In short, the argument for traditional authorship are quite simple: 1) Our earliest traditions (i.e., Papias, Irenaeus) claim that the gospels were written by Matthew (a disciple of Jesus), Mark (an interpreter of Peter), Luke (a companion of Paul), and John (the beloved disciple of Jesus). 2) There is no evidence that definitively and conclusively contradicts this early tradition. 3) Therefore, the tradition is mostly likely correct.
Consider the following quotes from Donald Guthrie's New Testament Introduction.
"To sum up, it may be said that there is no conclusive reason for rejecting the strong external testimony regarding the authorship of Matthew, although some difficulties arise from source hypotheses."*
"So strong is the early Christian testimony that Mark was the author of this gospel that we need do little more than mention this attestation."**
"The Apostle John. This, as has been seen, is the traditional view, which has much support for it in the internal evidence. Indeed, it may be said that there is no evidence which conclusively disproves it, in spite of much opposition to it."***These three statements nicely illustrate the late Dr. Guthrie's argument for traditional authorship. His position is that, without definitive and conclusive proof that early Christian testimony (read Papias, Irenaeus, et. al.) is incorrect, the traditional authors should hold.
A clarification is needed here. Dr. Guthrie introduces the concepts of internal and external evidence. Internal evidence refers to what we can determine from the text of the gospels themselves, and external evidence refers to what others (early Christians) have said about the text. For Guthrie, external evidence is the trump card that squashes all opposition. For Guthrie, external evidence is primary, and internal evidence is secondary. Consider this quote from his discussion of the authorship of Luke:
"It is against the background of the strong external evidence that the witness of the books themselves [i.e. internal evidence] must be considered."****This line of reasoning in my mind is faulty. Now, if Papias and Irenaeus were first century authors and their statements about the gospels were clear and unambiguous, that would be another matter. As it stands, Papias was writing 50-75 years after the gospels were written (not to mention the fact that the Papias quote is only preserved from Eusebius in the fourth century) and Irenaeus nearly 100 years after the writing of the gospels, I would actually reverse Guthrie's argument and claim that the internal evidence is primary, and the external evidence is only secondary.
Now, I want to be fair to the late Dr. Guthrie. He does deal at length with all of the evidence, and does so in a nuanced and fair manner. But, at the end of the day for Guthrie, no internal evidence could trump the external witness of second century Christians unless it were absolutely conclusive. So there it is. Guthrie provides one of the best scholarly attempts to argue for traditional gospel authorship. What do you think?
*Donald Guthrie, New Testament Introduction, (Downer's Grove, IL: Intervarsity Press, 1990), 53.
*** Ibid, 275.
**** Ibid, 115.