  Callias-type (or Dirac-Schr\"odinger) operators associated to abstract
semifinite spectral triples are introduced and their indices are computed in
terms of an associated index pairing derived from the spectral triple. The
result is then interpreted as an index theorem for a non-commutative analogue
of spectral flow. Both even and odd spectral triples are considered, and both
commutative and non-commutative examples are given.
