Oftentimes when we know a statement of the form
,
we reason as follows: ``We know that there is something for which
holds. Let
be a name for that something, so that
holds
for
. Then
'' where
is a derivation of some
conclusion
, presumably with the help of the assumption
that
holds for
, i.e., the premise
.
It is customary to refer to
as a witness, and to
the formula
as the witness premise.
No special assumptions about the witness
must be made.
It must serve only as a ``dummy''--any other name should
do just as well. In particular, the witness should not
occur in the final conclusion
.
This form of reasoning is captured in NDL with deductions
of the form