mirjana borisavljević-uvod u logiku ii deo

8/16/2019 Mirjana Borisavljević-Uvod u Logiku II Deo

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2/122


3/122

GCLC


4/122


5/122

N K

K N K K

Æ


6/122


7/122

J l

V x,y,z,...,x1, y1, z1,...,xn, yn, zn,...


8/122

{⊥, ∧, ∨, ⇒}

⊥

∧, ∨ ⇒

{∀, ∃} ∀ ∃

J

P

O

C

P

O

C

J

n

n ≥ 1

α,β,ρ,α1, β 1, ρ1


9/122

J

J

t1,...,tn f J n

f (t1,...,tn)

J P O C J J

J = {=, +, a, 0} = + a 0 J

a

0

V

J

+(0, a)

+(z, 0)

+(x, y)

+(+(+(x, z), 0), +(+(y, z), +(x, a)))

Æ

+(x, y)

(x+y)

J

(0 + a)

(z + 0)

(x + y)

(((x + z) + 0) + ((y + z) + (x + a)))

J

J

⊥

t1,...,tn ρ n J ρ(t1,...,tn)

A

B

(A ∧ B)

(A ∨ B)

(A ⇒ B)

A

x

∀xA ∃xA


10/122

0 + a

z + 0

x + y

((x + z) + 0) + ((y + z) + (x + a))

(A ∧ B) (A ∨ B) (A ⇒ B) A ∧ B A ∨ B A ⇒ B J

J = {α, ∗, c}

α

∗

c

ρ

ρ(x, y)

x ρ y

J

α(x, c) α(x ∗ y, (c ∗ z) ∗ x)

∀xα(c, x) ∀x∃yα(x, y)

∀y((α(c ∗ x, y) ∧ α(c, c)) ⇒ ∃xα(x, y))

∀x∃zα(x,y,z) ∀x∃y(x ∗ y = c) ∀xα(x ∗ y ∗ z, c)

J α α ∀x∃y(x∗y = c) J

=

J Æ

x ∗ y ∗ z (x ∗ y) ∗ z x ∗ (y ∗ z)

Æ

J l J

F

P f (F )

F P f (F )


11/122

A ∧ B ∈ P f (F ) A ∈ P f (F ) B ∈ P f (F )

A ∨ B ∈ P f (F )

A ∈ P f (F )

B ∈ P f (F )

A ⇒ B ∈ P f (F ) A ∈ P f (F ) B ∈ P f (F )

∀xA ∈ P f (F ) A ∈ P f (F )

∃xA ∈ P f (F ) A ∈ P f (F )

∀y((α(c ∗ x, y) ∧ α(c, c)) ⇒ ∃xα(x, y))

{∀y((α(c∗ x, y) ∧α(c, c)) ⇒ ∃xα(x, y)) (α(c ∗ x, y) ∧α(c, c)) ⇒ ∃xα(x, y) ∃xα(x, y)α(c ∗ x, y) ∧ α(c, c) α(c ∗ x, y) α(c, c) α(x, y)}

F

A

F

A

F

∀y((α(x, y)∨α(c, c)) ⇒ ∃x(α(x, y)∨α(c, c))) J F

α(x, y)∨α(c, c) α(x, y) α(c, c) F

{∀y((α(x, y)∨α(c, c)) ⇒ ∃x(α(x, y)∨α(c, c)))

α(x, y) ∨ α(c, c)

α(x, y)

α(c, c) (α(x, y) ∨ α(c, c))⇒∃x(α(x, y) ∨ α(c, c)) ∃x(α(x, y) ∨ α(c, c))}

F

F

F

F

D(F )

F

F A∧B A∨B A⇒B F D(F )

F

D(B)D(A)

D(A) A D(B) B

F ∀xA ∃xA F D(F )

F

D(A)

D(A) A


12/122

F

∀ y ((α(x, y) ∨ α(c, c)) ⇒ ∃ x(α(x, y) ∨ α(c, c)))

(α(x, y) ∨ α(c, c)) ⇒ ∃ x(α(x, y) ∨ α(c, c))

∃ x(α(x, y) ∨ α(c, c))

α(x, y) ∨ α(c, c)

α(c, c)α(x, y)

α(x, y) ∨ α(c, c)

α(c, c)α(x, y)

α(x, y) α(c, c) α(x, y) ∨ α(c, c) F

⇔ ¬

A

B

A ⇔ B =def (A ⇒ B) ∧ (B ⇒ A)

¬A =def A ⇒ ⊥

=def ⊥ ⇒ ⊥.

