‘Q‘ is the title Biblical scholars have given to an entirely hypothetical document which is sometimes believed to underlie the content which is common to Matthew and Luke and which isn’t found in Mark. The designation of ‘Q‘ is shorthand for the German word Quelle which means ‘source’. I heard one argument against Q once which went something like this: “the Church Fathers never mention its existence and seem completely oblivious to it, and given that it is entirely hypothetical (no trace evidence of it has ever been found in the manuscript tradition), it seems we don’t have any good reasons to believe Q exists.” Against this we might observe that something like ‘Q‘ is mentioned by a Church Father name Papias of Hierapolis.
Papias of Hierapolis makes a reference to “Matthew who composed the sayings.” Perhaps Q did exist as a document recording the sayings of Jesus, and perhaps it was a basic text which was used by Matthew, Luke, and perhaps it is even the source underlying what later evolved into the ‘Gospel of Thomas’ as it has come down to us today in the Coptic form at Nag Hammadi. I am tempted to think that the Gospel of Thomas went through (at least) three stages; the first original composition was a primitive sayings source, the second stage consisted of additions to some sayings, and the addition of some sayings altogether, and then finally the third stage consisted of a kind of ‘Gnostic’ redaction (though I won’t argue anything like that here, and in any case it’s just a pet theory).
However, it would make sense that Matthew composed a list of sayings, and perhaps it makes sense that Thomas-Christians took it seriously, since the Apostles were paired into pairs of two, and Matthew and Thomas are always named in the lists of the Apostles together (though in Matthew’s Gospel Matthew’s name comes after Thomas’, instead of before, and it adds the pejorative adjective ‘tax-collector’ to his name). If there was a sense in which both Matthean and Thomas communities understood some Q source to circulate under the authority of Matthew it would make sense for them to turn to it and keep it.
Moreover, imagine that Matthew did write Q, and then composed his Gospel using that source as a kind of ‘primary’ source – doesn’t that seem plausible if Matthew was really the author of not only the Gospel which we receive under his name, but also of a ‘sayings’ source according to Papias? Imagine you were Matthew and working up to composing a written account of the life of Jesus, you may well have committed many of Jesus’ sayings to writing as a memory aid before hand. Moreover, perhaps we could say the same about the Semeia source and the Gospel of John – that John recorded and took note of Jesus’ miracles in a certain order (thus “the first miracle…” [John 2:1-11] “the second miracle…” [John 4:54] and so on) for homiletic purposes, and then incorporated those notes into his Gospel.
Maybe, instead of imagining that the Gospels as we receive them are the first drafts of the Apostles, or imagining that the Gospels were in a state of constant evolution, expansion and redaction for a century and a half, we should just imagine that some Apostolic men understood it to be part of their calling to compose a written witness to the Gospel Kerygma, and this work they devoted themselves to for much of their lives. Thus the Gospels, as they left the hands of those Apostles or Apostolic men, were already in their 3rd or 5th (or whatever) editions at by the time of the Apostle’s death.