In VP, the interpretation of arguments depends on the verbal head. For this reason, is the verb changes, the interpretation of the argument (its theta role) also changes:
(1) I aim at George’s house (<goal>) vs. I stay at George’s house (<place>)
The interpretation of an adjunct, instead, remains the same even if the root verb is replaced:
(2) I aim at George’s house in the afternoon sv. I stay at George’s house in the afternoon
In (2), in fact, the adjunct in the afternoon refers to a moment in time in both sentences.
So, it seems that adjuncts are semantically independent, wheras arguments are not. Accordingly, adjuncts may be omitted from a sentence and be added to virtually any well formed sentence in which all the arguments of the verb are projected:
(3) I run in the afternoon, John puts his keys on the table in the afternoon, it rained in the afternoon…
Now, think of two-argument verbs such as eat, drink, think etc: they often occur with a single argument (the external one): I eat (an apple), I’m eating, I’m drinking/thinking. Note, however, that the omitted element is not a real adjunct. Let’s see why: a) it is not semantically independent from the verb; b) it can’t be freely added to any complete, well formed sentence:
(4) John eats an apple
(5) John eats
(6) *John puts his keys on the table an apple (vs. John puts his keys on the table in the afternoon).
In the afternoon is an adjunct, an apple (in (4)), instead, is an (optional) argument.
In conclusion, in order to decide whether an XP is an adjunct in a sentence S, optionality is not a sufficient requirement: it must also be possible to add this XP to a full sentence S’ in which all the main verb (H) ‘s arguments are projected. Obviously, XP’s internal and independent theta role should not be present in the theta grid of H, and tense specification should be compatible with XP (if XP is a time adverb, for instance).