Realizabilityfor

Realizabilityfor is a term that appears in discussions of realizability semantics to denote a variant or extension of realizability in which the interpretation of a formula is parameterized by a context or scope introduced by a “for” construction. It is not a widely standardized term with a single canonical definition in the literature, and its precise meaning can vary by author or by the formal setting.

In broad terms, realizability links logical statements to computational witnesses called realizers. Realizabilityfor extends this idea by making the interpretation depend on an environment, context, or target set of assumptions, effectively conditioning the realizability relation on a given context. This can help in extracting constructive content from proofs that operate under specific hypotheses or within a particular scope, by ensuring that realizers are appropriate for that context.

Formal approaches described under the banner of realizabilityfor typically introduce a family of interpretations ⟦A⟧Γ indexed by a context Γ. The realizability rules for logical connectives and quantifiers are then adapted to account for the context, yielding a parameterized soundness or extraction framework. The notion often aligns with related ideas such as parametric realizability, forcing-inspired interpretations, or realizability toposes, but without a universally agreed formalism.

Because realizabilityfor is not a standard, universally defined term, implementations and definitions vary. References to realizabilityfor usually point to context-sensitive realizability frameworks used to analyze proofs or extract programs under particular environmental constraints. See also realizability, constructive logic, proof theory, and type theory.