Logical Connectives: AND, OR, NOT, Implies and Iff

A proposition is a statement that is either true or false, but never both. "2 + 3 = 5" is a true proposition, "Delhi is the capital of France" is a false proposition, but "What time is it?" is not a proposition at all, because a question has no truth value.

Logical connectives combine simple propositions into larger ones, like plus and minus combine numbers. The five basic connectives are NOT, AND, OR, IMPLIES and IFF. Understanding exactly what each one means is the foundation of every proof you will ever read or write.

Key Concepts

• Proposition: a declarative sentence that is true or false.
• Negation (¬p, "not p"): flips the truth value of p.
• Conjunction (p ∧ q, "p and q"): true only when both p and q are true.
• Disjunction (p ∨ q, "p or q"): true when at least one of p, q is true. This is the inclusive or.
• Implication (p → q, "if p then q"): p is the hypothesis, q is the conclusion.
• Biconditional (p ↔ q, "p if and only if q"): true when p and q match.

How It Works

1. ¬p is true exactly when p is false, and false exactly when p is true.
2. p ∧ q is true only when p is true and q is true. In the other three cases it is false.
3. p ∨ q is false only when both p and q are false. In the other three cases it is true, including when both are true.
4. p → q is false only when p is true and q is false. When p is false it is vacuously true.
5. p ↔ q is true when p and q match (both true or both false) and false when they differ.

Worked Example

Let p be "It is raining" (true) and q be "I carry an umbrella" (false). Evaluate each compound statement:

• ¬p is false (p is true).
• p ∧ q is false (needs both true; q is false).
• p ∨ q is true (p is true).
• p → q: true hypothesis with false conclusion, so false.
• p ↔ q is false (values differ).

Common Misconceptions

• "OR" means "one or the other, but not both." In everyday speech, "tea or coffee" usually means pick one. In logic, OR is inclusive: p ∨ q is true when both are true.
• "If p then q" claims p causes q. It does not. It only rules out "p true, q false", so "if it rains, the ground is wet" stays true on a dry day.
• p → q is the same as its converse q → p. It is not. "If it rains, the ground is wet" is true, but "if the ground is wet, it rains" is false (the sprinkler may be on).

Real-World Applications

• Digital circuits: AND, OR and NOT gates follow exactly these truth rules; every processor is built from them.
• Programming: if conditions with &&, || and ! implement ∧, ∨ and ¬ directly.
• Databases: SQL WHERE clauses filter rows using AND, OR and NOT.

Logical connectives turn plain statements into precise compound claims: NOT flips a truth value, AND demands both parts, inclusive OR needs at least one, implication fails only on a true hypothesis with a false conclusion, and IFF needs both sides to match. Master these five rules and you can read any proof with confidence.