Definition 8.17 A variable of DF proposition is the variable of closed interval [0, 1] × [←, →]. We call that as DF proposition variable. We usually use lower case letters to express this.

For DF variable $\left(\overleftarrow{x},\overrightarrow{x}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right],$ the rules are as follows: (notes: $\left(\overleftarrow{x},\overrightarrow{x}\right)=\overleftarrow{x}\text{or}\overrightarrow{x},\max \left(\overleftarrow{x},\overrightarrow{x}\right)\triangleq \overrightarrow{x},\min \left(\overleftarrow{x},\overrightarrow{x}\right)\triangleq \overleftarrow{x})$

(1)Negation “—”.
For example: The negation of $\left(\overleftarrow{x},\overrightarrow{x}\right)\text{is}\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)},\mathrm{and}\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)}\triangleq \left(\left(\overleftarrow{1}-\overleftarrow{x}\right),\left(\overrightarrow{1}-\overrightarrow{x}\right)\right);$

(2)Disjunction “ V ”.
For example: The disjunction of $\left(\overleftarrow{x},\overrightarrow{x}\right)\mathrm{and}\left(\overleftarrow{y},\overrightarrow{y}\right)\text{is}\left(\overleftarrow{x},\overrightarrow{x}\right)\vee \left(\overleftarrow{y},\overrightarrow{y}\right)\triangleq$$\max \left(\left(\overleftarrow{x},\overrightarrow{x}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\right);$

(3)Conjunction “ ⋀ ”.
For example: The conjunction of $\left(\overleftarrow{x},\overrightarrow{x}\right)\mathrm{and}\left(\overleftarrow{y},\overrightarrow{y}\right)\text{is}\left(\overleftarrow{x},\overrightarrow{x}\right)\wedge \left(\overleftarrow{y},\overrightarrow{y}\right)\triangleq$$\min \left(\left(\overleftarrow{x},\overrightarrow{x}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\right);$

(4)Condition “ ← ”.
For example: $\left(\overleftarrow{x},\overrightarrow{x}\right)\to \left(\overleftarrow{y},\overrightarrow{y}\right)\iff \overline{\left(\overleftarrow{x},\overrightarrow{x}\right)}\vee \left(\overleftarrow{y},\overrightarrow{y}\right)\triangleq \max \left(\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)},\left(\overleftarrow{y},\overrightarrow{y}\right)\right);$

(5)Bi-condition “ ↔ ”.
$\begin{array}{l}\text{For example:}\left(\overleftarrow{x},\overrightarrow{x}\right)\leftrightarrow \left(\overleftarrow{y},\overrightarrow{y}\right)\\ \iff \left(\left(\overleftarrow{x},\overrightarrow{x}\right)\to \left(\overleftarrow{y},\overrightarrow{y}\right)\right)\wedge \left(\left(\overleftarrow{y},\overrightarrow{y}\right)\to \left(\overleftarrow{x},\overrightarrow{x}\right)\right)\\ \iff \left(\left(\overleftarrow{x},\overrightarrow{x}\right)\vee \left(\overleftarrow{y},\overrightarrow{y}\right)\right)\wedge \left(\left(\overleftarrow{y},\overrightarrow{y}\right)\vee \left(\overleftarrow{x},\overrightarrow{x}\right)\right)\\ \iff \min \left(\max \left(\overleftarrow{x},\overrightarrow{x}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\right),\max \left(\left(\overleftarrow{y},\overrightarrow{y}\right),\left(\overleftarrow{x},\overrightarrow{x}\right)\right).\end{array}$

For any DF proposition $\left(\overleftarrow{x},\overrightarrow{x}\right)P,\left(\overleftarrow{y},\overrightarrow{y}\right)Q,\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)P},\left(\overleftarrow{x},\overrightarrow{x}\right)P\vee \left(\overleftarrow{y},\overrightarrow{y}\right)Q,\left(\overleftarrow{x},\overrightarrow{x}\right)P\to$$\left(\overleftarrow{y},\overrightarrow{y}\right)Q$ is a DF proposition.

Definition 8.18 We can define dynamic fuzzy calculus logic formation as

(1)Single DF proposition is a well-formed formula.

(2)If $\left(\overleftarrow{x},\overrightarrow{x}\right)P$ is a well-formed formula, then $\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)P}$ also is well-formed formula.

(3)If $\left(\overleftarrow{x},\overrightarrow{x}\right)P\mathrm{and}\left(\overleftarrow{y},\overrightarrow{y}\right)Q$ is a well-formed formula, then
$\begin{array}{l}\left(\overleftarrow{x},\overrightarrow{x}\right)P\vee \left(\overleftarrow{y},\overrightarrow{y}\right)Q,\left(\overleftarrow{x},\overrightarrow{x}\right)P\wedge \left(\overleftarrow{y},\overrightarrow{y}\right)Q,\\ \left(\overleftarrow{x},\overrightarrow{x}\right)P\to \left(\overleftarrow{y},\overrightarrow{y}\right)Q,\left(\overleftarrow{x},\overrightarrow{x}\right)P\leftrightarrow \left(\overleftarrow{y},\overrightarrow{y}\right)Q\end{array}$ are also well-formed formula.

(4)A symbol string is a well-formed formula if and only if the symbol is constituted of variable coupling words, brackets and proposition which is obtained by finitely applying (1)(2) and (3).

