The Logic of Epistemic Inconsistency (LEI), is paraconsistent logic which sintatically distinguishes between sentences expressing irrefutable (monotonically deduced) "knowledge" from those expressing defeasible (nonmonotonically inferred) ones. It was conceived to tolerate contradiction among this last kind of sentences and to enable meaningful reasoning on these circumstances. The main motivation for its construction has been its use for nonmonotonic reasoning in association to a special default logic, named IDL (for Inconsistent Default Logic). A method which automatize LEI reasoning is given here. It is presented in the form of a proof method by semantic tableaux which can be shown sound and complete with respect to a first order version of LEI.