3 Logic and Proof

Every other chapter on this site eventually says “therefore.” Before any of them are worth trusting, it’s worth being explicit about what that word is allowed to mean in mathematics — because it means something much narrower than it does in ordinary argument.

3.1 What a proof actually is

A proof is a finite chain of statements, each of which is either an accepted starting point (an axiom, or something already proven) or follows from earlier statements by a small, fixed set of valid inference rules — most basically, modus ponens: from “if \(P\) then \(Q\)” and “\(P\),” conclude \(Q\). That’s it. No appeal to authority, no accumulation of supporting examples, no “it’s obviously true.” A statement supported by a thousand confirming examples is not proven — it’s conjectured, however strongly. This is mathematics’s actual epistemic advantage over the empirical sciences: a correct proof produces certainty conditional on the axioms, not probability updated by evidence. The condition matters and is easy to drop silently; it’s picked back up in Axioms and Definitions.

Two proof strategies are worth naming because they’ll appear constantly and look superficially different:

Direct proof — assume the hypotheses, chain forward via valid inferences, arrive at the conclusion.
Proof by contradiction — assume the hypotheses and the negation of the conclusion, derive a logical contradiction (some statement and its negation both provably true), and conclude the negated conclusion must be false. This works because classical logic accepts the law of excluded middle: every well-formed statement is true or false, no third option — a choice that turns out not to be forced (some logics reject it), but is the default this site uses throughout.

Neither strategy gets exercised in depth here — this site states results and explains what they mean far more often than it derives them from scratch. When a proof is included later, it’s because walking through it is the fastest way to understand why something is true, not because completeness demands it.

3.2 Where the rules themselves come from

The inference rules above didn’t arrive with mathematics already attached — they were extracted and formalized as their own object of study. Aristotle codified syllogistic reasoning in antiquity; mathematicians used informal deductive argument (most famously Euclid’s, built from explicit postulates) for roughly two thousand years without a symbolic notation for logic itself. Formal, symbolic mathematical logic — logic written as a calculus you can manipulate the way you manipulate algebra — is comparatively recent: George Boole’s algebraic treatment of logic (1854) and Gottlob Frege’s and Bertrand Russell’s work formalizing logical inference (roughly 1879–1910) are the usual reference points, and this is one of the clearest instances on this whole site of the historical graph and the formal graph pointing in opposite directions — correct proofs long preceded any formal theory of what made them correct.

3.3 Propositions and connectives

A proposition is a statement that is unambiguously true or false — “7 is prime” is one; “7 is a nice number” is not, absent a definition of “nice.” Propositions combine via a small vocabulary: and (\(\land\)), or (\(\lor\)), not (\(\neg\)), implies (\(\Rightarrow\)), and if and only if (\(\Leftrightarrow\)). Two of these are worth flagging because they diverge from everyday usage in ways that cause real confusion downstream:

\(P \Rightarrow Q\) is only false when \(P\) is true and \(Q\) is false. If \(P\) is false, \(P \Rightarrow Q\) is automatically true, regardless of \(Q\) — “vacuously true.” “If \(n\) is a unicorn, then \(n\) is purple” is, in this sense, a true statement, because no \(n\) is a unicorn. This isn’t a quirk; it’s the definition that makes proof by contradiction and universally quantified statements (“for all \(x\) in the empty set, …”) behave sensibly.
or (\(\lor\)) is inclusive by default: “\(P\) or \(Q\)” is true if either or both hold, unlike the “pick one” sense the word often carries in ordinary speech.

Quantifiers extend this to statements about many objects at once: \(\forall\) (“for all”) and \(\exists\) (“there exists”). “For every real number \(x\), there exists a real number \(y\) with \(y > x\)” is a single, precise claim built from two quantifiers and one inequality — and it’s a useful habit, whenever a sentence in later chapters feels slippery, to try restating it with explicit \(\forall\)/\(\exists\) until the ambiguity resolves.

3.4 Where this leads

Logic supplies the rules of inference; it doesn’t yet supply anything to reason about. The next chapter, Sets, introduces the first genuine mathematical object this site builds — and, in one of the sharper examples of the historical/formal gap from the map, a naive version of it that turns out to need the axiomatic method this chapter just described to survive its own paradoxes.