J l {⊥, ∧, ∨, ⇒}

A ⇔ B ¬A ⇔ ¬

F J

x

F

F

x

F x F

∀ ∃ ∀x ∃x x V x ∀x ∃x ∀ ∃

x

Æ

F

F

F

x ∀xA ∃xA

x F F


13/122

J = {


14/122

(3.3)

x

F x F

⇔

¬

A ⇔ B

¬A

x

A

B

A

x

A ⇔ B ¬A x A B A x A ⇔ B ¬A

x

F

x

F

x

F

x

F

0 + (z ◦ x) < x + y

Æ

F

x y z

F

∀x(0 < (x + y) ∧ ∃y(y


15/122

x

y

z

x

z

y

F

x z y

F

x

∀xA ∃xA x

F

x

F

SP (F )

F

SP (F )

F F

SP (F )

F

A ∧ B A ∨ B A ⇒B SP (F )

SP (A) ∪ SP (B)

F

∀xA

∃xA

SP (F ) SP (A)\{x} ⇔ ¬ SP (A ⇔ B) = SP (A) ∪ SP (B) SP (¬A) = SP (A)

F

SP (F )

x

F

∀xA ∃xA x ∈ SP (A) x F

Æ

F SP (F ) SP (F ) F

F

SP (F )

{y} {x, y} F y F z {y} {x, z}

F

x1,...,xn Æ ∀x1...∀xnF

F

F

F

F (x1,...,xn) x1,...,xn Æ F

x1,...,xn

F


16/122

z

z

Æ

J = {α, ∗, c}

F

∀yα(y ∗ c, x) ∨ α(x, c)

x

F x t

t

t

x

x

x

x

F

F x

(c ∗ c) ∗ c

∀yα(y ∗ c, (c ∗ c) ∗ c) ∨ α((c ∗ c) ∗ c, c)

F

x

z1 ∗ z2

∀yα(y ∗ c, z1 ∗ z2) ∨ α(z1 ∗ z2, c)

x F

y ∗ z

F

∀yα(y ∗ c, y ∗ z) ∨ α(y ∗ z, c)

y y ∗ z ∀

y

y ∗ z F

F

x F


17/122

x

F

∀yα(y ∗ c, x) ∀yA t

x

y ∀

F

x t

y

t

F ∀yA

x

A

t x F

t

x

F

x

F

t

x

s

t

txs

t

c

txs c

t

y

y=x txs y

t

x

txs s

t

f (t1,...,tn) t1,...,tn f J n txs f (t1

xs ,...,tn

xs )

x t F

F xt

F ⊥ ⊥xt ⊥

F

ρ(t1,...,tn) F xt ρ(t1

xt ,...,tn

xt )

F A ∧ B F xt Axt ∧ B

xt

F A ∨ B F xt Axt ∨ B

xt

F A ⇒ B F xt Axt ⇒ B

xt

F ∀xA ∃xA F xt F

F ∀yA x y

t x F F xt ∀yAxt

F ∃yA x y

t x F F xt ∃yAxt


18/122

x

F

x F x F

x

t

F

F

Æ

F J

S Æ

J S J

ρ J m

m S f J n

Sn

S

J

S

V

S

S

J ={α, ∗, c}

α

∗

c

N

V

J

c

J


19/122

α

= ∗ + : N2 → N c 1

(N, =, +, 1)

J = {α, ∗, c}

I = (S, I J ) J

J

S Æ

I J

ρ J

m

m

S

ρI ⊆ Sm

I J (ρ) = ρI

f J

n

f I n f I :S

n→ S I J (f ) = f I

c J

cI S I J (c) = cI

J I = (S, I J ) S I = (S, I J )

J

{ρ1,...,ρk, f 1,...,f m, c1,...,cn} k m n I = (S, I J )

(S, ρ1I ,...,ρkI , f 1I ,...,f mI , c1I ,...,cnI )

J = {α, ∗, c}

(N, =, +, 1)

Z

R

(Z, ≤, ·, −1) (R,


20/122

Æ

I ı = ( I , ı) = ((S, I J ), ı) Æ

J t J

I ı = ( I , ı) = ((S, I J ), ı)

t

S

ı

S

S

I J

I ı

S

S

t I ı=( I , ı) Æ

I = (S, I J ) J ı : V → S

I ı(t)

t x I ı(t) = ı(x)

t

c

I ı(t) = cI

t f (t1,...,tn) t1,...,tn f J n I ı(t) = f I (I ı(t1),...,I ı(tn))

J = {∗, c}

∗

c

x

c

(c ∗ x) ∗ y ((c ∗ x) ∗ c) ∗ y I ı Æ

(N, +, 1) J ı : V → N

ı(x) = 3 ı(y) = 5

N

I ı(x) = ı(x) = 3 I ı(c) = cI = 1

I ı((c ∗ x) ∗ y) = I ı(c ∗ x) + I ı(y) = (I ı(c) + I ı(x)) + I ı(y) = (1+ 3)+5 = 9

I ı(((c ∗ x) ∗ c) ∗ y) = I ı((c ∗ x) ∗ c) + I ı(y) = ((I ı(c) + I ı(x)) + I ı(c)) + I ı(y)

= ((1 + 3) + 1) + 5 = 10

Æ

I = (S, I J ) J x V

ı, : V → S ı(y) (y) y V y x ı x


21/122

ı x x V S

F

I ı = ( I , ı) Æ I = (S, I J ) J

ı : V → S I ı(F )

F ⊥ I ı(⊥) = 0

F ρ(t1,...,tn)

I ı(ρ(t1,...,tn)) = ρI (I ı(t1),...,I ı(tn))

F A ∧ B

I ı(A ∧ B) = min(I ı(A), I ı(B))

F A ∨ B

I ı(A ∨ B) = max(I ı(A), I ı(B))

