Crossword Genius

Vulgar upset after disposing of second quantity of beer (4)

Ross

I believe the answer is:

half

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'beer' is the definition.
Although both the answer and definition are singular nouns, I cannot understand how one could define the other.

'vulgar upset after disposing of second quantity of' is the wordplay.
'vulgar upset' becomes 'hal' (I can't justify this - if you can you should give a lot more credence to this answer).
'after disposing of' indicates named letters should be taken away.
'second' becomes 'mo' (both are short periods of time).
'quantity' becomes 'm' (m is a kind of quantity).
'm'+'of'='mof'
'mof' with 'mo' taken out is 'f'.
'hal'+'f'='HALF'

Can you help me to learn more?

(Other definitions for half that I've seen before include "Wife the better one (joc.)" , "Some football" , "shared round" , "Measure of beer" , "Fifty per cent" .)

I've seen this clue in The Independent.
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