (7.6)

*x* → *y* = max {1 − *x*, *y*} (7.7)

⇐⇒ (*p* → *q*) ∧ (*q* → *p*) (7.8)

∀*x*, *y* ∈ [0, 1], *x* ↔ *y* = min{*x* → *y*, *y* → *x*} (7.9)

*v*(¬*p*) = 1 − *v*(*p*) (Neg.) *v*(*p* → *q*) = min{1, 1 − *v*(*p*) + *v*(*q*)} (Impl.)

*p* → *q*

def

property (K).

*Kleene-Dienes many-valued logic*:

formulas (6.4), (6.5), and (6.6) above.

implication is as follows. For any *x*, *y* ∈ [0, 1],

Hence, the evaluation rule for equivalence is

**8. On Łukasiewicz' many-valued logic**

2 *Cf.* Rescher (Rescher, 1969), p.36, and 337.

set of truth values.2

are

Implication of Kleene-Dienes many-valued logic is defined by

Now, the connective *equivalence* is defined in the usual way:

*p* ↔ *q*

Kleene-Dienes many-valued logic with the set of truth values [0, 1].

general principle for S-implication is just the formula (7.6).

In **K**3, we have the following truth value evaluation rules for the connectives negation ¬, conjunction ∧, and disjunction ∨:

$$V(\neg p) = T - v(p)\_{\prime} \tag{7.1}$$

$$V(p \wedge q) = \min\{v(p), v(q)\},\tag{7.2}$$

$$V(p \lor q) = \max\{v(p), v(q)\}. \tag{7.3}$$

Two of these connectives can form a set of *primitive connectives*. One of the primitives must be negation. Hence, for example, the connectives negation and disjunction can be chosen as primitives, and all the other connectives can be defined by means of these primitive connectives. The alternative case for primitives are negation and conjunction. Hence, we can define the nonprimitive one by negation and the fact that disjunction and conjunction are dual (i.e., by using a suitable De Morgan's law). The implication defined by means of negation and disjunction is given by the formula (7.4).

We see immediately that there is a strong analogy between the basic operations of fuzzy sets (*cf.* Def. 2.2) and these three connectives of Kleene's 3-valued logic. In addition to this, an analogy can be found between Kleene's valuations and Zadeh's membership functions, too, although they have different ranges. Zadeh's comment above connects these two concepts "membership" (*μ* : *X* −→ **I**) and "valuation" (*v* : **Prop** −→ {*F*, *U*, *T*}) together. We can compare the set {*F*, *U*, *T*} with the set {0, 1 2 , 1} where *F* = 0, *U* = 1 2 , and *T* = 1. Hence, by analogy of the sets, we can understand some arithmetic operations in some evaluation rules.

Kleene defined the implication of his 3-valued logic, denoted by , analogously to material implication:

$$p \ni q \stackrel{\text{def}}{\iff} \neg p \lor q \tag{7.4}$$

hence, the evaluation rule of *p q* is

$$p \ni q = \max\{T - v(p), v(q)\}. \tag{7.5}$$

We may construct the truth tables according to the evaluation rules. Rescher tells that Kleene motivated the construction of his truth tables in terms of a mathematical application. He has in mind the case of a mathematical predicate *P* (i.e., a propositional function) of a variable *x* ranging over a domain *D* where "*P*(*x*)" is defined for only a part of this domain. For example, we might have the condition

$$P(\mathfrak{x}) \quad \text{iff} \quad 1 \le \frac{1}{\mathfrak{x}} \le 2.$$

Here *P*(*x*) will be:

Kleene presented his truth tables to formulate the rules of combination by logical connectives for such propositional functions. He writes:

"From this standpoint, the meaning of *Q* ∨ *R* is brought out clearly by the statement in words: *Q* ∨ *R* is true, if *Q* is true (here nothing is said about *R*) or if *R* is true (similarly); false, if *Q* and *R* are both false; defined only in these cases (and hence undefined, otherwise)."1

1 Kleene, *Introduction to Metamathematics* (1952).

Clearly, the algebraic approach to **K**3 is Kleene algebra, i.e., a 3-valued Zadeh algebra with the property (K).

*Kleene-Dienes many-valued logic*:

14 Will-be-set-by-IN-TECH

In **K**3, we have the following truth value evaluation rules for the connectives negation ¬,

Two of these connectives can form a set of *primitive connectives*. One of the primitives must be negation. Hence, for example, the connectives negation and disjunction can be chosen as primitives, and all the other connectives can be defined by means of these primitive connectives. The alternative case for primitives are negation and conjunction. Hence, we can define the nonprimitive one by negation and the fact that disjunction and conjunction are dual (i.e., by using a suitable De Morgan's law). The implication defined by means of negation

We see immediately that there is a strong analogy between the basic operations of fuzzy sets (*cf.* Def. 2.2) and these three connectives of Kleene's 3-valued logic. In addition to this, an analogy can be found between Kleene's valuations and Zadeh's membership functions, too, although they have different ranges. Zadeh's comment above connects these two concepts "membership" (*μ* : *X* −→ **I**) and "valuation" (*v* : **Prop** −→ {*F*, *U*, *T*}) together. We can

analogy of the sets, we can understand some arithmetic operations in some evaluation rules. Kleene defined the implication of his 3-valued logic, denoted by , analogously to material

def

We may construct the truth tables according to the evaluation rules. Rescher tells that Kleene motivated the construction of his truth tables in terms of a mathematical application. He has in mind the case of a mathematical predicate *P* (i.e., a propositional function) of a variable *x* ranging over a domain *D* where "*P*(*x*)" is defined for only a part of this domain. For example,

> 1 *x* ≤ 2.

*P*(*x*) iff 1 ≤

2 to 1,

Kleene presented his truth tables to formulate the rules of combination by logical connectives

"From this standpoint, the meaning of *Q* ∨ *R* is brought out clearly by the statement in words: *Q* ∨ *R* is true, if *Q* is true (here nothing is said about *R*) or if *R* is true (similarly); false, if *Q* and *R* are both false; defined only in these cases (and hence undefined,

*p q*

2 , 1} where *F* = 0, *U* = 1

2 , and *T* = 1. Hence, by

⇐⇒ ¬*p* ∨ *q*, (7.4)

*p q* = max{*T* − *v*(*p*), *v*(*q*)}. (7.5)

*V*(¬*p*) = *T* − *v*(*p*), (7.1) *V*(*p* ∧ *q*) = min{*v*(*p*), *v*(*q*)}, (7.2) *V*(*p* ∨ *q*) = max{*v*(*p*), *v*(*q*)}. (7.3)

conjunction ∧, and disjunction ∨:

and disjunction is given by the formula (7.4).

compare the set {*F*, *U*, *T*} with the set {0, 1

hence, the evaluation rule of *p q* is

(1) *true* if *x* lies within the range from 1

(2) *undefined* (or undetermined) if *x* = 0,

for such propositional functions. He writes:

1 Kleene, *Introduction to Metamathematics* (1952).

we might have the condition

(3) *false* in all other cases.

Here *P*(*x*) will be:

otherwise)."1

implication:

*Kleene-Dienes many-valued logic* is an extension of **K**3 into **K**ℵ1 having the set of truth values [0, 1]. The evaluation rules for conjunction, disjunction, and negation are the same as in the formulas (6.4), (6.5), and (6.6) above.

Implication of Kleene-Dienes many-valued logic is defined by

$$p \to q \stackrel{\text{def}}{\iff} \neg p \lor q \tag{7.6}$$

being in accordance with the implication of **K**3. This means that the evaluation rule for implication is as follows. For any *x*, *y* ∈ [0, 1],

$$\mathbf{x} \to \mathbf{y} = \max\left\{1 - \mathbf{x}\_{\prime}\mathbf{y}\right\} \tag{7.7}$$

Now, the connective *equivalence* is defined in the usual way:

$$p \leftrightarrow q \stackrel{\text{def}}{\Longleftrightarrow} (p \to q) \land (q \to p) \tag{7.8}$$

Hence, the evaluation rule for equivalence is

$$\forall \mathbf{x}, y \in [0, 1], \quad \mathbf{x} \leftrightarrow y = \min \{ \mathbf{x} \to y, y \to \mathbf{x} \} \tag{7.9}$$

The equations (6.4), (6.5), (6.6), (7.7), and (7.9) are the truth value evaluation rules for disjunction, conjunction, negation, implication, and equivalence, respectively, of Kleene-Dienes many-valued logic with the set of truth values [0, 1].

The implication operation of Kleene-Dienes many-valued logic is a typical example about a so-called *S-implication*. Another example is the implication operation of classical logic. The general principle for S-implication is just the formula (7.6).

### **8. On Łukasiewicz' many-valued logic**

We begin with Łukasiewicz' many-valued logic Łℵ1 having the closed unit interval [0, 1] as the set of truth values.2

As we know, Łukasiewicz chose the connectives of *negation* and *implication* as primitives. This is a remarkable difference, for example, between Kleene's logic and Łukasiewicz logic. Hence, the connection between standard fuzzy set theory and Łℵ1 cannot be seen immediately.

Let *v* be any valuation of Łℵ, then the truth value evaluation rules for negation and implication are

$$v(\neg p) = 1 - v(p) \tag{\text{Neg.}}$$

$$v(p \to q) = \min\{1, 1 - v(p) + v(q)\}\tag{\text{Impl.}}$$

2 *Cf.* Rescher (Rescher, 1969), p.36, and 337.

