Laws of Logic

Dec 15, 2023 permanent

The Laws of Logic are a set of fundamental principles which are the foundation of logical reasoning.

Let pp, qq, and rr be any three propositions.

Law Formula
Idempotent laws p∧p≡pp \land p ≡ p
p∨p≡pp \lor p ≡ p
Identity laws p∧true≡pp \land \text{true} ≡ p
p∨false≡pp \lor \text{false} ≡ p
Inverse laws p∧(¬p)≡falsep \land (\neg p) ≡ \text{false}
p∨(¬p)≡truep \lor (\neg p) ≡ \text{true}
Domination laws p∨true≡truep \lor \text{true} ≡ \text{true}
p∧false≡falsep \land \text{false} ≡ \text{false}
Commutative laws p∧q≡q∧pp \land q ≡ q \land p
p∨q≡q∨pp \lor q ≡ q \lor p
Double negation ¬(¬p)≡p\neg (\neg p) \equiv p
Associative laws p∧(q∧r)≡(p∧q)∧rp \land (q \land r) ≡ (p \land q) \land r
p∨(q∨r)≡(p∨q)∨rp \lor (q \lor r) ≡ (p \lor q) \lor r
Distributive laws p∧(q∨r)≡(p∧q)∨(p∧r)p \land (q \lor r) ≡ (p \land q) \lor (p \land r)
p∨(q∧r)≡(p∨q)∧(p∨r)p \lor (q \land r) ≡ (p \lor q) \land (p \lor r)
De Morgan's laws ¬(p∧q)≡¬p∨¬q\neg (p \land q) ≡ \neg p \lor \neg q
¬(p∨q)≡¬p∧¬q\neg (p \lor q) ≡ \neg p \land \neg q
Implication conversion law p→q≡¬p∨qp \rightarrow q ≡ \neg p \lor q
Contrapositive law p→q≡¬q→¬pp \rightarrow q ≡ \neg q \rightarrow \neg p
Reductio ad absurdum law p→q≡(p∧¬q)→falsep \rightarrow q ≡ (p \land \neg q) \rightarrow \text{false}