“That dog won’t hunt monsignor” is an old southern phrase that refers to a dog that is not capable of hunting. Today, it’s used to mean literarily that something is not relevant or someone’s opinion does not matter. The phrase originates from the southern United States. It is used to refer to a hopeless situation that is useless to pursue. Now, the phrase is used on broader terms, but it still has the same meaning.

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The quote is drawn from the 18th century phase, “the cock won’t fight”. In those days when there was cockfighting, the phrase is used to refer to a situation that won’t work. It is used to refer to a theory or plan that’s destined to fail. So, the phrase is used in that way to warn or offer advice on the situation. In this context, the saying “that dog won’t hunt monsignor” is used to mean the same thing.
Phrase derived directly from hunting dogs
Hunting dogs are usually goal getters that don’t stop until they find what they are looking for. When they perceive there is something around, they are always bent of exposing or digging out that particular thing. However, a hunting dog that doesn’t hunt is in a useless situation. It means it’s a waste of time taking the dog out for hunt. That is how the phrase was formed.
“That dog won’t hunt monsignor” used in other situations
Someone who wants to go into fruitless mission can be advised not to waste his/her time. The adviser can chip in the phrase as a direct pointer to the already doomed activity before it begins. The phrase only covers the “that dog won’t hurt”.
But the addition of monsignor is used to refer to the person the message is referring to. Monsignor is an Italian word used to describe clergy men or respected people in the society, just as sir is used.
Phrase used by Fry in the Futurama Episode
This statement was recently used by Fry in the futurama episode Fry utilized the word in the episode in accordance with its rightful meaning. The situation he used the word matches exactly the words he used to refer to the situation. In the episode, he meant to say the situation they are facing isn’t going to work as planned. He stressed that there is no need to depend on the situation because it would still be a fruitless mission.