[mathematical logic] propositional logic (propositional logic reasoning | formal structure of reasoning | inference law | additional law | simplification law | hypothetical reasoning | refusal | disju

One 、 The formal structure of reasoning

---

The formal structure of reasoning

Premise :

A

1

,

A

2

,

⋯
 

,

A

k

A_1 , A_2 , \cdots , A_k

A1​,A2​,⋯,Ak​

Conclusion :

B

B

B

The formal structure of reasoning is :

(

A

1

∧

A

2

∧

⋯

∧

A

k

)

→

B

(A_1 \land A_2 \land \cdots \land A_k) \to B

(A1​∧A2​∧⋯∧Ak​)→B

Two 、 The law of reasoning

---

The law of reasoning :

A

,

B

A,B

A,B There are two propositions , If

A

→

B

A \to B

A→B It's Yongzhen style , that

A

⇒

B

A \Rightarrow B

A⇒B ;

1、 Additional law

Additional law :

A

⇒

(

A

∨

B

)

A \Rightarrow (A \lor B)

A⇒(A∨B)

according to The law of reasoning ,

A

→

(

A

∨

B

)

A \to (A \lor B)

A→(A∨B) Implicative form yes Yongzhen style ;

Premise :

A

A

A

Conclusion :

A

∨

B

A \lor B

A∨B

A

A

A Yes. , that

A

∨

B

A \lor B

A∨B That's right. , The latter is an addition to the former

B

B

B ;

2、 The law of simplification

The law of simplification :

(

A

∧

B

)

⇒

A

( A \land B ) \Rightarrow A

(A∧B)⇒A ,

(

A

∧

B

)

⇒

B

( A \land B ) \Rightarrow B

(A∧B)⇒B

according to The law of reasoning ,

(

A

∧

B

)

→

A

( A \land B ) \to A

(A∧B)→A ,

(

A

∧

B

)

→

B

( A \land B ) \to B

(A∧B)→B Implicative form yes Yongzhen style ;

Premise :

A

∧

B

A \land B

A∧B

Conclusion :

A

A

A or

B

B

B

A

∧

B

A \land B

A∧B Yes. , that

A

A

A or

B

B

B That's right. , The latter is simplified on the basis of the former ;

3、 Hypothetical reasoning

Hypothetical reasoning :

(

A

→

B

)

∧

A

⇒

B

( A \to B ) \land A \Rightarrow B

(A→B)∧A⇒B

according to The law of reasoning ,

(

A

→

B

)

∧

A

→

B

( A \to B ) \land A \to B

(A→B)∧A→B Implicative form yes Yongzhen style ;

Premise :

A

→

B

A \to B

A→B ,

A

A

A

Conclusion :

B

B

B

This is a typical small three paragraph theory ;

4、 Reject

Reject :

(

A

→

B

)

∧

¬

B

⇒

¬

A

( A \to B ) \land \lnot B \Rightarrow \lnot A

(A→B)∧¬B⇒¬A

according to The law of reasoning ,

(

A

→

B

)

∧

¬

B

→

¬

A

( A \to B ) \land \lnot B \to \lnot A

(A→B)∧¬B→¬A Implicative form yes Yongzhen style ;

Premise :

A

→

B

A \to B

A→B ,

¬

B

\lnot B

¬B

Conclusion :

¬

A

\lnot A

¬A

It can be understood as a counter evidence ;

5、 Disjunctive syllogism

Disjunctive syllogism :

(

A

∨

B

)

∧

¬

A

⇒

B

( A \lor B ) \land \lnot A \Rightarrow B

(A∨B)∧¬A⇒B ,

(

A

∨

B

)

∧

¬

B

⇒

A

( A \lor B ) \land \lnot B \Rightarrow A

(A∨B)∧¬B⇒A

according to The law of reasoning ,

(

A

∨

B

)

∧

¬

A

→

B

( A \lor B ) \land \lnot A \to B

(A∨B)∧¬A→B ,

(

A

∨

B

)

∧

¬

B

→

A

( A \lor B ) \land \lnot B \to A

(A∨B)∧¬B→A Implicative form yes Yongzhen style ;

Premise :

A

∨

B

A \lor B

A∨B ,

¬

A

\lnot A

¬A

Conclusion :

B

B

B

(

A

∨

B

)

(A \lor B)

(A∨B) That's right. , among

A

A

A It's wrong. , that

B

B

B It must be right ;

(

A

∨

B

)

(A \lor B)

(A∨B) That's right. , among

B

B

B It's wrong. , that

A

A

A It must be right ;

Police often use reasoning methods to solve cases , Exclude suspects one by one ;

6、 Hypothetical syllogism

Hypothetical syllogism :

(

A

→

B

)

∧

(

B

→

C

)

⇒

(

A

→

C

)

( A \to B ) \land ( B \to C ) \Rightarrow ( A \to C )

(A→B)∧(B→C)⇒(A→C)

according to The law of reasoning ,

(

A

→

B

)

∧

(

B

→

C

)

→

(

A

→

C

)

( A \to B ) \land ( B \to C ) \to ( A \to C )

(A→B)∧(B→C)→(A→C) Implicative form yes Yongzhen style ;

Premise :

A

→

B

A \to B

A→B ,

B

→

C

B \to C

B→C

Conclusion :

A

→

C

A \to C

A→C

7、 Equivalent syllogism

Equivalent syllogism :

(

A

B

)

∧

(

B

C

)

⇒

(

A

C

)

( A \leftrightarrow B ) \land ( B \leftrightarrow C ) \Rightarrow ( A \leftrightarrow C )

(AB)∧(BC)⇒(AC)

according to The law of reasoning ,

(

(

A

B

)

∧

(

B

C

)

)

→

(

A

C

)

( ( A \leftrightarrow B ) \land ( B \leftrightarrow C ) ) \to ( A \leftrightarrow C )

((AB)∧(BC))→(AC) Implicative form yes Yongzhen style ;

Premise :

A

B

A \leftrightarrow B

AB ,

B

C

B \leftrightarrow C

BC

Conclusion :

A

C

A \leftrightarrow C

AC

8、 constructive dilemma

Equivalent syllogism :

(

A

→

B

)

∧

(

C

→

D

)

∧

(

A

∨

C

)

⇒

(

B

∨

D

)

( A \to B ) \land ( C \to D ) \land ( A \lor C ) \Rightarrow ( B \lor D )

(A→B)∧(C→D)∧(A∨C)⇒(B∨D)

according to The law of reasoning ,

(

(

A

→

B

)

∧

(

C

→

D

)

∧

(

A

∨

C

)

)

→

(

(

B

∨

D

)

)

( ( A \to B ) \land ( C \to D ) \land ( A \lor C ) ) \to ( ( B \lor D ) )

((A→B)∧(C→D)∧(A∨C))→((B∨D)) Implicative form yes Yongzhen style ;

Premise :

A

→

B

A \to B

A→B ,

C

→

D

C \to D

C→D ,

A

∨

C

A \lor C

A∨C

Conclusion :

B

∨

D

B \lor D

B∨D

Way of understanding :

A

A

A Is to develop the economy ,

B

B

B It's pollution

C

C

C Is not to develop the economy ,

D

D

D It's poverty

A

∨

B

A \lor B

A∨B Or develop the economy , Or do not develop the economy
The result is

B

∨

D

B \lor D

B∨D , Or produce pollution , Or endure poverty