  (Abridged) The hierarchical formation scenario for the stellar halo requires
the accretion and disruption of dwarf galaxies, yet low-metallicity halo stars
are enriched in alpha-elements compared to similar, low-metallicity stars in
dwarf spheroidal (dSph) galaxies. We address this primary challenge for the
hierarchical formation scenario for the stellar halo by combining chemical
evolution modelling with cosmologically-motivated mass accretion histories for
the Milky Way dark halo and its satellites. We demonstrate that stellar halo
and dwarf galaxy abundance patterns can be explained naturally within the LCDM
framework. Our solution relies fundamentally on the LCDM model prediction that
the majority of the stars in the stellar halo were formed within a few
relatively massive, ~5 x 10^10 Msun, dwarf irregular (dIrr)-size dark matter
halos, which were accreted and destroyed ~10 Gyr in the past. These systems
necessarily have short-lived, rapid star formation histories, are enriched
primarily by Type II supernovae, and host stars with enhanced [a/Fe]
abundances. In contrast, dwarf spheroidal galaxies exist within low-mass dark
matter hosts of ~10^9 Msun, where supernovae winds are important in setting the
intermediate [a/Fe] ratios observed. Our model includes enrichment from Type Ia
and Type II supernovae as well as stellar winds, and includes a
physically-motivated supernovae feedback prescription calibrated to reproduce
the local dwarf galaxy stellar mass - metallicity relation. We use
representative examples of the type of dark matter halos we expect to host a
destroyed ``stellar halo progenitor'' dwarf, a surviving dIrr, and a surviving
dSph galaxy, and show that their derived abundance patterns, stellar masses,
and gas masses are consistent with those observed for each type of system.
