Instrumental with, locatum with and the argument/adjunct distinction

The objects of the proposition 'with', as in 'I cut the bread with a knife' and 'I covered the ground with a blanket', have received conflicting analyses as to whether they are arguments or adjuncts of the verb. Utilizing a variety of semantic and syntactic diagnostics of argumenthood, I argue that the first kind of participant, an instrument, is an adjunct/modifier and the second kind of participant, what I call a locatum, is an argument/complement of the verbal head.