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logic-proofs/02-propositions.md

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+# 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]]