What is Propositional Logic?
∴Why This Matters
Every
if statement you write, every contract clause you read, and every argument you evaluate uses propositional logic. It's the foundation of clear thinking.Propositional logic is the study of propositions — statements that are either true or false — and how to combine them using logical connectives like "and", "or", and "not".
Core Concepts
✓ Propositions
- • "The sky is blue"
- • "2 + 2 = 5" (false, but still a proposition)
- •
user.isAdmin == true
✗ Not Propositions
- • "What time is it?"
- • "Close the door!"
- • "Wow, amazing!"
?Quick Check
Is 'x > 5' a proposition?
Logical Connectives
Conjunction (AND) — ∧
P ∧ Q is true only when both P and Q are true.
Truth TableP ∧ Q
| P | Q | P ∧ Q |
|---|---|---|
| F | F | |
| F | T | |
| T | F | |
| T | T |
Select T or F for each row
Disjunction (OR) — ∨
P ∨ Q is true when at least one of P or Q is true.
!Common Mistake
In everyday English, "or" often means "one or the other, but not both" (exclusive or). In logic, ∨ is inclusive.
Truth TableP ∨ Q
| P | Q | P ∨ Q |
|---|---|---|
| F | F | |
| F | T | |
| T | F | |
| T | T |
Select T or F for each row
Negation (NOT) — ¬
¬P flips the truth value.
Truth Table¬P
| P | ¬P |
|---|---|
| F | T |
| T | F |
Implication (IF...THEN) — →
P → Q is only false when P is true but Q is false.
iNote
Think of a promise: "If it rains, I'll bring an umbrella." You only break it if it rains AND you don't bring one.
Truth TableP → Q
| P | Q | P → Q |
|---|---|---|
| F | F | |
| F | T | |
| T | F | |
| T | T |
Select T or F for each row
Biconditional (IFF) — ↔
P ↔ Q is true when P and Q have the same truth value.
Truth TableP ↔ Q
| P | Q | P ↔ Q |
|---|---|---|
| F | F | |
| F | T | |
| T | F | |
| T | T |
Select T or F for each row