Kalkulus proposisional

Kalkulus proposisional adalah sistem formal untuk menyatakan rumus proposisi dan membuktikannya dengan cara menggabungkan rumus atomik dan operator logika.

Beberapa contoh operator logika adalah:

• $\mathrm{\neg }$ (negasi)
• $\wedge$ (konjungsi)
• $\vee$ (disjungsi)
• $\to$ (implikasi)
• $\leftrightarrow$ (ekuivalensi)

Bentuk-bentuk argumen
Nama	Sequent
Modus Ponens	$((p\to q)\wedge p)⊢q$
Modus Tollens	$((p\to q)\wedge \mathrm{\neg }q)⊢\mathrm{\neg }p$
Silogisme Hipotesis	$((p\to q)\wedge (q\to r))⊢(p\to r)$
Silogisme Disjungtif	$((p\vee q)\wedge \mathrm{\neg }p)⊢q$
Dilema Konstruktif	$((p\to q)\wedge (r\to s)\wedge (p\vee r))⊢(q\vee s)$
Dilema Destruktif	$((p\to q)\wedge (r\to s)\wedge (\mathrm{\neg }q\vee \mathrm{\neg }s))⊢(\mathrm{\neg }p\vee \mathrm{\neg }r)$
Dilema Bidireksi	$((p\to q)\wedge (r\to s)\wedge (p\vee \mathrm{\neg }s))⊢(q\vee \mathrm{\neg }r)$
Simplifikasi	$(p\wedge q)⊢p$
Konjungsi	$p,q⊢(p\wedge q)$
Penambahan	$p⊢(p\vee q)$
Komposisi	$((p\to q)\wedge (p\to r))⊢(p\to (q\wedge r))$
Teorema De Morgan	$\mathrm{\neg }(p\wedge q)⊢(\mathrm{\neg }p\vee \mathrm{\neg }q)$
Komutasi	$(p\vee q)⊢(q\vee p)$
Asosiasi	$(p\vee (q\vee r))⊢((p\vee q)\vee r)$
Distribusi	$(p\wedge (q\vee r))⊢((p\wedge q)\vee (p\wedge r))$
Dobel Negasi	$p⊢\mathrm{\neg }\mathrm{\neg }p$
Transposisi	$(p\to q)⊢(\mathrm{\neg }q\to \mathrm{\neg }p)$
Implikasi	$(p\to q)⊢(\mathrm{\neg }p\vee q)$
Ekuivalensi	$(p\leftrightarrow q)⊢((p\to q)\wedge (q\to p))$
Tautologi	$p⊢(p\vee p)$
Tertium non datur	$⊢(p\vee \mathrm{\neg }p)$
Non-Kontradiksi	$⊢\mathrm{\neg }(p\wedge \mathrm{\neg }p)$

Pustaka

• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.
• Chang, C.C., dan Keisler, H.J. (1973), Model Theory, North-Holland, Amsterdam, Netherlands.
• Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
• Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.
• Lambek, J. dan Scott, P.J. (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press, Cambridge, UK.
• Mendelson, Elliot (1964), Introduction to Mathematical Logic, D. Van Nostrand Company.

Pranala luar

• www.ltn.lv/~podnieks/mlog/ml2.htm
• www.fecundity.com/logic/