According to Wikipedia, the word bureaucracy came into use in France just before the French Revolution. (You can head off in a number of directions from there. I won't.)
Instead, I will tell you about an encounter with a woman at a counter which gave me an insight into the process of process.
After waking up at 4.15 am, being driven on a coach for six hours, changing into ski gear and renting skis for half a day, I followed the woman running our trip up to the ski pass desk. She had arranged a discount on half-day ski passes for everyone. Or so she thought.
WOMAN BEHIND COUNTER (WBC): We don't do half-day ski passes.
WOMAN RUNNING TRIP (WRT): But I've arranged a discount. Look. Twenty-five euros for half a day - for everyone in the group.
WBC: Where is this group?
WRT: They're not all here yet.
WBC: [PAUSE] We don't do half-day passes. You'll have to rent a full day. The full day is twenty-nine euros.
A man next to me, finding this unacceptable, stomped off to return his skis. I looked at the tariff on the wall.
ME: (TO WRT) Why don't we get a four-hour ski pass? There's only four hours left anyway. And it's 24 euros. [SHUFFLING FORWARD] (TO WBC) Two four-hour passes, please.
WBC: Of course. That's 24 euros each.
So, to answer the question: "Half a day" is never "a half day". They're completely different. You can tell that by looking at them.