According to most scholars, Mark was the first of the four gospels to be written. Matthew and Luke then relied upon Mark and a second written source called Q. What is the evidence for the existence of Q?
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Q is an entirely theoretical document that nevertheless seems likely to have existed if Mark was the the first written Gospel.
It's long been known that Matthew, Mark, and Luke share significant material and we know from internal evidence that Luke incorporated a variety of sources:
What are the other narratives1 he refers to? Matthew and Mark are obvious candidates as are Luke's unique sources for his genealogy, birth, and resurrection accounts. But as Beorn might have said, "This is the first time I've heard anyone call 3 or 4 many!"
The traditional view is that Matthew was the first written Gospel and that Mark summarized that account for easier consumption. More recently, however, scholars find the case for Markian priority stronger and therefore Matthew was an expansion of Mark. When all three synoptics share text (the triple-tradition), Luke invariably follows the Mark reading rather than the Matthew reading. This strongly suggests that Luke did not actually have access to Matthew as one of his sources.
But Luke does share material with Matthew. Many of these passages match nearly word-for-word. Consider, for instance Jesus' instruction to lay up treasure in heaven:
I've italicized the phrases that are strong parallels. Mark does not contain this saying, so Luke and Matthew could not have copied it from there. Therefore, if neither Luke nor Matthew copied from the other, they must have had a common source2 for this body of shared material. Since we have no documentary evidence of such a source, we simply label it with the German word for "source": Quelle.
Austin Farrer proposed a simpler hypothesis: Luke had access to both Mark and Matthew. Under this scenario, Luke would tend to follow Mark and supplement it with material from Matthew and from his own unique sources. Easily the most attractive element of the theory is that we can dispense with the Q proposal.
But the Farrer theory does not explain why or how Matthew and Luke could present contradictory evidence on key points of the story. What was Jesus' genealogy? Who was at the tomb when the women arrived? How many demon-possessed men or blind men did Jesus heal? How did Judas die? Why does Luke provide his own birth narratives and resurrection appearances without referencing Matthew's different accounts?
Farrer responded to similar objections:
That is certainly true, but we must follow that line of thought to discover what sort of edifice Luke was attempting to construct. Thankfully, Luke tells us in his prologue quoted above:
Assuming Theophilus had both Mark and Matthew in front of him, we can easily sympathize with his confusion. If we add in a few other traditions that we may or may not have, we can further empathize. But adding in Luke seems only to make the problem worse. As a historian, we would expect him to deal with the sources available to him and reconcile (or at least acknowledge) their apparent contradictions. Certainly Luke failed to harmonize the various accounts to the satisfaction of later readers.
Therefore, I find the simplicity of rejecting Q to be counter-balanced by the complexity of explaining Luke's purpose in writing his version of events. It seems more likely that Luke had many sources, but that Matthew was not among them. In this case, Luke succeeded in harmonizing Mark, Q, and Luke's unique sources of biographical information.
1. The NET Bible notes that Luke need not refer to written sources:
In terms of Q, this could imply that the source was oral rather than written. But most analysts find good reason to assume that Q was either written down or a product of careful rote memory.
2. One intriguing possibility is that Jesus was the common source. Although it's unlikely he would have written down his teaching, it's quite likely he required his disciples to memorize his version of the oral Torah. If Matthew really was the author of the gospel bearing his name, he could have simply penned (or dictated) the teaching from Jesus that he had memorized. Similarly, Luke could have transcribed the same material from some other disciple of Jesus. However, as Noah Snyder commented, "Matthew and Luke agree a lot in Greek, a common Aramaic source does not really explain that." This entire proposal is even more speculative than the Q hypothesis.
The Synoptics are very similar to each other, and it's almost universally agreed that this similarity is such that there had to be a literary relationship between them. That is, in many places the authors had access to one of the other Gospels, or that the authors of two gospels had a common written source.
There are two basic patterns which any solution to the Synoptic problem must explain:
There are two main theories which explain the pattern in the triple tradition:
Most scholars find the case for Markan priority to be very strong (and from what I've read I have to say I find the arguments very compelling), but people who subscribe to Griesbach theory hold the latter position. There's already a great answer explaining the arguments for Markan priority, so I won't repeat them.
Assuming Markan priority, how are we to explain the double tradition? Again there are two main possibilities:
Since those are essentially the only two possible explanations of the double traditions (assuming Markan priority) the main arguments for Q are arguments ruling out the first possibility. That is you argue that Luke couldn't possibly have had a copy of Matthew. Here are some of the arguments people make:
If you want to read more about this stuff, there's an introductory textbook available online: Mark Goodacre's The Synoptic Problem. I found it very readable. You should be warned that Goodacre is a proponent of Farrer theory and so disagrees with most scholars about whether Q exists. Nonetheless, he does present both sides of the argument at an introductory level. A shorter, but still thorough summary of the arguments for Markan priority and Q can be found at Peter Kirby's site. He find the arguments for Markan priority overwhelming, and leans towards the existence of Q.