What is the origin of the saying "carte blanche"?

It means to give full discretionary power, freedom or permission to do anything - from the original French term adopted into English, meaning a signed blank cheque for which the recipient decided the amount to be given, the translation meaning literally blank paper.

According to the author of www.Word-Detective.com , a second-generation professional etymologist & newspaper columnist,

"As one of the theories you heard implies, "carte blanche" is originally from the French, where it literally means "blank paper." The term "carte blanche" was probably of military origin, meaning an agreement of unconditional surrender submitted by the loser to the victor consisting of a sheet of paper blank except for the defeated commander's signature, signifying that the victor could fill in his own terms. "Carte blanche" first appeared in English in this literal sense around 1700, but by later in the 18th century was being used in its modern "do whatever is necessary" sense."

Literal meaning
Check owners are normally advised to specify the amount of the check before signing it. If created accidentally, blank checks can be extremely dangerous for their owner, because whoever obtains the check could write in any amount of money, and would legally be able to cash it (to the extent that the checking account contains such funds). The 1994 film Blank Check plays on such a situation.

One might give a blank check to a trusted agent for the payment of a debt where the writer of the check does not know the amount required, and it is not convenient or possible for the writer to enter the amount when it becomes known.

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Metaphoric meaning
The metaphor of the "blank check" is thus often used in politics. For example, in the US the Gulf of Tonkin Resolution has been called a "blank check," for it gave the President the power to "take all necessary measures" to prevent "aggression" in Southeast Asia. These powers were then used to escalate the Vietnam War. Many in Congress protested, but were helpless to effect change, for the Tonkin resolution's terms were too subjective to enforce.

This term was also used to describe how the Kaiser of Germany told Austria-Hungary officials that they could deal with Serbia however they wanted after Serbian Nationalists assassinated the heir to the throne of Austria-Hungary, Archduke Franz Ferdinand. This was immediately preceding World War I.

It may also be used in service fields. Customers may tell a company to treat the project as their own, which in essence is carte blanche. (To the extent the service meets normal expectations.)

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