A finite, ordered sequence of formulas in a formal system where each formula is either an axiom or follows from earlier formulas by an accepted rule of inference, and the sequence ends with the formula being established as proved.

Principle

A proof formalizes the justification that a statement is derivable under a chosen axiomatic basis and inference rules; finiteness and adherence to the system's rules are essential to guarantee the claim.

Demonstration

In propositional logic a simple proof of B from premises A→B and A is: (1) A→B (premise), (2) A (premise), (3) B (from 1 and 2 by Modus Ponens). The three-step finite sequence is a proof of B in that system.

Misapplication

Presenting persuasive or probabilistic argumentation as a formal proof; circular reasoning where the purported proof uses the formula to be proved as an unproven premise; relying on an unformalized step labelled 'obvious' that cannot be justified by the system's rules.

Consequence

A correct proof delivers a certificate of derivability: within a sound formal system it guarantees semantic validity of the proved formula relative to the system's semantics and can be checked mechanically or inspected by peers.

Reversal

Instead of constructing a derivation of a statement, one may construct a refutation (a proof of its negation) or produce a countermodel; the inversion exposes non-derivability or inconsistency.

Boundary

Applies only to finite sequences formed according to a specified formal system; it excludes empirical demonstrations, infinite-length derivations, informal explanations not reducible to the rules, and meta-theoretic arguments external to the system.

Semantic Tension

Tension arises between 'proof' as a fully formal syntactic certificate and the informal mathematical notion of proof that mixes intuition, diagrams, and rhetoric; the two share aims but differ in strictness and checkability.

Synthesis

A proof is a finite, rule-governed syntactic construction in a specified formal system that transforms axioms and premises into a concluded formula, yielding a verifiable guarantee of derivability relative to that system.