A relation between theories where a new theory is formed from a base theory by adding new symbols together with explicit defining axioms that uniquely characterize those symbols in terms of the base language, with the result that no new theorems in the original language are introduced (conservativity).

Principle

Each added symbol must come with a defining formula in the original language that determines its meaning (often via existence and uniqueness conditions), and the extension must be eliminable in the sense that any theorem about the original vocabulary provable in the extension was already provable in the base theory.

Demonstration

Introduce a function symbol f and add an axiom ∀x ∃!y φ(x,y) and the defining axiom ∀x∀y (f(x)=y ↔ φ(x,y)). Provided the base theory proves the existence‑and‑uniqueness statements or one treats f as a conservative definitional abbreviation, the extension does not yield new results in the old language.

Misapplication

Treating arbitrary axiom additions as definitional (for example adding existence axioms without definitional equivalence) or failing to verify eliminability; such moves can covertly increase strength, prove new sentences in the original language, or introduce unintended commitments.

Consequence

Definitional extensions enable modular theory development and notational convenience without changing the substantive content concerning the original symbols; they preserve consistency and allow elimination of definitions to recover original proofs.

Reversal

A non‑definitional (axiomatic) extension adds genuine new theoretical content and can prove statements in the original language that were unprovable before; reversing from definitional to axiomatic extensions increases expressive and proof‑theoretic strength.

Boundary

Applies to formal theories where explicit definitions can be stated; it does not cover conservative extensions achieved by indirect means that are not eliminable, nor does it include extensions that add existence claims without definitional equivalence or that rely on higher‑order resources.

Semantic Tension

Tension exists with notions like conservative extension, eliminability, and implicit definability: a definitional extension is a special kind of conservative extension with explicit eliminable definitions, but implicit definitions or non‑eliminable abbreviations blur the distinction.

Synthesis

A definitional extension adds symbols with explicit, eliminable definitions so that the extended theory is conservative over the base for the original language; it formalizes safe notation and modularity without altering the original theory's consequences.