This table presents various logical laws or properties along with their formulas. These laws are fundamental principles in propositional logic that govern how propositions interact with each other under different logical operations. Here’s a rewritten version of the table along with explanations and examples:
| No. | Logical Law | Formula | Explanation | Example |
|---|---|---|---|---|
| 1 | Law of Identity | p ∨ F ⇔ p | A proposition OR false is logically equivalent to the proposition itself. | If it’s raining OR false, it’s still raining. |
| p ∧ T ⇔ p | A proposition AND true is logically equivalent to the proposition itself. | If it’s raining AND true, it’s still raining. | ||
| 2 | Law of Null / Domination | p ∧ F ⇔ F | A proposition AND false is always false. | If it’s raining AND false, it’s always false. |
| p ∨ T ⇔ T | A proposition OR true is always true. | If it’s raining OR true, it’s always true. | ||
| 3 | Law of Negation | p ∨ ∼p ⇔ T | A proposition OR its negation is always true. | If it’s raining OR it’s not raining, it’s always true. |
| p ∧ ∼p ⇔ F | A proposition AND its negation is always false. | If it’s raining AND it’s not raining, it’s always false. | ||
| 4 | Law of Idempotent | p ∨ p ⇔ p | A proposition OR itself is logically equivalent to the proposition itself. | If it’s raining OR it’s raining, it’s still raining. |
| p ∧ p ⇔ p | A proposition AND itself is logically equivalent to the proposition itself. | If it’s raining AND it’s raining, it’s still raining. | ||
| 5 | Law of Involution | ∼(∼p) ⇔ p | Double negation of a proposition results in the original proposition. | It’s not the case that it’s not raining; it’s raining. |
| 6 | Law of Absorption | p ∨ (p ∧ q) ⇔ p | A proposition OR a conjunction involving itself simplifies to the proposition itself. | If it’s raining OR (it’s raining AND it’s cold), it’s still raining. |
| p ∧ (p ∨ q) ⇔ p | A proposition AND a disjunction involving itself simplifies to the proposition itself. | If it’s raining AND (it’s raining OR it’s cold), it’s still raining. | ||
| 7 | Law of Commutative | p ∨ q ⇔ q ∨ p | The order of propositions in a disjunction does not affect the result. | It’s raining OR it’s sunny is equivalent to it’s sunny OR it’s raining. |
| p ∧ q ⇔ q ∧ p | The order of propositions in a conjunction does not affect the result. | It’s raining AND it’s sunny is equivalent to it’s sunny AND it’s raining. | ||
| 8 | Law of Associative | p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r | The grouping of propositions in a disjunction does not affect the result. | (It’s raining OR (it’s sunny OR it’s windy)) is equivalent to ((it’s raining OR it’s sunny) OR it’s windy). |
| p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r | The grouping of propositions in a conjunction does not affect the result. | (It’s raining AND (it’s sunny AND it’s windy)) is equivalent to ((it’s raining AND it’s sunny) AND it’s windy). | ||
| 9 | Law of Distributive | p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) | Distribution of a disjunction over a conjunction. | It’s raining OR (it’s sunny AND it’s windy) is equivalent to (it’s raining OR it’s sunny) AND (it’s raining OR it’s windy). |
| p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r) | Distribution of a conjunction over a disjunction. | It’s raining AND (it’s sunny OR it’s windy) is equivalent to (it’s raining AND it’s sunny) OR ( |