The most common and most likely answer to this crossword clue is the 8 plus 10 letter word IMPRESSE (PREMISES)
I
M
P
R
E
S
S
E
(
P
R
E
M
I
S
E
S
)
Buddy explains!
The straight part
We know that "finishing propositions" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case finishing propositions = impresse (premises).The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.