F

A ⇒ B

I ı(A ⇒ B) = max(1 − I ı(A), I ı(B))

F ∀xA

I ı(∀xA)= 1 : V → S x ı I (A)= 1

I ı(∀xA)= 0

F ∃xA

I ı(∃xA)= 1 : V → S x ı I (A)= 1

I ı(∃xA)= 0

I ı=( I , ı) F I ı(F )=1 F I ı I ı F I ı(F ) = 0 F

I ı F

A∧B A∨B A⇒B

A

B

¬

⇔

Æ ¬A A⇔B I ı = ( I , ı) F

F

I ı() = 1 F ¬A I ı(¬A) = 1 − I ı(A)


22/122

F

A ⇔B

I ı(A⇔ B) =1 I ı(A) =I ı(B) I

ı(A ⇔B) = 0

x

F

I ı=( I , ı) I ı(F ) ı(x) I ı(x) I =( I , ) I

J

x

(x)

ı(x)

I (F ) I ı(F ) F

I ı = ( I , ı) I ı(F ) I = (S, I J ) J ı

I ı = ( I , ı) F I ı(F ) F ı Æ

∀xA

∃xA

J = {α, ∗, c}

∀xα(c, x) ∃xα(x, c) ∀xα(x, x ∗ c)

I ı ({0, 1, 2}, ≤, +, 0) J ı : V → {0, 1, 2}

I ı ∀x (0 ≤ x) ∀xA

I ı(∀xα(c, x)) = 1

I (α(c, x)) = 1 : V → {0, 1, 2} x ı x ı I α(c, x) 0 ≤ (x) (x) {0, 1, 2}

x ı I (α(c, x)) = 1

0 ≤ d

d ∈ {0, 1, 2}

0 ≤ 0

0 ≤ 1 0 ≤ 2 I ı(∀xα(c, x)) = 1

I (α(c, x)) = 1 x ı

0 ≤ 0 0 ≤ 1 0 ≤ 2

∀ ∀xA

I A x

∀xA

A

A


23/122

0 ≤ 0

0 ≤ 1

0 ≤ 2

∀xα(c, x)

I ı=(({0, 1, 2}, ≤, +, 0), ı) Æ ∀xα(c, x)

(({0, 1, 2}, ≤, +, 0), ı)

ı

I ı = ( I , ı) I J I = ({0, 1, 2}, ≤, +, 0) J

I ı = ( I , ı)

J

I ı = (({0, 1, 2}, ≤, +, 0), ı)

I ı ∃x (x ≤ 0)

I ı(∃xα(x, c))=1 ∃xA Æ x ı I (α(x, c)) = 1

x ı

I α(x, c) (x) ≤ 0 (x) {0, 1, 2} I ı(∃xα(x, c))= 1

d {0, 1, 2} d ≤ 0

0 ≤ 0 1 ≤ 0 2 ≤ 0 ∀ ∃ ∃xA A

x J

{0, 1, 2}

0 0 x ı x I 0(α(x, c))=1 x ı

I (α(x, c))=1 I ı(∃xα(x, c)) = 1 ∃xα(x, c)

I ı=(({0, 1, 2}, ≤, +, 0), ı)

I ı ∀x (x ≤ x + 0) ∀xα(x, x ∗ c)

I ı 0≤0 + 0

1 ≤ 1 + 0 2 ≤ 2 + 0 I ı(∀xα(x, x ∗ c)) = 1 ∀xα(x, x ∗ c)

I ı=(({0, 1, 2}, ≤, +, 0), ı)

I ı (N,


24/122

I ı(∀xα(c, x)) = 0 ∀xα(c, x)

I ı=((N,


25/122

F

J

I ı

= ( I , ı)=((S, I J ), ı)

I ı(F )= 1 F

F I ı = ( I , ı) = ((S, I J ), ı)

( I , ı) F ( I , ı) |= F

∀xα(x, x ∗ y) I ı2 = ((Z,


26/122

I ı(t) = f I (I ı(t1),...,I ı(tm)) = f I (I (t1),...,I (tm)) = I (t).

J

♦

I ı

I ı

I ı = ( I , ı) I = ( I , ) I = (S, I J ) J ı, : V → S x F J I ı(x) = I (x)

I ı(F ) = I (F ).

F

n

F

n = 0

F

F ⊥

I ı(⊥)=0=I (⊥) F ρ(t1,...,tm)

ρ

m

t1,...,tm

I ı I Æ

t1,...,tm

I ı(tl) = I (tl) tl 1 ≤ l ≤ m

I ı(F ) = ρI (I ı(t1),...,I ı(tm)) = ρI (I (t1),...,I (tm)) = I (F ).