**Proposition 8.1.** *Suppose x*, *y* ∈ [0, 1]*. Then De Morgan's laws hold for* min{*x*, *y*} *and* max{*x*, *y*}*.*

Standard Fuzzy Sets and some Many-Valued Logics 91

*Proof.* Consider the operation min(*x*, *y*) and max(*x*, *y*), where *x* and *y* are variables taking their values from the interval [0, 1]. Using the arithmetical formula for min operation (i.e., the

> = 2 − 1 − 1 + *x* + *y*− | 1 − 1 + *x* − *y* | 2

> = 1 − 1 + 1 − *x* − *y*+ | 1 − 1 + *x* − *y* | 2

From the formula (8.5), by replacing *x* by 1 − *x* and *y* by 1 − *y*, and then solving max(*x*, *y*), the

The formulas (8.5) and (8.6) show that DeMorgan laws hold for max and min, and they are

Łukasiewicz knew that the operations max and min are dual of each other. Actually, this property is easily found in the classical special case, i.e. using characteristic functions in presenting crisp sets. But the general proof for this is easily done by using the expressions (8.4) for max and min in such cases where a distance metric is defined in the universe of

Second, consider the connection between max and Łukasiewicz implication using Zadeh algebra (similar considerations are done in Mattila (Mattila, 2005), but the following

*Proof.* Consider disjunction operation *x* ∨ *y* = max(*x*, *y*). Because 0 ≤ *x*, *y* ≤ 1, using the

Ł *y*) →

Ł *y*, (8.7)

max(*x*, *y*)=(*x* →

= 1 − (1 − *x*)+(1 − *y*)+ | (1 − *y*) − (1 − *x*) | 2

= 1 − max{ (1 − *x*),(1 − *y*) }. (8.5)

max{*x*, *y*} = 1 − min{ 1 − *x*, 1 − *y*}. (8.6)

expression for min in (8.4)), we have

following formula follows:

dual of each other.

*where x* →

This completes the proof.

discourse. This always holds at least for real numbers.

proposition 8.2 is not completely proved).

Ł *y is Łukasiewicz implication.*

arithmetical formula (8.4) for max, we have

**Proposition 8.2.** *For all x*, *y* ∈ [0, 1]*,*

min{*x*, *y*} = *x* + *y*− | *x* − *y* |

2

By means of these connectives, Łukasiewicz defined the other connectives by the rules

$$p \lor q \stackrel{\text{def}}{\iff} (p \to q) \to q \tag{\text{Dist.}}$$

$$(p \land q \stackrel{\text{def}}{\iff} \neg(\neg p \lor \neg q) \tag{\text{Conj.}}$$

$$p \leftrightarrow q \stackrel{\text{def}}{\Longleftrightarrow} (p \to q) \land (q \to p) \tag{Eq.}$$

The truth value evaluation rules for these derived connectives are

$$\max\{v(p), v(q)\}\tag{8.1}$$

$$\min\{v(p), v(q)\} \tag{8.2}$$

1 − |*v*(*p*) − *v*(*q*)| for *p* ↔ *q* (8.3)

for any valuation *v* of Łℵ1 .

In Zadeh algebra we have the operations representing disjunction, conjunction, and negation as given. Negation in Zadeh algebra has the same construction as that in Łukasiewicz' logic Łℵ1 , so, we need not to do anything with it. Now our task is to derive algebraically the *implication* of Łℵ1 by means of these three other connectives. For this we use the operations of Zadeh algebra. Actually, we need only complementarity and max operations in our solution. After succeeding to solve this problem we know that standard fuzzy sets and Łℵ1 fits together completely, i.e., we can derive all the connectives of Łℵ1 in terms of Zadeh algebra. The final result is given in Proposition 8.3. This is the main task in this section.

Consider again the special case of Zadeh algebra (6.7)

$$\mathcal{L}\_{\aleph\_1} = \langle \mathbf{I}^{\mathbf{Prop}}, \mathbf{max}, \min, \neg, \mathbf{0}, \mathbf{1} \rangle.$$

From the considertations above, we know that

We observed in Section 3 that [0, 1] is a metric space with the natural metric distance (3.2)

$$d(\mathfrak{x}, y) = |\mathfrak{x} - y|\, \, \, \, \mathfrak{x} \, y \in [0, 1].$$

This formula satisfies the general definition of the concept *metric*. We need it in the following consideration where we manipulate expressions involving maxima and minima.

In manipulating maxima and minima, the consideration can sometimes be done easier by using the following expressions for max and min operations:

$$\max\{\mathbf{x}, y\} = \frac{\mathbf{x} + \mathbf{y} + |\mathbf{x} - \mathbf{y}|}{2}, \qquad \min\{\mathbf{x}, y\} = \frac{\mathbf{x} + \mathbf{y} - |\mathbf{x} - \mathbf{y}|}{2} \tag{8.4}$$

These formulas hold on the set of real numbers **R**, and especially on the unit interval [0, 1].

First, consider the case where the operations min and max are used in the form of the formulas (8.4), and ¬ is defined in the usual way: ¬*x* = 1 − *x*.

16 Will-be-set-by-IN-TECH

In Zadeh algebra we have the operations representing disjunction, conjunction, and negation as given. Negation in Zadeh algebra has the same construction as that in Łukasiewicz' logic Łℵ1 , so, we need not to do anything with it. Now our task is to derive algebraically the *implication* of Łℵ1 by means of these three other connectives. For this we use the operations of Zadeh algebra. Actually, we need only complementarity and max operations in our solution. After succeeding to solve this problem we know that standard fuzzy sets and Łℵ1 fits together completely, i.e., we can derive all the connectives of Łℵ1 in terms of Zadeh algebra. The final

Lℵ1 = �**IProp**, max, min, ¬, **0**, **1**�.

• The unary operation ¬ is a complementarity operation with the property of involution. We observed in Section 3 that [0, 1] is a metric space with the natural metric distance (3.2)

*d*(*x*, *y*) = |*x* − *y*| , *x*, *y* ∈ [0, 1]. This formula satisfies the general definition of the concept *metric*. We need it in the following

In manipulating maxima and minima, the consideration can sometimes be done easier by

These formulas hold on the set of real numbers **R**, and especially on the unit interval [0, 1]. First, consider the case where the operations min and max are used in the form of the formulas

2 , min{*x*, *y*} = *x* + *y* − |*x* − *y*|

2 (8.4)

consideration where we manipulate expressions involving maxima and minima.

⇐⇒ (*p* → *q*) → *q* (Disj.)

⇐⇒ ¬(¬*p* ∨ ¬*q*) (Conj.)

⇐⇒ (*p* → *q*) ∧ (*q* → *p*) (Eq.)

max{*v*(*p*), *v*(*q*)} for *p* ∨ *q*, (8.1) min{*v*(*p*), *v*(*q*)} for *p* ∧ *q*, (8.2) 1 − |*v*(*p*) − *v*(*q*)| for *p* ↔ *q* (8.3)

By means of these connectives, Łukasiewicz defined the other connectives by the rules

def

def

def

*p* ∨ *q*

*p* ∧ *q*

*p* ↔ *q*

result is given in Proposition 8.3. This is the main task in this section.

• Lℵ1 is a special Zadeh algebra, namely propositional algebra.

• The operations max and min are distributive to each others.

using the following expressions for max and min operations:

max{*x*, *y*} = *x* + *y* + |*x* − *y*|

(8.4), and ¬ is defined in the usual way: ¬*x* = 1 − *x*.

• The binary operations max and min are commutative and associative.

Consider again the special case of Zadeh algebra (6.7)

From the considertations above, we know that

The truth value evaluation rules for these derived connectives are

for any valuation *v* of Łℵ1 .

**Proposition 8.1.** *Suppose x*, *y* ∈ [0, 1]*. Then De Morgan's laws hold for* min{*x*, *y*} *and* max{*x*, *y*}*.*

*Proof.* Consider the operation min(*x*, *y*) and max(*x*, *y*), where *x* and *y* are variables taking their values from the interval [0, 1]. Using the arithmetical formula for min operation (i.e., the expression for min in (8.4)), we have

$$\min\{x, y\} = \frac{x + y - \lfloor x - y \rfloor}{2}$$

$$= \frac{2 - 1 - 1 + x + y - \lfloor 1 - 1 + x - y \rfloor}{2}$$

$$= 1 - \frac{1 + 1 - x - y + \lfloor 1 - 1 + x - y \rfloor}{2}$$

$$= 1 - \frac{(1 - x) + (1 - y) + \lfloor (1 - y) - (1 - x) \rfloor}{2}$$

$$= 1 - \max\{ (1 - x), (1 - y) \}. \tag{8.5}$$

From the formula (8.5), by replacing *x* by 1 − *x* and *y* by 1 − *y*, and then solving max(*x*, *y*), the following formula follows:

$$\max\{\mathbf{x}, y\} = 1 - \min\{1 - \mathbf{x}, 1 - y\}. \tag{8.6}$$

The formulas (8.5) and (8.6) show that DeMorgan laws hold for max and min, and they are dual of each other.

This completes the proof.

Łukasiewicz knew that the operations max and min are dual of each other. Actually, this property is easily found in the classical special case, i.e. using characteristic functions in presenting crisp sets. But the general proof for this is easily done by using the expressions (8.4) for max and min in such cases where a distance metric is defined in the universe of discourse. This always holds at least for real numbers.

Second, consider the connection between max and Łukasiewicz implication using Zadeh algebra (similar considerations are done in Mattila (Mattila, 2005), but the following proposition 8.2 is not completely proved).

