Bill is somewhat of an auto-antonym, since it can mean either a piece of paper which has positive monetary value (i.e. a note), or a piece of paper which has negative monetary value (though it only applies to a certain person). Maybe I'm reading too much into the usages, but it seems strange that the two meanings of bill are nearly opposite. Does anyone know why this is?
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Without doing too much research into this, the etymology of bill shows that the original usage was more like an official document at first.
This suggests that different people adopted the word to mean similar, but related, concepts. In fact, government currency is a form of money owed, in that (in the days of the Gold Standard) the government promised you the bill's face value in gold. Now it is more abstract but the terms are related, even if seemingly opposite.
In general, this situation can arise for two main reasons:
(1) a word that originally meant a narrower concept diversifies to cover related concepts or specific connotations, and gradually over time the precise meanings of these once strongly related concepts can further diverge over time;
(2) what were once two separate words could merge due to sound changes or other phonological processes.
In this case, as mentioned in the answer above, it appears that effectively situation (1) has arisen: "bill" expanded from its original meaning of "official document" to cover various different specific connotations of "document" today, be it the "bill" put up by a "bill poster", the "bill" you pay for your meal, a "bill" of rights, a "dollar bill" etc.
A "bill" is an evidence the party A owes money to party B.
The "bill" that arrives in the mail from say, the phone company, is evidence that you owe money to the phone company.
A "bill" in the currency sense, is evidence that the government owes money to you. If you give the bill (instead of writing a check) to the phone company, the government then owes money to the phone company.
(This came about because "money" used to be gold and silver. In the U.S. the 16th Amendment to the Constitution gave the Federal government the authority to issue paper money as a substitute for metal. So (most people turned in their gold and silver and received paper "bills" (IOUs) in return.