F

n

n

F

n

F

A ∧ B A ∨ B A ⇒ B ∀xA ∃xA

SP (A ∧ B)= SP (A ∨ B)= SP (A⇒B)= SP (A) ∪ SP (B) F A∧B A∨B A⇒B

A

B

SP (F )

A

B

I ı I F

I ı(A)= I (A) I ı(B) =I (B)


27/122

F

A ∧ B

I ı(F ) = min(I ı(A), I ı(B)) = min(I (A), I (B)) = I (F )

F A ∨ B

I ı(F ) = max(I ı(A), I ı(B)) = max(I (A), I (B)) = I (F )

F

A ⇒ B

I ı(F ) = max(1 − I ı(A), I ı(B)) = max(1 − I (A), I (B)) = I (F )

F

∀xA

∃xA SP (∀xA) = SP (∃xA) = SP (A)\{x}

A

SP (A)

SP (F )

x z

A

z = x

I ı(z) = I (z)

F

∀xA

I ı(∀xA) = 0 ı ı : V → S

ı x ı I ı(A)=0

: V → S

x ı(x) = (x) A

F

z

I ı I I ı(z) = I (z) I ı(x) = I (x) I ı I x z = x

I ı(z)=I ı(z)=I (z)=I (z)

I ı(A) = I (A) I

x I (A)= 0

I (∀xA) = 0 I (∀xA) = 0 I ı(∀xA) = 0

I ı(∀xA) = 0 I (∀xA) = 0

F ∀xA I ı(F )= I (F )

F ∃xA

F

∀xA

I ı(∃xA)=1 I (∃xA)=1

F ∃xA I ı(F )=I (F )

F J

♦

I ı = ( I , ı) = ((S, I J ), ı) F

Γ

I ı(F )= 1

I ı Γ


28/122

J I ı = ( I , ı) I = (S, I J ) J I J I ı = ( I , ı)

I ı = ( I , ı) I = (S, I J ) J I ı = ( I , ı) I = (S, I J ) J

I

F

J

I = (S, I J

)

J ı ı : V → S

I ı = ( I , ı) I ı(F ) = 1 I F F I I |= F

J = {α, ◦} α ◦

∀x∃yα(x, y) J (R,


29/122

I ı0(∀xα(x, x ◦ y))

x ı0

I (α(x, x ◦ y))=1

(x)≤ (x) + c0 x ı0 x (x)

d ≤ d + c0

d

I ı0(∀xα(x, x ◦ y)) = 1 ∀xα(x, x ◦ y)

I ı0 (N, ≤, +) J y c0 I ı0(y) = c0

d ≤ d + c0 d N

c0 N d ≤ d + c d c (N, ≤, +) J

I ı(∀xα(x, x ◦ y) ) = 1 I ı = ((N, ≤, +), ı)

(N, ≤, +)

ı ı : V →N

I ı1 = ((N, ≤, +), ı1) y c1 I ı1(y) = ı1(y) = c1 ∀xα(x, x ◦ y)

d

d ≤ d+c1 I ı(∀xα(x, x ◦ y)) I ı = ((N, ≤, +), ı)

d ≤ d + c c d

I ı(∀xα(x, x ◦ y))= 1

I ı=((N, ≤, +), ı) ∀xα(x, x◦y)

(N, ≤, +)

(N, ≤, +) |= ∀xα(x, x ◦ y)

J

({0, 1, 2}, ≤, +, 0) |= ∀xα(c, x)

({0, 1, 2}, ≤, +, 0) |= ∃xα(x, c)

({0, 1, 2}, ≤, +, 0) |= ∀xα(x, x ∗ c)

(N,


30/122

β (x) ∨ ¬β (x).

I =(S, β I )

J

ı:V →S

ı(x)=d

β I (d) ∨ ¬β I (d).β I (d)

¬β I (d) β I (d) ¬β I (d) β I (d)∨¬β I (d) I = (S, β I ) J ı :V →S β (x)∨¬β (x) I = (S, β I ) J

F

J

F |= F

Æ

p ∨ ¬ p β (x) ∨ ¬β (x) p ∨ ¬ p

p

β (x)

p1,...,pn Æ

F

F ( p1,...,pn) Æ A1,...,An F ( p1,...,pn) pk Ak k

1 ≤ k ≤ n F (A1,...,An)

F ( p1,...,pn) A1,...,An J F (A1,...,An)

F ( p) p

p∨¬ p p

β (x)

F (β (x))

β (x) ∨ ¬β (x)

F ( p)

F (β (x))

p ∨ ¬ p β (x)∨¬β (x)

Æ

v(F ( p1,...,pn)) v Æ I ı(F (A1,...,An)) I ı Æ


31/122

v

v(F ( p)) = v( p∨¬ p)

v

F ( p) p

v( p)

v(F ( p))=v( p ∨ ¬ p)

v(F ( p)) = v( p ∨ ¬ p) = max(v( p), 1 − v( p)).

v

p

v( p) v( p) v(F ( p))

v( p)

v( p)

1 − v( p) I ı(F (β (x))) = I ı(β (x) ∨ ¬β (x))

I ı I ı β (x)

I ı(β (x)) I ı(F (β (x))) = I ı(β (x) ∨ ¬β (x))

I ı(F (β (x))) = I ı(β (x) ∨ ¬β (x)) = max(I ı(β (x)), 1 − I ı(β (x))).

