Logic and Knowledge Representation P r o p o s i t i o n - - PowerPoint PPT Presentation

SLIDE 1

Logic and Knowledge Representation

G i

• v

a n n i S i l e n

• g

s i l e n

• @e

n s t . f r

T é l é c

• m

P a r i s T e c h , P a r i s

• D

a u p h i n e U n i v e r s i t y

P r

• p
• s

i t i

• n

a l L

• g

i c

4 M a y 2 1 8

SLIDE 2

SLIDE 3

T h i s i s a n i c e v a r i a t i

• n
• f

E p i m e n i d e s p a r a d

• x

, t h e s t

• r

y

• f

a C r e t a n s a y i n g “ A l l C r e t a n s a r e l i a r s ” .

SLIDE 4

Logic: a long history

SLIDE 5

Overview on (Western) logic

• G

r e e k L

• g

i c

– S

t

• i

c s

– A

r i s t

• t

l e

– l

• g

i c i n a r g u m e n t a t i

• n

– s

y l l

• g

i s m

SLIDE 6

Overview on (Western) logic

• G

r e e k L

• g

i c

– S

t

• i

c s

– A

r i s t

• t

l e

– l

• g

i c i n a r g u m e n t a t i

• n

– s

y l l

• g

i s m

• Me

d i e v a l a n d t r a d i t i

• n

a l l

• g

i c

– T

h

• m

a s A q u i n a s ( 1 2 2 5

• 1

2 7 4 )

– m

• d

a l l

• g

i c

– Wi

l l i a m

• f

O c k h a m ( 1 2 8 8

• 1

3 4 8 )

– l

a w s

• f

d e M

• r

g a n

– t

e r n a r y l

• g

i c

– L

• g

i c

• f

P

• r

t R

• y

a l

• A

n t

• i

n e A r n a u l d & P i e r r e N i c

• l

e ( 1 6 6 2 )

SLIDE 7

All bankers are athletes. No consultant is a banker. Therefore….

SLIDE 8

All bankers are athletes. No consultant is a banker. Therefore…. s

• m

e a t h l e t e i s n

• t

a c

• n

s u l t a n t ( ? ? ? )

SLIDE 9

All bankers are athletes. No consultant is a banker. Therefore….

• V

a l i d , a n d n

• t

t r i v i a l .

s

• m

e a t h l e t e i s n

• t

a c

• n

s u l t a n t !

SLIDE 10

All bankers are athletes. No consultant is a banker. Therefore….

B A

• V

a l i d , a n d n

• t

t r i v i a l .

C

s

• m

e a t h l e t e i s n

• t

a c

• n

s u l t a n t .

SLIDE 11

All bankers are athletes. No consultant is a banker. Therefore….

B A

• V

a l i d , a n d n

• t

t r i v i a l .

C

s

• m

e a t h l e t e i s n

• t

a c

• n

s u l t a n t .

SLIDE 12

All bankers are athletes. No consultant is a banker. Therefore….

B A

• V

a l i d , a n d n

• t

t r i v i a l .

C

s

• m

e a t h l e t e i s n

• t

a c

• n

s u l t a n t .

SLIDE 13

All bankers are athletes. No consultant is a banker. Therefore….

B A

• V

a l i d , a n d n

• t

t r i v i a l .

C

s

• m

e a t h l e t e i s n

• t

a c

• n

s u l t a n t .

SLIDE 14

• Mo

d e r n l

• g

i c

– D

e s c a r t e s , L e i b n i z

– G

e

• r

g e B

• l

e ( 1 8 4 8 )

– G

• t

t l

• b

F r e g e , B r e g r i fg s c h r i f t ( 1 8 7 9 )

• Q

u a n t i fj c a t i

• n

– C

h a r l e s P e i r c e

• R

e a s

• n

i n g a n d l

• g

i c

– G

u i s e p p e P e a n

• L
• g

i c a l A x i

• m

a t i z a t i

• n
• f

A r t i h m e t i c s

– B

e r t a n d R u s s e l l & A l f r e d N . Wh i t e h e a d , P r i n c i p i a M a t h e m a t i c a ( 1 9 2 5 )

• L
• g

i c a l A x i

• m

i z a t i

• n
• f

M a t h e m a t i c s

Overview on (Western) logic

SLIDE 15

Propositional logic

SLIDE 16

• a

l p h a b e t

– p

r

• p
• s

i t i

• n

a l s y m b

• l

s p 1 , p 2 , . . .

A language consists of symbols, ...

SLIDE 17

• a

l p h a b e t

– p

r

• p
• s

i t i

• n

a l s y m b

• l

s p 1 , p 2 , . . .

– c

• n

n e c t i v e s

• n

u l l a r y : T , ( t

• p

, b

• t

t

• m

) ⊥

• u

n a r y : ¬ ( n e g a t i

• n

)

• b

i n a r y : , , , , , , , , , ( a n d ,

• r

, ∧ ∨ ⊃ ⊂ ↑ ↓ ⊅ ⊄ ≡ ≠ i m p l i e s ,

• n

l y

• i

f , n a n d ( i n c

• m

p a t i b l e ) , n

• r

, n

• t
• i

m p l i e s , n

• t
• n

l y

• i

f , e q u i v a l e n t , n

• t
• e

q u i v a l e n t )

A language consists of symbols, ...

SLIDE 18

• s

e t A

• f

a t

• m

i c f

• r

m u l a s

– A

c

• n

t a i n s a l l p r

• p
• s

i t i

• n

a l s y m b

• l

s

– A

c

• n

t a i n s t h e n u l l a r y c

• n

n e c t i v e s T , ⊥

A language consists of a syntax (rules to aggregate symbols), ...

