  Effect algebras form an algebraic formalization of the logic of quantum
mechanics. For lattice effect algebras E we investigate a natural implication
and prove that the implication reduct of E is term equivalent to E. Then we
present a simple axiom system in Gentzen style in order to axiomatize the logic
induced by lattice effect algebras. For effect algebras which need not be
lattice-ordered we introduce a certain kind of implication which is everywhere
defined but whose result need not be a single element. Then we study effect
implication algebras and prove the correspondence between these algebras and
effect algebras satisfying the Ascending Chain Condition. We present an axiom
system in Gentzen style also for not necessarily lattice-ordered effect
algebras and prove that it is an algebraic semantics for the logic induced by
finite effect algebras.
