PTI 206 Logika
Semester I 2007/2008
Ratna
Deduksi
z
Definisi:
s :
≡
Socrates (filsuf Yunani kuno);
H
(
x
) :
≡
“
x
is human”;
M
(
x
) :
≡
“
x
mortal”
.
z
Premis:
H
(s)
Socrates manusia.
Deduksi
Kesimpulan valid yang dapat diambil:
z
H(s)
→
M(s)
[Instantiate universal.]
If Socrates is human then he is mortal.
z
¬
H(s)
∨
M(s)
Socrates is inhuman or mortal.
z
H(s)
∧
(
¬
H(s)
∨
M(s)) Socrates is human, and also either
inhuman or mortal.
z
(H(s)
∧ ¬
H(s))
∨
(H(s)
∧
M(s))
[Apply distributive law.]
z
F
∨
(H(s)
∧
M(s))
[Trivial contradiction.]
z
H(s)
∧
M(s)
[Use identity law.]
Contoh Lain
z
Definisi:
H
(
x
) :
≡
“
x
is human”;
M
(
x
) :
≡
“
x
is mortal”;
G
(
x
) :
≡
“
x
is a god”
z
Premis:
{
∀
x
(
H
(
x
)
→
M
(
x
)) (“Humans are mortal”) and
{
∀
x( G
(
x
)
→ ¬
M
(
x
)) (“Gods are immortal”).
Derivasi
z
∀
x
(
H
(
x
)
→
M
(
x
)) and (
∀
x G
(
x
)
→¬
M
(
x
).)
z
∀
x
(
¬
M
(
x
)
→¬
H
(
x
))
[Contrapositive.]
z
∀
x
([
G
(
x
)
→¬
M
(
x
)]
∧
[
¬
M
(
x
)
→¬
H
(
x
)])
z
∀
x
(
G
(
x
)
→¬
H
(
x
))
[Transitivity of
→
.]
z
∀
x
(
¬
G
(
x
)
∨ ¬
H
(
x
))
[Definition of
→
.]
z
∀
x
(
¬
(
G
(
x
)
∧
H
(
x
)))
[DeMorgan’s law.]
Derivasi
z
Universal Instantiation (UI)
{ Aturan bagaimana ∀ dieliminasi dg operasi Instansiasi
{ Ex. 1
{ Ex.2
∀x (cat(x) ⇒ hastail(x)) cat(Tom) ⇒ hastail(Tom)
( )
( )
A
S
A
x
x t
∀
( )
(
=
3−
2
)
⇒
4(
( )
4
=
4
3−
2
)
=
62
Derivasi
z
Derivasi dg Universal
Instantiation (UI)zEx.
H(x) :≡ “x is human”;
M(x) :≡ “x mortal”.
S :≡ Socrates (filsuf Yunani kuno); Prove : ∀x (H(x) ⇒ M(x)), H(S) ├ M(S)
zDerivation
1. ∀x (H(x) ⇒ M(x)) premise all humans are mortal 2. H(S) premise Socrates is human
3. (H(S) ⇒ M(S) Sx
Derivasi
z
Derivasi dg Universal
Instantiation (UI)zEx.
f(x,y) :≡ “x is the father of y”;
s(x,y) :≡ “x is the son of y”.
d(x,y) :≡ “x is the daughter of y”.
D :≡ Daug; P :≡ Paul
Prove : ∀x (f(D,x) ⇒ s(x,D) ∨ d(x,D)), f(D,P), ¬d(P,D) ├ s(P,D)
zDerivation
1. ∀x (f(D,x) ⇒ s(x,D) ∨ d(x,D)) premise 2. f(D,P) premise 3. ¬d(P,D) premise 4. f(D,P) ⇒ s(P,D) ∨ d(P,D) Sx
S
Derivasi
z
Universal Generalization (UG)
{ Aturan bagaimana ∀ digeneralisasi :Statement yg berlaku lokal menjadi statement yg berlaku global
{ Ex. 2
∀x (P(x))
∀x (P(x) ⇒ Q(Tom)
∀x (Q(x))
P(x) :≡ ‘x mhs TI’;
Q(x) :≡ ‘x menyukai programming’
( )
A
x
A
Derivasi
z
Derivasi dg Universal Generalization (UG)
zProve : ∀x P(x), ∀x (P(x) ⇒ Q(x)) ├ ∀x Q(x)
zDerivation
1. ∀x P(x) premise
2. ∀x (P(x) ⇒ Q(x)) premise 3. P(x) 1, Sx
x UI 4. P(x) ⇒ Q(x) 2, Sx
x UI 5. Q(x) 3,4 MP 6. ∀x Q(x) 5 UG
zProve : ∀x ∀yP(x,y) ├ ∀y ∀xP(x,y)
Derivasi
z
Existential Generalization (EG)
{ Aturan bagaimana ∃ digeneralisasi
{ Ex. 1
C :≡ ‘bibi Cordelia’;
P(x) :≡ ‘x berumur lebih dari 100 tahun’;
{ Ex.2
Setiap orang yang menang 1 milyar pasti kaya Mary menang 1 milyar
Ada orang yang kaya
( )
( )
A
x
A
S
tx∃
( )
( )
x
xP
C
P
Derivasi
z
Derivasi dg Existential Generalization (EG)
zEx.
W(x) :≡ “x memenangkan 1 milyar”;
R(x) :≡ “x orang yang kaya”.
M :≡ “Mary”;
Prove : ∀x (W(x) ⇒ R(x)), W(M) ├ ∃xR(x)
zDerivation
1. ∀x (W(x) ⇒ R(x)) premise
2. (W(M) ⇒ R(M) 1, Sx M
3. W(M) premise 4. R(M) 2,3 MP
Derivasi
z
Existential Instantiation (EI)
{ Aturan bagaimana ∃ dieliminasi
{ Ex. 1
P(x) :≡ ‘x does somersaults’;
∃xP(x) :≡ ‘somebody makes somersaults’;
{ Ex.2
Seseorang menang 1 milyar
Setiap orang yg memiliki 1 milyar pasti kaya Ada seseorang yang kaya
( )
( )
A
S
A
x
x t∃
( ) ( )
x
P
t
P
Derivasi
z
Derivasi dg Existential Instantiation (EI)
zEx.
W(x) :≡ “x memenangkan 1 milyar”;
R(x) :≡ “x orang yang kaya”. b :≡ “x”
Prove : ∀x (W(x) ⇒ R(x)), ∃x W(x) ├ ∃xR(x)
zDerivation
1. ∃x W(x) premise 2. W(b) 1, EI 3. ∀x (W(x) ⇒ R(x)) premise 4. W(b) ⇒ R(b) 3, Sx
