I have a problem identifying certain structures of the sentence; sometimes it is hard to tell whether I’m dealing with an argument or adjunct.
Adjunct is said to be optional;, that is, its omission will not change the meaning of the predicate. At the same time, the argument of the sentence is a mandatory syntactic unit that completes the meaning of the predicate.
What does the "meaning of the predicate" designate?
Does it simply mean, that with the complement being omitted the predicate becomes ambiguous?
Please consider my reasoning below:
Put the cheese back on the table. –> Should I put it on the table, in a purse or put it on as a hat and wear it to work?
Here, "back" appears to "complete" the meaning of the verb "put", whereas "on the table" can be omitted, because it doesn’t affect the predicate in any way, and therefore can be thought of as structurally dispensable.
He stood in silence –> "in silence" must be complementing and therefore mandatory (but I don’t know why).
I live in Bristol –> complement?
I’m running in the hallway of the Bristol University –> "in the hallway" – and "of the Bristol University" – both feel to be adjuncts;
The bag is under the table/round – "under the table" and "round" determine the state of the bag (i.e. whether its location or shape is in the focus)".
Am I getting it right? Does the argument help to identify the meaning of the predicate?
Let’s examine your first example a little more closely:
[1a] Put the cheese back on the table.
You point out that we can omit the prepositional phrase to get a perfectly sensible command to someone pilfering cheese:
[1b] Put the cheese back.
We can’t omit the adverb in 1b because
[1c] *Put the cheese
is ungrammatical. The verb to put licenses a mandatory destination.
But notice that we may omit the adverb in 1a if we keep the prepositional phrase:
[1d] Put the cheese on the table.
What’s required is thinking in larger syntactic units. What’s important is that put requires an adverbial phrase of place, which may be satisfied by back in 1b, by on the table in 1d, and by both in 1a.
The omission of an argument either makes the sentence ungrammatical because the argument is mandatory or it changes the meaning of the head, the syntactic unit governing the argument. (In your examples the head is the verb.)
I didn’t tell the full truth about 1c. There is a verb to put that takes a bare object, and in the case of 1c, you may imagine a track and field coach who hasn’t been able to find a shot put, but who fortunately has a ball of gouda that his team may practice with. But in that case, the put of 1c means to throw a heavy ball, while the put of the other sentences means to place.
You may see this in your example
[2a] I live in Bristol.
If you omit the prepositional phrase argument, you get the (barely) grammatical sentence
[2b] I live
but that means something along the lines of “I’m alive”, which is different from the live of 2, which means to reside.
There’s no ambiguity in either 2a or 2b, but the verbs means different things.