A formal proof is a finite, ordered sequence of well-formed formulas in a specified formal language and proof system such that each formula is either a premise, an axiom of the system, or is obtained from earlier formulas by a rule of inference of that system, and the final formula is the derived conclusion.

Principle

Derivability in a fixed formal system, according to its axioms and inference rules, is the criterion that determines whether a purported argument constitutes a formal proof.

Demonstration

Illustrative scenario — Situation: In a propositional language with axioms A1 and A2 and Modus Ponens as rule of inference, a prover lists (1) A1, (2) A2, (3) A1 → B, (4) B (from 1 and 3 by Modus Ponens). Recognition: Each step is either an axiom or follows by the system's rule. Action: The prover records the sequence as a derivation. Consequence: B is a theorem of the system because it appears as the final formula of a valid finite derivation.

Misapplication

Treating a persuasive or explanatory natural-language argument as a formal proof without providing its translation into the specified formal language and verification that each step follows the system's inference rules; the error is conflating pragmatic convincingness with syntactic derivability.

Consequence

When a statement has a formal proof in a sound system, it is syntactically derivable and thereby justified within that system; however, such derivability guarantees only what the system's axioms and rules permit and depends on the chosen language and axioms rather than on extra-system empirical claims.

Reversal

If the chosen proof system is unsound relative to a target semantics, or if the intended meaning of symbols differs from the system's formal interpretation, then a formal proof need not preserve truth under that external interpretation; conversely, some true semantic facts may lack a formal proof in an incomplete system.

Boundary

Clearly within: a Hilbert- or natural-deduction style derivation that lists each formula and justifies each step by an axiom or rule. Boundary case: a sketch that indicates the main inference pattern without checking every rule application. Clearly outside: an informal explanation that uses metaphors or appeals to intuition without explicit syntactic rules and formula-by-formula derivation.

Semantic Tension

Formal Proof ↔ Informal Mathematical Explanation — the formal proof prioritizes mechanical checkability and syntactic correctness, while informal explanation prioritizes insight, intuition, and pedagogy; translating between them can be nontrivial.

Synthesis

A formal proof converts argumentative correctness into a mechanically checkable sequence tied to a chosen system; its value depends both on the adequacy of the formalization and on the system's relationship to the intended semantics.