Shamelessly stolen from Hieromonk Silouan Thompson (from our Western ROCOR diocese) - ‘cause it’s a really good (and accurate) explanation (with a few nitpicky corrections/ edits by Fr. Iggy): Someone asked today whether in the Our Father we should be saying “Forgive us our trespasses as we forgive those who trespass against us,” or “our debts... our debtors” or “our sins... sin against us.”
The short answer: “Debts/ Debtors,” in St. Matthew’s Gospel, is the only literal translation for opheilḗmata/ opheilétais. It means what you owe and people who owe you. “Trespasses” is not anywhere in sight. “Trespasses” pops up in Tyndale’s 1526 translation without any antecedent.
St. Luke’s version is rather different: First he uses the totally different word amartías, so his text starts “Forgive us our failures/ errors” and he finishes with the participle opheílonti, so it’s “for we also forgive everybody who is owing us.”
Either way, the verb is not really “forgive.” Greek aphíēmi is to let something go, send away, pass by. If someone owes you a debt, you need to cancel it (Cf Mt 18:23ff). It’s often translated “Remit” which makes remission of sins and remission of cancer sound similar, and that’s not an inaccurate metaphor.
If either Matthew or Luke had used the words parábasis or paráptōma [transgression, violation, overstepping] then “trespasses” would be a great translation (Cf Rom 4:15). But instead Matthew says “debts” and Luke uses amartía, which as far back as 480 BC meant a failure or fault, an arrow that misses or falls short of its target (Cf Rom 3:23).
I expect “trespasses” has crept in from the next verse (Mt 6:14) where Christ explains “Forgive our debts as we forgive our debtors” using the word paraptṓmata, in effect equating a transgression or violation against you with a debt owed to you which you must cancel (Cf Mt 18:23ff).
Incidentally: We have texts from the first century* in which Matthew’s text is recited three times a day (with debts/debtors) but I’m not aware of any tradition of reciting Luke’s text.
Interesting fact about the Didache: It was known and quoted in the Church’s first few centuries, but because it wasn’t canonized as scripture it fell out of use and was lost . We only knew of some quotes and fragments until 1883 when Metropolitan Philotheos Bryennios published an 800-year-old copy that he had discovered in Istanbul.
That means that we don’t base anything on the Didache; nobody says we fast Wednesday and Friday *because* this document tells us to — it’s not a prescription for us, but it’s an ancient witness that confirms what we’re doing was normal back in the first century.
Also interesting: The version of the Our Father in the Didache is slightly different from the version in St. Matthew’s Gospel. No big changes - just spelling, and a few words use shorter forms. That suggests that Matthew’s gospel had not yet been written or anyway distributed and memorized, at the time this text was written down. The Our Father was already being recited from memory across the whole Church before anyone got a copy of Matthew in the mail. Kind of like how St. Paul records in 1 Cor 11:23ff, what he received and passed on [traditioned] ten or twenty years before any of the Gospels were written down. No wonder the Gospels seem to copy from one another; these accounts were already being memorized and passed along orally before the Apostles wrote them down.