nonexpressible

Nonexpressible is an adjective used in logic, philosophy of language, and related fields to describe something that cannot be adequately represented or named within a given expressive system, such as a formal language, a natural language, or a cognitive framework. A thing nonexpressible in a system is not necessarily nonexistent; rather, it lies beyond the available syntax or semantic resources to designate it unambiguously.

In formal logic, a property or relation is nonexpressible in a language L if there is no formula in L whose truth conditions coincide with the intended concept in all structures of interest. Classic results, such as limits shown by Gödelian phenomena and model-theoretic principles, illustrate that some features (for example, the finiteness or infinitude of a domain) are not expressible in first-order logic, while they may become expressible in stronger languages or higher-order logics.

In philosophy of language, nonexpressible concepts are distinguished from ineffable or inexpressible in ordinary speech; they concern the limits of a particular expressive system rather than a speaker’s communicative ability. Scholars examine how extending linguistic resources or adopting alternative formal frameworks changes what remains nonexpressible.

Relation to other notions: expressibility refers to what a system can express; definability concerns naming or characterizing objects within a theory; nonexpressibility is the negation relative to the system. Nonexpressibility often motivates the use of more powerful languages or different representations to capture certain concepts.

See also: expressibility, definability, ineffability, Gödel’s incompleteness theorems, model theory, formal languages.