Logical Connective
Definition
A logical connective is an operator that combines propositions to form new propositions.
Basic Connectives
| Connective | Symbol | Meaning |
|---|
| Negation | \(\neg P\) | NOT \(P\) |
| Conjunction | \(P \land Q\) | \(P\) AND \(Q\) |
| Disjunction | \(P \lor Q\) | \(P\) OR \(Q\) |
| Implication | \(P \implies Q\) | IF \(P\) THEN \(Q\) |
| Biconditional | \(P \iff Q\) | \(P\) IF AND ONLY IF \(Q\) |
Truth Tables
Negation
Conjunction
| \(P\) | \(Q\) | \(P \land Q\) |
|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction
| \(P\) | \(Q\) | \(P \lor Q\) |
|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Implication
| \(P\) | \(Q\) | \(P \implies Q\) |
|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note: \(P \implies Q\) is false only when \(P\) is true and \(Q\) is false.
Properties
De Morgan's Laws
\[\neg(P \land Q) \iff \neg P \lor \neg Q$$ $$\neg(P \lor Q) \iff \neg P \land \neg Q\]
Contrapositive
\[P \implies Q \iff \neg Q \implies \neg P\]