SLIDE 19

• s

e t A

• f

a t

• m

i c f

• r

m u l a s

– A

c

• n

t a i n s a l l p r

• p
• s

i t i

• n

a l s y m b

• l

s

– A

c

• n

t a i n s t h e n u l l a r y c

• n

n e c t i v e s T , ⊥

• s

e t P

• f

( w e l l

• f
• r

m e d ) p r

• p
• s

i t i

• n

a l f

• r

m u l a s

– P

c

• n

t a i n s a t

• m

i c f

• r

m u l a s

– i

f F i s i n P , t h e n ¬ F i s i n P

– i

f F a n d G a r e i n P , t h e n ( F

• G

) i s i n P , w h e r e

• i

s a b i n a r y c

• n

n e c t i v e (, , , , , , ∧ ∨ ⊃ ⊂ ↑ ↓ ⊅, , , ) . ⊄ ≡ ≠

– P

i s t h e s m a l l e s t s e t t h a t h a s t h e s e p r

• p

e r t i e s

( e q u i v a l e n t l y , t h e r e i s n

• t

h i n g i n P t h a t d

• e

s n

• t

s a t i s f y t h e s e p r

• p

e r t i e s )

A language consists of a syntax (rules to aggregate symbols), ...

SLIDE 20

A language consists of a semantic (rules to interpret its expressions)

• S

e m a n t i c s s h

• u

l d t e l l u s h

• w

t h e m e a n i n g

• f

t h e c

• n

s t i t u e n t p a r t s

• f

a d i s c

• u

r s e , a n d t h e i r m

• d

e

• f

c

• m

b i n a t i

• n

, d e t e r m i n e t h e

• v

e r a l l m e a n i n g .

SLIDE 21

A language consists of a semantic (rules to interpret its expressions)

• S

e m a n t i c s s h

• u

l d t e l l u s h

• w

t h e m e a n i n g

• f

t h e c

• n

s t i t u e n t p a r t s

• f

a d i s c

• u

r s e , a n d t h e i r m

• d

e

• f

c

• m

b i n a t i

• n

, d e t e r m i n e t h e

• v

e r a l l m e a n i n g .

• B

u t w h a t d

• w

e m e a n b y m e a n i n g ?

SLIDE 22

A language consists of a semantic (rules to interpret its expressions)

• S

e m a n t i c s s h

• u

l d t e l l u s h

• w

t h e m e a n i n g

• f

t h e c

• n

s t i t u e n t p a r t s

• f

a d i s c

• u

r s e , a n d t h e i r m

• d

e

• f

c

• m

b i n a t i

• n

, d e t e r m i n e t h e

• v

e r a l l m e a n i n g .

• B

u t w h a t d

• w

e m e a n b y m e a n i n g ? e x . “ t h e r e i s a d

• g

”

...that a dog is out there correspondence semantics

SLIDE 23

A language consists of a semantic (rules to interpret its expressions)

• S

e m a n t i c s s h

• u

l d t e l l u s h

• w

t h e m e a n i n g

• f

t h e c

• n

s t i t u e n t p a r t s

• f

a d i s c

• u

r s e , a n d t h e i r m

• d

e

• f

c

• m

b i n a t i

• n

, d e t e r m i n e t h e

• v

e r a l l m e a n i n g .

• B

u t w h a t d

• w

e m e a n b y m e a n i n g ? e x . “ t h e r e i s a d

• g

”

...that a dog is out there

“ t h e r e i s a d

• g

” i s t r u e

..that that proposition is true correspondence semantics truth-conditional semantics

SLIDE 24

A language consists of a semantic (rules to interpret its expressions)

• S

e m a n t i c s s h

• u

l d t e l l u s h

• w

t h e m e a n i n g

• f

t h e c

• n

s t i t u e n t p a r t s

• f

a d i s c

• u

r s e , a n d t h e i r m

• d

e

• f

c

• m

b i n a t i

• n

, d e t e r m i n e t h e

• v

e r a l l m e a n i n g .

• B

u t w h a t d

• w

e m e a n b y m e a n i n g ? e x . “ t h e r e i s a d

• g

”

...that a dog is out there

“ t h e r e i s a d

• g

” i s t r u e

..that that proposition is true ..that the locutor believes that.. correspondence semantics truth-conditional semantics cognitive semantics

SLIDE 25

Strange effects...

• “

a d

• g

i s a d

• g

”

• “

a d

• g

i s a m a m m a l ”

SLIDE 26

Strange effects...

• “

a d

• g

i s a d

• g

”

• “

a d

• g

i s a m a m m a l ”

TRUE TRUE Each sentence is assigned to a truth value

SLIDE 27

Strange effects...

• “

a d

• g

i s a d

• g

”

• “

a d

• g

i s a m a m m a l ”

TRUE TRUE

U n d e r t r u t h

• c
• n

d i t i

• n

a l s e m a n t i c s , t h e y h a v e t h e s a m e “ m e a n i n g ” !

Each sentence is assigned to a truth value

SLIDE 28

Strange effects...

• “

a d

• g

i s a d

• g

”

• “

a d

• g

i s a m a m m a l ”

TRUE TRUE

U n d e r t r u t h

• c
• n

d i t i

• n

a l s e m a n t i c s , t h e y h a v e t h e s a m e “ m e a n i n g ” !

