[mathematical logic] predicate logic (individual word | individual domain | predicate | full name quantifier | existence quantifier | predicate formula | exercise)

One . Predicate logic related concepts

1. Individual words

individual brief introduction :

• 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 ;

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2. The predicate

The predicate brief introduction :

• 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 :$F(x)$
• 4. Examples of relational property predicates :$F(x,y)$

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3. quantifiers

( 1 ) Full name quantifier

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 :$\forall x$

  x ;

• 4. Reading 2 :$\forall x(F(x))$

  F ;

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( 2 ) There are quantifiers

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 :$\exists x$

  x ;

• 4. Reading 2 :$\exists x(F(x))$

  F ;

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Two . Proposition symbolization skill

1. Two basic formulas ( important )

( 1 ) There is a quality F The individual of All have properties G

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 ) Existing nature F It also has nature G The individual of

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))

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2. Propositional symbolization skills

( 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 relationship The 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 ;

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( 2 ) Problem solving skills

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)))

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( 3 ) If and only if Predicate logic method

If and only if Predicate logic Symbolic method :

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 ;

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3. Predicate formula definition

Predicate formula definition :

• 1. The original predicate formula :

  n

  n

  n element The predicate It's a Predicate formula ;

• 2. Negative form : If $A$

  (¬A) It's also a predicate formula ;

• 3. Two predicate formulas Combine : If

  A

  ,

  B

  A, B

  (

  A

  ∧

  B

  )

  ,

  (

  A

  ∨

  B

  )

  ,

  (

  A

  →

  B

  )

  ,

  (

  A

  B

  )

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

  A,B It's a predicate formula , that

  (A∧B),(A∨B),(A→B),(AB) Four kinds of connectives Combined symbols , It's also a predicate formula ;

• 4. Predicate formula And quantifiers Combine : If $A$

  x , And

  ∃xA(x) It's also a predicate formula ;

• 5. Finite repetitions : A limited number of times Yes Predicate formula Use 1. ~ 4. Methods to deal with Got It's also Predicate formula ;

Predicate formula assembly :
1> After several times assemble Good combination The predicate formula of , perhaps Just written Single Predicate formula , Sure As primitive Predicate formula

S

S

S ;
2> stay The original predicate formula

S

S

S front add

¬

\lnot

¬ It's also a predicate formula , Notice the parentheses on the outside ; ( After combination This predicate formula can be regarded as the original predicate formula

S

S

S Use )
3> Use Conjunctions take Two The original predicate formula

S

S

S Connect , Whole Combine It's also Predicate formula ;( After combination This predicate formula can be regarded as the original predicate formula

S

S

S Use )
4> stay The original predicate formula

S

S

S front add Quantifier constraints

∀

x

A

(

x

)

\forall x A(x)

∀xA(x) , or

∃

x

A

(

x

)

\exist x A(x)

∃xA(x) , After combination It's also Predicate formula ;( After combination This predicate formula can be regarded as the original predicate formula

S

S

S Use )( Be careful Premise : Add quantifier constraints Individual words Can not be There are quantifier constraints )

4> step Of Be careful :
① Premise : The individual in this predicate , Not bound by quantifiers , If there is Constraints cannot be repeated ;

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3、 ... and . Proposition symbolization exercises

1. Simple quantifier Example

( 1 ) Examples of full quantifiers

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : Everyone eats ;

① Individual domain : Total individual domain ;

② Related nature or Relationship The predicate Definition :

• 1> $F(x)$

  x Is the person ;

• 2> $G(x)$

  x having dinner ;

③ Proposition symbolization :

∀

x

(

F

(

x

)

→

G

(

x

)

)

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

∀x(F(x)→G(x))

---

( 2 ) Full name quantifier Example 2

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : All the students in a class have studied calculus ;

① Individual domain : Total individual domain ;

② Related nature or Relationship The predicate Definition :

• 1> $F(x)$

  x Is a class of students ;

• 2> $G(x)$

  x Studied calculus ;

③ Proposition symbolization :

∀

x

(

F

(

x

)

→

G

(

x

)

)

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

∀x(F(x)→G(x))

---

( 3 ) There is quantifiers Example

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : Some people like sugar ;

answer :

① Individual domain : Total individual domain ;

