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Hanif Mandiratmaja

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Submitted By hanif12
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Logika Proposisi
Pertemuan 2
(Chapter 10 – Schaum, Set Theory)
(Chapter 3/4 – Schaum, Theory Logic)

Logika Proposisional
• Tujuan pembicaraan kali ini adalah untuk menampilkan suatu bahasa daripada kalimat abstrak yang disebut dengan logika proposisional • Dikenalkan teknik untuk menentukan apakah suatu kalimat abstrak yang di berikan itu valid atau contradictory dan apakah dua kalimat abstrak yg diberikan itu ekuivalen.

Logika Proposisional
• Menentukan kebenaran dan kesalahan daripada banyak kalimat konkrit hanya dengan melihat bentuk mereka.

Logika Proposisional
• Simbol-simbol dasar dan menunjukkan bagaimana mereka dikombinasikan untuk membentuk kalimat (abstrak) daripada logika proposisional. • Aturan sintaksis yang menjelaskan kombinasi apa daripada simbol-simbol yang diambil menjadi kalimat dalam bahasa tersebut.

Proposisi




Proposisi : suatu pernyataan yang mempunyai satu nilai benar atau salah (well-form format).
Simbol-simbol dibawah ini, yang disebut proposisi digunakan utk membangun suatu kalimat. Mereka adalah:
Simbol kebenaran : true dan false ( benar dan salah). Untuk menyingkat digunakan T = true atau B = benar , F = false atau S = salah.

Proposisi
• Simbol proposisional : p, q, r, p1, p2, ……… ( huruf kecil p, q, r, dan dari mereka dengan diberi indeks/ditambah dengan angka bilangan alam) Catatan : Ada beberapa buku yang menggunakan huruf besar P, Q, R, dan mereka diberi dengan indeks. Kalimat
• Kalimat dalam logika proposisional dibangun dari proposisi-proposisi dengan mengaplikasikan penghubung proposisional seperti : not, and, or, if-then, if-and-only-if
Kalimat dibentuk menurut aturan sbb :
• Setiap proposisi, yaitu suatu simbol kebenaran atau simbol proposisional, adalah suatu kalimat.
• Jika P suatu kalimat, maka begitu juga negasinya, yaitu (not P)
• Jika P dan Q kalimat, maka begitu juga konjungsinya, yaitu (P and Q)

Kalimat
• Jika P dan Q kalimat, maka begitu juga disjungsinya yaitu (P or Q)
• Jika P dan Q kalimat, maka begitu juga implikasinya yaitu (if P then Q) dimana P disebut anteseden dan Q disebut konsekuen
• Jika P dan Q kalimat, maka begitu juga bi-implikasi nya/ ekuivalensinya, yaitu : (P if and only if Q)
• Kalimat majemuk adalah kalimat yang dibentuk dari satu atau lebih dari satu kalimat tunggal.

Notasi
Bhs. Inggris

Bhs. Indonesia

Konvensional

And

Dan

∧ atau &

Or

Atau



Not

Tidak

∼ Atau ¬

If-then

Jika-maka

⊃ atau →

If-and only-if

Jika dan hanya jika

≡ atau ↔

Notasi
Bhs. Inggris

Bhs. Indonesia

Konvensional

NAnd

Tidak Dan

atau ↑

NOr

TIdak Atau



XOr

Exclusive Or

⊕ Atau ≠

Contoh Kalimat Proposisi
• Premis : Anda belajar rajin.
• Premis : Anda lulus ujian.
• Premis : Anda senang.
Bentuk proposisi
• P : Anda belajar rajin.
• Q : Anda lulus ujian.
• R : Anda senang.

Contoh Kalimat Proposisi
Bentuk argumen
• Jika P, maka Q
• Jika Q, maka R
• Jika P, maka R

Interpretasi
• Merupakan langkah penentuan benar atau salah untuk sekumpulan simbol-simbol proposisional.
• Misal suatu simbol P dapat diintrerpretasikan benar atau salah (tidak bisa dua-duanya).

Aturan Semantik
• Aturan yang mengatur penentuan interpretasi pada simbol-simbol proposisi.
• Interpretasi dari operator proposisi, dirangkum pada tabel kebenaran.
• Ekivalen
– Jika mempunyai tabel kebenaran yang sama.

Tabel Kebenaran - Not
P

~P

True

False

False

True

Tabel Kebenaran - And
P

Q

P∧Q

True

True

True

True

False

False

False

True

False

False

False

False

Tabel Kebenaran - Or
P

Q

P∨Q

True

True

True

True

False

True

False

True

True

False

False

False

Tabel Kebenaran – If-then
P

Q

P⇒Q

True

True

True

True

False

False

False

True

True

False

False

True

Tabel Kebenaran – If and only if
P

Q

P⇔Q

True

True

True

True

False

False

False

True

False

False

False

True

Hukum-Hukum Logika
• Silogisme
– Hipotesis (bentuk jika-maka, jika-maka)
– Disjungtif (bentuk atau dan tidak)

• Modus
– Ponens (bentuk jika-maka)
– Tollens (bentuk jika-maka, tidak)

Latihan

• Nyatakan argumen berikut dalam bentuk simbol proposisi dan bagaimana jika salah satu premisnya dinegasikan!
– Harga naik atau permintaan turun.
– Hari ini hujan dan sungai banjir.
– Jika Saya rajin belajar maka saya lulus ujian.
– Jika Saya puasa maka saya lapar.

A=Anda sakit flu.
B=Anda ujian.
C=Anda lulus.
Ubahlah ekspresi pernyataan dalam bahasa indonesia. (1). A → ¬B
(2). B → ¬C
(3). ¬B → C
(4). (A ∧ B) → C
(5). (A →¬C) ∨ (B →¬C)
(6). (A ∧ B) ∨ (¬A ∧ B)

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