**Proposition 8.2.** *For all x*, *y* ∈ [0, 1]*,*

$$\max(\mathbf{x}, y) = (\mathbf{x} \underset{\mathbf{L}}{\rightarrow} y) \underset{\mathbf{L}}{\rightarrow} y,\tag{8.7}$$

*where x* → Ł *y is Łukasiewicz implication.*

*Proof.* Consider disjunction operation *x* ∨ *y* = max(*x*, *y*). Because 0 ≤ *x*, *y* ≤ 1, using the arithmetical formula (8.4) for max, we have

The case 4 follows from the case 2 by Prop. 8.2 as follows. When we consider the equation (8.10) in the proof of Prop. 8.2, we find two min-stuctures corresponding to the evaluation rule of implication, so that one of them is an inside part of the whole formula. If we denote the inner min-structure min(1, 1 − *x* + *y*) by *z* then the outer min-structure is min(1, 1 − *z* + *y*), i.e., the min-structures are formally the same. The implication operations in (8.9) are situated

Standard Fuzzy Sets and some Many-Valued Logics 93

= (*x* → *y*)+(*y* → *x*) − |(*x* → *y*) − (*y* → *x*)| 2

= min(1, 1 − *x* + *y*) + min(1, 1 − *y* + *x*) 2 − min(1, 1 − *x* + *y*) − min(1, 1 − *y* + *x*) 2 = 4 − 2 |*x* − *y*| − |−2*x* + 2*y* − |*x* − *y*| + |*x* − *y*|| 4

4 = 1 − |*x* − *y*| .

These cases are similar to the truth value evaluation rules for connected formulas in Łℵ1 . Hence, if we want to use algebraic approach for Łℵ1 we need not necessarily to follow the mainstream described in Section 9 using the operations of MV-algebras for studying the

However, in the next section, we give also a very brief description about the alternative approach starting from the definition of general MV-algebra. It is the mainstream in this

We open another way a little for creating an algebra for Łukasiewicz logic. We adopt the definition and some properties of MV-algebras from Cignoli et. al. (Cignoli et al., 2000). Other

**Definition 9.1.** An *MV-algebra* is an algebra �*A*, ⊕, ¬, 0� with a binary operation ⊕, a unary

Ł *y*, by (Impl.).

in the same way. Hence, min(1, 1 − *x* + *y*) must be the evaluation rule of *x* →

*x* ↔ *y* = min{(*x* → *y*),(*y* → *x*)}

= 4 − 4 |*x* − *y*|

research topic, but a circuitous route in the case of Łukasiewicz logic.

sources are Bergmann (Bergmann, 2008) and Hájek (Hájek, 1998).

operation ¬ and a constant 0 satisfying the following equations:

A non-empty set *A* is the universe of the MV-algebra �*A*, ⊕, ¬, 0�.

**9. The relationship between Łukasiewicz logic and MV-algebras**

Hence, the connectives of Łℵ1 can be created by Zadeh algebra.

connections between the connectives in Łℵ1 .

(MV1) *x* ⊕ (*y* ⊕ *z*)=(*x* ⊕ *y*) ⊕ *z*

(MV6) ¬(¬*x* ⊕ *y*) ⊕ *y* = ¬(¬*y* ⊕ *x*) ⊕ *x*

(MV2) *x* ⊕ *y* = *y* ⊕ *x* (MV3) *x* ⊕ 0 = *x* (MV4) ¬¬*x* = *x* (MV5) *x* ⊕ ¬0 = ¬0

The case 5 is deduced as follows:

$$\begin{split} \max(\mathbf{x}, y) &= \min\{1, \max(\mathbf{x}, y)\} = \min\{1, \frac{\mathbf{x} + y + \mid \mathbf{x} - y\mid}{2}\} \\ &= \min\left\{1, \frac{2 - 1 - 1 + \mathbf{x} + 2y - y + \mid 1 - 1 + \mathbf{x} - y\mid}{2}\right\} \\ &= \min\left\{1, 1 - \frac{1 + (1 - \mathbf{x} + y) - \mid 1 - (1 - \mathbf{x} + y)\mid}{2} + y\right\} \\ &= \min\left\{1, 1 - \min(1, 1 - \mathbf{x} + y) + y\right\} \end{split} \tag{8.8}$$

On the other hand, in Łℵ1 disjunction is defined by(Disj.), i.e., by the formula

$$\mathbf{x} \lor \mathbf{y} = (\mathbf{x} \underset{\mathbf{L}}{\rightarrow} \mathbf{y}) \underset{\mathbf{L}}{\rightarrow} \mathbf{y} \tag{8.9}$$

When we apply the evaluation rule of implication (Impl.) to the right side of the equation (8.9) we get the equation

$$\mathbf{x} \lor y = (\mathbf{x} \to y) \underset{\mathbf{L}}{\to} y = \min\left\{1, 1 - \min(1, 1 - \mathbf{x} + y) + y\right\} = \max(\mathbf{x}, y) \tag{8.10}$$

by (8.8). Hence, the assertion (8.7) follows, and the proof is complete.

Of course, Łukasiewicz must have known the connection between maximum operation and his truth evaluation formula (Impl.) of the implication because without any knowledge about this, he would have not been sure that everything fits well together in his logic. But how he has inferred this is not known. Maybe, he has shown this in some special cases by truth tables with *n* truth values where *n* is finite.

The result of the proof of the formula (8.7) shows that from the join operation max of our algebra we deduce a formula that expresses the rule of Łukasiewicz' implication, and this formula is the truth value evaluation rule in Łℵ1 . Hence, we have shown that from our algebra (6.7) it is possible to derive similar rules as the truth value evaluation rules in Łℵ1 .

Hence, we may conclude our main result in a formal way:

**Proposition 8.3.** *If the cases*

*hold, then the other cases*

$$\begin{aligned} \text{3. } &\begin{aligned} \text{3. } &\begin{aligned} \text{x} \wedge y = \min(\text{x}, y); \\ \text{4. } &\begin{aligned} \text{x} \rightarrow y = \min(1, 1 - \text{x} + y); \\ \text{5. } &\begin{aligned} \text{x} \rightarrow y = (\text{x} \rightarrow y) \wedge (y \rightarrow \text{x}) = 1 - |\text{x} - y|. \end{aligned} \end{aligned} \end{aligned} \end{cases}$$

*can be derived based on Zadeh algebra (6.7).*

*Proof.* The case 3 follows from the case 2 by duality. (Actually, this operation already belongs to Zadeh algebra, and hence to Łℵ1 .)

The case 4 follows from the case 2 by Prop. 8.2 as follows. When we consider the equation (8.10) in the proof of Prop. 8.2, we find two min-stuctures corresponding to the evaluation rule of implication, so that one of them is an inside part of the whole formula. If we denote the inner min-structure min(1, 1 − *x* + *y*) by *z* then the outer min-structure is min(1, 1 − *z* + *y*), i.e., the min-structures are formally the same. The implication operations in (8.9) are situated in the same way. Hence, min(1, 1 − *x* + *y*) must be the evaluation rule of *x* → Ł *y*, by (Impl.).

The case 5 is deduced as follows:

18 Will-be-set-by-IN-TECH

1, 2 − 1 − 1 + *x* + 2*y* − *y*+ | 1 − 1 + *x* − *y* | 2

1, 1 − 1 + (1 − *x* + *y*)− | 1 − (1 − *x* + *y*) |

Ł *y*) →

= min { 1, 1 − min(1, 1 − *x* + *y*) + *y* } (8.8)

Ł *y* = min { 1, 1 − min(1, 1 − *x* + *y*) + *y* } = max(*x*, *y*) (8.10)

2 }

2 + *y*

Ł *y* (8.9)

max(*x*, *y*) = min{ 1, max(*x*, *y*) } = min{ 1, *x* + *y*+ | *x* − *y* |

On the other hand, in Łℵ1 disjunction is defined by(Disj.), i.e., by the formula

by (8.8). Hence, the assertion (8.7) follows, and the proof is complete.

Hence, we may conclude our main result in a formal way:

*x* ∨ *y* = (*x* →

When we apply the evaluation rule of implication (Impl.) to the right side of the equation (8.9)

Of course, Łukasiewicz must have known the connection between maximum operation and his truth evaluation formula (Impl.) of the implication because without any knowledge about this, he would have not been sure that everything fits well together in his logic. But how he has inferred this is not known. Maybe, he has shown this in some special cases by truth tables

The result of the proof of the formula (8.7) shows that from the join operation max of our algebra we deduce a formula that expresses the rule of Łukasiewicz' implication, and this formula is the truth value evaluation rule in Łℵ1 . Hence, we have shown that from our algebra

*Proof.* The case 3 follows from the case 2 by duality. (Actually, this operation already belongs

(6.7) it is possible to derive similar rules as the truth value evaluation rules in Łℵ1 .

= min

= min

we get the equation

*x* ∨ *y* = (*x* →

with *n* truth values where *n* is finite.

**Proposition 8.3.** *If the cases*

*4. x* → *y* = min(1, 1 − *x* + *y*)*;*

*5. x* ↔ *y* = (*x* → *y*) ∧ (*y* → *x*) = 1− | *x* − *y* |*,*

*can be derived based on Zadeh algebra (6.7).*

to Zadeh algebra, and hence to Łℵ1 .)

*1.* ¬*x* = 1 − *x; 2. x* ∨ *y* = max(*x*, *y*)*; hold, then the other cases 3. x* ∧ *y* = min(*x*, *y*)*;*

Ł *y*) →

$$\begin{split} x \leftrightarrow y &= \min\{ (\mathbf{x} \to y), (y \to \mathbf{x}) \} \\ &= \frac{(\mathbf{x} \to y) + (y \to \mathbf{x}) - |(\mathbf{x} \to y) - (y \to \mathbf{x})|}{2} \\ &= \frac{\min(1, 1 - \mathbf{x} + y) + \min(1, 1 - y + \mathbf{x})}{2} \\ &- \frac{\min(1, 1 - \mathbf{x} + y) - \min(1, 1 - y + \mathbf{x})}{2} \\ &= \frac{4 - 2|\mathbf{x} - y| - |-2\mathbf{x} + 2y - |\mathbf{x} - y| + |\mathbf{x} - y||}{4} \\ &= \frac{4 - 4|\mathbf{x} - y|}{4} = 1 - |\mathbf{x} - y|. \end{split}$$

Hence, the connectives of Łℵ1 can be created by Zadeh algebra.

