Extendedand

Extendedand is a generalized conjunction operator used in discussions of logic and formal languages to extend the classical boolean AND to work over richer truth-value domains. In many formulations it is defined as the binary operation that yields the greatest lower bound, or meet, of its operands within a given truth-value lattice or algebraic structure. In standard boolean logic, extendedand reduces to the familiar conjunction.

In non-classical logics, extendedand can take various forms depending on the chosen semantics. For a three-valued logic with values true, false, and unknown (ordered false ≤ unknown ≤ true), a common Kleene-style definition yields a table where false dominates, true only when both operands are true, and unknown propagates in combinations that do not force false. The precise results can vary by system, but the general idea is to capture a notion of conjunction that tolerates and propagates indeterminacy.

In fuzzy logic and related frameworks, extendedand is often implemented as a t-norm. The standard choice is extendedand(x, y) = min(x, y) for truth grades x, y in [0, 1], modeling conjunction as the intersection of degrees of truth. Other t-norms, such as the product or Łukasiewicz t-norm, provide alternative notions of how conjunction behaves with partial truth.

In programming and language design, extendedand may appear in libraries or domain-specific languages to express conjunction among predicates that yield non-boolean results or to support specific evaluation strategies. Because its exact definition depends on the underlying logic or algebra, extendedand is not a universal standard and is used variably across contexts.