##  [Consequence Conservation](/consequence-conservation-0) 

 Definition

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

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

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

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

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

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

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

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

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.