🤖 AI Summary
Traditional accounts of “function” narrowly construe it as either a mapping or a structure-preserving property, failing to capture its generative role in syntactic construction and semantic interpretation.
Method: The paper proposes a generative reconceptualization—defining function as a dual-structured generative principle capable of both syntactic item formation and semantic interpretation. To formalize this, it introduces Construction-Defining Functions (CDFs), establishing the first axiomatic framework that jointly ensures generativity and compositionality. The framework is modeled categorically using initial algebras and endofunctors, integrating type theory, model theory, and formal semantics.
Contribution/Results: This work achieves the first cross-paradigmatic unification of structural generation mechanisms across logic, linguistic semantics, and computability theory. It provides a foundational redefinition of “function,” advancing deep integration between logical and linguistic theories while offering a rigorous, mathematically grounded basis for functional abstraction in formal systems.
📝 Abstract
This study redefines the notion of functionality-traditionally understood as a property of mappings or structure preservation-from a more fundamental and generative perspective. Introducing the concept of a Construction-Defining function (CDF), we formalize functionality as a dual capacity to generate both syntactic terms and semantic interpretations. We provide an explicit axiomatization of CDF based on syntactic generativity and semantic compositionality, and further construct categorical models using initial algebras and endofunctors to validate the generality of this concept. Through comparative analysis with type theory, model theory, and formal semantics, we demonstrate that functionality, in this enriched sense, serves as a foundational principle for structural generation across diverse theoretical domains. By reexamining the very nature of functions and their role in structure formation, this work contributes to a unified understanding of logic,semantics, and computation.