These cases are similar to the truth value evaluation rules for connected formulas in Łℵ1 . Hence, if we want to use algebraic approach for Łℵ1 we need not necessarily to follow the mainstream described in Section 9 using the operations of MV-algebras for studying the connections between the connectives in Łℵ1 .

However, in the next section, we give also a very brief description about the alternative approach starting from the definition of general MV-algebra. It is the mainstream in this research topic, but a circuitous route in the case of Łukasiewicz logic.

### **9. The relationship between Łukasiewicz logic and MV-algebras**

We open another way a little for creating an algebra for Łukasiewicz logic. We adopt the definition and some properties of MV-algebras from Cignoli et. al. (Cignoli et al., 2000). Other sources are Bergmann (Bergmann, 2008) and Hájek (Hájek, 1998).

**Definition 9.1.** An *MV-algebra* is an algebra �*A*, ⊕, ¬, 0� with a binary operation ⊕, a unary operation ¬ and a constant 0 satisfying the following equations:

$$(\mathbf{M}\mathbf{V}\mathbf{1})\qquad \mathfrak{x}\oplus(\mathfrak{y}\oplus\mathfrak{z})=(\mathfrak{x}\oplus\mathfrak{y})\oplus\mathfrak{z}$$

$$(\mathsf{MV2})\qquad \mathfrak{x} \oplus \mathfrak{y} = \mathfrak{y} \oplus \mathfrak{x}$$

A non-empty set *A* is the universe of the MV-algebra �*A*, ⊕, ¬, 0�.

can be expressed in MV-algebra in the form (*cf.* Cignoli et al. (Cignoli et al., 2000), p. 78)

Standard Fuzzy Sets and some Many-Valued Logics 95

The equation (9.6) shows that between Łukasiewicz implication ant the operation ⊕ there is a similar connection as in S-implications which are defined by means of disjunction. But ⊕ is not disjunction in Łukasiewicz logic. In many cases ⊕ is interpreted as disjunction, but defined on the unit interval it gives different values as Łukasiewicz disjunction operation max. However, it is possible to define the operations max and min by means of the operations of MV-algebra, but then the result usually is a logic with additional operations having no reasonable interpretations (*cf.* for example, the logic Fuzzy*L* in Bergmann's book (Bergmann,

Wajsberg created an algebra, called by *Wajsberg algebra* (*W-algebra*, for short) which is know to serve an algebraic approach to Łukasiewicz infinite-valued logic. The following lemma (*cf.* Cignoli et al. (Cignoli et al., 2000), p. 83) gives the connection between Wajsberg algebras and

The binary operation of W-algebra is implication operation. In this algebra a unary operation is ¬ because it is needed to create the unit element 1. The zero element 0 belongs to this algebra because it implies the unit element by means of negation. Hence, the algebra is in a suitable form according to Łukasiewicz logic. Now, we have counterparts of the primitive connectives of Łukasiewicz logic as the operations of W-algebra. The other connectives can be created in

One consequence from this consideration is that in MV-algebras the operations max and min can be created by the operations of W-algebra, i.e., by the primitive connectives of Łukasiewicz

The main problem we considered here is to find connections between standard fuzzy sets and Łukasiewicz logic Łℵ1 and to find a suitable algebra for it, especially, because the primitive connectives are negation and implication. In De Morgan algebras the counterparts for the logical connectives disjunction, conjunction, and negation appear as the algebraic operations. It cannot immediately be seen how Łukasiewicz implication, that belongs to the primitive connectives, are derived from the disjunction (max) and negation (¬*x* = 1 − *x*, *x* ∈ **I**). We have done it here using a special De Morgan algebra, namely, Zadeh algebra. Hence, the connection between standard fuzzy sets and Łukasiewicz logic Łℵ1 becomes clear. The key result, where Łukasiewicz implication is derived algebraically from disjunction and negation,

Kleene's logic is considered because of its close connection to standard fuzzy sets, already motivated by Zadeh. The sections 2, 3, 4, 6, and 8 gives the method we have used for creating

def

= ¬*x* ⊕ *y and* 1 def

= ¬0*. Then* �*A*, →, 1� *is a*

2008). Some comments on Fuzzy*L* is given in (Mattila, 2010)).

the similar way in W-algebra as Łukasiewicz has introduced them.

**Lemma 9.1.** *Let A be an MV-algebra, and put x* → *y*

logic by means of the evaluation rule of (9.6).

whence,

MV-algebras.

**10. Conclusion**

is given in Proposition 8.3.

our results.

*W-algebra.*

*x* → *y* = ¬*x* ⊕ *y* (9.6)

*x* ⊕ *y* = ¬*x* → *y*. (9.7)

In particular, axioms (MV1) - (MV3) state that �*A*, ⊕, 0� is an *abelian monoid*.

Given an MV-algebra *A* and a set *X*, the set *AX* of all functions *f* : *X* −→ *A* becomes an MV-algebra if the operations ⊕ and ¬ and the element 0 are defined pointwise. It is obvious that the unit interval [0, 1] is an MV-algebra. The continuous functions from [0, 1] into [0, 1] form a subalgebra of the MV-algebra [0, 1] [0,1] .

On each MV-algebra *A* we define the constant 1 and the operations � and � as follows:

1 def = ¬0 , (9.1)

$$\mathfrak{x} \ominus \mathfrak{y} \stackrel{\text{def}}{=} \neg(\neg \mathfrak{x} \oplus \neg \mathfrak{y}) \, , \tag{9.2}$$

$$
\mathfrak{x} \ominus \mathfrak{y} \stackrel{\text{def}}{=} \mathfrak{x} \ominus \neg \mathfrak{y}.\tag{9.3}
$$

An MV-algebra is nontrivial is and only if 0 �= 1. The following identities are immediate consequences of (MV4):

$$(\mathbf{M}\mathbf{V}\mathbf{\bar{\mathbf{\bar{\mathbf{\bar{\mathbf{\bar{\mathbf{\bar{\mathbf{\bar{\mathbf{\bar{\mathbf{\mathbf{\bar{\mathbf{\mathbf{\bar{\mathbf{\mathbf{\mathbf{\bar{\mathbf{\mathbf{\mathbf{\cdot}}}}}}}}}}}}}}}}}}}}}}}}}$$
}}

(MV8) *x* ⊕ *y* = ¬(¬*x* � ¬*y*).

Axioms (MV5) and (MV6) can now be written as:

(MV5� ) *x* ⊕ 1 = 1 , (MV6� ) (*x* � *y*) ⊕ *y* = (*y* � *x*) ⊕ *x* .

Setting *y* = ¬0 in (MV6) we obtain

$$(\mathsf{MV9})\qquad \mathfrak{x} \oplus \neg \mathfrak{x} = 1\_{\mathsf{M}}$$

In the MV-algebra �[0, 1], ⊕, ¬, 0� we have

$$\mathbf{x} \odot \mathbf{y} = \max(0, \mathbf{x} + \mathbf{y} - 1) \tag{9.4}$$

$$
\pi \ominus y = \max(0, \pi - y) \tag{9.5}
$$

*Notation*: Following common usage, we consider the ¬ operation more binding than any other operation, and the � operation more binding than ⊕ and �.

Consider the question about the connection between Łukasiewicz implication and operations in MV-algebra.

Given an MV-algebra �*A*, ⊕, ¬, 0� and a set *X*, the set

$$A^X = \{ f \mid f: X \longrightarrow A \}$$

becomes an MV-algebra if the operations ⊕ and ¬ and the element 0 are defined pointwice (Cignoli et al. (Cignoli et al., 2000), p. 8). To define 0 pointwice means here that the result is a constant function **0** : *x* �→ 0 for any *x* in the universe of that algebra.

Further, Łukasiewicz implication

$$x \to y \stackrel{\text{def}}{=} \min\{1, 1 - x + y\}$$

can be expressed in MV-algebra in the form (*cf.* Cignoli et al. (Cignoli et al., 2000), p. 78)

$$
\mathfrak{x} \to \mathfrak{y} = \neg \mathfrak{x} \oplus \mathfrak{y} \tag{9.6}
$$

whence,

20 Will-be-set-by-IN-TECH

Given an MV-algebra *A* and a set *X*, the set *AX* of all functions *f* : *X* −→ *A* becomes an MV-algebra if the operations ⊕ and ¬ and the element 0 are defined pointwise. It is obvious that the unit interval [0, 1] is an MV-algebra. The continuous functions from [0, 1] into [0, 1]

An MV-algebra is nontrivial is and only if 0 �= 1. The following identities are immediate

*Notation*: Following common usage, we consider the ¬ operation more binding than any other

Consider the question about the connection between Łukasiewicz implication and operations

*AX* = { *f* | *f* : *X* −→ *A*}

becomes an MV-algebra if the operations ⊕ and ¬ and the element 0 are defined pointwice (Cignoli et al. (Cignoli et al., 2000), p. 8). To define 0 pointwice means here that the result is a

= min{1, 1 − *x* + *y*}

= ¬0 , (9.1)

= ¬(¬*x* ⊕ ¬*y*), (9.2)

= *x* � ¬*y* . (9.3)

*x* � *y* = max(0, *x* + *y* − 1) (9.4) *x* � *y* = max(0, *x* − *y*) (9.5)

[0,1] .