I ı β (x) I ı(β (x)) I ı(β (x)) I ı(β (x) ∨ ¬β (x)) I ı(β (x)) I ı I ı(β (x)) 1 − I ı(β (x))

v(F ( p))

v( p)

I ı(F (β (x))) I ı(β (x))

v(F ( p)) = h(v( p))

I ı(F (β (x))) = h(I ı(β (x)))

h : {0, 1} → {0, 1} h(z) = max(z, 1 − z) v

I ı

F ( p1,...,pn)

F (A1,...,An)

v(F ( p1,...,pn)) v( p1),...,v( pn) v

I ı(F (A

1,...,A

n))

I ı(A

1),...,I

ı(A

n)

I ı

h : {0, 1}n → {0, 1}

v(F ( p1,...,pn)) = h(v( p1),...,v( pn))

I ı(F (A1,...,An)) = h(I ı(A1),...,I ı(An))

F ( p1,...,pn) F (A1,...,An)

F ( p1,...,pn) A1,...,An F (A1,...,An)

A1,...,An J

I ı = ( I , ı) I = (S, I J )

J

I ı(F (A1,...,An))

F ( p1,...,pn) v p1,...,pn v( p1),...,v( pn)


32/122

h

{0, 1}n → {0, 1}

v(F ( p1

,...,pn

)) v( p1

),...,v( pn

)

v(F ( p1,...,pn)) = h(v( p1),...,v( pn)).

F (A1,...,An) I ı I ı(F (A1,...,An)) I ı(A1),...,I ı(An)

h

v(F ( p1,...,pn)) v( p1),...,v( pn)

I ı(F (A1,...,An)) = h(I ı(A1),...,I ı(An)).

I ı = ( I , ı)

I ı(F (A1,...,An)) A1,...,An I ı(A1),...,I ı(An)

v

p1,...,pn

F ( p1,...,pn)

v( pk) I ı(Ak) k 1 ≤ k ≤ n

F ( p1,...,pn)

p1,...,pn v

1 = v(F ( p1,...,pn)) = h(v( p1),...,v( pn))

F (A1,...,An) I ı = ( I , ı) I ı(F (A1,...,An))

I ı(F (A1,...,An)) = h(I ı(A1),...,I ı(An))

= h(v( p1),...,v( pn)) v( pk) = I ı(Ak) 1 ≤ k ≤ n

= v(F ( p1,...,pn)) = 1.

I ı = ( I , ı)

F (A1,...,An) F (A1,...,An) F (A1,...,An)

♦

Æ

modus ponens

MP

A

A ⇒ B B

MP

Æ

modus ponens

MP

A

A⇒B B

M P

A

A ⇒ B B

A A⇒B


33/122

I ı = ( I , ı) I ı(A) = 1 I ı(A ⇒ B) = 1

I ı

1 = I ı(A ⇒ B) = max(1 − I ı(A), I ı(B)) = max(0, I ı(B)) = I ı(B)

I ı(B) I ı B

♦

MP

Æ generalizacije Gen

Gen

Gen

A ∀xA

I ı = ( I , ı) A A

I x ı A I (A) = 1

I ı(∀xA) 1

I ı I ı(∀xA)=1 ∀xA

♦

Æ Æ

Gen

∀xA A

∀xA

I ı I ı(A) = 1 I ı ∀xA I ı(∀xA) = 1 I ı x ı I (A)=1

0 0(x) = ı(x) x 0(x) = ı(x) y A I 0(y) = I ı(y) I ı(A) I 0(A) I ı(A) I ı I ı(A) = 1 A Gen

A ∀xA


34/122

Æ

x1

xn

F

F ∀x1...∀xnF

x1,...,xn

n = 1

F ∀x1F

n − 1 x1,...,xn−1

F ∀x1...∀xn−1F

F n x1,...,xn

F

∀xnF

∀xnF x1 xn−1 n − 1

∀xnF ∀x1...∀xn−1∀xnF

F ∀x1...∀xnF

♦

F

F

F

F

F

x1,...,xn Æ

F ∀x1...∀xnF

F

A

B

A ⇔ B

A

B

A ⇔ B

A

B

J

A ⇔ B

|= A ⇔ B

A

B

I ı = ( I , ı) I ı(A) = I ı(B)


35/122

A

B

A⇔B |= A ⇔B

I ı = ( I , ı) I ı(A ⇔ B) = 1 I ı(A ⇔ B) = 1 A⇔B

I ı(A) I ı(B) I ı(A) = I ı(B)

I ı I ı(A) = I ı(B)

I ı(A⇔B) I ı I ı(A⇔B) A ⇔ B

A

B

J

≡ ≡

A ≡ B A B

≡

J

|= A ⇔ B |= B ⇔ C |= A ⇔ C

|= A ⇔ B

|= B ⇔ C

I ı(A)= I ı(B) I ı(B)= I ı(C ) I ı

I ı(A) = I ı(C ) I ı A ⇔ C |= A ⇔ C

|= A ⇔ B |= C ⇔ D |= A ⇒ C |= B ⇒ D

I ı(A) = I

ı(B)

I ı(C ) = I

ı(D)

I ı(A ⇒ C ) = 1 I ı

I ı

1=I ı(A⇒C )= max(1−I ı(A), I ı(C ))= max(1−I ı(B), I ı(D))= I ı(B⇒D). B⇒D

A

B J

|= A ⇔ B |= ¬A ⇔ ¬B |= A ⇔ B C J

∧ |= (C ∧ A) ⇔ (C ∧ B) ∧ |= (A ∧ C ) ⇔ (B ∧ C )

