A Boolean-valued function is a function of type $f : X \to \mathbb{B}$ , where $X$ is an arbitrary set and where $\mathbb{B}$ is a Boolean domain.

A Boolean function (of finite arity $k$ ) is a Boolean-valued function of type $f : \mathbb{B}^k \to \mathbb{B}$ .

In some contexts a Boolean-valued function may be referred to as a predicate or a proposition. In other contexts these terms may be used to describe the syntactic entities that denote or express Boolean-valued functions, or that have Boolean-valued functions among their canonical or intended models. When necessary to avoid confusion, terms like propositional expression or predicate formula may be used to distinguish signs from their objects.