totalFalse

TotalFalse is a term used in logic and computer science to denote a total function that always returns the boolean value false. It is the constant-false function across its domain and is often discussed alongside totalTrue, the function that always returns true.

Formally, if A is a type, totalFalse : A -> Bool is defined by totalFalse(a) = False for every a in A. Because it provides an output for every input, it is a total function; the predicate it represents is the characteristic function of the empty subset of A.

In programming, totalFalse is realized by a function that ignores its argument and yields false. Examples include Haskell: totalFalse :: a -> Bool; totalFalse _ = False; and many imperative languages where a function can be defined as returning the boolean false regardless of the input. This contrasts with partial functions that may be undefined for some inputs.

Uses and interpretation: totalFalse serves as a baseline or corner case in logic, type theory, and program verification. It encodes vacuous truth and can be used to test error handling, pattern matching exhaustiveness, or to represent a predicate that is never satisfied. Because its output does not depend on the input, it cannot distinguish elements of the domain.

See also: constant function; vacuous truth; boolean algebra; total functions vs partial functions.