∨

|= (C ∨ A) ⇔ (C ∨ B)

∨

|= (A ∨ C ) ⇔ (B ∨ C )

⇒ |= (C ⇒ A) ⇔ (C ⇒ B) ⇒ |= (A ⇒ C ) ⇔ (B ⇒ C ) ⇔ |= (C ⇔ A) ⇔ (C ⇔ B) ⇔ |= (A ⇔ C ) ⇔ (B ⇔ C )

|= A ⇔ B |= ∀xA ⇔ ∀xB


36/122

|= A ⇔ B

|= ∃xA ⇔ ∃xB

|= A ⇔ B A B

I ı = ( I , ı)=((S, I J ), ı) I ı(A) = I ı(B) I ı I ı(∀xA) = I ı(∀xB)

I ı

I ı(∀xA)

I ı(∀xB) I (A) I (B) : V → S x ı

A

B Æ

I x ı

I ı

I ı(∀xA) = 1

x ı I (A) = 1

x ı I (B) = 1 I (A) = I (B)

I ı(∀xB) = 1

∀xA ∀xB ∀xA ⇔ ∀xB

♦

∀x∀yA ⇔ ∀y∀xA

∃x∃yA ⇔ ∃y∃xA

∀xA ⇔ ∀yAxy y A

∃xA ⇔ ∃yAxy y A

∀x∀yA ⇔ ∀y∀xA C ⇔D ∀x∀yA ∀y∀xA

I ı(∀x∀yA)=0 I ı(∀y∀xA)=0 I ı=((S, I

J ), ı)

I ı(∀x∀yA)=0


37/122

0 0 x ı I 0(∀yA)=0

0

0 y 0

I 0(A)=0

x 0 I (∀xA)=0

1 1 x 0 1(x) = ı(x)

I 1(∀xA)=0 1 y ı y 0, 0, 1

I ı(∀y∀xA)=0

I ı(∀y∀xA)=0 I ı(∀x∀yA)=0

I ı=((S, I J ), ı)

I ı(∀x∀yA)=0 I ı(∀y∀xA)=0

∀x∀yA

∀y∀xA

∀x∀yA ⇔ ∀y∀xA

I ı=((S, I J ), ı)

I ı(∃x∃yA) = 1 I ı(∃y∃xA) = 1

∃x∃yA ∃y∃xA ∃x∃yA ⇔ ∃y∃xA

A Axy y

A I ı = ((S, I J ), ı)

0 0 x ı 0 0 y ı I 0(A)=I 0(A

xy)

0 0x ı x 0

0(z) = ı(z) z x

x

s ∈ S 0(x) = s ı(x) 0 0 y ı 0(z) = ı(z) = 0(z) z

x

y

0(x) = ı(x) 0(y)=s= 0(x) 0 y y A I 0(A)=I 0(A

xy)

0 0yı 0 0 x ı

I 0(Axy)=I 0(A)

I ı = ((S, I J ), ı)

I ı(∀xA) = I ı(∀yAxy)

I ı = ((S, I J ), ı)

I ı(∀xA) = 0

x ı I (A)=1


38/122

0 0 x ı I 0(A)=0

0

0 y ı

I 0(Axy)=0

y ı I (Axy)=1

I ı(∀yAxy) = 0

y

A ∀xA

∀yAxy A y

∀xA ⇔ ∀yAxy

I ı=((S, I J ), ı)

A

y

I ı(∃xA) = 1 I ı(∃yAxy) = 1

A

y

∃xA ⇔ ∃yAxy

(3) (4)

F ∀xA ∃xA x F

F

∀xA

∃xA

¬∀xA

¬∃xA

¬(A ∧ B) ⇔ (¬A ∨ ¬B) ¬(A ∨ B) ⇔ (¬A ∧ ¬B)

¬∀xA ⇔ ∃x¬A ¬∃xA ⇔ ∀x¬A

¬∀xA ⇔ ∃x¬A

¬∀xA ∃x¬A I ı=((S, I

J ), ı)

I ı(¬∀xA) = 1

I ı(∀xA) = 0

x ı I (A)=1

0 0 x ı I 0(A)=0

0 0 x ı I 0(¬A)=1

I ı(∃x¬A) = 1


39/122

¬∀xA

∃x¬A

¬∀xA ⇔ ∃x¬A

|= ¬∀xA ⇔ ∃x¬A

A

¬∀xA ⇔ ∃x¬A

A ¬A

|= ¬∀x¬A ⇔ ∃x¬¬A

¬¬ p ⇔ p A ¬¬A ⇔ A ∃x¬¬A ⇔ ∃xA

¬∀x¬A ⇔ ∃x¬¬A

∃x¬¬A ⇔ ∃xA

|= ∃xA ⇔ ¬∀x¬A (∗)

|= ¬∃xA ⇔ ¬¬∀x¬A

¬¬∀x¬A ⇔ ∀x¬A

|= ¬∃xA ⇔ ∀x¬A

¬∀xA ⇔ ∃x¬A

¬¬∀xA ⇔ ¬∃x¬A

∀xA ⇔ ¬¬∀xA ∀xA ⇔ ¬∃x¬A

|= ∀xA ⇔ ¬∃x¬A (∗∗)

(∗) (∗∗) ∃ ∀ ∀ ∃

∃xA =def ¬∀x¬A.

