A relation between formal theories that holds when each theory can be obtained from the other by adding explicit definitions for new symbols so that the two theories are intertranslatable without changing the content of sentences in the original vocabularies.

Principle

Two theories are definitionally equivalent when there exist explicit, truth-preserving definitions that translate every nonlogical symbol of each theory into the language of the other and render their axioms mutual consequences under those definitions.

Demonstration

Take a first-order theory T that uses a binary symbol R and an alternative presentation T' that replaces R by a symbol S together with a defining axiom S(x,y) ↔ φ_R(x,y) where φ_R is a formula in T's language. If conversely R can be defined in T' by a formula φ_S, then T and T' are definitionally equivalent; their theorems about the original vocabulary coincide after unfolding definitions.

Misapplication

Treating mere mutual interpretability or having the same models as sufficient for definitional equivalence. Mutual interpretability can be weaker and fail to provide explicit definitions that eliminate added symbols, so assuming equivalence from interpretability alone is a common error.

Consequence

When theories are definitionally equivalent one may freely substitute one presentation for the other in proofs, model constructions, and applications that only involve the shared vocabulary; the two presentations count as notational variants of the same theory.

Reversal

The inverse situation is two theories that prove the same sentences in a given language (conservative equivalence) but where no explicit definitions exist to eliminate new symbols; such theories are not definitionally equivalent despite empirical agreement on those sentences.

Boundary

Applies to formal theories where one permits explicit definitional extension rules (introducing abbreviating symbols with exact defining formulas). It excludes weaker relations such as mere mutual interpretability, Morita equivalence in some senses, or semantic equivalence that lacks explicit definitional clauses; it also presumes an accepted notion of admissible definitions in the background logic.

Semantic Tension

Competes with notions like bi-interpretability and categorical equivalence: bi-interpretability allows mutual translations up to isomorphism of interpretations, while definitional equivalence demands explicit eliminative definitions. The tension centers on whether syntactic eliminability or semantic intertranslatability is the right criterion for 'sameness'.

Synthesis

Definitional equivalence is the syntactic notion that two theories are the same theory up to the introduction and elimination of defined symbols: explicit definitions convert one vocabulary into the other so that each theory becomes a notational variant of the other.