A mapping between languages, formulas, or structures such that whenever a formula is true in a source interpretation or model, its image under the mapping is true in the corresponding target interpretation or model (with respect to the specified semantics).

Principle

Truth preservation requires that the mapping commute with satisfaction: for every source model M and formula φ, if M ⊨ φ then mapped(M) ⊨ mapped(φ). The direction is important—truth need only be preserved from source to target unless the mapping is additionally truth‑reflecting.

Demonstration

A homomorphism of relational structures that sends elements and relations so that atomic formulas satisfied in the source are also satisfied in the target provides a truth‑preserving mapping for atomic satisfaction; the standard translation of modal formulas into first‑order formulas is truth‑preserving for pointed Kripke models in the intended way.

Misapplication

Assuming that a truth‑preserving mapping is invertible or that it preserves entailment and provability in both directions; also assuming preservation of degrees of truth in nonclassical logics without checking the semantics used.

Consequence

Truth preservation allows one to transfer models, countermodels, and satisfiability results from the source to the target, establishing sound embeddings and enabling reuse of satisfiability and model‑building arguments in the target framework.

Reversal

The converse notion is truth‑reflection: a mapping is truth‑reflecting if truth of the image implies truth of the preimage. A mapping that is both truth‑preserving and truth‑reflecting yields a truth‑equivalence between source and target.

Boundary

Depends on the semantics: truth must be defined in both source and target and the mapping must be specified on both formulas and models; the property does not automatically imply preservation of proof-theoretic features or entailment relations unless those are included in the specification.

Semantic Tension

Tension exists between truth preservation and consequence preservation: a mapping can preserve truth of single formulas without preserving logical consequence between sets of formulas, and it can preserve truth under one semantics but fail under another (classical vs. many‑valued, for example).

Synthesis

A truth‑preserving mapping is a semantic embedding that guarantees that truth in the source carries over to truth in the target; it is a directed semantic guarantee that supports sound model transfer but must be paired with other properties to secure fuller equivalence.