Detailed explanation of predicate logic

In order to solve the shortcomings of propositional logic, predicate logic was introduced

Predicate logic further analyzes simple propositions to find out the relationship between the described objects and objects.
For example: " $2$ is an even number" can be generalized to "$x$ is an even number”

predicate

Individual constants and individual variables

individual constant

Symbols used to represent specific or specific individuals are called individual constants , commonly used lowercase letters $a,b,c,\cdots$ means

For example: " 2 22 2in "$2\; is\; an\; even\; number"$$2$ is the individual constant

individual variable

definition

Variables used to represent any individual are called individual variables , commonly used lowercase letters $x,y,z,\cdots$ means

For example: " xxxxin "$x\; is\; an\; even\; number"$$x$ is the individual variable

area of ​​study

The value range of an individual variable is called the domain of discourse or individual domain of the variable.

total individual domain

The set of all individuals that can be represented by all individual variables is called the total individual domain

definition

A pattern that describes the characteristics of a single individual (unary predicate) or the relationship between multiple individuals (multiple predicates) is called a predicate
, such as " $\cdots$ is an even number" is the predicate

1. A unary predicate
   represents the characteristics of an individual, represented by an expression consisting of a capital letter expressing the individual characteristics and an individual constant or variable, such as $P(a)$、$P(x)$
2. Binary predicate
   represents the relationship between two individuals, expressed by an expression composed of a capital letter expressing the relationship between the two individuals and two individual constants or variables, such as $P(x,y)$
3. $n-$ ary predicate
   expressesnnThe relationship between$n$ individuals is expressed as nThe capital letters and nn$of\; the\; relationship\; between\; n$ individuals$An\; expression\; composed\; of\; n$ individual constants or variables, such as$P(x_1,x_2,\cdots ,x_n)$

The predicate can be regarded as a propositional function , assuming $n$ -ary predicate$P(x_2,x_2,\cdots ,x_n)$，其中 $x_1\in D_1,x_2\in D_2,\cdots ,x_n\in D_n$，则 $P(x_1,x_2,\cdots ,x_n)$ can be regarded as starting from the set$D_1\times D_2\times \cdots \times D_n$to the set { T , F } \{T,F\}{ T,The mapping of$F$$\}$
is as shown below:

It can be seen from the definition of predicate that predicate $P(x_1,x_2,\cdots ,x_n)$ is just a function, so it has no true or false values. Only by substituting each individual variable into the determined individual constant in the corresponding individual domain can we obtain a proposition with a definite true or false value.

Propositions can be seen as special forms of predicates
when $n=0$ , predicate$P$ degenerates into a proposition

characteristic predicate

Before understanding characteristic predicates, you need to understand quantization

introduction

For example, propositions: "All people are mortal", "Some people are not afraid of death",
suppose $H\left(x\right)$ means "$x$ is a person",$D\left(x\right)$ means "$x$ is mortal",$F\left(x\right)$ means "$x$ is not afraid of death"
will be symbolized below

• If the domain of discussion is all human beings, the symbolic result is $\forall xD(x)$、$\exists xF(x)$
• If the domain of discussion is the domain of all individuals, the symbolic result is
  (1) "Everyone is mortal" which can be equivalently expressed as "For any $x$ , if$x$ is a person, then$x$ is mortal"
  so the symbolic result is$\forall x[H(x)\to D(x)]$
  (2) "Some people are not afraid of death" can be equivalently expressed as "there is$x$，$x$ is a person, and$x$ is not afraid of death"
  so the symbolic result is$\exists x[H(x)\wedge D(x)]$

For the example above, $H\left(x\right)$ is called a characteristic predicate and is used to qualify$x$ is a person

definition

Characteristic predicates are used to limit the domain of discourse to individuals that satisfy the predicate.

rule

When adding characteristic predicates to formulas, the following two rules must be met:

• For universal quantifiers, the characteristic predicate is added as the antecedent of the conditional
• For existential quantifiers, the characteristic predicate is added as the conjunction of the conjunction

quantifier

Only predicates cannot express propositions such as "all people can breathe" and "some rational numbers are natural numbers". Quantifiers need to be introduced.

