# Propositions and Predicates

**Source:** MIT 6.1200J Lecture 01

## Proposition
**Definition:** A statement that is either True or False.

### Examples
- **True:** 2 + 3 = 5
- **False:** 2 + 3 = 6

### Non-examples (not propositions)
- "Hello" — not a statement
- "Who are you?" — a question, not a declarative statement

## Predicate
**Definition:** A proposition whose truth depends on variables.

### Examples
- P(n) = "n² + n + 41 is prime" where n ∈ ℕ
- Q(x, y) = "x + y = 5" where x, y ∈ ℝ

### Notation
- ∀ (for all): ∀n ∈ ℕ. P(n)
- ∃ (there exists): ∃n ∈ ℕ. P(n)

## Implication (A ⇒ B)
**Key point:** A ⇒ B is NOT about causation or time ordering.

Truth table:
| A | B | A ⇒ B |
|---|---|-------|
| T | T | T     |
| T | F | F     |
| F | T | **T** |
| F | F | T     |

**Note:** "If and only if" (A ⟺ B) means (A ⇒ B) AND (B ⇒ A)

## Related
- [[01-what-is-a-proof|What is a Proof?]]
- [[03-axioms-and-godel|Axioms & Gödel's Theorem]]