Гладкий А.В. Введение в современную логику
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x
x X
X ∨¬X
X
X ∨¬X
¬(X & ¬X)
R
F
X
1
X ∨R ≡ R
2
X ∨F ≡ X
3
X & R ≡ X
4
X & F ≡ F
5
X → R ≡ R
6
F → X ≡ R
X
R X & R
X
X & R X
F → X
6
2,3
1,4,5,6
7,8
(¬(X & ¬X → Y ) ∨ X)&(Y ∨ Z ∨¬Z)
X X & ¬X → Y
6
X & ¬X
¬(X & ¬X → Y ) ∨X X
2
Y ∨Z ∨¬Z
1
X &(Y ∨Z ∨¬Z) X
3
(X ∨ Y ∨ X)&(X ∨ Y )
X ∨Y
7 8
(¬X ∨¬Z ∨X ∨Y )&(X ∨¬Z ∨X ∨Y )&(X ∨¬Z ∨Z ∨Y )
X → (Y & ¬Y → Z)
(X
1
→ (X
2
→ ...(X
n
→ Y ∨¬Y ) ...)) & Z
5 6
R → X X → F
∀x∀y∀z(A(x, y)&A(y, z) → A(x, z)) A(x, y)
x<y
∀x∃yF(x, y)
F (x, y)
x<y
x>y
y
x
y x
¬∃xA(x, y) →¬B(y)
A(x, y)
x y
B(x)
x
y
y
y
A(x, y)
x y
y x
B(x)
x
y
∀x∀y∀z(A(x, y)&A(y,z) → A(x, z))
A(x, y)
x y
x y z x y y z
x z x
y x y z
x y y z x z
∀x∀y∀z∀t(S(x, y, z)&S(y, x,t) → E(z, t)
S(x, y, z) E(x, y)
x + y = z x = y
x + y = z x −y = z
≡
¬(F (x)&G(x)) ≡¬F (x) ∨¬G(x)
F G
F
0
G
0
x
x
0
¬(F (x)&G(x))
¬(F
0
(x
0
)&G
0
(x
0
)) ¬F (x) ∨¬G(x)
¬F
0
(x
0
) ∨¬G
0
(x
0
)
¬(F
0
(x)&G
0
(x))
¬F
0
(x) ∨¬G
0
(x) x
¬(F (x)&G(x)) ¬F (x) ∨¬G(x)
¬(X & Y ) ¬X ∨¬Y
X Y F (x) G(x)
∀x(F (x)&G(x)) ≡∀xF (x)&∀xG(x)
F (x) G(x) F
0
(x) G
0
(x)
F
0
(x)&G
0
(x)
F
0
(x) G
0
(x) ∀xF
0
(x)
∀xG
0
(x)
∃x(F (x)&G(x)) ∃xF (x)&∃xG(x)
F (x) G(x)
F G
x x
∃x(F (x) ∨G(x)) ≡∃xF (x) ∨∃xG(x)
F (x) G(x)
F
0
(x) G
0
(x)
F
0
(x) ∨ G
0
(x)
F
0
(x) G
0
(x)
∃xF
0
(x) ∃xG
0
(x)
∀x(F (x) ∨G(x)) ∀xF (x) ∨∀xG(x)
∀x∃yF(x, y) ∃y∀xF(x, y)
F (x, y)
x<y
∀x∃y(F (x)&G(y)) ≡∃y∀x(F (x)&G(y))
& ∨
∀x(F (x) → G(x)) ∀xF (x) →∀xG(x)?
∀ ∃
1
∀x∀yF(x, y) ≡∀y∀xF (x, y)
2
∃x∃yF(x, y) ≡∃y∃xF (x, y)
1
F
2
1
¬∀xF (x) ≡∃x¬F (x)
2
¬∃xF (x) ≡∀x¬F (x)
1
F
F
0
¬∀xF
0
(x)
∀xF
0
(x) F
0
¬F
0
x ∃x¬F (x)
¬∀xF
0
(x) ∀xF
0
(x)
F
0
(x) ¬F
0
(x)
∃x¬F
0
(x)
2
P (x)
{a
1
,...a
n
} ∀xP(x)
P (a
1
)&...& P (a
n
) ∃xP (x) P (a
1
) ∨...∨P (a
n
)
x
1
& ∀xF (x) ≡∀x( & F (x))
2
∨∀xF (x) ≡∀x( ∨F (x))
3
& ∃xF (x) ≡∃x( & F (x))
4
∨∃xF (x) ≡∃x( ∨F (x))
F
F
0
0
0
& ∀xF
0
(x)
0
∀xF
0
(x) ∀xF
0
(x) F
0
(x)
0
& F
0
(x)
∀x(
0
& F
0
(x))
0
& ∀xF
0
(x)
0
∀xF
0
(x)
0
& F
0
(x)
F
0
(x)
0
& F
0
(x)
∀x(
0
& F
0
(x))
1
∀xF (x) ≡∀yF (y)
2
∃xF (x) ≡∃yF (y)
∀xF
0
(x) ∀yF
0
(y)
F
0
∃xF
0
(x) ∃yF
0
(y)
x y
∀x(F (x) →∃yG(x, y)) ≡∀x(¬F (x) ∨∃yG(x, y)) ≡
≡∀x∃y(¬F (x) ∨G(x, y)) ≡∀x∃y(F (x) → G(x, y))
y ¬F (x)
y
¬∀x¬F (x)&¬∀xG(x) ≡∃x¬¬F (x)&∃x¬G(x) ≡
≡∃xF (x)&∃y¬G(y) ≡∃x(F (x)&∃y¬G(y)) ≡
≡∃x∃y(F (x)&¬G(y))
1
x ∃x¬G(x) y
x y
¬∃x∀yF(x, y) ∨∀xG(x) ≡
≡∀x¬∀yF (x, y) ∨∀zG(z) ≡∀x∃y¬F(x, y) ∨∀zG(z) ≡
≡∀x∃y(¬F (x, y) ∨∀zG(z)) ≡
≡∀x∃y∀z(¬F (x, y) ∨G(z))
2
x ∀xG(x) z
1
x y z
¬∀x∃y∀z(L(x, z) → C(y, z)) ≡
≡∃x¬∃y∀z(L(x, z) → C(y, z)) ≡
≡∃x∀y¬∀z(L(x, z) → C(y, z)) ≡
≡∃x∀y∃z¬(L(x, z) → C(y, z))
1 2 1
∃x∀y∃z(L(x, z)&¬C(y, z))
x
y
z
L(x, z) C(y, z)
z x z
y
∀xF (x) →∃yG(x, y)
(¬∀xF (x, y ) →∃yG(y,z)) & ¬∃tH(t)
¬∃x(¬∀y∃zF(x, y, z) ∨¬∃xG(x, y, z))
x
∀x
∃x