On each MV-algebra *A* we define the constant 1 and the operations � and � as follows:

1 def

*x* � *y* def

*x* � *y* def

In particular, axioms (MV1) - (MV3) state that �*A*, ⊕, 0� is an *abelian monoid*.

form a subalgebra of the MV-algebra [0, 1]

consequences of (MV4):

(MV8) *x* ⊕ *y* = ¬(¬*x* � ¬*y*).

Setting *y* = ¬0 in (MV6) we obtain

Further, Łukasiewicz implication

) *x* ⊕ 1 = 1 ,

(MV9) *x* ⊕ ¬*x* = 1 ,

in MV-algebra.

Axioms (MV5) and (MV6) can now be written as:

) (*x* � *y*) ⊕ *y* = (*y* � *x*) ⊕ *x* .

In the MV-algebra �[0, 1], ⊕, ¬, 0� we have

operation, and the � operation more binding than ⊕ and �.

constant function **0** : *x* �→ 0 for any *x* in the universe of that algebra.

*x* → *y*

def

Given an MV-algebra �*A*, ⊕, ¬, 0� and a set *X*, the set

(MV7) ¬1 = 0 ,

(MV5�

(MV6�

$$
\mathfrak{x} \oplus \mathfrak{y} = \neg \mathfrak{x} \to \mathfrak{y}.\tag{9.7}
$$

The equation (9.6) shows that between Łukasiewicz implication ant the operation ⊕ there is a similar connection as in S-implications which are defined by means of disjunction. But ⊕ is not disjunction in Łukasiewicz logic. In many cases ⊕ is interpreted as disjunction, but defined on the unit interval it gives different values as Łukasiewicz disjunction operation max. However, it is possible to define the operations max and min by means of the operations of MV-algebra, but then the result usually is a logic with additional operations having no reasonable interpretations (*cf.* for example, the logic Fuzzy*L* in Bergmann's book (Bergmann, 2008). Some comments on Fuzzy*L* is given in (Mattila, 2010)).

Wajsberg created an algebra, called by *Wajsberg algebra* (*W-algebra*, for short) which is know to serve an algebraic approach to Łukasiewicz infinite-valued logic. The following lemma (*cf.* Cignoli et al. (Cignoli et al., 2000), p. 83) gives the connection between Wajsberg algebras and MV-algebras.

**Lemma 9.1.** *Let A be an MV-algebra, and put x* → *y* def = ¬*x* ⊕ *y and* 1 def = ¬0*. Then* �*A*, →, 1� *is a W-algebra.*

The binary operation of W-algebra is implication operation. In this algebra a unary operation is ¬ because it is needed to create the unit element 1. The zero element 0 belongs to this algebra because it implies the unit element by means of negation. Hence, the algebra is in a suitable form according to Łukasiewicz logic. Now, we have counterparts of the primitive connectives of Łukasiewicz logic as the operations of W-algebra. The other connectives can be created in the similar way in W-algebra as Łukasiewicz has introduced them.

One consequence from this consideration is that in MV-algebras the operations max and min can be created by the operations of W-algebra, i.e., by the primitive connectives of Łukasiewicz logic by means of the evaluation rule of (9.6).

### **10. Conclusion**

The main problem we considered here is to find connections between standard fuzzy sets and Łukasiewicz logic Łℵ1 and to find a suitable algebra for it, especially, because the primitive connectives are negation and implication. In De Morgan algebras the counterparts for the logical connectives disjunction, conjunction, and negation appear as the algebraic operations. It cannot immediately be seen how Łukasiewicz implication, that belongs to the primitive connectives, are derived from the disjunction (max) and negation (¬*x* = 1 − *x*, *x* ∈ **I**). We have done it here using a special De Morgan algebra, namely, Zadeh algebra. Hence, the connection between standard fuzzy sets and Łukasiewicz logic Łℵ1 becomes clear. The key result, where Łukasiewicz implication is derived algebraically from disjunction and negation, is given in Proposition 8.3.

Kleene's logic is considered because of its close connection to standard fuzzy sets, already motivated by Zadeh. The sections 2, 3, 4, 6, and 8 gives the method we have used for creating our results.

**5**

*Mexico*

**Parametric Type-2 Fuzzy Logic Systems**

The use of Fuzzy Logic Systems (FLS) for control applications has increased since they became popular from 80's. After Mendel in 90's showed how uncertainty can be computed in order to achieve more robust systems, Type-2 Fuzzy Logic Systems (T2FLS) are in the

At same time, Batyrshin et al demonstrated that parametric conjunctions can be useful for tuning a FLS in order to achieve better performance beyond the set parameter tuning. In signal processing and system identification, this fact let the designer to add freedom degrees

This chapter presents the parametric T2FLS and shows that this new FLS is a very useful option for sharper approximations in control. In order to verify the advantages of the parametric T2FLS, it is used the Ball and Plate System as a testbench. This study case helps us to understand how a parametric conjunction affects the controller behavior in measures like response time or overshoot. Also, this application let us observe how the controller

A Parametric Type-2 Fuzzy Logic Systems (PT2FLS) is a general FLS which can be fully adjusted through a single or multiple parameters in order to achieve a benefit in its general performance. It means that a PT2FLS has several options to adjust set parameters (i.e. membership function parameters), rule parameters and output set parameters. Fig. 1 shows

In this figure the Defuzzification stage comprises the Output Processing Block and the Defuzzifier as Mendel stated in (Karnik, Mendel et al. 1999). For Interval Type-2 Fuzzy Logic Systems (IT2FLS) this block represents only the centroid calculation for example considering the WM Algorithm (Wu and Mendel 2002). As it can be seen, a dashed arrow

A general Fuzzy System is a function where all input variables are mapped to the output variables according to the knowledge base defined by rules. Rule Set represents the

crosses every stage; this means that every stage is tunable for optimization purposes.

focus of researchers and recently they became a new research topic.

the structure of a PT2FLS which it is almost equal to a general T2FLS.

**1. Introduction**

to adjust a general FLS.

works in noise presence.

**2. Parametric T2FLS**

configuration of the T2FLS.

Arturo Tellez, Heron Molina, Luis Villa,

Elsa Rubio and Ildar Batyrshin

*IPN, CIC, Mexico City,*

The section 9 tells very briefly how the others consider this topic. That way is different and alternative to ours. MV-algebra is quite general, and many algebras, like Boolean algebras and also De Morgan algebras belong to its scope. The reader may become familiar with this topic, for example, by reading the Bergmann's, Cignoli's et. al, and Hájek's books mentioned in References. A lot of other material is available, too.

Our alternative way we have considered the topic here, is not totally new, because these things are considered in (Mattila, 2004), (Mattila, 2005), and (Mattila, 2010), but our key result, Proposition 8.3 is. Proposition 8.2 is the core of this result. It makes the connection of max operation and Łukasiewicz inmplication clear by means of Zadeh algebra. Using the expressions (8.4) for the operations max and min is usually not used in general, but it makes the consideration easy. As we can see from the used references, De Morgan algebras have already been well known relatively long time having, in the long run, different alternative names, like "quasi Boolean algebras", and "soft algebras".

H. Rasiowa has considered *implicative algebras* and *implication algebras* in her book ((Rasiowa, 1974)). Hence, the future research policy may be based on these algebras. Also, the connections between implicative/implication algebras and De Morgan algebras or MV-algebras restricted to many-valued or modal logics are included to the future research.

### **11. References**

## **Parametric Type-2 Fuzzy Logic Systems**

Arturo Tellez, Heron Molina, Luis Villa, Elsa Rubio and Ildar Batyrshin *IPN, CIC, Mexico City, Mexico*

### **1. Introduction**

22 Will-be-set-by-IN-TECH

96 Fuzzy Logic – Algorithms, Techniques and Implementations

The section 9 tells very briefly how the others consider this topic. That way is different and alternative to ours. MV-algebra is quite general, and many algebras, like Boolean algebras and also De Morgan algebras belong to its scope. The reader may become familiar with this topic, for example, by reading the Bergmann's, Cignoli's et. al, and Hájek's books mentioned

Our alternative way we have considered the topic here, is not totally new, because these things are considered in (Mattila, 2004), (Mattila, 2005), and (Mattila, 2010), but our key result, Proposition 8.3 is. Proposition 8.2 is the core of this result. It makes the connection of max operation and Łukasiewicz inmplication clear by means of Zadeh algebra. Using the expressions (8.4) for the operations max and min is usually not used in general, but it makes the consideration easy. As we can see from the used references, De Morgan algebras have already been well known relatively long time having, in the long run, different alternative

H. Rasiowa has considered *implicative algebras* and *implication algebras* in her book ((Rasiowa, 1974)). Hence, the future research policy may be based on these algebras. Also, the connections between implicative/implication algebras and De Morgan algebras or MV-algebras restricted to many-valued or modal logics are included to the future research.

