[mathematical logic] predicate logic (judge whether the first-order predicate logic formula is true or false | explain | example | predicate logic formula type | forever true | forever false | satisfi

One 、 Judge whether the predicate logic formula is true or false ( semantics )

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Predicate logic grammar And semantics :

grammar : The above two sections explain Predicate logic Formula , how Write the formula according to the statement , yes grammar Category ;

semantics : How about the formula written Determine whether it is true or false , Belong to semantics Category ;

Determine whether the formula is true or false :

• Propositional logic : In propositional logic , By assigning values to propositional arguments , And calculate according to the rules of connectives , Finally get the true value , This process is called assignment ;
• First order predicate logic : In first-order predicate logic , Use “ explain ” Method , Determine whether a formula is true or false ;

Two 、 Predicate logic “ explain ”

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explain :

Given Predicate logic The formula

A

A

A , The formula

A

A

A from Individual words , The predicate , quantifiers form ;

Individual domain : Appoint The formula

A

A

A Of Individual domain by It is known that Individual domain

D

D

D ;

Individual words : Use specific Individual constant yuan replace

A

A

A Medium Individual words ;

function : Use Specific functions , replace

A

A

A Medium Function arguments ;

The predicate : Use specific The predicate , replace

A

A

A Medium Predicate argument ;

After performing the above operations , You can get

A

A

A One of the formulas “ explain ” ;

assignment And explain :

assignment : assignment yes For propositional logic Propositional argument take

0

,

1

0 , 1

0,1 True or false ;

explain : explain yes to Individual words In the individual domain Specify which individual , to The predicate Specify a specific nature or relationship , to quantifiers Appoint Individual domain Determine its scope , To determine the Individual words , The predicate , quantifiers , You can determine the truth of the formula ;

Given a Predicate logic The formula , Give a explain , Can Determine whether it is true or false ;

The same Predicate logic The formula , There can be Different interpretations ;

• individual Appoint Different individual
• The predicate Appoint Different Nature or relationship
• quantifiers Use different Individual domain Explain ;

3、 ... and 、 Predicate logic “ explain ” Example

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Given First order predicate logic The formula

A

A

A by

∀

x

(

F

(

x

)

→

G

(

x

)

)

\forall x ( F(x) \to G(x) )

∀x(F(x)→G(x)) , There are several explanations ;

Explain a :

Individual domain : Set of real numbers ;

F

(

x

)

F(x)

F(x) :

x

x

x It's a reasonable number ;

G

(

x

)

G(x)

G(x) :

x

x

x It's a score ;

At this point, the formula

A

A

A It can be explained as : Rational numbers can be expressed as fractions ;

At this time, the corresponding proposition of this explanation is True proposition ;

Explain two :

Individual domain : Total individual domain ;

F

(

x

)

F(x)

F(x) :

x

x

x Is the person ;

G

(

x

)

G(x)

G(x) :

x

x

x Hair is black ;

At this point, the formula

A

A

A It can be explained as : Everyone has black hair ;

At this time, the corresponding proposition of this explanation is False proposition ;

Four 、 Predicate logic formula type

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Predicate logic The formula , With the explanation , You can judge the type of formula ;

Predicate logic Formula types are divided into Yongzhen style , Permanent falsehood , Satisfiability , Equivalent formula etc. ;

• Yongzhen style : The formula $A$
• Permanent falsehood : The formula $A$
• Satisfiability : The formula $A$
• Equivalency : If $A$

  AB It's Yongzhen style , The formula