② Related nature or Relationship The predicate Definition :

• 1> $F(x)$

  x Is the person ;

• 2> $G(x)$

  x Like sugar ;

③ Proposition symbolization :

∃

x

(

F

(

x

)

∧

G

(

x

)

)

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

∃x(F(x)∧G(x))

Another symbolic method : Sugar can also be called an individual :

① Individual domain : Total individual domain

② The predicate : nature / Relationship Definition :

• $F(x)$
• $G(y)$
• $H(x,y)$

③ Proposition symbolization :

∃

x

(

F

(

x

)

∧

G

(

x

)

∧

H

(

x

,

y

)

)

\exist x (F(x) \land G(x) \land H(x, y))

∃x(F(x)∧G(x)∧H(x,y))

---

2. The position of quantifiers is different Resulting symbolization The results are different

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : Men run faster than women ;

1> The way One :

① Individual domain : Total individual domain ;

② Related nature or Relationship The predicate Definition :

• 1> $F(x)$

  x It's men ;

• 2> $G(y)$

  y It's a woman ;

• 3> $H(x,y)$

  y Run fast ;

③ Proposition symbolization :

∀

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)))

The propositional symbol has an equivalent form :

∀

x

∀

y

(

F

(

x

)

∧

G

(

y

)

→

H

(

x

,

y

)

)

)

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

∀x∀y(F(x)∧G(y)→H(x,y)))

This proposition is false , But it does not prevent us from symbolizing it ;

Symbolic analysis :
① take

∀

y

(

G

(

y

)

→

H

(

x

,

y

)

)

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

∀y(G(y)→H(x,y)) Independent analysis , First Whole Propositions are all in

∀

x

\forall x

∀x Scope , here Has the following properties , All women , All men run faster than women ; Think of it as an independent proposition

A

A

A ;
② The following analysis

∀

x

(

F

(

x

)

→

A

)

∀x(F(x)→ A)

∀x(F(x)→A) , For all men Come on , As long as it's a man , There are proposition

A

A

A The nature of ;

2> The way Two :

① Individual domain : Total individual domain ;

② Related nature or Relationship The predicate Definition :

• 1> $F(x)$

  x It's men ;

• 2> $G(x)$

  x It's a woman ;

• 3> $H(x,y)$

  y Run fast ;

③ Proposition symbolization :

∀

x

∀

y

(

F

(

x

)

∧

G

(

x

)

→

H

(

x

,

y

)

)

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

∀x∀y(F(x)∧G(x)→H(x,y))

This proposition is false , But it does not prevent us from symbolizing it ;

Symbolic analysis :
take

F

(

x

)

∧

G

(

x

)

F(x) \land G(x)

F(x)∧G(x) As a whole

A

A

A , namely

x

x

x It's men ,

y

y

y It's a woman , For all

x

,

y

x, y

x,y There is a quality

A

A

A , that

x

,

y

x, y

x,y At the same time, it has properties or Relationship

H

(

x

,

y

)

H(x,y)

H(x,y) ;

---

3. belt perhaps Of Proposition symbolization

( 1 ) belt perhaps Of Proposition symbolization

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : Every student in a class has a computer perhaps He has a friend who owns a computer ;

answer :

① Individual domain : All the students in a class

② Individual nature or Relationship Predicate definition :

• 1> $F(x)$

  x There's a computer ;

• 2> $G(x,y)$

  y It's a friend. ;

③ Propositional symbols :

∀

x

(

F

(

x

)

∨

∃

y

(

F

(

y

)

∧

G

(

x

,

y

)

)

)

\forall x ( F(x) \lor \exist y ( F(y) \land G(x , y) ) )

∀x(F(x)∨∃y(F(y)∧G(x,y)))

analysis :
1> Individual domain definition : Individual domain As the “ All the students in a class ” ;

2> The outermost quantifier is determined : They all have properties “ Every student in a class has a computer perhaps He has a friend who owns a computer ” , therefore The outermost layer has to be Full name quantifier

∀

x

(

A

(

x

)

)

\forall x (A(x))

∀x(A(x)) , Now let's start to analyze

A

(

x

)

A(x)

A(x) ;

3> Between the two properties is perhaps The relationship between : Two properties are used

