A syntactic mapping from formulas and proofs of one theory into another that aims to preserve logical consequence, provability, or other structural proof-theoretic relations so that derivations in the source correspond to derivations in the target.

Principle

A translation is specified by a rule assigning to each symbol, formula, or proof-step of the source theory a target-language counterpart, typically preserving entailment: if Γ ⊢_S φ then τ(Γ) ⊢_T τ(φ), where τ is the translation function between languages of S and T.

Demonstration

Gödel's negative translation sends each classical formula to an intuitionistic counterpart by systematically inserting double negations; this translation preserves provability in the sense that if classical logic proves φ then intuitionistic logic proves the translated τ(φ), enabling transfer of some proof-theoretic information between systems.

Misapplication

Assuming that any translation that maps theorems to theorems constitutes an equivalence of theories. A translation may preserve provability in one direction while failing to be invertible or to respect other desiderata (complexity, intended models), so treating it as full equivalence misreads its scope.

Consequence

A faithful translation permits comparison of expressive power and consistency strength, transfers of proofs and consistency results, and construction of interpretations; it formalizes how one theory can be embedded in another for syntactic analysis.

Reversal

A failure of translation is the inability to find a systematic mapping that preserves consequence, indicating genuine incommensurability between inferential resources of the two theories or a need to widen the target language or change the translation criteria.

Boundary

Translations are syntactic devices and may not preserve semantic features such as model-theoretic properties, intended interpretations, or computational complexity. The adequacy of a translation depends on what preservation criteria are required (provability, entailment, model preservation, etc.).

Semantic Tension

Tension lies between translation and interpretation: translations act on syntactic objects and aim to preserve derivability, whereas semantic interpretations act on models and structures; a translation may exist without a corresponding model-theoretic interpretation and vice versa.

Synthesis

Theory translation is the formal procedure of mapping symbols, formulas, and proofs from one theory into another to preserve inferential structure, enabling syntactic comparison, embedding, and transfer of results subject to the chosen preservation criteria.