Denotesx

Denotesx is a formal framework and notation for representing the denotation of expressions across linguistics, philosophy of language, and computer science. It treats the denotation of an expression as a typed value in a designated interpretation domain and provides compositional rules for building complex denotations from simpler ones. The framework is designed to be implementable in software while remaining suitable for theoretical analysis.

Core idea and syntax: In denotesx, every expression E has a denotation denoted as denotesx(E, context) within a given world or context. Denotations are typed, and composition follows the principles of lambda calculus: applying a function to an argument yields a new denotation. The approach supports both extensional denotation, where results are observable sets or individuals, and intensional denotation, where results depend on possible worlds or contexts through a world parameter.

Origins and scope: Denotesx emerged from discussions in formal semantics and type-theoretic programming communities as a lightweight, implementable alternative to fully abstract model-theoretic treatments. It aims to provide a uniform method for mapping expressions to their meanings while accommodating context-sensitivity and scope.

Applications: The framework is used for educational demonstrations of semantic composition, rapid prototyping of semantic parsers, and lightweight symbolic reasoning about expressions. In natural language processing, denotesx can underpin semantic parsing pipelines; in programming languages, it supports symbolic evaluation and formal verification of expressions.

Example: denotesx("the cat", w) yields the unique individual in world w that is a cat if such an individual exists; denotesx("cat", w) yields the set of cats in world w. Denotesx operations can be extended with world parameters, type annotations, and context variables as needed.

See also: denotational semantics, lambda calculus, Montague grammar, type theory.