The property of a translation, mapping, or extension between logical systems that guarantees entailments (consequences) in one system correspond to entailments in the other, so no consequences are spuriously added or lost under the translation.

Principle

A mapping is consequence-conserving if for every theory T and formula φ in the relevant language fragment, T entails φ in the source system exactly when the translated theory entails the translated formula in the target system, or at least in one direction specified (preservation or reflection).

Demonstration

Example: a conservative extension of a theory adds new symbols and axioms but does not change consequences in the original language; any sentence in the original signature that was provable before remains provable and no new sentence in the original signature becomes provable solely due to the extension.

Misapplication

Claiming consequence conservation when the mapping only preserves truth of atomic formulas or fails for quantified or negated formulas; confusing preservation in one direction (soundness) with full conservation (equivalence of consequences).

Consequence

When consequence conservation holds, translations between systems are reliable for reasoning: proofs and refutations in the source can be carried to the target without introducing spurious theorems in the source language, enabling modular development and safe extension of theories.

Reversal

The failure of consequence conservation results in addition or loss of entailments: a translation might introduce new consequences (unsound extension) or lose consequences (incomplete translation), undermining equivalence of theories across systems.

Boundary

Consequence conservation is often relative to a fragment of language (e.g., sentences in a shared signature, existential formulas, or syntactic derivability) and may not hold uniformly for all formula classes, models, or proof systems.

Semantic Tension

Tension exists between truth-preservation, provability-preservation, and consequence-conservation: a mapping may preserve satisfiability or models without preserving derivability, or preserve one class of formulas but not another, so precise criteria matter.

Synthesis

Consequence conservation formalizes the idea that a translation or extension does not change the inferential content of a theory for the formulas of interest; it provides the criterion for safe translation, conservative extension, and faithful comparison of logical systems.