  The sufficient-component cause framework assumes the existence of sets of
sufficient causes that bring about an event. For a binary outcome and an
arbitrary number of binary causes any set of potential outcomes can be
replicated by positing a set of sufficient causes; typically this
representation is not unique. A sufficient cause interaction is said to be
present if within all representations there exists a sufficient cause in which
two or more particular causes are all present. A singular interaction is said
to be present if for some subset of individuals there is a unique minimal
sufficient cause. Empirical and counterfactual conditions are given for
sufficient cause interactions and singular interactions between an arbitrary
number of causes. Conditions are given for cases in which none, some or all of
a given set of causes affect the outcome monotonically. The relations between
these results, interactions in linear statistical models and Pearl's
probability of causation are discussed.
