  The existence of stars with masses up to $2 M_{\odot}$ and the hints of the
existence of stars with radii smaller than $\sim 11$ km seem to require, at the
same time, a stiff and a soft hadronic equation of state at large densities. We
argue that these two apparently contradicting constraints are actually an
indication of the existence of two families of compact stars: hadronic stars
which could be very compact and quark stars which could be very massive. In
this respect, a crucial role is played, in the hadronic equation of state, by
the delta isobars whose early appearance shifts to large densities the
formation of hyperons. We also discuss how recent experimental information on
the symmetry energy of nuclear matter at saturation indicate, indirectly, an
early appearance of delta isobars in neutron star matter.
