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Materi Logika Matematika



Logika Matematis adalah ilmu yang mempelajari secara sistematis
kaidah-kaidah penalaran yang absah (valid). Logika Matematis dikenal
juga sebagai Logika Simbolik yang memanfaatkan simbol-simbol matematis
untuk mempresentasikan bahasa alamiah manusia. Dalam dunia ilmu
dikenal dua penalaran yaitu Penalaran Deduktif dan Penalaran
Induktif.

¤Logika Simbolik terdiri dari dua cabang utama, yaitu :
A. Logika Proposisi; dan
B. Logika Predikat.

A. Logika Proposisi terdiri dari :
1. Proposisi Atomik yaitu proposisi yang tidak memuat proposisi lain
sebagai komponennya.
Contoh:
- Ir. Soekarno adalah Presiden RI yang pertama.
- 2+3=5
2. Proposisi Majemuk adalah proposisi yang dibentuk dari
proposisi-proposisi atomik yang dirangkaikan dengan menggunakan
Operasi Logis.
Contoh:
Saya menyukai matematika atau bahasa inggris. (Menggunakan operasi
logis disjungsi)

¤Operasi Logis dalam Logika Matematika terdiri dari:
1. Operasi Uner, yaitu operasi logis yang hanya melibatkan satu
proposisi atomik. Negasi dari suatu proposisi merupakan operasi uner
dalam logika matematika.
2. Operasi Biner, yaitu operasi yang melibatkan dua proposisi atomik.
Konjungsi, Disjungsi, Implikasi, dan Ekivalensi merupakan operasi
biner dalam logika matematika.

¤Kelima operasi logis di atas akan dibahas secara Sintatik dan
Semantik. Pembahasan secara Sintatik akan menjelaskan bagaimana aturan
untuk membentuk proposisi yang menggunakan operasi logis sedangkan
pembahasan secara semantik akan menguraikan bagaimana nilai kebenaran
dari proposisi yang telah dibentuk.

Seperti yang telah dijelaskan, logika matematis memanfaatkan
simbol-simbol sehingga disebut logika simbolik. Suatu pernyataan yang
disimbolkan atau dilambangkan dengan variabel-variabel dinamakan
sebagai Bentuk Proposisi. Variabel proposisi itu ialah huruf yang
melambangkan suatu pernyataan, biasanya digunakan huruf, p, q, dan r.
Jika semua variabel proposisi dalam bentuk proposisi disubsitusikan
dengan suatu proposisi tertentu (yang mempunyai nilai kebenaran
tertentu), maka akan dihasilkan suatu nilai kebenaran dan dapat
disusun dalam suatu tabel kebenaran. Ada tiga pembahasan bentuk
proposisi mengenai Nilai Kebenaran jika kita melakukan subsitusi oleh
suatu proposisi tertentu.

1. Tautologi, yaitu suatu bentuk proposisi yang selalu menghasilkan
nilai benar untuk setiap subsitusi yang mungkin ke dalam
variabel-variabelnya.
2. Kontradiksi, yaitu kebalikan dari tautologi karena untuk setiap
subsitusi yang mungkin nilai kebenaranya selalu salah.
3. Kontigensi yaitu untuk setiap subsitusi variabel-valiabel yang
mungkin maka bentuk proposisi yang dihasilkan adalah bernilai benar
atau salah.

B. Logika Predikat
Tidak semua penarikan kesimpulan yang sah dapat dilakukan dengan
Logika Proposisi karena logika proposisi hanya memandang proposisi
sebagai suatu unit tanpa memperhatikan susunan internalnya.
Contoh:
Premis 1: Semua Mahasiswa mempunyai telepon genggam.
Premis 2: Fredi adalah mahasiswa.
Kesimpulan: Fredi mempunyai telepon genggam.

Akal sehat kita mengatakan bahwa penalaran tersebut absah.
Perhatikan di bawah ini, jika proposisi/pernyataannya dipandang
sebagai suatu unit dan misalkan disimbolkan dengan p, q, dan r maka :

Premis1 : P
Premis 2: q
Kesimpulan: r

Dalam tabel kebenaran, mudah diperlihatkan bahwa "Jika p dan q maka r"
bukanlah suatu Tautologi.

Logika predikat ini digunakan ketika proposisi atomik tidak dipandang
sebagai unit tetapi memperhatikan susunan internalnya ( subjek dan
predikat) yang bersifat umum atau khusus dalam melakukan penyimpulan.
Dalam contoh di atas, premis 1 bersifat umum karena menggunakan kata
semua mahasiswa dan premis 2 bersifat khusus karena menyatakan Fredi
sebagai salah satu mahasiswa. Selanjutnya, dari segi stuktur internal
premis 1 terdiri dari dua bagian, yaitu subjek (mahasiswa) dan bagian
predikat (mempunyai telepon genggam). Logika inilah yang
mengakomodasi struktur internal dari proposisi-proposisinya.

Logika Predikat terdiri dari dua pembahasan yaitu Proposisi Umum
(Universal) dan Proposisi Khusu (Eksistensial).

Selain dari pada itu, dalam logika matematika kita dituntut untuk
dapat membuktikan suatu Tautologi dengan menggunakan kaidah inferensi
atau bentuk yang telah dibuktikan kebenarannya dan kita harus juga
dapat melakukan suatu pembuktian yang dinamakan dengan Metode
Pembuktian dalam Matematis.

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