Bergmann, M. (2008). *An Introduction to Many-Valued and Fuzzy Logic*, Cambridge University Press, New York, Melbourne, Madrid,Cape Town, Singapore, São Paulo, Delhi. Cignoli, L. & D'Ottaviano, M. & Mundici, D. (2000). *Algebraic Foundations of Many-valued Reasoning*, Kluwer Academic Publishers, Dordrecht, Boston, London. Hájek, P. (1998). *Metamathematics of Fuzzy Logic*, Kluwer Academic Publishers, Dordrecht,

Lowen, R. (1996). *Fuzzy Set Theory. Basic Concepts, Techniques and Bibliography*, Kluwer

Mattila, J. K. (2004). Zadeh algebras as a syntactical approach to fuzzy sets, *Current Issues in Data and Knowledge Engineering*, Springer-Verlag, Warszawa, Poland, pp. 343–349. Mattila, J. K. (year 2005). On łukasiewicz modifier logic, *Journal of Advanced Computational*

Mattila, J. K. (2009). Many-valuation, modality, and fuzziness, *in* Seising R. (ed.), *Views on*

Mattila, J. K. (2010). On Łukasiewicz' infinite-Valued logic and Fuzzy*L*, *in*, *KES 2010*

Negoit ˘a, C. V. & Ralescu, D. A., (1975). *Applications of Fuzzy Sets to Systems Analysis*,

Rescher, N. (1969). *Many-Valued Logic*, McGraw-Hill, New York, St. Louis, San Francisco,

*Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisms and*

*Knowledge-Based Intelligent Information and Engineering Systems*, Part IV, LNAI 6279,

Academic Publishers, Dordrecht, Boston, London.

Springer-Verlag, Berlin, Heidelberg, pp. 108–115.

London, Sydney, Toronto, Mexiko, Panama. Zadeh, L. A. (1965). Fuzzy Sets, *Information and Controll* 8.

*Intelligence and Intelligent Informatics* Vol. 9(No. 5): 506–510.

*Applications*, Springer-Verlag, Berlin, Heidelberg, pp. 271–300.

Rasiowa, H. (1974). *An Algebraic Approach to non-classical Logics*, North-Holland.

in References. A lot of other material is available, too.

names, like "quasi Boolean algebras", and "soft algebras".

**11. References**

Boston, London.

URL: *www.fujipress.jp*

Birkhäuser, Basel, Stuttgart.

The use of Fuzzy Logic Systems (FLS) for control applications has increased since they became popular from 80's. After Mendel in 90's showed how uncertainty can be computed in order to achieve more robust systems, Type-2 Fuzzy Logic Systems (T2FLS) are in the focus of researchers and recently they became a new research topic.

At same time, Batyrshin et al demonstrated that parametric conjunctions can be useful for tuning a FLS in order to achieve better performance beyond the set parameter tuning. In signal processing and system identification, this fact let the designer to add freedom degrees to adjust a general FLS.

This chapter presents the parametric T2FLS and shows that this new FLS is a very useful option for sharper approximations in control. In order to verify the advantages of the parametric T2FLS, it is used the Ball and Plate System as a testbench. This study case helps us to understand how a parametric conjunction affects the controller behavior in measures like response time or overshoot. Also, this application let us observe how the controller works in noise presence.

### **2. Parametric T2FLS**

A Parametric Type-2 Fuzzy Logic Systems (PT2FLS) is a general FLS which can be fully adjusted through a single or multiple parameters in order to achieve a benefit in its general performance. It means that a PT2FLS has several options to adjust set parameters (i.e. membership function parameters), rule parameters and output set parameters. Fig. 1 shows the structure of a PT2FLS which it is almost equal to a general T2FLS.

In this figure the Defuzzification stage comprises the Output Processing Block and the Defuzzifier as Mendel stated in (Karnik, Mendel et al. 1999). For Interval Type-2 Fuzzy Logic Systems (IT2FLS) this block represents only the centroid calculation for example considering the WM Algorithm (Wu and Mendel 2002). As it can be seen, a dashed arrow crosses every stage; this means that every stage is tunable for optimization purposes.

A general Fuzzy System is a function where all input variables are mapped to the output variables according to the knowledge base defined by rules. Rule Set represents the configuration of the T2FLS.

$$\begin{aligned} F^l &= \left[ \overline{F}^l; \underline{F}^l \right] = \left[ \overline{\mu}\_{\tilde{A}\_l^l} (\boldsymbol{\chi}\_l); \,\underline{\mu}\_{\tilde{A}\_l^l} (\boldsymbol{\chi}\_l) \right] \\\\ \overline{F}^l &= \overline{\mu}\_{\tilde{A}\_1^l} (\boldsymbol{\chi}\_1) \wedge \overline{\mu}\_{\tilde{A}\_2^l} (\boldsymbol{\chi}\_2) \wedge \dots \wedge \overline{\mu}\_{\tilde{A}\_m^l} (\boldsymbol{\chi}\_m) \\\\ \underline{F}^l &= \underline{\mu}\_{\tilde{A}\_1^l} (\boldsymbol{\chi}\_1) \wedge \underline{\mu}\_{\tilde{A}\_1^l} (\boldsymbol{\chi}\_2) \wedge \dots \wedge \underline{\mu}\_{\tilde{A}\_m^l} (\boldsymbol{\chi}\_m) \end{aligned}$$

$$\overline{F}^l = \mathbf{T}\left(\overline{\mu}\_{\vec{A}\_1^l}(\mathbf{x}\_1), \overline{\mu}\_{\vec{A}\_2^l}(\mathbf{x}\_2), \dots, \overline{\mu}\_{\vec{A}\_m^l}(\mathbf{x}\_m); \mathbb{p}\_r\right) = \mathbf{T}\left(\overline{\mu}\_{\vec{A}\_l^l}(\mathbf{x}\_l); \mathbb{p}\_r^l\right) \tag{1}$$

$$\underline{F}^l = \mathbf{T}\left(\underline{\mu}\_{\tilde{A}\_1^l}(\mathbf{x}\_1), \underline{\mu}\_{\tilde{A}\_2^l}(\mathbf{x}\_2), \dots, \underline{\mu}\_{\tilde{A}\_m^l}(\mathbf{x}\_m); \mathbf{p}\_r\right) = \mathbf{T}\left(\underline{\mu}\_{\tilde{A}\_l^l}(\mathbf{x}\_l); \mathbf{p}\_r^l\right) \tag{2}$$

$$\overline{\mu}\_{\mathcal{B}\_1^{\ell}}(\mathbf{y}) = \sqcup\_{\mathbf{x} \in \mathcal{X}} \left( \mathbf{T} \left( \overline{\mu}\_{\mathcal{A}\_l^{\ell}}(\mathbf{x}\_l); \mathbb{p}\_r \right) \right) \tag{3}$$

$$\underline{\mu}\_{\mathcal{B}\_1^{\ell}}(\mathcal{y}) = \sqcup\_{\mathbf{x} \in \mathcal{X}} \left( \mathbf{T} \left( \underline{\mu}\_{\mathcal{A}\_l^{\ell}}(\mathbf{x}\_l); \mathbb{p}\_r \right) \right) \tag{4}$$

$$T(a,b) = T\_{ll}(a,b) \text{ if } (a,b) \in D\_{lj}; t, j \in G \tag{5}$$

$$T\_{l,l} = T\_{l,l} \tag{6}$$

$$T(a,b,p) = \begin{cases} T\_{11}(a,b), & (a \le p) \land (b \le p) \\ T\_{21}(a,b), & (a > p) \land (b \le p) \\ T\_{12}(a,b), & (a \le p) \land (b > p) \\ T\_{22}(a,b), & (a > p) \land (b > p) \end{cases} \tag{7}$$

$$T(a,b,p) = \begin{cases} T\_{11}(a,b), & (a \le p) \land (b \le p) \\ T\_{21}(a,b), & (a > p) \land (a \le 1-p) \land (b \le p) \\ T\_{12}(a,b), & (a > 1-p) \land (b \le p) \\ T\_{12}(a,b), & (a \le p) \land (b > p) \land (b \le 1-p) \\ T\_{22}(a,b), & (a > p) \land (a \le 1-p) \land (b > p) \land (b \le 1-p) \\ T\_{32}(a,b), & (a > 1-p) \land (b > p) \land (b \le 1-p) \\ T\_{13}(a,b), & (a \le p) \land (b > p) \land (b \le 1-p) \\ T\_{23}(a,b), & (a > p) \land (a \le 1-p) \land (b > p) \\ T\_{33}(a,b), & (a > p) \land (b > p) \end{cases} \tag{8}$$

$$H\_{p-2} \xrightarrow{p} \begin{bmatrix} 1 & p & t-p \\ D\_{12} & D\_{22} & D\_{32} \\ D\_{13} & D\_{23} & D\_{33} \\ & H\_{12} & D\_{12} & D\_{23} \\ H\_{12} & H\_{21} & D\_{22} & D\_{33} \end{bmatrix} t-p$$

$$H\_{p-1} \xrightarrow[h \to H\_{p-1}]{H\_{p-1} \xrightarrow[h \to H\_{p-1}]{H\_{p-1} \xrightarrow[h \to H\_{p-1}]}} 1 \quad H\_{p-2} \xrightarrow[h \to H\_{p-2}]{H\_{p-1} \xrightarrow[h \to H\_{p-1}]} 1$$

Parametric Type-2 Fuzzy Logic Systems 105

ball, over the axis X. If it is assumed that the velocity and the acceleration of the ball are constant at shorter values of �, it is possible to estimate the next position of ball with

NA NM Z PM PA

NA NM Z PM PA PA NM NM NM PA PA PA Z NA NA Z PA PA PM NA NA NA PM PM PA NA NA NM Z PM

��(���) and finally the desired position �� and the estimated error �(���).

**Error**

Rubio-Espino et al. 2010) used for the PT2FLC purposes.

Fig. 7. IT2FLS Simulator for B&P System

negative" (NA). Rule set is described in Table 1.

corresponds to every axis.