∨

\lor

∨ Connect , Namely

B

(

x

)

B(x)

B(x) ( “ There's a computer ” ) and

C

(

x

)

C(x)

C(x) ( “ There is a friend who has a computer ” ) , Current symbols :

∀

x

(

B

(

x

)

∧

C

(

x

)

)

\forall x (B(x) \land C(x))

∀x(B(x)∧C(x)) ;

4> “ There's a computer ” : Expressed as

F

(

x

)

F(x)

F(x) ; Current symbols :

∀

x

(

F

(

x

)

∧

C

(

x

)

)

\forall x (F(x) \land C(x))

∀x(F(x)∧C(x)) ;

5> “ There is a friend with a computer ” ( This is more complicated ) :
① First Make up One Student

y

y

y , This

y

y

y On behalf of the friend who has a computer ;
② Then determine the quantifier :" There is one " Obviously, there are quantifiers

∃

y

\exist y

∃y ( If you use full weighing words , Everyone in that class is his friend ) ;
③ For this Fictitious

y

y

y The requirement is ,

y

y

y Two conditions are met at the same time , “a. There are computers ” “b.

x

,

y

x,y

x,y It's a friend. ” , Therefore use

∧

\land

∧ Connect them , Finally expressed as

F

(

y

)

∧

G

(

x

,

y

)

F(y) \land G(x , y)

F(y)∧G(x,y) ;
④ The symbol of this sentence is :

∃

y

(

F

(

y

)

∧

G

(

x

,

y

)

)

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

∃y(F(y)∧G(x,y)) ;

6> The final symbol is :

∀

x

(

F

(

x

)

∨

∃

y

(

F

(

y

)

∧

G

(

x

,

y

)

)

)

\forall x ( F(x) \lor \exist y ( F(y) \land G(x , y) ) )

∀x(F(x)∨∃y(F(y)∧G(x,y))) ;

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( 2 ) belt Or the proposition Example 2

Proposition symbolization :
In a class Every Student perhaps been Beijing , Or have been Shanghai

answer :

Proposition symbolization result :

① Individual domain : All students in a class

② Individual nature or Relationship Predicate definition :

• 1> $F(x)$

  x Have been to Beijing ;

• 2> $G(x)$

  x Have been to Shanghai ;

③ Propositional symbols :

∀

x

(

F

(

x

)

∨

G

(

x

)

)

\forall x ( F(x) \lor G(x))

∀x(F(x)∨G(x))

analysis :

1> Individual domain quantifiers analysis :

∀

x

\forall x

∀x refer to All of a class Student Medium every last , be-all Student ;

2>

F

(

x

)

∨

G

(

x

)

F(x) \lor G(x)

F(x)∨G(x) Reading : Express

x

x

x been Beijing perhaps been Shanghai ;

3>

∀

x

(

F

(

x

)

∨

G

(

x

)

)

\forall x ( F(x) \lor G(x))

∀x(F(x)∨G(x)) Reading : All the students , Or have been to Beijing , Or have been to Shanghai , One of them must be chosen , And You can only choose one ;

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4. Complex proposition Example

( 1 ) Symbolization of complex propositions

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : There is a student $x$

  z Not a good friend ;

Topic analysis :

• 1. Individual domain analysis : The individuals involved in the proposition are Student , that take Individual domain Set to All the students ;
• 2. Nature and relationship analysis :
  • ① “ For all two different students ” : It's about Two different students , Therefore need Define a The predicate , Express The two students are Different or same ;
  • ② "$x$

    y It's a good friend " : involves Two Student yes perhaps No A good friend , therefore Here we need to define a predicate , Express Two students yes perhaps No A good friend ;

• 3. Topic frame analysis :
  • ① Quantifier constraints : " There is a student $x$

    z Come on " You can write The outermost Of Quantifier constraints ,

    x,y,z Describe the relationship between ;

  • ② " If $x$
    • a> proposition $x$

      A :" If

    • b> proposition $y$

      B :" that

    • c> proposition $A\to B$

      A,B The relationship between :

    z Not a good friend ; " : This proposition It can be used implication Conjunctions To said ;

answer :

Proposition symbolization result :

① Individual domain : All the students

② Individual nature or Relationship Predicate definition :

