  We demonstrate in the context of the minisuperspace model consisting of a
closed Friedmann-Robertson-Walker universe coupled to a scalar field that
Vilenkin's tunneling wavefunction can only be consistently defined for
particular choices of operator ordering in the Wheeler-DeWitt equation. The
requirement of regularity of the wavefunction has the particular consequence
that the probability amplitude, which has been used previously in the
literature in discussions of issues such as the prediction of inflation, is
likewise ill-defined for certain choices of operator ordering with Vilenkin's
boundary condition. By contrast, the Hartle-Hawking no-boundary wavefunction
can be consistently defined within these models, independently of operator
ordering. The significance of this result is discussed within the context of
the debate about the predictions of semiclassical quantum cosmology. In
particular, it is argued that inflation cannot be confidently regarded as a
"prediction" of the tunneling wavefunction, for reasons similar to those
previously invoked in the case of the no-boundary wavefunction. A synthesis of
the no-boundary and tunneling approaches is argued for.