Each sentence is assigned to a truth value

t r u t h

• c
• n

d i t i

• n

a l s e m a n t i c s i s p r

• n

e t

• l
• g

i c s

• l

i p s i s m

SLIDE 29

Truth space and functions

• T

r u t h s p a c e : T v = { T, F}

SLIDE 30

Truth space and functions

• T

r u t h s p a c e : T v = { T, F}

• T

r u t h f u n c t i

• n

s :

– 2

N u l l a r y f u n c t i

• n

s : T, F

SLIDE 31

Truth space and functions

not

T F F T

• T

r u t h s p a c e : T v = { T, F}

• T

r u t h f u n c t i

• n

s :

– 2

N u l l a r y f u n c t i

• n

s : T, F

– 1

U n a r y f u n c t i

• n

s : T v T v →

SLIDE 32

Truth space and functions

and

T T T T F F F T F F F F

• r

T T T T F T F T T F F F

imp

T T T T F F F T T F F T

not

T F F T

• T

r u t h s p a c e : T v = { T, F}

• T

r u t h f u n c t i

• n

s :

– 2

N u l l a r y f u n c t i

• n

s : T, F

– 1

U n a r y f u n c t i

• n

s : T v T v →

– 1

6 B i n a r y f u n c t i

• n

s : T v x T v T v →

SLIDE 33

Connecting syntax with semantics

• B
• l

e a n v a l u a t i

• n

, a f u n c t i

• n

m a p p i n g p r

• p
• s

i t i

• n

s t

• t

r u t h v a l u e s : v : P T v →

– v

( ) = ⊤ T

– v

( ) = ⊥ F

– v

( ¬ X ) = n

• t

v ( X )

– v

( X

• Y

) = v ( X )

• v

( Y )

• s

y n t a c t i c c

• n

n e c t i v e s

¬

∧ ∨ , ⊃ →

… .

• s

e m a n t i c c

• n

n e c t i v e s not

and

• r

implies, ⇒ … .

SLIDE 34

syntax vs semantics

l a n g u a g e

• b

j e c t s a t

• m

s a n d f

• r

m u l a s

“ a ”

“ w

• r

l d ”

• b

j e c t s , h e r e t r u t h v a l u e s

“ b ” T F

valuations

SLIDE 35

syntax vs semantics

l a n g u a g e

• b

j e c t s a t

• m

s a n d f

• r

m u l a s “ w

• r

l d ”

• b

j e c t s , h e r e t r u t h v a l u e s

“ a ∧ b ” T F

valuations

? “ a ” “ b ”

SLIDE 36

syntax vs semantics

l a n g u a g e

• b

j e c t s a t

• m

s a n d f

• r

m u l a s “ w

• r

l d ”

• b

j e c t s , h e r e t r u t h v a l u e s

T F

valuations

?

and

T T T T F F F T F F F F

“ a ∧ b ” “ a ” “ b ” F and T

SLIDE 37

syntax vs semantics

l a n g u a g e

• b

j e c t s a t

• m

s a n d f

• r

m u l a s “ w

• r

l d ”

• b

j e c t s , h e r e t r u t h v a l u e s

T F

valuations

F and T “ a ∧ b ” “ a ” “ b ”

SLIDE 38

syntax vs semantics 2

l a n g u a g e

• b

j e c t s s y m b

• l

s a n d f

• r

m u l a s “ w

• r

l d ”

• b

j e c t s , h e r e c

• u

n t i n g i t e m s

interpretation

“ 3 + 5 ” “ 3 ” “ 5 ” “ 8 ” plus

SLIDE 39

syntax vs semantics 2

l a n g u a g e

• b

j e c t s s y m b

• l

s a n d f

• r

m u l a s “ w

• r

l d ”

• b

j e c t s , h e r e c

• u

n t i n g i t e m s

interpretation

“ 3 + 5 ” “ 3 ” “ 5 ” “ 8 ” plus

l a n g u a g e

• p

e r a t

• r

“ w

• r

l d ”

• p

e r a t

• r

SLIDE 40

syntax vs semantics 3

% backward chaining rule interpreter is_true(P) :- fact(P). is_true(P) :- if Condition then P, is_true(Condition). is_true(P1 and P2) :- is_true(P1), is_true( P2). is_true(P1 or P2) :- is_true(P1) ; is_true( P2). % symbols of DSL and priority :- op(800, fx, if). :- op(700, xfx, then). :- op(300, xfy, or). :- op(200, xfy, and). % knowledge base If cloud then rain. If rain then wet. If sprinkler then wet. fact(sprinkler).

l a n g u a g e

• b

j e c t s ? l a n g u a g e

• p

e r a t

• r

s ? “ w

• r

l d ”

• b

j e c t s ? “ w

• r

l d ”

• p

e r a t

• r

s ?

SLIDE 41

Tautologies & co.

SLIDE 42

Tautology, satisfiability, consequence

• A

p r

• p
• s

i t i

• n

a l f

• r

m u l a F i s a t a u t

• l
• g

y i f v ( F ) = T f

• r

a n y B

• l

e a n v a l u a t i

• n

v

SLIDE 43

Tautology, satisfiability, consequence

• A

p r

• p
• s

i t i

• n

a l f

• r

m u l a F i s a t a u t

• l
• g

y i f v ( F ) = T f

• r

a n y B

• l

e a n v a l u a t i

• n

v F ( x , y , z , . . . ) T

x y z

T F T F T F

...

