The property that whenever a non-logical symbol (relation, function, or constant) is implicitly definable by a theory — meaning any two models of the theory that agree on the base language also agree on the interpretation of that symbol — then there exists an explicit definition of that symbol by a formula in the original language (i.e., an explicit defining formula).

Principle

The organising idea is that semantic uniqueness (implicit definability) should be convertible into a syntactic specification (an explicit formula): if the theory forces a unique interpretation of a symbol up to agreement on the base vocabulary, that interpretation can be captured by a formula using the base vocabulary.

Demonstration

Example: Beth's theorem shows that classical first-order logic satisfies this property: if a predicate symbol is implicitly defined by a first-order theory, then there is a first-order formula (in the smaller language) that explicitly defines it.

Misapplication

Assuming Beth definability automatically in logics where it fails (some modal, intuitionistic or extended first-order logics lack the Beth property) or confusing implicit definability (semantic uniqueness) with mere conservative extension or eliminability without producing an explicit formula.

Consequence

When the Beth property holds one can replace implicit definitions by explicit ones, which streamlines axiomatizations, supports elimination of auxiliary symbols, and links to interpolation and uniform interpolation phenomena in the logic.

Reversal

Failure of the Beth property yields phenomena where a concept is uniquely determined by a theory but no formula in the base language denotes it; such failures indicate a gap between semantic determinacy and syntactic expressibility.

Boundary

Concerns definability of non-logical symbols relative to a base language; it does not by itself assert algorithmic constructibility of the explicit definition nor does it apply to meta-theoretic notions outside the language of the theory.

Semantic Tension

Tension arises between semantic implicitness and syntactic explicitness and between Beth property and expressive extensions: stronger expressive resources may make explicit definability easier, while weaker or constructive logics may separate implicit and explicit definability.

Synthesis

Beth definability property ties semantic uniqueness to syntactic definability: when it holds, any concept forced uniquely by a theory can be given an explicit formula in the original language, closing the gap between model-theoretic determination and proof-theoretic specification.