A logical statement is a declarative sentence that is either true or false, never both. Logical connectives—AND, OR, NOT, IF-THEN—combine simple statements into compound statements, allowing us to express complex logical relationships precisely. Mastering these connectives is the foundation for all formal reasoning.
Mathematics is built on precise claims, and the first tool of precision is identifying what can be true or false. A statement (also called a proposition) is a declarative sentence with a definite truth value: "17 is prime" is true; "the square root of 4 is 3" is false. By contrast, "is 17 prime?" (a question) and "let x be a number" (a command) are not statements — they have no truth value. The discipline of logic begins by demanding that every sentence we reason about be a statement in this strict sense.
Logical connectives combine simple statements into compound ones. The conjunction "P AND Q" (written P ∧ Q) is true only when both P and Q are true simultaneously. The disjunction "P OR Q" (written P ∨ Q) is true when at least one is true — mathematical "or" is inclusive, unlike the everyday "either/or" which excludes the both-true case. The negation "NOT P" (written ¬P) flips the truth value: if P is "x > 5," then ¬P is "x ≤ 5." Each connective has a precise definition that a truth table captures exhaustively — listing every combination of truth values for the component statements and the resulting truth value of the compound.
The most important connective for mathematical reasoning is implication: "IF P THEN Q" (written P → Q). It asserts that whenever P is true, Q must also be true. Crucially, P → Q is false only when P is true but Q is false — a true hypothesis leading to a false conclusion is the only way to violate a conditional claim. When P is false, P → Q is vacuously true regardless of Q. This surprises newcomers: "If the moon is made of cheese, then 2 + 2 = 5" is logically true, because the hypothesis is false. The implication made a promise only about what happens when P holds — and P never holds.
These four connectives (AND, OR, NOT, IF-THEN) are enough to express any logical relationship precisely. Every mathematical theorem is ultimately a statement — often an implication — built from simpler components. The rules for manipulating these connectives (like the equivalence P → Q ≡ ¬P ∨ Q, or De Morgan's laws) let you transform statements while preserving truth, which is exactly what a mathematical proof does. Mastering the meaning of each connective, especially the surprising cases of vacuous truth and inclusive or, prevents the logical errors that invalidate proofs before they even begin.
This is a foundational topic with no prerequisites.
No prerequisites — this is a starting point.