F u n c t i

• n

a l v i e w : a n y c

• n

fj g u r a t i

• n
• f

i n p u t s b r i n g s t h e s a m e

• u

t c

• m

e

T

i n p u t s ( a s s

• c

i a t e d t

• f

a c t

• r

s )

SLIDE 44

Tautology, satisfiability, consequence

• A

p r

• p
• s

i t i

• n

a l f

• r

m u l a F i s a t a u t

• l
• g

y i f v ( F ) = T f

• r

a n y B

• l

e a n v a l u a t i

• n

v

• A

s e t S

• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s i s s a t i s fj a b l e i f s

• m

e v a l u a t i

• n

v

i

m a p s e v e r y m e m b e r

• f

S t

• T:

– v

i

( F ) = T f

• r

a l l F

• f

S .

SLIDE 45

Tautology, satisfiability, consequence

• A

p r

• p
• s

i t i

• n

a l f

• r

m u l a F i s a t a u t

• l
• g

y i f v ( F ) = T f

• r

a n y B

• l

e a n v a l u a t i

• n

v

• A

s e t S

• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s i s s a t i s fj a b l e i f s

• m

e v a l u a t i

• n

v

i

m a p s e v e r y m e m b e r

• f

S t

• T:

– v

i

( F ) = T f

• r

a l l F

• f

S . F

1

( x , y , z , . . . ) T F

2

( x , y , z , . . . ) F

3

( x , y , z , . . . ) . . . T T

x y z

S

F u n c t i

• n

a l v i e w : T h e r e i s a c

• n

fj g u r a t i

• n
• f

i n p u t s m a k i n g a l l

• u

t p u t s

T.

SLIDE 46

Tautology, satisfiability, consequence

• A

p r

• p
• s

i t i

• n

a l f

• r

m u l a F i s a t a u t

• l
• g

y i f v ( F ) = T f

• r

a n y B

• l

e a n v a l u a t i

• n

v

• A

s e t S

• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s i s s a t i s fj a b l e i f s

• m

e v a l u a t i

• n

v

i

m a p s e v e r y m e m b e r

• f

S t

• T:

– v

i

( F ) = T f

• r

a l l F

• f

S . F

1

( x , y , z , … )

∧

F

2

( x , y , z , … )

∧ .

. . T

x y z

S = { F

1 ,

, F

2

, . . . }

F u n c t i

• n

a l v i e w : T h e r e i s a c

• n

fj g u r a t i

• n
• f

i n p u t s m a k i n g

T t

h e

• u

t p u t

• f

t h e c

• n

j u n c t i

• n
• f

t h e f

• r

m u l a i n S .

SLIDE 47

Tautology, satisfiability, consequence

• S

C i s c a l l e d ⊨ s e m a n t i c c

• n

s e q u e n c e : i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

l l e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C

SLIDE 48

Tautology, satisfiability, consequence

• S

C i s c a l l e d ⊨ s e m a n t i c c

• n

s e q u e n c e : i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

l l e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C

F

1

( x , y , z , . . . ) T F

2

( x , y , z , . . . ) F

3

( x , y , z , . . . ) . . . T T

x y z

S C ( x , y , z , . . . ) T

F u n c t i

• n

a l v i e w : A l l c

• n

fj g u r a t i

• n

s m a k i n g

Tt

h e

• u

t p u t s

• f

S m a k e C t r u e .

SLIDE 49

Tautology, satisfiability, consequence

• S

C i s c a l l e d ⊨ s e m a n t i c c

• n

s e q u e n c e : i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

l l e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C

F

1

( x , y , z , . . . ) T F

2

( x , y , z , . . . ) F

3

( x , y , z , . . . ) . . . T T

x y z

S C ( x , y , z , . . . ) T

F u n c t i

• n

a l v i e w : A l l c

• n

fj g u r a t i

• n

s m a k i n g

Tt

h e

• u

t p u t s

• f

S m a k e C t r u e . not the inverse !!

SLIDE 50

Tautology, satisfiability, consequence

• S

C i s c a l l e d ⊨ s e m a n t i c c

• n

s e q u e n c e : i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

l l e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C
• ⊨ C

d e n

• t

e s t h e f a c t t h a t C i s a t a u t

• l
• g

y . T C ( x , y , z , . . . ) T ⊤

SLIDE 51

Tautology, satisfiability, consequence

• S

C ⊨ i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

n y e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C
• ⊨ C

C i s a t a u t

• l
• g

y . E x e r c i c e s ( 1 ) :

– S

h

• w

t h a t X i s a t a u t

• l
• g

y

• n

l y i f ( X  ⊤) i s a t a u t

• l
• g

y

– S

h

• w

t h a t (

• (

X

∧ Y

)

≡

(

• X

∨

• Y

) ) i s a t a u t

• l
• g

y

SLIDE 52

Tautology, satisfiability, consequence

• S

C ⊨ i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

n y e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C
• ⊨ C

C i s a t a u t

• l
• g

y . E x e r c i c e s ( 2 ) :

– (

e x f a l s

• q

u

• d

l i b e t s e q u i t u r ) . i f A ,

• A

 S , t h e n f

• r

a n y X : S X . ⊨

– (

m

• n
• t
• n

i c i t y ) . i f S X , t h e n S ⊨  { Y } X ⊨

SLIDE 53

Tautology, satisfiability, consequence

• S

C ⊨ i f a v a l u a t i

• n

a s s i g n s t h e v a l u e T t

• a

n y e l e m e n t

• f

S , t h e n i t w i l l a s s i g n T t

• C
• ⊨ C

C i s a t a u t

• l
• g

y . E x e r c i c e s ( 3 ) :

– S

h

• w

t h a t S C e n t a i l s t h a t ⊨ S  {

• C

} i s n

• t

s a t i s fj a b l e .