∀xA =def ¬∃x¬A.

∀xA(x) ⇒ A(t)

A(t)

A(x)

x t


40/122

x

A(x)

A(t) A x

t

∀xA(x) ⇒ A(t)

(∀x∃yα(x, y)) ⇒ ∃yα(y, y)

J = {α}

α

∀xA(x) ⇒ A(t) A ∃yα(x, y) t y A(t) ∃yα(y, y)

(∀x∃yα(x, y)) ⇒ ∃yα(y, y)

I ı = ( I , ı)

((N,


41/122

t

x

A

( p ⇒ q ) ⇔ (¬q ⇒ ¬ p) ∀x¬A(x) ¬Axt

|= (∀x¬A(x) ⇒ ¬Axt ) ⇔ (¬¬Axt ⇒ ¬∀x¬A(x))

∀x¬A(x) ⇒ ¬Axt ¬¬Axt ⇒¬∀x¬A(x) ¬¬A

xt ⇒¬∀x¬A(x)

¬¬Axt ⇔Axt ∃xA(x)⇔¬∀x¬A(x)

|= Axt ⇒ ∃xA(x)

∀x(A(x) ∧ B(x)) ⇔ (∀xA(x) ∧ ∀xB(x))

∀x(A(x) ∧ B) ⇔ (∀xA(x) ∧ B) x /∈ SP (B)

∀x(B ∧ A(x)) ⇔ (B ∧ ∀xA(x)) x /∈ SP (B)

∃x(A(x) ∧ B(x)) ⇒ (∃xA(x) ∧ ∃xB(x))

∃x(A(x) ∧ B) ⇔ (∃xA(x) ∧ B) x /∈ SP (B)

∃x(B ∧ A(x)) ⇔ (B ∧ ∃xA(x)) x /∈ SP (B)

∀x(A(x)∧B(x)) ∀xA(x) ∧ ∀xB(x)

I ı=((S, I J ), ı)

I ı(∀x(A(x) ∧ B(x))) = 1

: V → S x ı I (A(x) ∧ B(x)) = 1

: V → S x ı min(I (A(x)), I (B(x))) = 1

: V → S x ı I (A(x)) = 1 I (B(x)) = 1

I ı(∀xA(x)) = 1 I ı(∀xB(x)) = 1

min(I ı(∀xA(x)), I ı(∀xB(x))) = 1

I ı(∀xA(x) ∧ ∀xB(x)) = 1.

∀x(A(x)∧B) ∀xA(x)∧B

I ı = ((S, I J ), ı)

I ı(∀x(A(x) ∧ B)) = 1

: V → S

x ı I (A(x) ∧ B) = 1

: V → S x ı min(I (A(x)), I (B)) = 1 : V → S x ı I (A(x)) = 1 I (B) = 1

I ı(∀xA(x)) = 1 I ı(B) = 1

min(I ı(∀xA(x)), I ı(B)) = 1

I ı(∀xA(x) ∧ B) = 1.


42/122


43/122

∃

∧

∃x(A(x) ∧ B(x)) (∃xA(x) ∧ ∃xB(x))

(∃xA(x) ∧ ∃xB(x)) ⇒ (∃x(A(x) ∧ B(x))) A(x) B(x)

α(x)

β (x)

α

β

∃x(α(x) ∧ β (x))

(∃xα(x) ∧ ∃xβ (x))

J

I ı=((S, I J ), ı) S

N

αI β I

I ı(∃xα(x) ∧ ∃xβ (x)) = 1

I ı(∃x(α(x) ∧ β (x))) = 0 I ı((∃xα(x) ∧ ∃xβ (x)) ⇒ (∃x(α(x) ∧ β (x))))

∃x(A(x) ∨ B(x)) ⇔ (∃xA(x) ∨ ∃xB(x))

∃x(A(x) ∨ B) ⇔ (∃xA(x) ∨ B) x /∈ SP (B)

∃x(B ∨ A(x)) ⇔ (B ∨ ∃xA(x)) x /∈ SP (B)

(∀xA(x) ∨ ∀xB(x)) ⇒ ∀x(A(x) ∨ B(x))

∀x(A(x) ∨ B) ⇔ (∀xA(x) ∨ B) x /∈ SP (B)

∀x(B ∨ A(x)) ⇔ (B ∨ ∀xA(x)) x /∈ SP (B)

I ı

I ı(∃x(A(x) ∨ B(x))) = 0 I ı(∃xA(x) ∨ ∃xB(x)) = 0,

(1)

I ı

I ı(∃x(A(x) ∨ B)) = 0 I ı((∃xA(x)) ∨ B) = 0,

(2)

(2)

(2)

∨

∧

(4)

I ı(∀xA(x) ∨ ∀xB(x))=1 I ı(∀x(A(x) ∨ B(x)))=0 I ı

I ı

I ı(∀x(A(x) ∨ B)) = 1 I ı((∀xA(x)) ∨ B) = 1,

(5)


44/122

(5)

(5) ∨

(2)

(3)

(1)

