A bidirectional relation between formal languages or systems indicating that there exist translations in both directions which preserve specified semantic or syntactic properties, so that formulas and their intended meanings can be carried back and forth up to the chosen notion of equivalence.

Principle

Intertranslatability requires a pair of translations T: L1→L2 and S: L2→L1 such that compositions S∘T and T∘S recover original expressions up to an agreed equivalence (equality, logical equivalence, provable equivalence) and preserve the targeted properties (truth, derivability, satisfiability).

Demonstration

Two presentations of the same theory (for example, two syntactic variants of arithmetic or two equational presentations of a universal algebra) are intertranslatable: there are mechanical translations that map axioms and theorems back and forth, preserving provability and theoremhood up to renaming or definitional expansion.

Misapplication

Assuming intertranslatability implies identity of all meta-properties; for instance, believing mutual translations guarantee identical proof complexity, decidability behavior, or that every semantic nuance is preserved without qualification.

Consequence

Intertranslatability establishes a strong form of equivalence: the languages can serve interchangeably for specification and proof, results transfer in both directions, and metatheoretical properties (when the translations preserve them) may be ported between systems.

Reversal

A failure of intertranslatability means a one-way embedding exists but no faithful reverse mapping; this asymmetry signals a genuine difference in expressive or structural resources between the systems.

Boundary

Depends on which properties must be preserved and on the acceptable notion of equivalence; intertranslatability may hold syntactically but fail semantically, or vice versa, and may exclude translations that are non-effective or that rely on external or nonconstructive resources.

Semantic Tension

Competes with the weaker notion of relative expressiveness: two languages can be mutually interpretable (intertranslatable) yet differ in complexity, succinctness, or convenience of expression, creating tension between formal equivalence and pragmatic difference.

Synthesis

Intertranslatability is mutual expressibility implemented by effective forward and backward translations that preserve the agreed semantic or syntactic features, giving a bidirectional equivalence class of languages while leaving open secondary differences like complexity or brevity.