– S

h

• w

t h e r e c i p r

• c

a l .

SLIDE 54

Tautology, satisfiability, consequence

C i s a s e m a n t i c c

• n

s e q u e n c e

• f

S , i . e . S C ⊨ i f a n d

• n

l y i f S  {

• C

} i s n

• t

s a t i s fj a b l e

• C

e n t r a l r e s u l t : t h e c

• n

j u n c t i

• n
• f

t h e f

• r

m u l a s i n S a n d t h e

• C

i s a n a n t i l

• g

y

• r

c

• n

t r a d i c t i

• n

( f a l s e f

• r

a l l i n p u t s )

SLIDE 55

Replacement theorem

• G

i v e n F ( P ) , f

• r

m u l a w i t h a n y

• c

c u r r e n c e s

• f

s y m b

• l

P

if (X Y) is a tautology, then ≡ (F(X) F(Y)) is a tautology as well. ≡

SLIDE 56

Replacement theorem

• G

i v e n F ( P ) , f

• r

m u l a w i t h a n y

• c

c u r r e n c e s

• f

s y m b

• l

P

if (X Y) is a tautology, then ≡ (F(X) F(Y)) is a tautology as well. ≡

P r

• f

.

I f ( X Y ) i s a t a u t

• l
• g

y , t h e n v ( X ) = v ( Y ) f

• r

a n y e v a l u a t i

• n

≡ v , b u t t h e n a l s

• v

( F ( X ) ) = v ( F ( Y ) ) . A s v ( F ( X ) ) = v ( F ( Y ) ) f

• r

a n y v , ( F ( X ) F ( Y ) ) i s a t a u t

• l
• g

y . ≡

SLIDE 57

Replacement theorem

• G

i v e n F ( P ) , f

• r

m u l a w i t h a n y

• c

c u r r e n c e s

• f

s y m b

• l

P

if (X Y) is a tautology, then ≡ (F(X) F(Y)) is a tautology as well. ≡

E x e r c i c e s :

– (

d

• u

b l e n e g a t i

• n

) S h

• w

t h a t ( X

≡

• X

) i s a t a u t

• l
• g

y

– (

m

• d

u s p

• n

e n s ) S h

• w

t h a t Y i s a t a u t

• l
• g

y i f X a n d ( X  Y ) a r e t a u t

• l
• g

i e s

SLIDE 58

Normal Forms (CNF and DNF)

SLIDE 59

Rewriting of formulas

• A

n y n u m b e r m a y b e c

• m

p u t e d a s

– p

r

• d

u c t

• f

s u m s , e . g . 8 = ( 1 + 1 ) * ( 2 + 2 )

– s

u m s

• f

p r

• d

u c t s , e . g . 8 = ( 2 * 1 ) + ( 2 * 1 ) + ( 2 * 1 ) + ( 2 * 1 )

SLIDE 60

Rewriting of formulas

• A

n y n u m b e r m a y b e c

• m

p u t e d a s

– p

r

• d

u c t

• f

s u m s , e . g . 8 = ( 1 + 1 ) * ( 2 + 2 )

– s

u m s

• f

p r

• d

u c t s , e . g . 8 = ( 2 * 1 ) + ( 2 * 1 ) + ( 2 * 1 ) + ( 2 * 1 )

• I

n t h e a l g e b r a i c i n t e r p r e t a t i

• n
• f

b

• l

e a n l

• g

i c ,

– c

• n

j u n c t i

• n

∧s

t a n d s f

• r

p r

• d

u c t *

– d

i s j u n c t i

• n

∨

s t a n d s f

• r

a d d i t i

• n

+

SLIDE 61

Rewriting of formulas

• A

n y f

• r

m u l a m a y b e r e w r i t t e n a s

– c

• n

j u n c t i

• n
• f

d i s j u n c t i

• n

s ( C N F )

– d

i s j u n c t i

• n
• f

c

• n

j u n c t i

• n

s ( D N F )

• I

n t h e a l g e b r a i c i n t e r p r e t a t i

• n
• f

b

• l

e a n l

• g

i c ,

– c

• n

j u n c t i

• n

∧s

t a n d s f

• r

p r

• d

u c t *

– d

i s j u n c t i

• n

∨

s t a n d s f

• r

a d d i t i

• n

+

SLIDE 62

Rewriting of formulas

• A

n y f

• r

m u l a m a y b e r e w r i t t e n a s

– c

• n

j u n c t i

• n
• f

d i s j u n c t i

• n

s ( C N F )

– d

i s j u n c t i

• n
• f

c

• n

j u n c t i

• n

s ( D N F ) a b F

• F

T T T F T F F T F T T F F F T F

D N F

F

= ( a

∧

b )

∨ (

a

∧b

)

∨ (

• a

∧

• b

) C N F

F

=

• D

N F

• F

=

• (

a

∧

• b

) =

• a

∨ b F

• r

i n s t a n c e f r

• m

t r u t h t a b l e s :

SLIDE 63

Conjunctive normal form

• T

h e c

• n

j u n c t i v e n

• r

m a l f

• r

m ( C N F ) r e w r i t e s a n y p r

• p
• s

i t i

• n

a l f

• r

m u l a a s a c

• n

j u n c t i

• n
• f

c l a u s e s .

