For what it’s worth, I think Christopher Tuckett’s article, The Current State of the Synoptic Problem (2008, given at the Oxford Conference on the Synoptic Problem; republished in Foster, Paul; et al., eds. (2011). New Studies in the Synoptic Problem: Oxford Conference, April 2008 . Bibliotheca Ephemeridum Theologicarum Lovaniensium. 239 . pp. 9–50) is well worth reading. Tuckett favors the two-source hypothesis, but is very fair-minded towards those who disagree with him. He writes:
In one sense, one could say that where we are now in relation to the Synoptic Problem is not very different from where we were 100 years ago. Then, as now, some form of the Two Source Theory (2ST) ruled/rules the roost. Certainly in relation to the theory of Markan Priority (MP), there is widespread (but by no means universal) acceptance of such a theory, at least in some shape or form, to explain the agreements between the gospels in triple tradition passages. There may well be greater doubt about the other main plank of the 2ST [two-source theory - VJT], viz. the Q hypothesis; though again it is probably fair to say that some form of Q theory, or at least a theory that Matthew-Luke agreements which are not due to common dependence on Mark are due to use of some kind of common tradition(s), still commands widespread (but again not universal) acceptance.
Personally, I’m not sure what I think about Q, but Tuckett makes a telling observation, drawing upon the work of Derrenbacker:
One of the points regularly made by defenders of the Q hypothesis is to refer to the phenomenon of order. Many have claimed that, if Luke used Mark and Matthew as sources (i.e. the FGH [Farrer-Goulder Hypothesis]), he must have used them in radically different ways: he appears to have stuck fairly close to the Markan order, but changed the relative order of the Matthean material he decided to include at almost every point. Hence Luke’s general approach to his sources on the hypothesis seems inconsistent. For the GH [Griesbach Hypothesis], the problem may be slightly different, but still present: Luke is using Matthew alone, but it is not clear, at least in general terms, why Luke would have chosen to keep closely to Matthew’s order in part of the tradition (the material which, according to the GH, would be later used by Mark as well), and be very free elsewhere. Further, for both hypotheses, such a radical reordering of one of his sources by Luke would be highly unusual in relation to the general procedure adopted by writers at the time using sources since writers of the time appear to have tended not to engage in any substantial reordering.
By contrast, a more coherent and/or consistent pattern of redactional activity might emerge on the Q hypothesis (especially if Q were a single source). Luke may have preserved the order of the Q material largely unchanged, operating with Q in a very similar way to the way he used Mark. Matthew may have made more changes to the order, but this may have been part of a more general redactional strategy of collecting together related teaching material into his five great teaching “blocks” (a strategy which few would deny in general terms). Thus, it is claimed, a Q hypothesis would make a more coherent picture of the overall phenomenon of the agreements and disagreements in order between the three synoptic gospels than a model which postulates Luke’s use of Matthew.
Finally, on the question of Markan Priority, here’s what the late Fr. Raymond E. Brown wrote on pages 164-165 of his work, An Introduction to the New Testament (Anchor/Doubleday, 1997):
"(1) As explained in Chapter 6, This INTRODUCTION works with the thesis that Matt and Luke used Mark. Yet for many centuries the dominant view was Augustine’s thesis that Mark was little more than an epitome of Matt; and recently attention has been given to the (modified) Griesbach hypothesis wherein Mark drew on Matt (p. 113 above). It is instructive to test the theological consequences of positing Marcan dependence on the other Synoptics. For instance, Mark would have omitted the Lord’s Prayer and the four beatitudes that Matt and Luke agree upon. As for christology, if Mark was written after Matt and drew on it, at a period when the title “God” for Jesus was becoming more common, Mark 10:17-18 would have complicated Matt 19:16-17 by gratuitously introducing an objection to giving Jesus a title that belonged to God alone. Mark 6:5 would have introduced the idea that Jesus could not do miracles at Nazareth, changing the statement of Matt 13:58 that he did none.