**Tilt Change**

Table 1. Optimal Rule Set of B&P System with T1FLC described in (Moreno-Armendariz,

It is proved that B&P System is a decoupled system over its two axes (Moreno-Armendariz, Rubio-Espino et al. 2010). So, (10) are similar for the axis Y. Fig 5 shows that B&P system block has two inputs and two outputs for our control purposes; so, every in-out pair

T1FLC proposed by (Moreno-Armendariz, Rubio-Espino et al. 2010) has two inputs and one output. Every variable has 5 FS (Fig. 6) associated to linguistic variables "high positive" (PA), "medium positive" (PM), "zero or null" (Z), "medium negative" (NM), and "high

The T1FLC controls the tilt of plate using the information that the FPGA takes form the camera and calculates the current position using (10) under perfect environment conditions. But what happens when some external forces (e.g. weather) complicate the system stability? Some equivalent phenomena may be introduced to the plate. For example, the illuminating

in axis Y. Servomotors perform the adequate tilt over both axes. Every tilt value is calculated by its corresponding PT2FLC using the error position and the position change. Position change is the differential of the feedback of the plant, i.e. the current position.

It is noteworthy that PT2FLC hardware has not been implemented and tested for this application. Only simulations are performed in order to show all advantages of the use of PT2FLC in control applications. Mechanical model proposed in (Moreno-Armendariz, Rubio-Espino et al. 2010) has the characteristic of designing and testing new and improved controllers, which it is a suitable future work, because of the flexibility of FPGA.

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -9.81 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ -6.1313 \times 10^4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -9.81 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -6.1313 \times 10^4 & 0 & 0 & 0 \end{bmatrix}$$

$$B = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 5.6618 \times 10^4 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 5.6618 \times 10^4 \end{bmatrix} \tag{9}$$

$$C = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{bmatrix}$$

$$D = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{bmatrix}$$

$$x' = Ax + Bu$$

$$y = Cx$$

The characteristics of this B&P System (9) is a linearized state-space model, the same as described in (Moreno-Armendariz, Rubio-Espino et al. 2010). With (10), it can be calculated the current velocity, acceleration and position in axis x.

$$\begin{aligned} v(k) &= \frac{\{\mathbf{x}(k) - \mathbf{x}(k-1)\}}{T} \\ a(k) &= v(k) - v(k-1) \\ \mathbf{x}\_e(k+1) &= \mathbf{x}(k) + \frac{v(k)}{T} + \frac{a(k)T^2}{2} \\ e(k+1) &= \mathbf{x}\_d - \mathbf{x}\_e(k+1) \end{aligned} \tag{10}$$

The vision system described in (Moreno-Armendariz, Rubio-Espino et al. 2010) uses a sampling time � (50ms), which captures and processes a single image in that period. After the vision system process the image, FPGA calculates the current position of the ball in axis X, �(�), where � is the current sample. Once it is known the position, it is possible to find the current velocity component �(�) and the current acceleration component �(�) of the 104 Fuzzy Logic – Algorithms, Techniques and Implementations

in axis Y. Servomotors perform the adequate tilt over both axes. Every tilt value is calculated by its corresponding PT2FLC using the error position and the position change. Position

It is noteworthy that PT2FLC hardware has not been implemented and tested for this application. Only simulations are performed in order to show all advantages of the use of PT2FLC in control applications. Mechanical model proposed in (Moreno-Armendariz, Rubio-Espino et al. 2010) has the characteristic of designing and testing new and improved

change is the differential of the feedback of the plant, i.e. the current position.

controllers, which it is a suitable future work, because of the flexibility of FPGA.

� � � � � � �

the current velocity, acceleration and position in axis x.

� 0 100 0 000 0 0 −9.81 0 0 0 0 0 0 001 0 000 −6.1313 × 10� 000 0 000 0 000 0 100 0 0 0 0 0 0 −9.81 0 0 000 0 001 0 0 0 0 −6.1313 × 10� 000�

> � 0 0 0 0 0 0 5.6818 × 10� 0 0 0 0 0 0 0 0 5.6818 × 10��

�=�10000000 00001000�

> � = [0] �� = �� + �� � = ��

�(�) = ��(�) − �(�−1)� �

�(�+1) = �� − ��(�+1)

��(�+1) = �(�) +

The characteristics of this B&P System (9) is a linearized state-space model, the same as described in (Moreno-Armendariz, Rubio-Espino et al. 2010). With (10), it can be calculated

> �(�) � +

The vision system described in (Moreno-Armendariz, Rubio-Espino et al. 2010) uses a sampling time � (50ms), which captures and processes a single image in that period. After the vision system process the image, FPGA calculates the current position of the ball in axis X, �(�), where � is the current sample. Once it is known the position, it is possible to find the current velocity component �(�) and the current acceleration component �(�) of the

� � � � � � �

� = (9)

(10) �(�) = �(�) − �(�−1)

�(�)�� 2

� � � � � � �

� =

� � � � � � � ball, over the axis X. If it is assumed that the velocity and the acceleration of the ball are constant at shorter values of �, it is possible to estimate the next position of ball with ��(���) and finally the desired position �� and the estimated error �(���).

Table 1. Optimal Rule Set of B&P System with T1FLC described in (Moreno-Armendariz, Rubio-Espino et al. 2010) used for the PT2FLC purposes.

Fig. 7. IT2FLS Simulator for B&P System

It is proved that B&P System is a decoupled system over its two axes (Moreno-Armendariz, Rubio-Espino et al. 2010). So, (10) are similar for the axis Y. Fig 5 shows that B&P system block has two inputs and two outputs for our control purposes; so, every in-out pair corresponds to every axis.

T1FLC proposed by (Moreno-Armendariz, Rubio-Espino et al. 2010) has two inputs and one output. Every variable has 5 FS (Fig. 6) associated to linguistic variables "high positive" (PA), "medium positive" (PM), "zero or null" (Z), "medium negative" (NM), and "high negative" (NA). Rule set is described in Table 1.

The T1FLC controls the tilt of plate using the information that the FPGA takes form the camera and calculates the current position using (10) under perfect environment conditions. But what happens when some external forces (e.g. weather) complicate the system stability? Some equivalent phenomena may be introduced to the plate. For example, the illuminating

Parametric Type-2 Fuzzy Logic Systems 107

Fig. 8. Second approximation of PT2FLC modifying the FOU of sets

optimal set distribution.

Fig. 9. Parametric Fuzzy Conjunction using (p) −Monotone Sum with parameter p = 0.25

In first experiments, (Fig. 6) it was re-adjusted the FOU of every set, leaving the set distribution intact, so it was found that only for a very thin FOU in every input set it is gotten a good convergence without overshoot and other phenomena. But, what is the sense of having a very short FOU like T1FS if they will not capture the associated uncertainties of the system? So, there should be a way of tuning the PT2FLC without changing this initial

variation due to light incidence over the plate, an unbalanced motor tied to the plate, a low quality image sensor or some interference noise added to the processed image, may be introduced as external disturbances.

Table 2. Phenomena associated with the FOU of every set in system

In initial experiments, noise-free optimization is performed and similar results are achieved in order to compare it with T1FLC. For noise tests it is only considered an unbalanced motor tied to the plate that makes it tremble while a sine trajectory is performed, analyzing a single axis. This experiment helps us to verify the noise-proof ability of the T2FLC.

### **4. Experimental results**

FS distribution, i.e., FS shape parameters may arise several characteristic phenomena that expert must take into account when designing applied-to-control fuzzy systems, so-called Fuzzy Logic Controller (FLC). As described in (Moreno-Armendariz, Rubio-Espino et al. 2010), authors found an optimal FS distribution where FLC shows a great performance in 3.8 seconds. However, when it is used this same configuration some phenomena arises when it is introduced T2FS.

Starting from the initial optimal set distribution and without considering any possible noise influence, it was tested several configurations modifying every set FOU, starting from a T1FS (without FOU) and increasing it as much as possible; or starting from a very wide FOU and collapsing it until it becomes a T1FS. Some phenomena are related to them as described in Table 2, but in general, when it is introduced a T2FS a certain level of overshoot is found, no matter which variable was modified; so, if every variable has a T2FS, then the expert has to deal with the influence of nonlinear aggregation of overshoot, steady-state error or offset (SSE) and ripple, when tuning a PT2FLC, which might be a complicated task.

For every experiment it was used an implemented simulator for IT2FLS. With some instructions it can be constructed any parametric IT2FLS and expert may choose from set shape, several parametric conjunctions and defuzzification options (Fig. 7).

106 Fuzzy Logic – Algorithms, Techniques and Implementations

variation due to light incidence over the plate, an unbalanced motor tied to the plate, a low quality image sensor or some interference noise added to the processed image, may be

which set FOUs are increasing from zero. Yes No No

In initial experiments, noise-free optimization is performed and similar results are achieved in order to compare it with T1FLC. For noise tests it is only considered an unbalanced motor tied to the plate that makes it tremble while a sine trajectory is performed, analyzing a single

FS distribution, i.e., FS shape parameters may arise several characteristic phenomena that expert must take into account when designing applied-to-control fuzzy systems, so-called Fuzzy Logic Controller (FLC). As described in (Moreno-Armendariz, Rubio-Espino et al. 2010), authors found an optimal FS distribution where FLC shows a great performance in 3.8 seconds. However, when it is used this same configuration some phenomena arises when it

Starting from the initial optimal set distribution and without considering any possible noise influence, it was tested several configurations modifying every set FOU, starting from a T1FS (without FOU) and increasing it as much as possible; or starting from a very wide FOU and collapsing it until it becomes a T1FS. Some phenomena are related to them as described in Table 2, but in general, when it is introduced a T2FS a certain level of overshoot is found, no matter which variable was modified; so, if every variable has a T2FS, then the expert has to deal with the influence of nonlinear aggregation of overshoot, steady-state error or offset (SSE) and ripple, when tuning a PT2FLC, which might be a

For every experiment it was used an implemented simulator for IT2FLS. With some instructions it can be constructed any parametric IT2FLS and expert may choose from set

shape, several parametric conjunctions and defuzzification options (Fig. 7).

