A relation between predicates, concepts, or theories in which each can be defined inside the other by formulas of a fixed surrounding language, so that for any object or structure the concept holds exactly when its definition in terms of the other does.

Principle

If A and B are interdefinable, then there exist formulas φ(x, p) and ψ(x, q) using the primitives of the ambient language such that φ defines A from B and ψ defines B from A; the relation is symmetric and captures mutual expressibility rather than mere implication.

Demonstration

In first-order arithmetic, the order relation < on natural numbers is interdefinable with the successor function S when the language already contains zero and addition: one can give a formula using S that expresses <, and conversely describe S in terms of < and arithmetic axioms in some expansions.

Misapplication

Treating interdefinability as if it required identical syntactic formulas rather than mutual definability within the ambient language, or assuming interdefinability implies computational equivalence without checking definability constraints.

Consequence

When accepted, interdefinability lets one replace one vocabulary by the other in proofs and definitions, transferring theorems and enabling conservative extensions or translations between theories while preserving intended models.

Reversal

If one reverses interdefinability to mere definability (A definable from B but not vice versa), the symmetry and the ability to substitute vocabularies collapses and one gets a strictly weaker expressive relationship.

Boundary

Applies only relative to a specified surrounding language and class of formulas; it excludes cases where definitions require new nonlogical symbols, nonconstructive existence assumptions, or resources outside the allowed syntactic/semantic framework.

Semantic Tension

Interdefinability competes with bi-interpretability and conservative extension: bi-interpretability demands structure-preserving translations between theories, whereas conservative extension asks only that one theory add no new theorems in the old language; all are related but distinct.

Synthesis

Interdefinability is the symmetric, language-relative notion that two concepts or theories can each be expressed exactly in terms of the other by formulas allowed in the ambient setting, enabling vocabulary exchange while respecting the chosen syntactic and semantic constraints.