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late 14c., "praised;" mid-15c., "assigned as a due share;" late 15c., "permitted," past participle adjective from allow.
early 14c., allouen, "to commend, praise; approve of, be pleased with; appreciate the value of;" also, "take into account or give credit for," also, in law and philosophy, "recognize, admit as valid" (a privilege, an excuse, a statement, etc.). From late 14c. as "sanction or permit; condone;" in business use from early 15c.
The Middle English word is from Anglo-French alouer, Old French aloer, alloiier (13c.) "allot, apportion, bestow, assign," from Latin allocare (see allocate). This word in Old French was confused and ultimately merged with aloer; alloer "to praise, commend," from Latin allaudare, adlaudare, compound of ad- "to" (see ad-) + laudare "to praise" (see laud). From the first word came the sense preserved in allowance as "money granted;" from the second came its meaning "permission based on approval."
Between the two primary significations there naturally arose a variety of uses blending them in the general idea of assign with approval, grant, concede a thing claimed or urged, admit a thing offered, permit, etc., etc. [OED].Related: Allowed; allowing.