type

universal quantifier

Chinese expression "for any $x$ ” can be written as$\forall x$ , where$\forall$ is calledthe universal quantifier,$x$ is called a quantifier$\forall$ 'saction variableorguidance variable

For example: $\forall xP\left(x\right)$ means “for all$x$ has$P(x)$”

existential quantifier

Chinese expression "there is a certain $x$ ” can be written as$\exists x$ , where$\exists$ is calledan existential quantifier,$x$ is called a quantifier$\exists$ ’saction variableorguidance variable

如：$\exists xP\left(x\right)$ means “there is a certain$x$ satisfies$P(x)$”

Quantify

definition

In the predicate $P\left(x\right)$ is preceded by the universal quantifier$\forall x$ or existential quantifier$\exists x$ is calledthe individual variable$x$ is quantified by a universal quantifier or an existential quantifier, if it is$P$ specifies the specific meaning, which is$x$ specifies the domain of discussion, then$\forall xP(x)$ 或 $\exists xP(x)$ 成为一个具有真假值的命题

量化后所得命题的真值与个体变元的论域有关，如 $\exists x(x=3)$；如果论域为自然数则为真，如果论域为负整数则为假
为统一表述，若不加说明，则默认都为全总个体域

量化的命题形式表示

如果论域是有限集合，则对于某一个体变元的量化可以用命题形式表示

设论域 D = { a 1 , a 2 , ⋯   , a n } D=\{a_1,a_2,\cdots,a_n\} D={ a1​,a2​,⋯,an​}，则有

• $\forall xP(x)\iff P(a_1)\wedge P\left(a_2\right)\wedge \cdots \wedge P\left(a_n\right)$
• $\exists xP(x)\iff P\left(a_1\right)\vee P\left(a_2\right)\vee \cdots \vee P\left(a_n\right)$

Jurisdiction

Before understanding the scope of quantifiers, you need to understand the predicate formula first

definition

In the predicate formula, the scope of the quantifier is called the scope of the quantifier , also known as the scope of the quantifier.

• If the quantifier is followed by only an atomic predicate formula, the scope of the quantifier is the atomic predicate formula,
  such as: $\forall$ x in$\forall \; x$$P$$($$x$$)$$\forall$ The domain of$x$$P(x)$
• If the quantifier is followed by brackets, the area represented by the brackets is the scope of the quantifier.
  For example: $\exists x(P(x)\to$∃ x \exists xin$Q$$($$x$$))$The domain of$\exists$$x$$(P(x)\to Q(x))$
• If multiple quantifiers appear next to each other, the following quantifier and its scope are the scope of the previous quantifier.
  For example: $\forall x\exists yP(x,y)$ 中的 $\exists$ The domain of$y$$P(x,y)$； ∀ x \forall x The domain of$\forall$$x$$\exists yP(x,y)$

constraint appears

definition

In the quantifier $\forall x$ 或 $\exists xx$ within the jurisdiction ofxAll occurrences of$x$ are called constrained occurrences

constraint variables

The individual variables in which a constraint occurs are called constraint variables

free variable

Individual variables that do not appear as constraints are called free variables

For example: predicate formula $\forall xP(x,$xxin$y$$)$$x$ is the constraint variable;$y$ free variable

predicate formula

atomic formula

A single predicate without connectives and quantifiers is called the atomic formula
of predicate calculus , such as: $P(x_1,x_2,\cdots ,x_n)(n\ge 0)$

Because the proposition is the predicate $n=A\; special\; form\; of\; 0$ , so that individual propositional constants and propositional variables are atomic formulas of the predicate calculus

definition

Predicate formula , also known as the well-formed formula of predicate logic , the following is its recursive definition

