  We present a correlated \textit{and} gate which may be used to propagate
uncertainty and dependence through Boolean functions, since any Boolean
function may be expressed as a combination of \textit{and} and \textit{not}
operations. We argue that the \textit{and} gate is a bivariate copula family,
which has the interpretation of constructing bivariate Bernoulli random
variables following a given Pearson correlation coefficient and marginal
probabilities. We show how this copula family may be used to propagate
uncertainty in the form of probabilities of events, probability intervals, and
probability boxes, with only partial or no knowledge of the dependency between
events, expressed as an interval for the correlation coefficient. These results
generalise previous results by Fr\'echet on the conjunction of two events with
unknown dependencies. We show an application propagating uncertainty through a
fault tree for a pressure tank. This paper comes with an open-source Julia
library for performing uncertainty logic.
