The Language of Mathematics

Intro to Proofs · Axiom Academy

In ordinary speech a sentence can be vague. In mathematics, every statement is forced to land on exactly one of two values — true or false — and that is what makes proof possible. No "maybe": every statement is true or false Everyday language tolerates "kind of", "usually", "it depends." Mathematics does not. A mathematical statement is a sentence that is definitely true or definitely false — and nothing else. That single rule, called bivalence , is the bedrock the whole subject is built on. Watch the checker move down the list. As it reaches each statement, that statement commits — sliding into the TRUE column or the FALSE column. Notice there is no third place for it to go. Five statements, five verdicts — and not one of them gets to sit on the fence. That definiteness is the raw material of every proof. Joining statements: (and), (or), (not) Real statements get combined. Pick a connective, then set the truth of each input and watch how the combined statement is forced to a single value. Here P is " 5 > 3 " and Q is " 5 < 10 " — both happen to be true, but you can flip them to see every case. The gate computes; it never guesses. is true only when both are; is true when at least one is; just flips P . One false input can sink an entire AND — which is exactly why every line of a proof has to hold. The truth table: every possibility, all at once

This is the written version of the interactive lesson above. See the full Intro to Proofs course.