[mathematical logic] equivalent calculus and reasoning calculus of predicate logic (individual word | predicate | quantifier | predicate logic formula | two basic formulas | proposition symbolization

Reference blog :

• 【 Mathematical logic 】 Predicate logic ( Individual words | Individual domain | The predicate | Full name quantifier | There are quantifiers | Predicate formula | exercises )
• 【 Mathematical logic 】 Predicate logic ( First order predicate logic formula | Example )
• 【 Mathematical logic 】 Predicate logic ( Judge whether the first-order predicate logic formula is true or false | explain | Example | Predicate logic formula type | Yongzhen style | Permanent falsehood | Satisfiability | Equivalent formula )
• 【 Mathematical logic 】 Predicate logic ( Basic equivalence of predicate logic | Eliminate quantifier equivalents | The quantifier negates the equivalent | The scope of quantifier is shrinking and expanding | The equivalent of quantifier distribution )
• 【 Mathematical logic 】 Predicate logic ( The toe in paradigm | The method of toe in normal form transformation | Basic equivalence of predicate logic | Name change rules | Predicate logic inference law )

One 、 Predicate logic related concepts

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1、 Individual words

Individual words :

① individual source : First order predicate logic in , take Atomic proposition Divide into The subject and Predicate , Here we have Individual words And The predicate Of Concept ;

② individual Concept : take Independent object , Concrete things , Abstract things ( Concept ) be called individual or Individual words ;

③ individual Argument : Use

a

,

b

,

c

a,b,c

a,b,c Represents an individual argument ;

④ individual Changyuan : Use

x

,

y

,

z

x, y, z

x,y,z Represents individual constant ;

⑤ Individual domain Concept : individual Argument The value of be called Individual domain ;

⑥ Individual domain Value : Individual domain Sure Value Finite set or Infinite set ;

⑦ Total individual domain : Everything in the universe Composed of Individual domain be called Total individual domain ;

A proposition is a statement , The declarative sentence is composed of The subject , Predicate , The object form , The subject object is the individual , Predicate is predicate ;

Predicate logic from individual , The predicate , quantifiers form ;

2、 The predicate

The predicate :

① Predicate concept : Will represent Individual nature or Relationship between each other Of word be called The predicate ;

② The predicate indicates : Use

F

,

G

,

H

F, G, H

F,G,H Representation predicate Changyuan or Argument ;

③ The individual property predicate indicates :

F

(

x

)

F(x)

F(x) Express

x

x

x have nature

F

F

F , Such as

F

(

x

)

F(x)

F(x) Express

x

x

x It's black ;

④ Examples of relational property predicates :

F

(

x

,

y

)

F(x, y)

F(x,y) Express

x

,

y

x, y

x,y have Relationship F , Such as :

F

F

F

G

(

x

,

y

)

G(x, y)

G(x,y) Express

x

x

x Greater than

y

y

y ;

There are quantifiers :Exist Medium E Turn left and right, then turn upside down ;

① Language correspondence : Corresponding Natural language in “ There is one ” , “ There is a ” , “ yes , we have ” etc. ;

② Representation : Using symbols

∃

\exist

∃ Express ;

③ Reading 1 :

∃

x

\exist x

∃x Represents the individual domain There are

x

x

x ;

④ Reading 2 :

∃

x

(

F

(

x

)

)

\exist x( F(x) )

∃x(F(x)) Express , In the individual domain There is

x

x

x Have the quality of

F

F

F ;

3、 quantifiers

Full name quantifier :Any Medium A Upside down ;

① Language correspondence : Corresponding Natural language in “ arbitrarily ” , “ be-all ” , “ every last ” etc. ;

② Representation : Using symbols

∀

\forall

∀ Express ;

③ Reading 1 :

∀

x

\forall x

∀x Represents the individual domain be-all

x

x

x ;

④ Reading 2 :

∀

x

(

F

(

x

)

)

\forall x( F(x) )

∀x(F(x)) Express , All in the individual domain

x

x

x All have properties

F

F

F ;

Reference blog : 【 Mathematical logic 】 Predicate logic ( Individual words | Individual domain | The predicate | Full name quantifier | There are quantifiers | Predicate formula | exercises )

Two 、 First order predicate logic formula

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Propositional formula : Basic proposition ( Propositional constant / Argument ) and A number of Conjunctions Form a finite length string ;

① Single Propositional argument / Propositional constant Is a proposition formula ;

② If

A

A

A Is a proposition formula , be

(

¬

A

)

(\lnot A)

(¬A) It is also a propositional formula ;

③ If

A

,

B

A,B

A,B Is a proposition formula , be

(

A

∧

B

)

,

(

A

∨

B

)

,

(

A

→

B

)

,

(

A

B

)

(A \land B) , (A \lor B), (A \to B), (A \leftrightarrow B)

(A∧B),(A∨B),(A→B),(AB) It is also a propositional formula ;

④ A limited number of times application ① ② ③ Formed symbol string Is a proposition formula ; ( Infinite times cannot )

First order predicate logic formula : stay Propositional formula On the basis of , Add a condition :

