I don't know if the phrase has currency outside of where I grew up, but back in northeastern PA, people let their dogs out to "do their DUTY." When Nature calls, dog and man must obey.
I used to be much fonder of Immanuel Kant's categorical imperative — that’s the fundamental moral law that he recognized — than I am now. He conceives the moral life in terms of DUTY rather than of love, and, with the pious soul of a Prussian, he goes so far as to say that if you enjoy doing the right thing, it has no moral value. Only when you do your DUTY in the teeth of what repels you do you earn any Kantian points. That is quite the opposite of what Thomas Aquinas says. We grow in virtue by delighting in the good, so that when we are fully blessed, we will express our freedom in doing naturally what the good thing to do is. We will be fulfilled not in gritting our teeth and sweating, but in delight: and we will do the good as freely and inevitably as, to use the image that Dante puts on the lips of Thomas, a river that flows downhill to the sea.
Still, I think it's unfortunate that you don't hear much about DUTY these days. The reason why we have freedom of religion is not because the Founders thought that every man's strongly held opinion was sacred. Many strongly held opinions are stupid and worthless. It was that they knew that man had a DUTY to God. Religion, properly speaking, is a system of DUTIES. That perhaps comes across better in German than in English. In English, you go to a SERVICE, and these days you are apt to think that you are the one being SERVED, with a sermon and some music you like and coffee and doughnuts afterward. But the German for it is GOTTESDIENST, the SERVICE that belongs to GOD, that is FOR GOD; EIN DIENER is a SERVANT (cf. Scots and archaic English THANE, the servant of a lord; both are cousins of Greek TEKNON, a CHILD or a SERVANT). People have to fulfill their DUTIES, and that is why we owe them the liberty to do so.
The right is a right because it is a responsibility to fulfill, as a lot of people wiser than I am have said. Hence it is egregiously wrong to compel people to violate those DUTIES, either by compelling them to treat them with contempt (as if you might compel a Quaker to bear arms in wartime, or an evangelical florist to make a display for Gomorrah), or by preventing them from fulfilling them in ways that do not break the peace. It is an INALIENABLE freedom, coterminous with man, because man is man and not God, and man is man and not some beast, of whose doings, as Milton says, "God takes no account." The dog does his DUTY by nature, and faces no reckoning.
The word comes to us as so many thousands of others do, from Latin through the Norman French. The Latin was DEBITUS (which became English DEBT, also through French, spelled DETTE in Chaucer's time, because the B had vanished; cf. Italian DETTA; but the pedants later inserted the etymological B, to make the word look like its origin). DEBITUS itself, or DEBERE, the infinitive, was itself a contraction: DE + HABERE, to KEEP AWAY or GIVE AWAY. You OWE it to someone else, that is. It is no longer your OWN.
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