1. Basic clause: Atomic formulas are predicate formulas
2. induction clause
   • Young $A$ is a predicate formula, then$\neg A$ is a predicate formula
   • Young $A,B$ is a predicate formula, then$A\wedge B$、$A\vee B$、$A\to B$、$A\leftrightarrow B$ 是谓词公式
   • 若 $A$ 是谓词公式，且 $A$ 中的个体变元 $x$ 未被量词量化，则量化后 $\forall xA(x)$ 和 $\exists xA(x)$ 是量词公式
3. 极小性条款：只有有限次地应用条款 1 和条款 2 生成的表达式才是谓词公式

由定义知所有的命题公式都是谓词公式

子公式

$B$ 是谓词公式 $A$ 的子公式的定义如下

• $B$ 是 $A$ 的连续段
• $B$ 是谓词公式

赋值

对谓词公式赋值需要完成以下操作：

1. 指定论域 $E$
2. 指定谓词符的含义
3. Specify definite propositions for propositional arguments
4. Specify individuals on the universe of discussion for free variables

Note : Constraint variables do not need to be specified

Eishin official

Generalization of Equivalence and Implication Formulas in Propositional Logic

Apply the substitution rule to the equivalence or implication in propositional logic and substitute it with the predicate formula in predicate logic. The resulting formula is the equivalence or implication of the predicate formula.

For example: proposition formula equivalence $P\to Q\iff \neg P\vee The\; generalization\; of\; Q$ in predicate logic is
$\forall xP(x)\to \exists xQ(x)\iff \neg \forall xP(x)\vee \exists xQ\left(x\right)$

Negation law of quantifiers

• $\neg \forall xP\left(x\right)\iff \exists x\neg P\left(x\right)$
  means “not all$x$ all satisfies$P\left(x\right)$ ” can be said to be “there is$x$ satisfies non-$P(x)$”
• $\neg \exists xP(x)\iff \forall x\neg P\left(x\right)$ means “ xx
  does not exist$"x$ satisfies $P(x)" can be said to be "for all$x$ all satisfies non-$P\left(x\right)$”

Laws of Expansion and Contraction of Quantifier Scope

• $\forall x(P(x)\wedge Q)\iff \forall xP(x)\wedge Q$
  $\exists x(P(x)\wedge Q)\iff \exists xP(x)\wedge Q$
  $\forall x\left(P\right(x)\vee Q)\iff \forall xP\left(x\right)\vee Q$
  $\exists x\left(P\right(x)\vee Q)\iff \exists xP\left(x\right)\vee Q$

• $\forall xP(x)\to Q\iff \exists x(P(x)\to Q)$
  $\exists xP(x)\to Q\iff \forall x(P(x)\to Q)$

  proving process

  Prove below that $\forall xP\left(x\right)\to Q\iff \exists x\left(P\right(x)\to Q)$
  $$\begin{array}{rl}\forall xP(x)\to Q& \iff \neg \forall xP(x)\vee Q\\ & \iff \exists x\neg P(x)\vee Q\\ & \iff \exists x(\neg P(x)\vee Q)\\ & \iff \exists x(P(x)\to Q)\end{array}$$

• $Q\to \forall xP(x)\iff \forall x(Q\to P(x))$
  $Q\to \exists xP(x)\iff \exists x(Q\to P(x))$

distributive law of quantifiers

• $\forall x(P(x)\wedge Q(x))\iff \forall xP(x)\wedge \forall xQ(x)$
  $\exists x(P(x)\vee Q(x))\iff \exists xP(x)\vee \exists xQ(x)$

