Truthorder

Truthorder is a term used in logic and computer science to describe a partial order on truth values that expresses the degree of truth carried by a value. It is most often discussed in the context of multi-valued and paraconsistent logics, where truth values extend beyond classical true and false. In these systems, truthorder is one of the lattice structures defined on the set of truth values, the other common one being the information (or knowledge) order. The truth order organizes values according to increasing truth content, while the information order organizes them by the amount of information they convey.

A canonical setting for discussing truthorder is Belnap’s four-valued logic, which uses the values true (t), false (f), both true and false (b), and neither true nor false (n). In the truth order, these values are arranged to reflect increasing commitment to truth, with more truth-bearing values positioned higher in the lattice. The information order, in contrast, emphasizes the amount of information present, allowing n and b to interact in ways that are not purely about truth instead of information.

Truthorder has applications in reasoning about inconsistent or incomplete data, such as databases, knowledge bases, and certain artificial intelligence systems. It provides a formal framework for combining and comparing statements when data may be simultaneously true and false or partially unknown. The concept also appears in discussions of various many-valued logics and their lattice-theoretic semantics.

See also: Belnap four-valued logic, paraconsistent logic, lattice theory, many-valued logic.