  The model-dependence of the relation between the inelastic and various
minimum-bias proton-proton cross sections is analyzed, paying a special
attention to the sensitivity of minimum-bias triggers to diffractive
collisions. Concentrating on the trigger selections of the ATLAS experiment,
the measured cross sections are compared to predictions of a number of hadronic
Monte Carlo models used in the cosmic ray field. It is demonstrated that the
ATLAS results are able to discriminate between different models and between
certain theoretical approaches for soft multi-particle production. On the other
hand, the strong model-dependence of the selection efficiency of the
minimum-bias triggers prevents one from inferring high mass diffraction rate
from the discussed data. Moreover, the measured cross sections prove to be
insensitive to the production of low mass diffractive states in proton-proton
collisions. Consequently, a reliable determination of the total inelastic cross
section requires forward proton tracking by a dedicated experiment.
