命題邏輯 Logical Equivalences 邏輯等價, Rules of Inference 推理規則

2021-11-26 约 673 字预计阅读 2 分钟

Logical Equivalences

\begin{matrix} \begin{array}{ccc} Equivalence & Name \\ \begin{matrix} p \land ⊤ \equiv p \\ p \lor ⊥ \equiv p \end{matrix} & Identity laws \\ \begin{matrix} p \lor ⊤ \equiv ⊤ \\ p \land ⊥ \equiv ⊥ \end{matrix} & Domination laws \\ \begin{matrix} p \lor p \equiv p \\ p \land p \equiv p \end{matrix} & Idempotent or tautology laws \\ \begin{matrix} \neg (\neg p) \equiv p \end{matrix} & Double negation law \\ \begin{matrix} p \lor q \equiv q \lor p \\ p \land q \equiv q \land p \end{matrix} & Commutative laws \\ \begin{matrix} (p \lor q) \lor r \equiv p \lor (q \lor r) \\ (p \land q) \land r \equiv p \land (q \land r) \end{matrix} & Associative laws \\ \begin{matrix} p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \\ p \land (q \lor r) \equiv (p \land q) \lor (p \land r) \end{matrix} & Distributive laws \\ \begin{matrix} \neg (p \land q) \equiv \neg p \lor \neg q \\ \neg (p \lor q) \equiv \neg p \land \neg q \end{matrix} & De Morgan's laws \\ \begin{matrix} p \lor (p \land q) \equiv p \\ p \land (p \lor q) \equiv p \end{matrix} & Absorption laws \\ \begin{matrix} p \lor \neg p \equiv ⊤ \\ p \land \neg p \equiv ⊥ \end{matrix} & Negation laws \end{array} \end{matrix}

\begin{matrix} \begin{aligned} p \to q & \equiv \neg p \lor q \\ p \to q & \equiv \neg q \to \neg p \\ p \lor q & \equiv \neg p \to q \\ p \land q & \equiv \neg (p \to \neg q) \\ \neg (p \to q) & \equiv p \land \neg q \\ (p \to q) \land (p \to r) & \equiv p \to (q \land r) \\ (p \to q) \lor (p \to r) & \equiv p \to (q \lor r) \\ (p \to r) \land (q \to r) & \equiv (p \lor q) \to r \\ (p \to r) \lor (q \to r) & \equiv (p \land q) \to r \end{aligned} \end{matrix}

\begin{matrix} \begin{aligned} p \leftrightarrow q & \equiv (p \to q) \land (q \to p) \\ p \leftrightarrow q & \equiv \neg p \leftrightarrow \neg q \\ p \leftrightarrow q & \equiv (p \land q) \lor (\neg p \land \neg q) \\ \neg (p \leftrightarrow q) & \equiv p \leftrightarrow \neg q \end{aligned} \end{matrix}

\begin{matrix} \begin{array}{cc} R u l e o f I n f e r e n c e & N a m e \\ \begin{matrix} p \\ \underset{―}{p \to q} \\ ∴ q \end{matrix} & Modus ponens \\ \begin{matrix} \neg q \\ \underset{―}{p \to q} \\ ∴ \neg p \end{matrix} & Modus tollens \\ \begin{matrix} p \to q \\ \underset{―}{q \to r} \\ ∴ p \to r \end{matrix} & Hypothetical syllogism \\ \begin{matrix} p \lor q \\ \neg p \\ \overset{―}{∴ q} \end{matrix} & Disjunctive syllogism \\ \begin{matrix} p \\ \overset{―}{∴ p \lor q} \end{matrix} & Addition \\ \begin{matrix} \underset{―}{p \land q} \\ ∴ p \end{matrix} & Simplification \\ \begin{matrix} p \\ q \\ \overset{―}{∴ p \land q} \end{matrix} & Conjunction \\ \begin{matrix} p \lor q \\ \neg p \lor r \\ \overset{―}{∴ q \lor r} \end{matrix} & Resolution \end{array} \end{matrix}

\begin{matrix} \begin{array}{cc} R u l e o f I n f e r e n c e & N a m e \\ \begin{matrix} \forall x P (x) \\ ∴ P (c) \end{matrix} & Universal instantiation \\ \begin{matrix} P (c) f o r a n a r b i t r a r y c \\ ∴ \forall x P (x) \end{matrix} & Universal generalization \\ \begin{matrix} \exists x P (x) \\ ∴ P (x) f o r s o m e e l e m e n t c \end{matrix} & Existential instantiation \\ \begin{matrix} P (x) f o r s o m e e l e m e n t c \\ ∴ \exists x P (x) \end{matrix} & Existential generalization \end{array} \end{matrix}

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