nLab boolean-valued function

A boolean-valued function is a function from any set to the boolean domain $\{\bot, \top\}$.

In classical logic, the definable boolean-valued functions on a type $X$ correspond precisely to predicates on $X$. Assuming the law of excluded middle, the boolean-valued functions on $X$ correspond precisely to the subsets of $X$; even in constructive mathematics, they corresond to the decidable subsets of $X$.