  In the DGP model, the ``self-accelerating'' solution is plagued by a ghost
instability, which makes the solution untenable. This fact as well as all
interesting departures from GR are fully captured by a four-dimensional
effective Lagrangian, valid at distances smaller than the present Hubble scale.
The 4D effective theory involves a relativistic scalar \pi, universally coupled
to matter and with peculiar derivative self-interactions. In this paper, we
study the connection between self-acceleration and the presence of ghosts for a
quite generic class of theories that modify gravity in the infrared. These
theories are defined as those that at distances shorter than cosmological,
reduce to a certain generalization of the DGP 4D effective theory. We argue
that for infrared modifications of GR locally due to a universally coupled
scalar, our generalization is the only one that allows for a robust
implementation of the Vainshtein effect--the decoupling of the scalar from
matter in gravitationally bound systems--necessary to recover agreement with
solar system tests. Our generalization involves an internal ``galilean''
invariance, under which \pi's gradient shifts by a constant. This symmetry
constrains the structure of the \pi Lagrangian so much so that in 4D there
exist only five terms that can yield sizable non-linearities without
introducing ghosts. We show that for such theories in fact there are
``self-accelerating'' deSitter solutions with no ghost-like instabilities. In
the presence of compact sources, these solutions can support spherically
symmetric, Vainshtein-like non-linear perturbations that are also stable
against small fluctuations. [Short version for arxiv]
