## 研究领域

•universal system of reasoning: reasoning based on symbols+calculations

•**Areas**: (1) set theory, (2) proof theory, (3) recursion theory, (4) model theory, and their foundation (5) propositional logic and predicate logic

```
（1）集合论，（2）证明论，（3）递归论，（4）模型论及其基础（5）命题逻辑和谓词逻辑
```

**Our focus**: propositional logic and predicate logic

## proposition

A **proposition** **( 命题)** is a declarative sentence that is either true or false.

•Lower-case letters represent propositions: p,q,r,…

•**truth value** **( 真值)**: The truth value of p is true (

**T**) if p is a true proposition. The truth value of p is false (

**F**) if p is a false proposition.

**Simple Proposition**(**简单命题**) : cannot be broken into 2 or more propositions

•√2 " is irrational".

**Compound Proposition**(**复合命题**): not simple

•2 is rational and √2 is irrational.

**Propositional Constant**(**命题常项**): a concrete proposition (truth value fixed)

•Every even integer n>2 is the sum of two primes.

**Propositional Variable**(**命题变项**): a variable that represents any proposition

•Lowercase letters denote proposition variables: p,q,r,s,…

•Truth value is not determined until it is assigned a concrete proposition

**Propositional Logic**(**命题逻辑**): the area of logic that deals with propositions

## 运算

### Negation (¬)

**DEFINITION:** Let p be any proposition.

• The **negation** (**否定**) of p is the statement “It is not the case that p”

• notation: ¬p; read as “not p”

• **truth table** (**真值表**):

p | ¬p |
---|---|

T |
F |

F |
T |

### Conjunction (∧)

**DEFINITION:** Let p,q be any propositions.

•The **conjunction** (**合取**) of p and q is the statement “p and q”

•Notation: p∧q; read as “p and q”

•Truth table:

p | q | p∧q |
---|---|---|

T |
T |
T |

T |
F |
F |

F |
T |
F |

F |
F |
F |

### Disjunction (∨)

**DEFINITION:** Let p,q be any propositions.

•The **disjunction** (**析取**) of p and q is the statement “p or q”

•Notation: p∨q; read as “p or q”

•Truth table:

p | q | p∨q |
---|---|---|

T |
T |
T |

T |
F |
T |

F |
T |
T |

F |
F |
F |

### Implication (→)

**DEFINITION**: Let p,q be any propositions.

•The conditional statement p→q (蕴涵) is the proposition “if p, then q.”

•p: hypothesis*;* q*:* conclusion*;* read as “p implies q”, or “if p, then q”

•Truth table:

p | q | p→q |
---|---|---|

T |
T |
T |

T |
F |
F |

F |
T |
T |

F |
F |
T |

### Bi-Implication (↔)

**DEFINITION**: Let p,q be any propositions.

•The biconditional statement p↔q (等值) is the proposition “p if and only if q.”

•read as “p if and only if q”

•Truth table:

p | q | p↔q |
---|---|---|

T |
T |
T |

T |
F |
F |

F |
T |
F |

F |
F |
T |

### Well-Formed Formulas (合式公式)

**DEFINITION**: recursive definition of **well-formed formulas** (**WFFs**)

- propositional constants (
**T, F**) and propositional variables are WFFs - If A is a WFF, then ¬A is a WFF
- If A,B are WFFs, then (A∧B), (A∨B), (A→B), (A↔B) are WFFs
- WFFs are results of finitely many applications of ①, ② , and ③