“Some claim that Matthean priority and Marcan dependence support traditional Roman Catholic positions, but Mark’s presentation of Mary and Peter becomes all the more difficult if the evangelist knew Matt and/or Luke. Mark would have deliberately omitted the infancy narratives of Matt and Luke, even the details in which they both agree, including the conception of Jesus through the Holy Spirit. Mark would have consciously added two items lacking in Matt and Luke pertinent to Mary, namely, that Jesus’ own family thought he was “beside himself” (3:19b-21) and that he received no honor from his own relatives (6:4). As for the Marcan view of Peter and the apostles, Mark would have deliberately omitted both Matt 16:16-19 that makes Peter the rock on which the church was built, and Luke 22:31-34 that has Peter strengthening his brothers after his own failure. (Even though those are not passages shared by both Matt and Luke, Mark can scarcely not have noticed the impact of omitting such positive passages.) Mark would have deliberately omitted the promise of Jesus to the disciples in Matt 19:28 and Luke 22:29-30 whereby they would sit on thrones judging the twelve tribes of Israel. Mark 4:38 would have made the disciples more rude to Jesus than they were in Matt 8:25. Using a book with the Gospels in parallel columns, readers are invited to test other examples of Marcan thought and procedure in the Griesbach hypothesis.”
And on the reliability of Papias, here’s a relevant footnote (36) from Matthew Ferguson’s online essay, Why Scholars Doubt the Traditional Authors of the Gospels (2013):
A minority of mainstream scholars have defended the 2nd century attributions and “eyewitness” status of the canonical Gospels, perhaps most notably Richard Bauckham in Jesus and the Eyewitnesses . (Although, even Bauckham does not think that the disciple Matthew or John the son of Zebedee authored the final versions of the gospels attributed to them.) Bauckham’s study deals less with defending the authorship of the titular names affixed to the Gospels, as much as presenting arguments that eyewitnesses lie behind the traditions and sources in the Gospels.
And here’s another footnote (28):
A common apologetic slogan about the church fathers’ attributions is that they allegedly “universally agreed” upon Matthew, Mark, Luke, and John, and thus were not speculating about the authors. However, this is only true of the later church fathers from the latter half of the 2nd century CE onward into the 3rd and 4th centuries–such as Irenaeus, Tertullian, Origen, and Eusebius–whereas the earliest references to the texts treat them anonymously (likewise discussed in footnote 20 above). Ignatius (c. 105-115 CE), Polycarp (c. 110-140 CE), the Epistle of Barnabas (80-120 CE), and the Didache (c. 50-120 CE), for example, allude to or quote the Gospels anonymously. As NT scholar Bart Ehrman explains, “in any of the writings of the Apostolic Fathers–ten proto-orthodox writers, most of them from the first half of the second century”–the Gospels are not identified by their traditional names, but are treated anonymously. Later, Justin Martyr (c. 150-160 CE) refers to the Gospels collectively as the “Memoirs of the Apostles.” Later still, Irenaeus (c. 175-185 CE) finally refers to the four Gospel canon with the names of its traditional authors.
What about Papias? Ferguson argues that the question of his reliability hinges on which John he knew:
It should be noted that even Richard Bauckham ( Jesus and the Eyewitnesses , pp. 452-463) argues that both Papias and Polycarp were hearers of a separate “elder” John, and not John the son of Zebedee. As James McGrath explains in “Which John? The Elder, the Seer, and the Apostle,” by the time of Irenaeus in the late-2nd century CE, different figures in the early church named “John”–such as John the son of Zebedee, John of Patmos, and John the Presbyter (or, “elder” John)–had become conflated in their identities. One of the reasons for differentiating John the Elder from John the son of Zebedee, McGrath explains, is the fact that “Papias mentioned his efforts to find out what a variety of key figures, including John the apostle, said (using the past tense), and also what Aristion and John the Elder say (using the present tense).” This change in tense suggests that Papias is referring to John the son of Zebedee as a past figure, whereas the John the Elder was a subsequent figure, alive during his own day. Due to this ambiguity, and the general unreliability of the available sources, it is untenable that either Papias or Polycarp knew John the son of Zebedee, or any of the twelve disciples.
Food for thought. Cheers.