SLIDE 64

Conjunctive normal form

• T

h e c

• n

j u n c t i v e n

• r

m a l f

• r

m ( C N F ) r e w r i t e s a n y p r

• p
• s

i t i

• n

a l f

• r

m u l a a s a c

• n

j u n c t i

• n
• f

c l a u s e s .

• A

c l a u s e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l s y m b

• l

s p

• s

s i b l y w i t h n e g a t i

• n

. I t i s n

• t

e d a s [a,b,c].

SLIDE 65

Conjunctive normal form

• T

h e c

• n

j u n c t i v e n

• r

m a l f

• r

m ( C N F ) r e w r i t e s a n y p r

• p
• s

i t i

• n

a l f

• r

m u l a a s a c

• n

j u n c t i

• n
• f

c l a u s e s .

• A

c l a u s e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l s y m b

• l

s p

• s

s i b l y w i t h n e g a t i

• n

. I t i s n

• t

e d a s [a,b,c].

• A

c

• n

j u n c t i

• n
• f

c l a u s e s i s n

• t

e d <C1,C2,C3>.

SLIDE 66

Conjunctive normal form

• T

h e c

• n

j u n c t i v e n

• r

m a l f

• r

m ( C N F ) r e w r i t e s a n y p r

• p
• s

i t i

• n

a l f

• r

m u l a a s a c

• n

j u n c t i

• n
• f

c l a u s e s .

• A

c l a u s e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l s y m b

• l

s p

• s

s i b l y w i t h n e g a t i

• n

. I t i s n

• t

e d a s [a,b,c].

• A

c

• n

j u n c t i

• n
• f

c l a u s e s i s n

• t

e d <C1,C2,C3>. p r

• p
• s

i t i

• n

a l f

• r

m u l a C N F ( a ¬ ( b c ) ) ⊃ ⊃ < [ ¬ a , b ] , [ ¬ a , ¬ c ] >

SLIDE 67

Evaluation with CNF

• E

v a l u a t i

• n

s a r e p e r f

• r

m e d a s f

• l

l

• w

s :

– v

( [X1,X2,…,Xn]) = F i f a n d

• n

l y i f v ( X i ) = F f

• r

a l l I

– v

( <C1,C2,…,Cm>) = T i f a n d

• n

l y i f v ( C i ) = T f

• r

a l l I

– e

m p t y c l a u s e : v ( [ ] ) = F

– e

m p t y c

• n

j u n c t i

• n

: v ( < > ) = T.

SLIDE 68

Transforming a formula to CNF

• T

h e a l g

• r

i t h m t h a t c

• n

v e r t s a p r

• p
• s

i t i

• n

a l f

• r

m u l a i n t

• C

N F p r

• c

e e d s s e q u e n t i a l l y w i t h t h e s e s t e p s :

r e p l a c e < . . . [ . . . β . . . ] . . . > b y < . . . [ … β1 , β2 . . . ] . . . > r e p l a c e < . . . [ . . . a . . . ] . . . > b y < . . . [ . . . a1 . . . ] , [ . . . a2 . . . ] . . . > r e p l a c e < . . . [ . . . ¬ ¬ a . . . ] . . . > b y < . . . [ . . . a . . . ] . . . > a-f

• r

m u l a a1 a2 ( X Y ) ∧ X Y ¬ ( X Y ) ∨ ¬ X ¬ Y ¬ ( X Y ) ⊃ X ¬ Y . . . β-f

• r

m u l a β1 β2 ( X Y ) ∨ X Y ¬ ( X Y ) ∧ ¬ X ¬ Y ( X Y ) ⊃ ¬ X Y . . .

SLIDE 69

Example

• T

r a n s f

• r

m ( ( A  ( B  C ) )  ( ( A  B )  ( A  C ) ) ) t

• C

N F

• S

h

• w

t h a t i t i s a t a u t

• l
• g

y .

SLIDE 70

Automatic proof methods

SLIDE 71

Resolution method

• A

s e q u e n c e i s t h e c

• n

j u n c t i

• n
• f

l i n e s .

• A

l i n e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

SLIDE 72

Resolution method

• A

s e q u e n c e i s t h e c

• n

j u n c t i

• n
• f

l i n e s .

• A

l i n e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

• F
• r

a p r

• p
• s

i t i

• n

a l f

• r

m u l a X c

• m

p

• s

e d

• n

t h e l i n e L , t h e g r

• w

t h

• f

t h e s e q u e n c e c

• n

s i s t s

• f

:

– I

f X i s

• f

t y p e β, r e p l a c e i t w i t h β1 , β2 .

– I

f X i s

• f

t y p e a, c r e a t e t w

• n

e w l i n e s L 1 a n d L 2 , r e c

• p

y t h e l i n e L r e p l a c i n g a w i t h a1 a n d a2 r e s p e c t i v e l y .

SLIDE 73

Resolution method

• A

s e q u e n c e i s t h e c

• n

j u n c t i

• n
• f

l i n e s .

• A

l i n e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

• F
• r

a p r

• p
• s

i t i

• n

a l f

• r

m u l a X c

• m

p

• s

e d

• n

t h e l i n e L , t h e g r

• w

t h

• f

t h e s e q u e n c e c

• n

s i s t s

• f

:

– I

f X i s

• f

t y p e β, r e p l a c e i t w i t h β1 , β2 .

– I

f X i s

• f

t y p e a, c r e a t e t w

• n

e w l i n e s L 1 a n d L 2 , r e c

• p

y t h e l i n e L r e p l a c i n g a w i t h a1 a n d a2 r e s p e c t i v e l y .

