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# Predicate Logic(谓词逻辑)

### Predicate谓词

describe the property of the subject term (in a sentence)

• A predicate is a function from a domain of individuals to {𝐓,𝐅}
• 𝒏-ary  predicate𝒏元谓词: a predicate on 𝑛 individuals𝐼: “is an integer” // unary𝐺: “is greater than” //binary
• Predicate constant谓词常项: a concrete predicate // 𝐼,𝐺
• Predicate variable谓词变项: a symbol that represents any predicate

### Individual个体词

the object you are considering (in a sentence)

“√(1+2√(1+3√(1+⋯)) )  is an integer”
“𝑒^𝜋 is greater than 𝜋^𝑒”

• Individual Constant个体常项: √(1+2√(1+3√(1+⋯)) ) ,  𝑒^𝜋, 𝜋^𝑒
• Individual Variable个体变项: 𝑥,𝑦,𝑧
• Domain个体域: the set of all individuals in consideration

### Propositional function命题函数

**$𝑃(𝑥_1,…,𝑥_𝑛)$, where 𝑃 is an 𝑛-ary predicate **

𝑃(𝑥,𝑦):“𝑥 is greater than 𝑦”
𝑃(𝑥,𝑦) gives a proposition when we assign values to 𝑥,𝑦
𝑃(𝑒^𝜋,𝜋^𝑒) is a proposition (a true proposition)
𝑃(𝑥,𝑦) is not a proposition

EXAMPLE:

​ 𝑝:“Alice’s father is a doctor”; 𝑞:“Bob’s father is a doctor”

• Individuals: Alice’s father, Bob’s father; Predicate 𝐷: “is a doctor”
• 𝑝=𝐷(Alice′ s father), 𝑞=𝐷(Bob′ s father)

### Function of Individuals:  a map on the domain of individuals

• 𝑓(𝑥)=𝑥’s father
• 𝑝=𝐷(𝑓(Alice));𝑞=𝐷(𝑓(Bob))

### Universal Quantifier全称量词

#### DEFINITION:

Let $P(x)$ be a propositional function. The universal quantification全称量化 of $P(x)$ is “$P(x)$ for all x in the domain”.

notation: $∀x P(x)$; read as “for all $x$ $P(x)$” or “for every $x$ $P(x)$”

“∀” is called the universal quantifier全称量词

“∀x P(x)” is true iff P(x) is true for every x in the domain

“∀x P(x)” is false iff there is an x_0 in the domain such that $P(x_0)$ is false

Counterexample反例: an x_0 such that P(x_0) is false

### Existential Quantifier

#### DEFINITION:

Let P(x) be a propositional function. The existential

​    quantification存在量化 of P(x) is “there is an x in the domain

​    such that P(x)”

• notation: ∃x P(x); read as “for some x P(x)” or “there is an x s.t. P(x)”
• “∃” is called the existential quantifier****存在量词
• “∃x P(x)” is true iff there is an x in the domain such that P(x) is true
• “∃x P(x)” is false iff P(x) is false for every x in the domain

#### EXAMPLE:

$P(x):“x^2-x+1=0”$

•“$∃x P(x)$”  is false when D=R and is true when D=C

REMARK: If the domain is empty, then "∃x P(x)" is false for any P.

REMARK: if not stated, the individual can be anything.

### Binding Variables and Scope 绑定范围和变量

#### DEFINITION:

An individual variable x is bound约束的 if a quantifier (∀, ∃) is used on x; otherwise, x is said to be free自由的.

• ∃x(x+y=1)
• x is bound and y is free
• scope辖域 of a quantifier: the part of a formula to which a quantifier is used
• the scope of ∃x in ∃x(x+y=1) is (x+y=1)

#### Predicate Logic谓词逻辑:

the area of logic that deals with predicates and quantifiers (a.k.a. predicate calculus)处理谓词和量词的逻辑区域（又称谓词演算)

• predicate logic is an extension of propositional logic

## Well-Formed Formulas

### Elements that may appear in Well-Formed Formulas合式公式

• Propositional constants: T,F, p,q,r,…（命题常数）
• Propositional variables: p,q,r,…  （命题变量）
• Logical Connectives: ¬,∧,∨,→, ↔（逻辑连接词）
• Parenthesis: (, )（插入语）
• Individual constants: a,b,c,…(个体常项)
• Individual variables: x,y,z,…（个体变项）
• Predicate constants: P,Q,R,…（命题常项）
• Predicate variables: P,Q,R,…（命题变项）
• Quantifiers: ∀, ∃（量词）
• Functions of individuals: f,g,…（个体函数）

### DEFINIITON: well-formed formulas合式公式/formulas

1. propositional constants, propositional variables, and propositional functions without connectives are WFFs（无连接词的命题常数，命题变量和命题函数是WFF）
2. If A is a WFF, then ¬A is also a WFF（如果A是WFF，则¬A也是WFF）
3. If A,B are WFFs and there is no individual variable x which is bound in one of A,B but free in the other, then (A∧B), (A∨B), (A→B), (A↔B) are WFFs.（如果A，B是WFF，并且没有单个变量x绑定到A，B中的一个而在另一个中自由，则（A∧B），（A∨B），（A→B），（A↔ B）是WFF。）
4. If A is a WFF with a free individual variable x, then ∀x A, ∃x A are WFFs.（如果A是具有自由个体变量x的WFF，则∀xA，∃xA是WFF。）
5. WFFs can be constructed with.（WFF由1到4构造而成）

#### Precedence

∀, ∃ have higher precedence than ¬, ∧, ∨, → , ↔

### From Natural Language to WFFs

#### The Method of Translation

1. Introduce symbols to represent propositional constants, propositional variables, individual constants, individual variables, predicate constants, predicate variables, functions of individuals
2. Construct WFFs with 1-4 such that WFFs reflect the real meaning of the natural language