• 1> $F(x,y)$

  y It's a good friend ;

• 2> $G(x,y)$

  y It's the same ;

③ Propositional symbols :

∃

x

∀

y

∀

z

(

(

¬

G

(

y

,

z

)

∧

F

(

x

,

y

)

∧

F

(

x

,

z

)

)

→

¬

F

(

y

,

z

)

)

\exist x \forall y \forall z ( ( \lnot G(y, z) \land F(x,y) \land F(x, z) ) \rightarrow \lnot F(y, z) )

∃x∀y∀z((¬G(y,z)∧F(x,y)∧F(x,z))→¬F(y,z))

analysis :

1> Quantifier analysis :

∃

x

∀

y

∀

z

\exist x \forall y \forall z

∃x∀y∀z Corresponding In the title " There is a student

x

x

x, For all two different students

y

y

y and

z

z

z Come on "

2>

(

¬

G

(

y

,

z

)

∧

F

(

x

,

y

)

∧

F

(

x

,

z

)

)

( \lnot G(y, z) \land F(x,y) \land F(x, z) )

(¬G(y,z)∧F(x,y)∧F(x,z)) analysis : This sentence corresponds to “ Two different students

y

y

y and

z

z

z Come on , If

x

x

x And

y

y

y It's a good friend , also

x

x

x and

z

z

z It's also a good friend ” At the same time satisfy this Three conditions ;

3>

¬

F

(

y

,

z

)

\lnot F(y, z)

¬F(y,z) analysis : Corresponding to the result “ that

y

y

y and

z

z

z Not a good friend ” ;

4> At the same time satisfy 3 Conditions Then exit the result :

(

¬

G

(

y

,

z

)

∧

F

(

x

,

y

)

∧

F

(

x

,

z

)

)

→

¬

F

(

y

,

z

)

( \lnot G(y, z) \land F(x,y) \land F(x, z) ) \rightarrow \lnot F(y, z)

(¬G(y,z)∧F(x,y)∧F(x,z))→¬F(y,z) ;

5> Add quantifier constraints Get the final result :

∃

x

∀

y

∀

z

(

(

¬

G

(

y

,

z

)

∧

F

(

x

,

y

)

∧

F

(

x

,

z

)

)

→

¬

F

(

y

,

z

)

)

\exist x \forall y \forall z ( ( \lnot G(y, z) \land F(x,y) \land F(x, z) ) \rightarrow \lnot F(y, z) )

∃x∀y∀z((¬G(y,z)∧F(x,y)∧F(x,z))→¬F(y,z)) ;

---

( 2 ) Individual domain changes situation Of Two kinds of analysis

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : In a class Some students have been to Beijing

answer :

( 1 ) Method One ( Individual domain by All students in a class ) :

Proposition symbolization result :

① Individual domain : All students in a class

② Individual nature or Relationship Predicate definition :

• 1> $F(x)$

  x Have been to Beijing ;

③ Propositional symbols :

∃

x

(

F

(

x

)

)

\exist x ( F(x) )

∃x(F(x))

analysis : Just write it directly , Some students , Use There are quantifiers

∃

x

\exist x

∃x Express ,

∃

x

(

F

(

x

)

)

\exist x( F(x) )

∃x(F(x)) Express Some students have been to Beijing ;

( 1 ) Method Two ( Individual domain by Total individual domain ) :

Proposition symbolization result :

① Individual domain : Total individual domain

② Individual nature or Relationship Predicate definition :

• 1> $F(x)$

  x Have been to Beijing ;

• 2> $G(x)$

  x Is a class of students ;

③ Propositional symbols :

∃

x

(

F

(

x

)

∧

G

(

x

)

)

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

∃x(F(x)∧G(x))

analysis :

∃

x

(

F

(

x

)

∧

G

(

x

)

)

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

∃x(F(x)∧G(x))

1> Individual domain analysis : Individual domain by Total individual domain , that

∃

x

\exist x

∃x Namely There is something , The attribute of this thing is something in the universe ;

2>

F

(

x

)

∧

G

(

x

)

F(x) \land G(x)

F(x)∧G(x) : Sure Reading by There is something , That is, students in a class , Have been to Beijing ;

3> Complete interpretation of :

