multiplevalued

Multiple-valued, or multi-valued, describes concepts that can take more than one value for a given input or proposition. The term is used across several disciplines to distinguish from single-valued, deterministic notions.

In mathematics and computer science, a multi-valued function assigns a set of values to each input rather than a single output. This contrasts with ordinary functions and is central to set-valued analysis and nondeterministic models. Examples include the square root function in the complex domain, which yields two values, and inverse trigonometric functions that can be multi-valued unless a branch is chosen. To work with single values, mathematicians use branches and branch cuts or interpret multi-valued functions on Riemann surfaces, where each point has a single-valued representation.

In logic, multi-valued logic extends classical true/false logic with additional truth values, such as unknown, both true and false, or other modalities. Systems like Kleene’s three-valued logic or Łukasiewicz’s many-valued logics model reasoning under uncertainty or inconsistency. Multi-valued logics are used in informal reasoning about incomplete information, program semantics, and knowledge representation.

In information systems, a multi-valued attribute can hold several values for a single entity, which complicates database design. Normalization techniques address multi-valued dependencies to preserve data integrity, leading to higher normal forms such as the fourth normal form (4NF).

Overall, the concept of multiple-valued highlights flexibility and non-determinism across mathematical, logical, and data-driven contexts. Related topics include multi-valued logic, multi-valued functions, and multivalued attributes.