  proving process

  Prove below that $\forall x(P(x)\wedge Q(x))\iff \forall xP\left(x\right)\wedge \forall xQ\left(x\right)$
  $\exists x\left(P\right(x)\vee Q(x))\iff \exists xP\left(x\right)\vee \exists xQ\left(x\right)$ can be proved in the same way. Suppose
  the domain of discussion is$D$
  当 $\forall x(P(x)\wedge When\; Q(x))$ is true, that is, for all$a\in D$ has$P(a)\wedge Q\left(a\right)$ below∴
  $\therefore P(a)$、$Q\left(a\right)$ are all true
  $\therefore \forall xP(x)$、$\forall xQ\left(x\right)$ are all true
  $\therefore \forall xP(x)\wedge \forall xQ\left(x\right)$ is true
  when$\forall x(P(x)\wedge When\; Q(x))$ is false, there exists$a\in D$ 使 $P(a)\wedge Q\left(a\right)$ if∴
  $\therefore P(a)$、$At\; least\; one\; of\; Q\left(a\right)$ is false. Let us assume that$P\left(a\right)$ is false, then$\forall xP\left(x\right)$ is false
  $\therefore xP(x)\wedge xQ\left(x\right)$ is false
  so$\forall x(P(x)\wedge Q(x))$ 与 $\forall xP(x)\wedge \forall xQ(x)$ 真值相同
  即 $\forall x(P(x)\wedge Q(x))\iff \forall xP(x)\wedge \forall xQ(x)$

• $\forall xP(x)\vee \forall xQ(x)\Rightarrow \forall x(P(x)\vee Q(x))$
  $\exists x(P(x)\wedge Q(x))\Rightarrow \exists xP(x)\wedge \exists xQ(x)$

  proving process

  Prove below that $\forall xP(x)\vee \forall xQ(x)\Rightarrow \forall x(P(x)\vee Q(x))$
  $\exists x\left(P\right(x)\wedge Q(x))\Rightarrow \exists xP\left(x\right)\wedge \exists xQ\left(x\right)$ can be proved in the same way
  . Let’s assume that the domain of discussion is$D$
  当 $\forall xP(x)\vee When\; \forall xQ\left(x\right)$ is true,$\forall xP(x)$、 ∀ x Q ( x ) \forall xQ(x) At least one of$\forall$$x$$Q$$($$x$$)$
  is true . Let$\forall xP\left(x\right)$ is true, that is, for all$a\in D$ , both have$P(a)$ 为真
  $\therefore P(a)\vee Q(x)$ 为真
  $\therefore \forall x(P(x)\vee Q(x))$ 为真
  由肯定前件法得 $\forall xP(x)\vee \forall xQ(x)\Rightarrow \forall x(P(x)\vee Q(x))$
  当 $\forall x(P(x)\vee Q(x))$ is true, that is, for all$a\in D$ , both have$P(a)\vee Q\left(a\right)$ is true
  but it does not necessarily have$\forall xP(x)\vee \forall xQ\left(x\right)$ is true
  becauseD = { a , b } D=\{a,b\}D={ a,b}，$P\left(a\right)$ is true,$Q\left(a\right)$ is false;$P\left(b\right)$ is false,Q ( b ) Q(b)When$Q$$($$b$$)$$\forall x(P(x)\vee Q(x))$ is true, but$\forall xP(x)\vee \forall xQ\left(x\right)$ is false

• $\forall x(P(x)\to Q(x))\Rightarrow \forall xP(x)\to \forall xQ(x)$
  $\forall x\left(P\right(x)\leftrightarrow Q(x))\Rightarrow \forall xP\left(x\right)\leftrightarrow \forall xQ(x)$

  proving process

  对于 $\forall x\left(P\right(x)\to Q(x))\Rightarrow \forall xP\left(x\right)\to \forall xQ(x)$
  这里用否定后件法证明
  不妨设论域为 $D$
  当 $\forall xP(x)\to \forall xQ(x)$ 为假时，$\forall xP(x)$ 为真、$\forall xQ(x)$ 为假
  因此，存在 $a\in D$ 使 $P(a)$ 为真、$Q\left(a\right)$ is false.
  Therefore,$\forall x(P(x)\to Q(x))$ is false

  对于 $\forall x\left(P\right(x)\leftrightarrow Q(x))\Rightarrow \forall xP(x)\leftrightarrow \forall xQ(x)$
  $$\begin{array}{rl}\forall x(P(x)\leftrightarrow Q(x))& \iff \forall x[(P(x)\to Q(x))\wedge (Q(x)\to P(x))]\\ & \iff \forall x(P(x)\to Q(x))\wedge \forall x(Q(x)\to P(x))\\ & \Rightarrow (\forall xP(x)\to \forall xQ(x))\wedge (\forall xQ(x)\to \forall xP(x))\\ & \iff \forall xP(x)\leftrightarrow \forall xQ(x)\end{array}$$

