古典論理
classical logic、標準論理、standard logic
古典命題論理 (classical propositional logic)
公理、始式 :\frac{}{A\vdash A}(I)
Cut :\frac{\Gamma\vdash\Delta,D\quad D,\Pi\vdash\Lambda}{\Gamma,\Pi\vdash\Delta,\Lambda}(Cut)
構造規則
弱化 (增 W) (weakening) :
\frac{\Gamma\vdash\Delta}{D,\Gamma\vdash\Delta}(WL),
\frac{\Gamma\vdash\Delta}{\Gamma\vdash\Delta,D}(WR) 轉置 (換 P) (permutation) :
\frac{\Gamma,C,D,\Pi\vdash\Delta}{\Gamma,D,C,\Pi\vdash\Delta}(PL),
\frac{\Gamma\vdash\Delta,C,D,\Lambda}{\Gamma\vdash\Delta,D,C,\Lambda}(PR) 縮約 (減 C) (contraction) :
\frac{D,D,\Gamma\vdash\Delta}{D,\Gamma\vdash,\Delta}(CL),
\frac{\Gamma\vdash\Delta,D,D}{\Gamma\vdash\Delta,D}(CR) 論理規則
\neg:\frac{\Gamma\vdash\Delta,D}{\neg D,\Gamma\vdash\Delta}(\neg L),\frac{D,\Gamma\vdash\Delta}{\Gamma\vdash\Delta,\neg D}(\neg R)
\land:\frac{C,\Gamma\vdash\Delta}{C\land D,\Gamma\vdash\Delta}(\land L_1),\frac{D,\Gamma\vdash\Delta}{C\land D,\Gamma\vdash\Delta}(\land L_2),\frac{\Gamma\vdash\Delta,C\quad\Gamma\vdash\Delta,D}{\Gamma\vdash\Delta,C\land D}(\land R)
\lor:\frac{C,\Gamma\vdash\Delta\quad D,\Gamma\vdash\Delta}{C\lor D,\Gamma\vdash\Delta}(\lor L),\frac{\Gamma\vdash\Delta,C}{\Gamma\vdash\Delta,C\lor D}(\lor R_1),\frac{\Gamma\vdash\Delta,D}{\Gamma\vdash\Delta,C\lor D}(\lor R_2)
\to:\frac{\Gamma\vdash\Delta,C\quad D,\Pi\vdash\Lambda}{C\to D,\Gamma,\Pi\vdash\Delta,\Lambda}(\to L),\frac{C,\Gamma\vdash\Delta,D}{\Gamma\vdash\Delta,C\to D}(\to R)
眞理の木 (truth tree)
意味論
眞理値表
眞理値表| A | B | A∧B | A∨B | A→B | ¬A |
| T | T | T | T | T | F |
| T | F | F | T | F | |
| F | T | F | T | T | T |
| F | F | F | F | T | |
Karnaugh 圖 (Karnaugh map)
Veitch 圖
二分決定圖 (binary decision diagram; BDD)