The most common and most likely answer to this crossword clue is the 7 plus 8 letter word SPECIAL PLEADING
I've seen this crossword clue in The Guardian
S
P
E
C
I
A
L
P
L
E
A
D
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More clues leading to the result SPECIAL PLEADING
Buddy explains!
The straight part
We know that "not a logical way to win an argument" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not a logical way to win an argument = special pleading.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.