Mainstream scholarship widely supports the view that the authors of Matthew and Luke shared a common source, since lost, in contemporary discourse called "Q".
Yet it is easily plausible that the common source between them is simply whichever of them occurred first, such as to be then used by the other.
Indeed, the synoptic "problem" appears easy to resolve through the proposal that Mark appeared first, followed by Matthew, which referred to Mark, and that Matthew was in turn followed by Luke, which referred to both predecessors. Such a proposal is offered here not definitely to reject other possibilities, only to suggest that the supposed "problem" might not accurately be portrayed as a serious or difficult dilemma.
(The Mark-Matt-Luke chronology with maximal dependency among them and no shared extinct source is known as the Farrer-Goulder Hypothesis.)
Why are the contrasted features of the three texts argued to be problematic, and why do so many suggest that Q provides the best available solution?
I appreciate the link to the article by Peter Kirby. Although I cannot evaluate how much it is representative of the wider scholarship, it is certainly more concise, precise, and directed than the other summary accounts I have seen.
Unfortunately, I fail to be persuaded, though I welcome further commentary that might clarify the steel-man representation, that is, the most compelling possible argument, for the position.
I would share my thoughts on the Kirby summary:
Currently, the Wikipedia article on Q, like other accounts I have seen, frames the hypothesized source as substantially a list of sayings and teachings. In contrast, the summary includes references to the genealogy, nativity, and resurrection accounts, the former two notably creating a glaring contrast between Matthew and Luke. A source centered on teachings during adult life would not likely resolve questions surrounding birth and death. Of course Q plausibly could be framed as including these details. At issue is not the possible scope of the document, but rather that any such unresolved ambiguity threatens dialectal consistency between facts in the premises and inferences in the conclusion. Any particular framing and argumentation must resolve this ambiguity outright, and preserve consistency throughout.
The stance largely betrays an urgency to depict the authors' intention and skills, especially Luke's, as elevating above other objectives factual precision and rigor, such that the authors would have been unwilling to make any alteration that detracts from providing the reader with a comprehensive and accurate account of events. Yet plainly all three synoptic texts, replete with emotional poignancy, rhetorical structure, and poetical presentation, are, in addition to anything else, works of creative literature. As the arguments apparently rest on the belief that the authors pursued a fact-centered method, they collapse unless the possibility can be excluded that an author based decisions on his subjective perception of how they contribute to the artistic literary quality the resulting text. Even as the view is common that the synoptic texts are not fact-centered accounts, the argumentation variously dismisses or ignores the relevance of artistic tastes or motives. The controversy need not entail whether the texts might describe real events, only whether they might also express artistic literary intent as a competing concern.
The argumentation entails an onus-shifting fallacy, wherein the perceived lack of clear indication of an author's motives and rationale for usage of two extant sources in a discernibly consistent and systematic way is claimed to support the conclusion that the author did not rely on those two sources. If Luke had reasons to weave together two sources and embellish them in some particular fashion, then those reasons simply may have died with Luke. Equally, even he might have been unable to articulate such reasons were he asked while alive. Any or indeed every scholar's inability to conceive of Luke's reasons for making some choice cannot lead to a conclusion about whether he made that choice. Given that inferring motives and rationale is central to the critical practice of studying text, it may be tempting to presume that every one indeed may be reliably inferred, but ultimately the number of questions that one might hope to resolve long after the fact will vastly exceed the number that one accurately can resolve. Confidence that anyone can reasonably know why another did all that he did will surely generate erroneous results.
The discussion omits consideration of the intrinsic probability that the particular, hypothetical text was indeed created and then subsequently lost. While surely many documents have perished, in fact most that ever were created, the Q hypothesis asserts the occurrence of a text carrying details on a particular subject and in possession by particular authors, yet that was not preserved, nor even mentioned in extant text, while still other texts in possession of the same individuals were preserved. Each of these particular constraints lowers the probability that the entire sequence of events occurred as described, and augments the accumulated demand on the strength of evidence to support the hypothesis. Weighed against the vast uncertainty in the speculation supporting the conclusion, the argumentation would seem to fall short of such demand.