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:

• ${\displaystyle \lnot }$ (negasi)
• ${\displaystyle \land }$ (konjungsi)
• ${\displaystyle \lor }$ (disjungsi)
• ${\displaystyle \rightarrow }$ (implikasi)
• ${\displaystyle \leftrightarrow }$ (ekuivalensi)

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

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/