##  [Consequence-Preserving Mapping](/consequence-preserving-mapping-0) 

 Definition

A transformation between syntactic systems, sets of formulas, or models that preserves logical consequence: whenever a formula follows from a set of premises in the source (Γ ⊨ φ or Γ ⊢ φ), the images of those premises entail the image of the conclusion in the target (mapped(Γ) ⊨ mapped(φ) or mapped(Γ) ⊢ mapped(φ)).

 

 

 

 

 

 





## Principle

Principle

Consequence preservation means the mapping respects the consequence relation of interest (semantic entailment or syntactic derivability). It typically demands that proofs or semantic entailments in the source can be translated into proofs or entailments in the target, often by a mechanically specified mapping of rules and formulas.

 

 

 

 

 





## Demonstration

Demonstration

A syntactic translation that maps each inference rule of a proof system into a derivable schema in another system provides a consequence‑preserving mapping; for example, an embedding of one proof calculus into another that converts source proofs into target proofs preserves consequence in the syntactic sense.

 

 

 

 

## Misapplication

Misapplication

Assuming that consequence preservation implies truth preservation on individual formulas, or conflating preservation of classical entailment with preservation of nonmonotonic or defeasible consequence relations without verifying the target's consequence notion.

 

 

 

 

 





## Consequence

Consequence

Consequence‑preserving mappings enable transfer of theorems, proof obligations, and correctness arguments across formalisms; they support modular reasoning, reuse of derivations, and establishing conservativity or embedding results between systems.

 

 

 

 

## Reversal

Reversal

The converse notion is consequence reflection: if mapped(Γ) entails mapped(φ) implies Γ entails φ, then the mapping reflects consequences and—when both hold—yields consequence equivalence between source and target.

 

 

 

 

 





## Boundary

Boundary

Applies only relative to the specified notion of consequence (semantic vs syntactic, monotonic vs nonmonotonic); it may require adjusting the target proof system or enriching premises to recover completeness, and it excludes mappings that preserve only satisfiability or single‑formula truth but not entailment among sets.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between consequence preservation and proof‑theoretic constraints: a mapping may preserve entailment but blow up proof length or complexity, or it may preserve entailment only at cost of changing the intended proof semantics (e.g., classical to constructive).

 

 

 

 

 





## Synthesis

Synthesis

A consequence‑preserving mapping is a structure‑respecting translation that carries logical consequence from a source to a target, enabling sound transfer of derivability and theoremhood while requiring careful specification of which notion of consequence is being preserved.