Full Idea
There are constitutive and causal notions of supervenience. Kim claims that mental events have no causal role, and merely supervene on brain events which do (which implies epiphenomenalism). But it seems obvious that mind is caused by brain.
Gist of Idea
If mind-brain supervenience isn't causal, this implies epiphenomenalism
Source
John Searle (The Rediscovery of the Mind [1992], Ch. 5.V)
Book Reference
Searle,John R.: 'The Rediscovery of the Mind' [MIT 1999], p.125
A Reaction
Personally I think the whole discussion is doomed to confusion because it is riddled with a priori dualism. There is no all-or-nothing boundary between 'mind' and 'brain'. Kim's views have changed.