If

A

A

A It's the formula , be

∀

x

A

\forall x A

∀xA and

∃

x

A

\exist x A

∃xA Is also a formula

Related concepts of first-order predicate logic formula : With

∀

x

A

\forall x A

∀xA ,

∃

x

A

\exist x A

∃xA Take the formula as an example ;

Guide arguments :

∀

,

∃

\forall , \exist

∀,∃ After quantifier

x

x

x be called Guide arguments

Jurisdiction :

A

A

A be called The scope of the corresponding quantifier ;

Constraints appear : stay

∀

x

\forall x

∀x ,

∃

x

\exist x

∃x Jurisdiction

A

A

A in ,

x

x

x All appearances are constrained , This is called constraint occurrence ;

Free to appear : Jurisdiction

A

A

A in , It is not the argument of the constraint , Are free to appear ;

Reference blog : 【 Mathematical logic 】 Predicate logic ( First order predicate logic formula | Example )

3、 ... and 、 Two basic formulas

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1、 Formula 1

In the individual domain all There is a quality

F

F

F Of individual , all have nature

G

G

G ;

Use predicate logic to express :

①

F

(

x

)

F(x)

F(x) :

x

x

x Have the quality of

F

F

F ;
②

G

(

x

)

G(x)

G(x) :

x

x

x Have the quality of

G

G

G ;
③ The proposition is symbolized as :

∀

x

(

F

(

x

)

→

G

(

x

)

)

\forall x ( F(x) \rightarrow G(x) )

∀x(F(x)→G(x))

2、 Formula 2

Individual domain in Existence has nature

F

F

F At the same time, it has the nature of

G

G

G The individual of ;

Use predicate logic to express :

①

F

(

x

)

F(x)

F(x) :

x

x

x Have the quality of

F

F

F ;
②

G

(

x

)

G(x)

G(x) :

x

x

x Have the quality of

G

G

G ;
③ The proposition is symbolized as :

∃

x

(

F

(

x

)

∧

G

(

x

)

)

\exist x ( F(x) \land G(x) )

∃x(F(x)∧G(x))

Four 、 Propositional symbolization skills

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1、 Propositional Symbolization Method

Propositional Symbolization Method :

① Write individual fields : The first Individual domain Write clearly , namely indicate

∀

x

\forall x

∀x , representative Everything , If it's everything , Then it must be indicated that it is the total individual domain ;

② Write a relational predicate : Use

F

,

G

,

H

F , G , H

F,G,H indicate Individual nature or Relationship ;

③ Propositional symbols : take Proposition symbolization result Indicate the , It's best to bring a detailed explanation ;

2、 Predicate logical combination

from Full name quantifier or There are quantifiers Individual words The predicate Combined Predicate logic , You can also think of it as One Predicate logic

F

(

x

)

F(x)

F(x) or

G

(

x

,

y

)

G(x, y)

G(x,y) parts Combine again ;

as follows Predicate logic :

∀

x

(

F

(

x

)

→

∀

y

(

G

(

y

)

→

H

(

x

,

y

)

)

)

\forall x (F(x) \rightarrow \forall y ( G(y) \rightarrow H(x,y) ))

∀x(F(x)→∀y(G(y)→H(x,y)))

among

∀

y

(

G

(

y

)

→

H

(

x

,

y

)

)

\forall y ( G(y) \rightarrow H(x,y) )

∀y(G(y)→H(x,y)) It's already assembled Predicate logic , Now think of it as a nature , perhaps Predicate logic unit

A

A

A , Combine again more complex and Gigantic Predicate logic , Get the following :

∀

x

(

F

(

x

)

→

A

)

\forall x (F(x) \rightarrow A)

∀x(F(x)→A)

therefore , Above Predicate logic After deployment , You get the first

∀

x

(

F

(

x

)

→

∀

y

(

G

(

y

)

→

H

(

x

,

y

)

)

)

\forall x (F(x) \rightarrow \forall y ( G(y) \rightarrow H(x,y) ))

∀x(F(x)→∀y(G(y)→H(x,y)))

3、 If and only if predicate logic

If and only if Predicate logic Symbolization :

( 1 ) The third variable : Make sure to introduce The third party The variable of ;

( 2 ) nature or Relationship positive Deduce : The general pattern is

① For all

x

x

x And There is a

y

y

y Yes A certain quality or relationship ,
② For all

x

x

x and be-all

z

z

z There is a certain nature or relationship ;
③

y

y

y And

z

z

z Having equal attributes ;

( 3 ) nature or Relationship Back to back : The general pattern is

① For all

x

x

x And There is a

y

y

y Yes A certain quality or relationship ,
②

y

y

y And be-all

z

z

z There is another property or Relationship , Generally equal or Unequal Relationship ,
③ Can be launched

x

x

x and

z

z

z Yes perhaps No, some nature or Relationship ;

5、 ... and 、 Examples of propositional symbolization

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Reference resources : 【 Mathematical logic 】 Predicate logic ( Individual words | Individual domain | The predicate | Full name quantifier | There are quantifiers | Predicate formula | exercises ) 3、 ... and . Proposition symbolization exercises