The Laws of Propositional Logic

The Laws of Propositional Logic

Theorem 1 (The Laws of Propositional Logic): Let $P$, $Q$, and $R$ be statements. Let $t$ be a formula that is a tautology and let $f$ be a formula that is a contradiction. Then:
1. The Law of Double Negation: $\neg \neg P \Leftrightarrow P$.
2. The Associative Law for Conjunction: $P \vee (Q \vee R) \Leftrightarrow (P \vee Q) \vee R$.
3. The Associative Law for Disjunction: $P \wedge (Q \wedge R) \Leftrightarrow (P \wedge Q) \wedge R$.
4. The Commutative Law for Conjunction: $P \vee Q \Leftrightarrow Q \vee P$.
5. The Commutative Law for Disjunction: $P \wedge Q \Leftrightarrow Q \wedge P$.
6. The First Distributive Law: $P \vee (Q \wedge R) \Leftrightarrow (P \vee Q) \wedge (P \vee R)$.
7. The Second Distributive Law: $P \wedge (Q \vee R) \Leftrightarrow (P \wedge Q) \vee (P \wedge R)$.
8. The Idempotent Law for Conjunction: $P \vee P \Leftrightarrow P$.
9. The Idempotent Law for Disjunction: $P \wedge P \Leftrightarrow P$.
10. The Identity Law for Tautologies: $P \wedge t \Leftrightarrow P$.
11. The Identity Law for Contradictions: $P \vee f \Leftrightarrow P$.
12. The Inverse Law for Tautologies: $P \vee \neg P \Leftrightarrow t$.
13. The Inverse Law for Contradictions: $P \wedge \neg P \Leftrightarrow f$.
14. The Domination Law for Tautologies: $P \vee t \Leftrightarrow t$.
15. The Domination Law for Contradictions: $P \wedge f \Leftrightarrow f$.
16. De Morgan's Law 1: $\neg (P \vee Q) \Leftrightarrow \neg P \wedge \neg Q$.
17. De Morgan's Law 2: $\neg (P \wedge Q) \Leftrightarrow \neg P \vee \neg Q$.

All of the laws of propositional logic described above can be proven fairly easily by constructing truth tables for each formua and comparing their values based on the corresponding truth assignments.

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