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GCLC
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N K
K N K K
Æ
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J l
V x,y,z,...,x1, y1, z1,...,xn, yn, zn,...
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{⊥, ∧, ∨, ⇒}
⊥
∧, ∨ ⇒
{∀, ∃} ∀ ∃
J
P
O
C
P
O
C
J
n
n ≥ 1
α,β,ρ,α1, β 1, ρ1
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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
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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 )
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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
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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
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J = {
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(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
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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
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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
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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
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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
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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,
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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
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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)
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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
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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,
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I ı(∀xα(c, x)) = 0 ∀xα(c, x)
I ı=((N,
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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,
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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)
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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
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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,
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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,
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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
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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)
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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
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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
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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)
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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
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|= 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
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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
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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
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¬∀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
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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,
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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.
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∃
∧
∃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)
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(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
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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))
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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