• A

r e s

• l

u t i

• n

c

• n

s i s t s

• f

c

• n

c a t e n a t i n g t w

• l

i n e s w h e r e X a n d

• X

a r e s e p a r a t e d ,

• m

i t t i n g a l l

• c

c u r r e n c e s

• f

t h e s e l a s t t w

• f
• r

m u l a s . T h e n e w l i n e i s c a l l e d t h e r e s

• l

v i n g c l a u s e

• f

t h e

• t

h e r t w

• .

SLIDE 74

Resolution method

• A

s e q u e n c e i s t h e c

• n

j u n c t i

• n
• f

l i n e s .

• A

l i n e i s a d i s j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

• F
• r

a p r

• p
• s

i t i

• n

a l f

• r

m u l a X c

• m

p

• s

e d

• n

t h e l i n e L , t h e g r

• w

t h

• f

t h e s e q u e n c e c

• n

s i s t s

• f

:

– I

f X i s

• f

t y p e β, r e p l a c e i t w i t h β1 , β2 .

– I

f X i s

• f

t y p e a, c r e a t e t w

• n

e w l i n e s L 1 a n d L 2 , r e c

• p

y t h e l i n e L r e p l a c i n g a w i t h a1 a n d a2 r e s p e c t i v e l y .

• A

r e s

• l

u t i

• n

c

• n

s i s t s

• f

c

• n

c a t e n a t i n g t w

• l

i n e s w h e r e X a n d

• X

a r e s e p a r a t e d ,

• m

i t t i n g a l l

• c

c u r r e n c e s

• f

t h e s e l a s t t w

• f
• r

m u l a s . T h e n e w l i n e i s c a l l e d t h e r e s

• l

v i n g c l a u s e

• f

t h e

• t

h e r t w

• .
• A

p r

• f

b y r e s

• l

u t i

• n
• f

F i s a s e q u e n c e d e r i v e d f r

• m

<

• F

> a n d c

• n

t a i n i n g a n e m p t y c l a u s e [ ] .

SLIDE 75

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 76

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 77

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 78

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b f . [ ¬ A , B ] r e w r i t i n g d

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 79

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b f . [ ¬ A , B ] r e w r i t i n g d g . [ ¬ B , C ] r e w r i t i n g e

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 80

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b f . [ ¬ A , B ] r e w r i t i n g d g . [ ¬ B , C ] r e w r i t i n g e h . [ ¬ C ] d e v e l

• p

m e n t

• f

c i . [ A ] d e v e l

• p

m e n t

• f

c

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 81

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b f . [ ¬ A , B ] r e w r i t i n g d g . [ ¬ B , C ] r e w r i t i n g e h . [ ¬ C ] d e v e l

• p

m e n t

• f

c i . [ A ] d e v e l

• p

m e n t

• f

c j . [ B ] r e s

• l

v i n g f a n d i

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 82

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b f . [ ¬ A , B ] r e w r i t i n g d g . [ ¬ B , C ] r e w r i t i n g e h . [ ¬ C ] d e v e l

• p

m e n t

• f

c i . [ A ] d e v e l

• p

m e n t

• f

c j . [ B ] r e s

• l

v i n g f a n d i k . [ C ] r e s

• l

v i n g g a n d j

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 83

Example

a . [ ¬ ( ( ( A B ) ∧ ( B  C ) ) ¬ ( ¬ C ∧A ) ) ] n e g a t i

• n
• f

t a r g e t b . [ ( ( A B ) ∧( B C ) ) ] d e v e l

• p

m e n t

• f

a c . [ ( ¬ C ∧A ) ] d e v e l

• p

m e n t

• f

a d . [ ( A B ) ] d e v e l

• p

m e n t

• f

b e . [ ( B C ) ] d e v e l

• p

m e n t

• f

b f . [ ¬ A , B ] r e w r i t i n g d g . [ ¬ B , C ] r e w r i t i n g e h . [ ¬ C ] d e v e l

• p

m e n t

• f

c i . [ A ] d e v e l

• p

m e n t

• f

c j . [ B ] r e s

• l

v i n g f a n d i k . [ C ] r e s

• l

v i n g g a n d j l . [ ] r e s

• l

v i n g h a n d k

P r

• f

u s i n g t h e r e s

• l

u t i

• n

m e t h

• d

t h a t : ( ( A  B ) ∧( B  C ) ) 

• (
• C

∧A )

SLIDE 84

Prolog and resolution

a. b. c :- a, b. ?- c. < [ a ] , [ b ] , [ c ,

¬

a ,

¬

b ] , [

¬

c ] >

C N F

SLIDE 85

Complexity (time or space)

• S

A T p r

• b

l e m

( c h e c k w h e t h e r a b

• l

e a n e x p r e s s i

• n

i s s a t i s fj a b l e )

– g

e n e r a l c a s e : N P

• c
• m

p l e t e

– w

i t h H

• r

n c l a u s e s : P

SLIDE 86

• A

t r e e r e p r e s e n t s t h e d i s j u n c t i

• n
• f

b r a n c h e s .

• A

b r a n c h r e p r e s e n t s a c

• n

j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

Tableaux method

SLIDE 87

• A

t r e e r e p r e s e n t s t h e d i s j u n c t i

• n
• f

b r a n c h e s .

• A

b r a n c h r e p r e s e n t s a c

• n

j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

• F
• r

a p r

• p
• s

i t i

• n

a l f

• r

m u l a X c

• m

p

• s

e d

• n

t h e b r a n c h B , t h e g r

• w

t h

• f

t h e t r e e c

• n

s i s t s

• f

:

– I

f X i s

• f

t y p e a, a d d a1 t h e n a2 a t t h e e n d

• f

B .

