The phrase "when pigs fly" is an adynaton—a figure of speech so hyperbolic that it describes an impossibility. The implication of such a phrase is that the circumstances in question (the adynaton, and the circumstances to which the adynaton are being applied) will never occur. Because of the historical importance of the pig industry to the city, prominent in the local iconography of Cincinnati are such events as the Big Pig Gig and the Flying ig Marathon.

The phrase is often used for humorous effect, to scoff at over-ambtion. There are numerous variations on the theme; when an individual with a reputation for failure finally succeeds, onlookers may sarcastically claim to see a flying pi. ("Hey look! A flying pig!") Other variations on the phrase include "And pigs will fly," this one in retort to an outlandish statement.
An example occurs in the film The Eagle Has Landed: an Irish secret agent working for the Nazis replies to a German general speaking of Germany's shortly winning World War II, "Pigs may ly, general, but I doubt it!" Later, when the Irishman sees German soldiers parachuting before an attack, he says to himself, "Mother of *#d! Flying pigs!"
An identical phrase, used to express impossibilities, exists in Romanian, Când o zbura porcul, literally meaning "When the pig shall fly"; an equivalent also implying an animal is La Paștele cailor, literally: "on horses' Easter". Similar phrases in English include "when hell freezes over", the Latin expression "to the Greek calends," and "and monkeys might fly out of my butt", popuarized in Wayne's World skits and movies. They are examples of adynata. In Finnish, the expression "kun lehmät lentävät" (when cows fly) is used because of its alliteration.
The idiom is apparently derived from a centuries-old Scottish proverb, though some other references to pigs flying or pigs with wings are more famous. At least one appears in the works of Lewis Carroll:
"Thinking again?" the Duchess asked, with another dig of her sharp little chin.
"I've a right to think," said Alice sharply, for she was beginning to feel a little worried.
"Just about as much right," said the Duchess, "as pigs have to fly...." — Alice's Adventures in Wonderland, Chapter 9.
The band, Pink Floyd, had a hit with a song called "Pigs on the Wing". It is from the album Animals.

American literature author John Steinbeck was told by his professor that he would be an author when pigs flew. When he eventually became a novelist, he started to print every book he wrote with the insignia "Ad astra per alas porci" (to the stars on the wings of a pig).
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With sufficient thrust, pigs fly just fine.