  We consider the behaviour of logarithmic differential forms on arrangements
and multiarrangements of hyperplanes under the operations of deletion and
restriction, extending early work of G\"unter Ziegler. The restriction of
logarithmic forms to a hyperplane may or may not be surjective, and we measure
the failure of surjectivity in terms of commutative algebra of logarithmic
forms and derivations. We find that the dual notion of restriction of
logarithmic vector fields behaves similarly but inequivalently. A main result
is that, if an arrangement is free, then any arrangement obtained by adding a
hyperplane has the "dual strongly plus-one generated" property. One application
is another proof of a main result of a paper by the first author characterizing
when adding a hyperplane to a free arrangement remains free. A further
application is to resolve two conjectures due to Ziegler, which we defer to a
sequel.