∃

x

(

F

(

x

)

∧

G

(

x

)

)

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

∃x(F(x)∧G(x)) , Sure Reading by There is something , That is, students in a class , Have been to Beijing ;

---

( 3 ) If and only if Transformation problem

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : Everyone has only one good friend

answer :

Proposition symbolization result :

① Individual domain : All of you

② Individual nature or Relationship Predicate definition :

• 1> $F(x,y)$

  x,y It's a good friend ;

• 2> $G(x,y)$

  x,y equal ;

③ Propositional symbols One :

∀

x

∃

y

∀

z

(

(

F

(

x

,

y

)

∧

¬

G

(

y

,

z

)

)

→

¬

F

(

x

,

z

)

)

\forall x \exist y \forall z ( ( F(x,y) \land \lnot G(y, z) ) \rightarrow \lnot F(x,z) )

∀x∃y∀z((F(x,y)∧¬G(y,z))→¬F(x,z))

analysis : Everyone has only one good friend , here

x

,

y

x ,y

x,y Already a good friend , If there is one

z

z

z And

y

y

y It's not equal , that

x

,

z

x,z

x,z Must not be a good friend ;
Quantifier analysis :
For all

x

x

x , There is one.

y

y

y It's his friend , be-all

z

z

z And

x

x

x It's a good friend , that This

z

z

z Namely

y

y

y ;

④ Propositional symbol two :

∀

x

∃

y

∀

z

(

(

F

(

x

,

y

)

∧

F

(

x

,

z

)

)

→

G

(

y

,

z

)

)

\forall x \exist y \forall z ( ( F(x,y) \land F(x, z) ) \rightarrow G(y,z) )

∀x∃y∀z((F(x,y)∧F(x,z))→G(y,z))

analysis : Everyone has only one good friend , If

x

,

y

x,y

x,y It's a good friend ,

x

,

z

x,z

x,z It's a good friend , that

y

,

z

y,z

y,z Definitely equal ;
Quantifier analysis :
For all

x

x

x , There is one.

y

y

y It's his friend , be-all

z

z

z And

x

x

x It's a good friend , that This

z

z

z Namely

y

y

y ;

If and only if Predicate logic Symbolic method :

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 ;

---

( 4 ) Use Full name quantifier and There are quantifiers Two forms Make propositional symbolization

subject :

• 1. requirement : Proposition symbolization :
• 2. Proposition content : Not all animals are cats

answer :

Proposition symbolization result ( The whole quantifier ) : This way Belong to Positive answer ;

① Individual domain : Total individual domain Everything in the universe

② Individual nature or Relationship Predicate definition :

• 1> $F(x)$

  x yes animal ;

• 2> $G(x)$

  x yes cat ;

③ Propositional symbols One :

¬

(

∀

x

(

F

(

x

)

→

G

(

x

)

)

)

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

¬(∀x(F(x)→G(x)))

analysis : Proposition is “ Not all animals are cats ” , Here we begin to disassemble the proposition :
1> Extract negation : Extract the non by

¬

\lnot

¬ , The negative proposition is “ Not all animals are cats ” ;
2> Write “ Not all animals are cats ” proposition : namely Anything of animal nature , All have yes cat The nature of , Here, it is symbolized as

∀

x

(

F

(

x

)

→

G

(

x

)

)

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

∀x(F(x)→G(x)) ;
3> final result :

¬

(

∀

x

(

F

(

x

)

→

G

(

x

)

)

)

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

¬(∀x(F(x)→G(x))) ;

Proposition symbolization result ( There are quantifiers ) : This way Belong to Answer sideways ;

Transforming propositions : There are some animals It's not a cat ;

① Individual domain : Total individual domain Everything in the universe

② Individual nature or Relationship Predicate definition :

• 1> $F(x)$

  x yes animal ;

• 2> $G(x)$

  x yes cat ;

③ Propositional symbols One :

∃

x

(

F

(

x

)

∧

¬

G

(

x

)

)

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

∃x(F(x)∧¬G(x))

∃

x

(

F

(

x

)

∧

¬

G

(

x

)

)

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

∃x(F(x)∧¬G(x)) analysis : There is something , Its satisfaction is the nature of animals , At the same time satisfy It's not a cat The nature of ;

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