– I

f X i s

• f

t y p e β, c r e a t e a n

• d

e a n d t w

• n

e w b r a n c h e s B 1 , B 2 a t t h e e n d

• f

B , a d d β1 a n d β2 r e s p e c t i v e l y .

Tableaux method

SLIDE 88

Tableaux method

• A

t r e e r e p r e s e n t s t h e d i s j u n c t i

• n
• f

b r a n c h e s .

• A

b r a n c h r e p r e s e n t s a c

• n

j u n c t i

• n
• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s .

• F
• r

a p r

• p
• s

i t i

• n

a l f

• r

m u l a X c

• m

p

• s

e d

• n

t h e b r a n c h B , t h e g r

• w

t h

• f

t h e t r e e c

• n

s i s t s

• f

:

– I

f X i s

• f

t y p e a, a d d a1 t h e n a2 a t t h e e n d

• f

B .

– I

f X i s

• f

t y p e β, c r e a t e a n

• d

e a n d t w

• n

e w b r a n c h e s B 1 , B 2 a t t h e e n d

• f

B , a d d β1 a n d β2 r e s p e c t i v e l y .

• A

b r a n c h i s c l

• s

e d i f X a n d

• X

a p p e a r .

• A

t r e e i s c l

• s

e d i f a l l i t s b r a n c h e s a r e c l

• s

e d .

• A

p r

• f

t r e e f

• r

F i s a c l

• s

e d t r e e g r e w f r

• m

{

• F

} .

SLIDE 89

Syntaxic consequence

• S

X , i f X c a n b e p r

• v

e n f r

• m

S . ⊢

• ⊢ X

, i f X a d m i t s a p r

• f

.

• d

e d u c t i

• n

t h e

• r

e m : S  { X } Y i f a n d

• n

l y i f S ( X ⊢ ⊢  Y )

SLIDE 90

Syntaxic consequence

• S

X , i f X c a n b e p r

• v

e n f r

• m

S . ⊢

• ⊢ X

, i f X a d m i t s a p r

• f

. ( X i s s a i d t h e

• r

e m )

• P

r

• f

t h e m

• d

u s p

• n

e n s : { P , ( P  Q ) } Q ⊢

{ ( P  Q ) } ( P ⊢  Q ) t r i v i a l { ( P  Q ) }  { P } Q ⊢ f

• r

d e d u c t i

• n

t h e

• r

e m

SLIDE 91

Syntaxic consequence

• S

X , i f X c a n b e p r

• v

e n f r

• m

S . ⊢

• ⊢ X

, i f X a d m i t s a p r

• f

. ( X i s s a i d t h e

• r

e m )

• (

( P  ( Q  R ) )  ( Q  ( P  R ) ) ) i s a t h e

• r

e m :

{ ( P  ( Q  R ) , P , Q } R ⊢ a p p l y i n g t w i c e m

• d

u s p

• n

e n s { ( P  ( Q  R ) , Q } ( P ⊢  R ) d e d u c t i

• n

t h e

• r

e m { ( P  ( Q  R ) } ( Q ⊢  ( P  R ) ) d e d u c t i

• n

t h e

• r

e m ⊢ ( ( P  ( Q  R ) ( Q  ( P  R ) ) ) d e d u c t i

• n

t h e

• r

e m

SLIDE 92

Soundness, completeness

L e t F b e a n y p r

• p
• s

i t i

• n

a l f

• r

m u l a a n d S a s e t

• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s ( a l s

• c

a l l e d a x i

• m

s ) ,

• T

h e l

• g

i c a l s y s t e m i s S

• u

n d : i f S F , t h e n S F ⊢ ⊨ ( a l l t h a t c a n b e p r

• v

e n i s t r u e , b u t t h e r e m a y b e t r u e p r

• p
• s

i t i

• n

s u n p r

• v

e n )

• T

h e l

• g

i c a l s y s t e m i s C

• m

p l e t e : i f S F , t h e n S F ⊨ ⊢ ( w h a t e v e r i s t r u e c a n b e p r

• v

e n , b u t t h e r e m a y p r

• f

r e t u r n i n g f a l s e p r

• p
• s

i t i

• n

s )

SLIDE 93

Consistency

L e t F b e a n y p r

• p
• s

i t i

• n

a l f

• r

m u l a a n d S a s e t

• f

p r

• p
• s

i t i

• n

a l f

• r

m u l a s ( a l s

• c

a l l e d a x i

• m

s ) ,

• A

l

• g

i c a l s y s t e m i s C

• n

s i s t e n t : i f S F , t h e n S F ⊢ ⊬

SLIDE 94

Gödel's incompleteness theorems

if S is a logical system which contain elementary arithmetic, then S is incomplete

t h e r e a r e p r

• p
• s

i t i

• n

s t h a t c a n b e n e i t h e r p r

• v

e d , n e i t h e r d i s p r

• v

e d

SLIDE 95

Gödel's incompleteness theorems

if S is a logical system which contain elementary arithmetic, then S is incomplete

t h e r e a r e p r

• p
• s

i t i

• n

s t h a t c a n b e n e i t h e r p r

• v

e d , n e i t h e r d i s p r

• v

e d

if S is a logical system which contain elementary arithmetic, then S Consistent(S) ⊬

a s y s t e m c a n n

• t

p r

• f

i t s

• w

n c

• n

s i s t e n c y