# 笔记 逻辑学导论 Week2

### 1Propositional Logic 命题逻辑

Roughly speaking, a proposition is a possible condition of the world that is either true or false

### 2语法syntax

• 简单句 simple sentences
简单句在书写时由字符、数字和下划线构成，第一个字符需要小写
简单句采取的是原子符号（atomic symbol）的形式，又称为命题常量（proposition constants）
简单句表达了有关世界的简单事实 express simple facts about the world.
• 复合句  compound sentences
复合句表达的是组成其的所有简单句之间的逻辑关系 express logical relationships between the simpler sentences of which they are composed.

• negation
p)           ~p
• conjunction
(p ∧ q)    p % q
• disjunction
(p ∨ q)    p | q
• implication
(p ⇒ q)    p => q
• equivalence
(p ⇔ q)   p <=> q

1. ¬
2. ∧∨
3. ⇒ ⇔

### 3语义semantic

truth assignment for Propositional Logic is a function assigning a truth value to each of the proposition constants of the language.

pi = 1
qi = 0
ri = 1

φ ¬φ
1 0
0 1
φ ψ φ ∧ ψ
1 1 1
1 0 0
0 1 0
0 0 0
φ ψ φ ∨ ψ
1 1 1
1 0 1
0 1 1
0 0 0
φ ψ φ ⇒ ψ
1 1 1
1 0 0
0 1 1
0 0 1
φ ψ φ ⇔ ψ
1 1 1
1 0 0
0 1 0
0 0 1
evaluation method 求值方式：用0或1来替代

pi = 1
qi = 0
ri = 1

(p ∨ q) ∧ (¬ q ∨ r)

(1 ∨ 0) ∧ (¬ 0 ∨ 1)
1 ∧ (¬ 0 ∨ 1)
1 ∧ (1 ∨ 1)
1 ∧ 1
1

### 4 Satisfaction 满足

Satisfaction is the opposite of evaluation.

### 5命题语句的逻辑属性Logical Properties of Propositional Sentences

1. valid  对于所有的真值指派都满足语句
2. unsatisfiable 一些真值指派可以满足语句，而另一些则不可以
3. contingent  所有真值指派都不能满足语句

### 6 命题蕴含Propositional Entailment

p q p p ∨ q
1 1 1 1
1 0 1 1
0 1 0 1
0 0 0 0

Unsatisfiability Theorem

Δ |= φ if and only if Δ ∪ {¬φ} is unsatisfiable.