The main law of DFL is as follows:

(1)Idempotent law
$\begin{array}{l}\left(\overleftarrow{x},\overrightarrow{x}\right)A\vee \left(\overleftarrow{x},\overrightarrow{x}\right)A=\left(\overleftarrow{x},\overrightarrow{x}\right)A\\ \left(\overleftarrow{x},\overrightarrow{x}\right)A\wedge \left(\overleftarrow{x},\overrightarrow{x}\right)A=\left(\overleftarrow{x},\overrightarrow{x}\right)A\end{array}$

(2)Law of commutation
$\begin{array}{l}\left(\overleftarrow{x},\overrightarrow{x}\right)A\vee \left(\overleftarrow{y},\overrightarrow{y}\right)B=\left(\overleftarrow{y},\overrightarrow{y}\right)B\vee \left(\overleftarrow{x},\overrightarrow{x}\right)A\\ \left(\overleftarrow{x},\overrightarrow{x}\right)A\wedge \left(\overleftarrow{y},\overrightarrow{y}\right)B=\left(\overleftarrow{y},\overrightarrow{y}\right)B\wedge \left(\overleftarrow{x},\overrightarrow{x}\right)A\end{array}$

(3)Law of association
$\begin{array}{l}\left(\overleftarrow{x},\overrightarrow{x}\right)A\vee \left(\left(\overleftarrow{y},\overrightarrow{y}\right)B\vee \left(\overleftarrow{c},\overrightarrow{c}\right)C\right)=\left(\left(\overleftarrow{x},\overrightarrow{x}\right)A\vee \left(\overleftarrow{y},\overrightarrow{y}\right)B\right)\vee \left(\overleftarrow{c},\overrightarrow{c}\right)C\\ \left(\overleftarrow{x},\overrightarrow{x}\right)A\wedge \left(\left(\overleftarrow{y},\overrightarrow{y}\right)B\wedge \left(\overleftarrow{c},\overrightarrow{c}\right)C\right)=\left(\left(\overleftarrow{x},\overrightarrow{x}\right)A\wedge \left(\overleftarrow{y},\overrightarrow{y}\right)B\right)\wedge \left(\overleftarrow{c},\overrightarrow{c}\right)C\\ \end{array}$

(4)Absorption law
$\begin{array}{l}\left(\overleftarrow{x},\overrightarrow{x}\right)A\vee \left(\left(\overleftarrow{y},\overrightarrow{y}\right)B\wedge \left(\overleftarrow{x},\overrightarrow{x}\right)A\right)=\left(\overleftarrow{x},\overrightarrow{x}\right)A\\ \left(\overleftarrow{x},\overrightarrow{x}\right)A\wedge \left(\left(\overleftarrow{y},\overrightarrow{y}\right)B\vee \left(\overleftarrow{x},\overrightarrow{x}\right)A\right)=\left(\overleftarrow{x},\overrightarrow{x}\right)A\end{array}$

(5)De Morgan Rule
$\begin{array}{l}\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)A\wedge \left(\overleftarrow{y},\overrightarrow{y}\right)B}=\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)A}\vee \overline{\left(\overleftarrow{y},\overrightarrow{y}\right)B}\\ \overline{\left(\overleftarrow{x},\overrightarrow{x}\right)A\vee \left(\overleftarrow{y},\overrightarrow{y}\right)B}=\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)A}\wedge \overline{\left(\overleftarrow{y},\overrightarrow{y}\right)B}\end{array}$

(6)Operational Rule of Constant
$\begin{array}{l}A\vee \left(\overleftarrow{x},\overrightarrow{x}\right)A=A\\ A\vee \left(\overleftarrow{x},\overrightarrow{x}\right)A=\left(\overleftarrow{x},\overrightarrow{x}\right)A\end{array}$

Definition 8.19 Assume P is a DFL formula; if for all the variables in P there are $T\left(p\right)\ge \left(\overleftarrow{a},\overrightarrow{a}\right),\left(\overleftarrow{a},\overrightarrow{a}\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right],$ then we call that the DFL formula P is constant true formula of $\left(\overleftarrow{a},\overrightarrow{a}\right).$

Especially, when $T\left(P\right)\ge \left(\overleftarrow{\frac{1}{2}},\overrightarrow{\frac{1}{2}}\right),$ we call the above formula P as a DF constant true formula. When $T\left(P\right)<\left(\overleftarrow{\frac{1}{2}},\overrightarrow{\frac{1}{2}}\right),$ we call the formula P as a DF constant false formula.

We call variable $\left(\overleftarrow{x},\overrightarrow{x}\right)\mathrm{and}\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)}$ as word (note as L), and the disjunction of word L1 ⋁ L2 ⋁ L3 ⋁ ... Lr as clause; we note C, the conjunction of word L1 ⋀ L2 ⋀ L3 ⋀...⋀ Lr as word group, and we note that ø

Theorem 8.12 The necessary and sufficient condition of clause C is DF true is clause C contains variable pair $\left(\left(\overleftarrow{x},\overrightarrow{x}\right),\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)}\right).$ The necessary and sufficient condition of word group ø is DF false is word group ø contains variable pair $\left(\left(\overleftarrow{x},\overrightarrow{x}\right),\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)}\right).$

Definition 8.20 If DF formula f represents disjunction of word group øi, i = 1, 2, 3,..., p, namely:

$f={\varphi }_1\vee {\varphi }_2\vee \dots \vee {\varphi }_p,$

then we call the formula as the disjunction normal form of f.

If the DF formula f represents conjunction of clause, namely,

$f=C_1\Lambda C_2\Lambda \dots C_p,$

then we call the formula as the conjunction normal form of f.

Theorem 8.13 The necessary and sufficient condition of disjunction normal form P = ø1⋀ ø2⋀ ... ⋀ øi is false is all word group are DF false.

The necessary and sufficient condition of conjunction normal form Q = C1 ⋁ C2 ⋁ ... ⋁Cp is DF true is all clause Ci are DF true.