A semantic study of the first-order predicate logic with uncertainty involved

Abstract

In this paper, we provide a semantic study of the first-order predicate logic for situations involving uncertainty. We introduce the concepts of uncertain predicate proposition, uncertain predicate formula, uncertain interpretation and degree of truth in the framework of uncertainty theory. Compared with classical predicate formula taking true value in $\{0,1\}$, the degree of truth of uncertain predicate formula may take any value in the unit interval $[0,1]$. We also show that the uncertain first-order predicate logic is consistent with the classical first-order predicate logic on some laws of the degree of truth.

Appendix: Classical logic

The section recalls some basic concepts and results about classical propositional logic and first-order predicate logic.

Let $X$ be a formula containing propositions $p_{1}, p_{2},\ldots ,p_{n}$. Then there is a Boole function $f: \{0,1\}^{n}\rightarrow \{0,1\}$ such that $T(X)=1$ if and only if $f(x_{1}, x_{2},\ldots ,x_{n})=1$ where $x_i=T(p_i)$ for $i=1,2,\ldots ,n$. For simplicity, we denote the Boole function of formula $X$ as $f_{X}$.

Definition (a)

A formula $X$ is called a tautology, denoted by $\models X$, if $f_{X}(x_{1},x_{2},\ldots , x_{n})=1$ for all $(x_{1},x_{2},\ldots ,x_{n})\in \{0,1\}^{n}.$

Definition (b)

A formula $X$ is said to be contradiction, denoted by $\models \lnot X$, if $f_{X}(x_{1},x_{2},\ldots ,x_{n})=0$ for all $(x_{1},x_{2},\ldots ,x_{n})\in \{0,1\}^{n}.$

Definition (c)

Formulae $X$ and $Y$ are called semantically equivalent, denoted by $X \equiv Y$, if $f_{X}(x_{1},x_{2},$ $\cdots ,x_{n})=f_{Y}(x_{1},x_{2},\ldots ,x_{n})$ for all $(x_{1},x_{2},\ldots ,x_{n})\in \{0,1\}^{n}.$

Definition (d)

Let $X$ be a formula containing propositions $p_{1}, p_{2},\ldots ,p_{n}$. It is said to be a disjunctive normal form if

$$\begin{aligned} X&= (Q_{11}\wedge Q_{12}\wedge ...\wedge Q_{1n})\vee (Q_{21}\wedge Q_{22}\wedge \cdots \wedge Q_{2n})\vee \cdots \vee \\&(Q_{m1}\wedge Q_{m2}\wedge \cdots \wedge Q_{mn}), \end{aligned}$$

where $Q_{ij}$ is either $p_{j}$ or $\lnot p_{j}$ for $i=1,2,\ldots ,m,j=1,2,\ldots ,n.$

Theorem (a)

Let $X$ be a formula containing propositions $p_{1}, p_{2},\ldots ,p_{n}$. Then it is semantically equivalent to a disjunctive normal form as follows:

$$\begin{aligned} \bigvee _{f_{X}(x_{1},x_{2},\ldots ,x_{n})=1,(x_{1},x_{2},\ldots ,x_{n})\in \{0,1\}^{n}} Q_{x_1}\cap Q_{x_2}\cap \cdots \cap Q_{x_n}, \end{aligned}$$

where for each $(x_{1},x_{2},\ldots ,x_{n})\in \{0,1\}^{n}$ with $f_{X}(x)=1$, $Q_{x_i}=p_{i}$ if $x_{i}=1$ and $Q_{x_i}=\lnot p_{i}$ if $x_{i}=0$.

The disjunctive normal form of $X$ is denoted by $G(X)$. For example, $G(p_{1}\wedge p_{2}\rightarrow p_{3})=(\lnot p_{1}\wedge p_{2}\wedge \lnot p_{3})\vee ( p_{1}\wedge \lnot p_{2}\wedge \lnot p_{3})\vee (\lnot p_{1}\wedge \lnot p_{2}\wedge \lnot p_{3}) \vee (p_{1}\wedge p_{2}\wedge \lnot p_{3})\vee (\lnot p_{1}\wedge p_{2}\wedge \lnot p_{3})\vee (p_{1}\wedge \lnot p_{2}\wedge \lnot p_{3})\vee (\lnot p_{1}\wedge \lnot p_{2}\wedge \lnot p_{3})$.