**Experiment Overshoot SSE Ripple**

No No Yes

Yes No No

No Yes No

introduced as external disturbances.

are wide as much as it can be possible.

are wide as much as it can be possible.

**4. Experimental results**

is introduced T2FS.

complicated task.

When all sets in every variable are T1, except the variable

When all sets in input variable error are T2 and their FOU are decreasing until they become T1. All other variable sets

When all sets in input variable change are T2 and their FOU are decreasing until they become T1. All other variable sets are wide as much as it can be possible.

When all sets in output variable tilt are T2 and their FOU are decreasing until they become T1. All other variable sets

Table 2. Phenomena associated with the FOU of every set in system

axis. This experiment helps us to verify the noise-proof ability of the T2FLC.

Fig. 8. Second approximation of PT2FLC modifying the FOU of sets

Fig. 9. Parametric Fuzzy Conjunction using (p) −Monotone Sum with parameter p = 0.25

In first experiments, (Fig. 6) it was re-adjusted the FOU of every set, leaving the set distribution intact, so it was found that only for a very thin FOU in every input set it is gotten a good convergence without overshoot and other phenomena. But, what is the sense of having a very short FOU like T1FS if they will not capture the associated uncertainties of the system? So, there should be a way of tuning the PT2FLC without changing this initial optimal set distribution.

Parametric Type-2 Fuzzy Logic Systems 109

**shoot SSE Ripple**

No No No

No No Yes

Yes No No

No Yes No

No No Yes

**Parameter Description Over**

non-parametric conjunctions.

parameter value of rule 18.

7b, 19c These rules increase or decrease the offset of

8b This rule help to stretch the ripple slightly but

Fig. 10. Rule 18 parameter distribution for 43 experiments

conjunctions also.

overshoot.

These rules have no influence with the final response, so their parameter values might be any. These rules may be quantified using

These rules have a very slight influence with the final response. Some of them reduce the ripple, but they are negligible. These rules may be quantified using non-parametric

These rules have a very positive influence with the final response, especially the

the final response, but could add some

also might be useful to reduce small ripple.

Table 3. Phenomena associated with the rule operator of every implication in inference

**Rule**

1, 2, 5, 6, 10, 16, 17, 20, 21, 22, 25

3, 4, 9, 11, 12,

13b, 14c, 18a, 23b, 24b

15

In second experiments, it was moved the FOU of every set in every variable and found a very close approximation of time response as described in Fig. 8. This configuration has wider FOU in every input and output variable as much as necessary (with uniform spread) for supporting variations in error until 0.0075 radians, in change until 0.01 radians per second and in tilt until 0.004 radians, all around the mean of every point of its corresponding set and variable.

As it can be seen, every set exhibits a wider FOU and its time response has increased over 5 seconds. Also, some overshoot and ripple are present, but reference is reached, so SSE is eliminated. This is the first best approximation using the same optimal distribution of sets, although it does not mean that there could not be any other set distribution for this application.

As it is sated in (Batyrshin, Rudas et al. 2009), a parametric operator may help to tune a T1FLC through the inference step, so every rule of the knowledge base related with the implication of the premises might be a parametric conjunction. In third experiments, it is used commutative (�) −monotone sum of conjunctions (11), where it is assigned to every section the following conjunctions: ��� = �� is the drastic intersection, ��� = ��� = �� is the product and ��� = �� is the minimum, using (7) as follows:

$$T(\mathbf{x}, \mathbf{y}, p) = \begin{cases} T\_d(\mathbf{x}, \mathbf{y}), \ (\mathbf{x} \le p) \land (\mathbf{y} \le p) \\ T\_p(\mathbf{x}, \mathbf{y}), \ [(\mathbf{x} > p) \land (\mathbf{y} \le p)] \lor [(\mathbf{x} \le p) \land (\mathbf{y} > p)] \\ T\_m(\mathbf{x}, \mathbf{y}), \ (\mathbf{x} > p) \land (\mathbf{y} > p) \end{cases} \tag{11}$$

In (10), it is possible to assure that when parameter �=0 then the conjunction in (11) will have a minimum t-norm behavior, but when parameter �=1, it will be a drastic product tnorm behavior as it can be seen in Fig. 9. If � has any other value between the interval (0,1), then it will have a drastic, product o minimum t-norm behavior depending on the membership values of operands. Resulting behavior of this monotone sum might help to diminish the fuzzy implication between two membership degrees of premises and therefore to reduce the resulting overshoot of system and then reach the reference faster. Now another task is to choose the values of every parameter of conjunctions.

Moreover the optimal FS distribution, it is used the same rule set of (Moreno-Armendariz, Rubio-Espino et al. 2010) as shown in Table 1 in order to show that any T1FLC can be extended to a PT2FLC. So, � = 25 rules define the T1FLC configuration, it means that there are 25 parametric conjunctions and therefore 25 parameters. When searching for an optimal value of every �, it is recommended to use an optimization algorithm in order to obtain optimal values and the resulting waste of time when calculating them manually.

According to (11), the initial values of ��, make the conjunctions to behave like min, i.e.

$$\mathbb{p}\_r = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$$

It is proposed some values when optimization was performed with heuristics to get optimal rule parameters, i.e.

$$\mathfrak{p}\_r = \{0, 0.25, 0.25, 0, 0, 0, 0.2, 0, 0, 1, 0, 1, 0, 0, 0.2, 0, 0, 0, 0, 0.25, 0.25, 0\} \tag{12}$$

108 Fuzzy Logic – Algorithms, Techniques and Implementations

In second experiments, it was moved the FOU of every set in every variable and found a very close approximation of time response as described in Fig. 8. This configuration has wider FOU in every input and output variable as much as necessary (with uniform spread) for supporting variations in error until 0.0075 radians, in change until 0.01 radians per second and in tilt until 0.004 radians, all around the mean of every point of its

As it can be seen, every set exhibits a wider FOU and its time response has increased over 5 seconds. Also, some overshoot and ripple are present, but reference is reached, so SSE is eliminated. This is the first best approximation using the same optimal distribution of sets, although it does not mean that there could not be any other set distribution for this

As it is sated in (Batyrshin, Rudas et al. 2009), a parametric operator may help to tune a T1FLC through the inference step, so every rule of the knowledge base related with the implication of the premises might be a parametric conjunction. In third experiments, it is used commutative (�) −monotone sum of conjunctions (11), where it is assigned to every section the following conjunctions: ��� = �� is the drastic intersection, ��� = ��� = �� is the

��(�, �), [(���) ∧ (���)] ∨ [(���) ∧ (���)]

(11)

��(�, �), (���) ∧ (���)

��(�, �), (���) ∧ (���)

In (10), it is possible to assure that when parameter �=0 then the conjunction in (11) will have a minimum t-norm behavior, but when parameter �=1, it will be a drastic product tnorm behavior as it can be seen in Fig. 9. If � has any other value between the interval (0,1), then it will have a drastic, product o minimum t-norm behavior depending on the membership values of operands. Resulting behavior of this monotone sum might help to diminish the fuzzy implication between two membership degrees of premises and therefore to reduce the resulting overshoot of system and then reach the reference faster. Now another

Moreover the optimal FS distribution, it is used the same rule set of (Moreno-Armendariz, Rubio-Espino et al. 2010) as shown in Table 1 in order to show that any T1FLC can be extended to a PT2FLC. So, � = 25 rules define the T1FLC configuration, it means that there are 25 parametric conjunctions and therefore 25 parameters. When searching for an optimal value of every �, it is recommended to use an optimization algorithm in order to obtain

optimal values and the resulting waste of time when calculating them manually.

According to (11), the initial values of ��, make the conjunctions to behave like min, i.e.

�� = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] It is proposed some values when optimization was performed with heuristics to get optimal

�� = [0,0.25,0.25,0,0,0,0,0.2,0,0,1,0,1,0,1,0,0,0.2,0,0,0,0,0.25,0.25,0] (12)

product and ��� = �� is the minimum, using (7) as follows:

task is to choose the values of every parameter of conjunctions.

�(�, �, �) = �

corresponding set and variable.

application.

rule parameters, i.e.

Table 3. Phenomena associated with the rule operator of every implication in inference

Fig. 10. Rule 18 parameter distribution for 43 experiments

Parametric Type-2 Fuzzy Logic Systems 111

Suppose a PT2FLC where it is only modified the parameter value of rule 18 and a set of parameters that can be spread randomly around the mean of its value ��(18) = 0.7. For this experiment, it was performed 43 iterations in order to show how the variation of ��(18)

Table 4 shows some results about the transient when trying to reach a tilt = 0.125 rads. Other phenomena can be analyzed for all 43 iterations. Also, in Fig.12 it can be seen that overshoot is attenuated drastically when ��(18) → 1, if it is only modified this rule. Time response (rise time, peak time and settling time) is also compromised due to parametric conjunctions. It can be seen also that drastic attenuation of overshoot occurs for ��(18) ≲ 0.7. Greater values do not affect it meaningfully. As it can be seen in (12), rule parameter proposed as the optimal for rule 18 is near to 1, which might be different with other configurations. This is because of the influence of the rest of rule parameters. However, this optimal configuration does not compromise the response time but it does eliminate the

affects the overshoot attenuation and also other phenomena (Fig. 10-11).

Fig. 13. Final approximation of T2FLC modifying rule parameters

Fig. 14. Comparison of response between T1FLC and parametric T2FLC when reference is a

overshoot completely.

noisy sine signal

Fig. 11. Transient response for several values of parameter of rule 18

Table 4. Transient characteristics for parameter variation of rule 18

Fig. 12. Histograms for transient measures (overshoot, rise time, peak time and settling time) for rule parameter 18