(4) ∀ ∃

∨

∃

∨

∃x(A(x) ∨ B(x)) ∃xA(x) ∨ ∃xB(x) ∀x(A(x) ∨ B(x)) ∀xA(x) ∨ ∀xB(x) (∀x(A(x) ∨ B(x)))⇒(∀xA(x) ∨ ∀xB(x)) A(x)

B(x)

α(x)

β (x)

α

β

(∀x(α(x) ∨ β (x)))⇒(∀xα(x) ∨ ∀xβ (x)) (∃xα(x)∧∃xβ (x)) ⇒ (∃x(α(x)∧β (x))) I ı=((S, I

J ), ı)

S

N

αI β I

∀x(A(x)⇒B(x)) ⇒ (∀xA(x)⇒∀xB(x))

∀x(A(x)⇒B) ⇔ (∃xA(x)⇒B) x /∈ SP (B)

∀x(B⇒A(x)) ⇔ (B⇒∀xA(x)) x /∈ SP (B)

∃x(A(x)⇒B(x)) ⇒ (∀xA(x)⇒∃xB(x))

∃x(A(x)⇒B) ⇔ (∀xA(x)⇒B) x /∈ SP (B)

∃x(B⇒A(x)) ⇔ (B⇒∃xA(x)) x /∈ SP (B)

∀x(A(x)⇒B(x)) ⇒ (∃xA(x)⇒∃xB(x))

I ı(∀x(A(x) ⇒ B(x))) = 1

I ı(∀xA(x) ⇒ ∀xB(x)) = 0 I ı I ı = ((S, I

J ), ı)

I ı(∀x(A(x) ⇒ B(x))) = 1 I ı(∀xA(x) ⇒ ∀xB(x)) = 0

: V → S x ı I (A(x) ⇒ B(x)) = 1

I ı(∀xA(x)) = 1 I ı(∀xB(x)) = 0

: V → S

x ı I (A(x) ⇒ B(x))=1 I (A(x))=1

0 : V → S 0 x ı I 0(B(x))=0

0 : V → S 0 x ı I 0(A(x) ⇒ B(x))=1

I 0(A(x) ⇒ B(x)) = 0

∀x(A(x) ⇒ B) ∃xA(x) ⇒ B I ı=((S, I

J ), ı)

I ı(∀x(A(x) ⇒ B)) = 0

0 : V → S 0 x ı I 0(A(x) ⇒ B) = 0

0 : V → S 0 x ı I 0(A(x))=1 I 0(B) = 0


45/122

I ı(∃xA(x)) = 1 I ı(B) = 0

I ı(∃xA(x) ⇒ B) = 0.

(2)

I ı(∃x(A(x)⇒B(x)))=1 I ı(∀xA(x)⇒∃xB(x))=0

I ı=((S, I J ), ı)

I ı(∃x(A(x) ⇒ B (x))) I ı(∀xA(x) ⇒ ∃xB(x)) 1

I ı(∃x(A(x) ⇒ B(x))) = 1

0:V → S 0xı I 0(A(x) ⇒ B(x)) = 1

0: V → S 0xı max(1−I 0(A(x)), I 0(B(x)))=1

0: V → S 0xı

1 − I 0(A(x)) I 0(B(x)) 1

0: V → S 0xı

I 0(A(x)) 0 I 0(B(x)) 1

I 0(A(x)) = 0 I 0(B(x))=1 I ı(∀xA(x)⇒∃xB(x)) 1

I 0(A(x)) = 0 I ı(∀xA(x)) = 0

I ı(∀xA(x) ⇒ ∃xB(x)) = max(1 − 0, I ı(∃xA(x))) = 1

I 0(B(x)) = 1 I ı(∃xB(x)) = 1

I ı(∀xA(x) ⇒ ∃xB(x)) = max(1 − I ı(∀xA(x)), 1) = 1

(2)

(2)

(4)

F

∃zα(z) ∧ ∀x∃yβ (x, y)

α β

F

F

F F 1

∀x(∃zα(z) ∧ ∃yβ (x, y))


46/122

F 1 ⇔ F ∀x(B ∧ A(x)) ⇔ (B ∧ ∀xA(x))x /∈SP (B) ∃x(A(x) ∧ B) ⇔ (∃xA(x) ∧ B) x /∈SP (B)

F 1 ∃zα(z) ∧ ∃yβ (x, y)

∃z(α(z) ∧ ∃yβ (x, y))

F 1 F 2

∀x∃z(α(z) ∧ ∃yβ (x, y)).

∃x(B ∧A(x)) ⇔ (B ∧∃xA(x)) x /∈ SP (B)

α(z) ∧ ∃yβ (x, y) F 2 ∃y(α(z) ∧ β (x, y)) F 2

F

F 3

∀x∃z∃y(α(z) ∧ β (x, y)).

F 3 ∀x∃z∃yC C α(z) ∧ β (x, y) F 3

n > 0 Q1x1...QnxnA

Qi 1 ≤ i ≤ n ∀ ∃ A

F J F pf

|= F ⇔ F pf

F J

F {∧, ∨, ¬}

Æ

F

F

F

∧

∨

¬ F

n

mirjana borisavljević-uvod u logiku ii deo

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