• $\exists x(P(x)\to Q(x))\iff \forall xP(x)\to \exists xQ(x)$
  $\exists xP(x)\to \forall xQ(x)\Rightarrow \forall x(P(x)\to Q(x))$

  proving process

  下面证明 $\exists x(P(x)\to Q(x))\iff \forall xP(x)\to \exists xQ(x)$
  $$\begin{array}{rl}\exists x(P(x)\to Q(x))& \iff \exists x(\neg P(x)\vee Q(x))\\ & \iff \exists x\neg P(x)\vee \exists xQ(x)\\ & \iff \neg \forall xP(x)\vee \exists xQ\left(x\right)\\ & \iff \forall xP(x)\to \exists xQ(x)\end{array}$$

  下面证明 $\exists xP(x)\to \forall xQ(x)\Rightarrow \forall x(P(x)\to Q(x))$
  $$\begin{array}{rl}\exists xP(x)\to \forall xQ(x)& \iff \forall x\neg P(x)\vee \forall xQ\left(x\right)\\ & \Rightarrow \forall x(\neg P(x)\vee Q(x))\\ & \iff \forall x(P(x)\to Q(x))\end{array}$$

multi-weighted lexicon

• $\forall x\forall yP(x,y)\iff \forall y\forall xP(x,y)$
  $\exists x\exists yP(x,y)\iff \exists y\exists xP(x,y)$
• $\exists x\forall yP(x,y)\Rightarrow \forall y\exists xP(x,y)$
  $\forall x\exists yP(x,y)\Rightarrow \exists y\exists xP(x,y)$

The two variables can be visually demonstrated by the following figure:

Inference theory of predicate logic

inference rules

The inference rules of propositional logic also apply to predicate logic, but in the reasoning process of predicate logic, it is sometimes necessary to eliminate or introduce quantifiers

Eliminate quantifier

When eliminating quantifiers, exist designation comes first and then full name designation

There is a specified rule

There is an existential specification , abbreviated as ES, as follows
$$\frac{\exists xP(x)}{\therefore P(a)}$$
Among them, $P$ is the predicate,$a$ is in the domain of discourse such thatP ( a ) P(a)Individuals where$P$$($$a$$)\; is\; true$

The meaning of this rule is: if $\exists xP\left(x\right)$ is true, then there is an individual constant$a$ ,from$P\left(a\right)$ is true

Full name designation rules

The full name specification (universal specification) , abbreviated as US, is as follows
$$\frac{\forall xP(x)}{\therefore P\left(y\right)}$$
Among them, $P$ is the predicate,$y$ is a free variable

The meaning of this rule is: if $\forall xP\left(x\right)$ is true, then for any individual constant$a$ , both have$P\left(a\right)$ is true

Introduce quantifiers

There are promotion rules

There is an existential generalization rule , abbreviated as EG, as follows
$$\frac{P(a)}{\therefore \exists xP(x)}$$
Among them, $P$ is the predicate,$a$ is in the domain of discourse such thatP ( a ) P(a)Individuals where$P$$($$a$$)\; is\; true$

The meaning of this rule is: if there is an individual constant $a$ ,from$P\left(a\right)$ is true, then$\exists xP\left(x\right)$ is true

Full name promotion rules

The full name is universal generalization , abbreviated as UG, as follows
$$\frac{C\Rightarrow P(y}{\therefore C\Rightarrow \forall xP(x)}$$
Among them, $\Gamma$ is the conjunction of known axioms and premises,$P$ is the predicate,$y$ is a free variable

The meaning of this rule is: if $\Gamma$ can be derived for any individual$y$ has$P\left(y\right)$ is true, then$\forall xP\left(x\right)$ is true

reference

[1] Discrete Mathematics Xi'an University of Electronic Science and Technology Press Second Edition
[2] CSDN Blog