Matthew assigned the name to the Gospel of Matthew.
Mark assigned the name to the Gospel according to Mark.
Luke assigned the name to the Gospel of Luke;
and John assigned the name to the Gospel of John.
If they did not then they certainly vouched for their own authorship. As soon as there was more than one written gospel account it was necessary to distinguish easily between the two of them with a name.
I am a Reformed Calvinist Baptist. One of the best books I have read in recent years (come to think of it, ever read) is by the Roman Catholic scholar Dr Brant Pitre, "The Case For Jesus", 2016. This is aimed at the lay reader. (If you want you can read my review along with many other 5 star reviews on Amazon.co.uk.)
Brant Pitre lays to rest the possibility that the Gospels were originally anonymous. I quote:
First, there is a striking absense of any anonymous Gospel manuscripts. This is because they don't exist. Not even one. The reason this is so significant is that one of the most basic rules in the study of New Testament manuscripts (a practice known as textual criticism) is that you go back to the earliest and best Greek copies to see what they actually say. Not what you wish they said, but what they actually say. When it comes to the titles of the Gospels, not only the earliest and best manuscripts, but all of the ancient manuscripts - without exception, in every language - attribute the four Gospels to Matthew, Mark, Luke, and John.
There is one book in the NT that really was given anonymously... the letter to the Hebrews. Dr Pitre demonstrates to us what happened in this case: from the 2nd to the 4th century the copies only had as the title "To the Hebrews". Starting in the fifth century additions to the title began: one manuscript adds "written from Rome", another "written from Italy", another "written from Italy by Timothy", another "written from Rome by Paul to those in Jerusalem", and finally "written in Hebrew from Italy anonymously by Timothy"(!) But in the case of the Gospels there is no deviation from the four names attributed. Unanimous indicates not anonymous.
Just one more slightly shortened quote, else Brant will start chasing me for theft:
The Anonymouos Scenario is Incredible....Think about it for a minute. According to the theory of anonymous Gospels, the Gospel of Matthew was "originally" the gospel according to nobody. It was copied by hand, and recopied and recopied, and circulated throughout the Roman Empire for decades....[Likewise with the other three Gospels.] Then sometime in the early second century AD, the exact same titles were supposedly added to not one, not two, not three, but all four of these very different, anonymous books. Moreover, this attribution of authorship supposedly took place even though by the second century the four Gospels had already been spread throughout the Roman Empire: in Galilee, Jerusalem, Syria, Africa, Egypt, Rome, France, and so on, wherever copies were found.
When it came to the false gnostic inspired "gospels", the authors thought it necessary to attribute their "gospel" creation to one of the Apostles, eg "the Gospel of Thomas", to try to add to their credibility. So why were two of these four gospels attributed to relatively insignificant authors, ie "Mark" and "Luke", if the titles were added to try "add credibility" as some claim?
I can't resist one more quote (really sorry Brant, I hope you get more purchases from this).
Before I quote, you obviously appreciate that some manuscripts of a Gospel will not have a name of the author simply because only a portion of the middle of the said gospel remains extant. To have a name you must have "page 1", as it were, of the gospel where the name is going to be written. In some cases the name is given (probably again) at the end of the gospel, and the beginning has deteriorated (so we do not know if the name was put at the front as well as at the end).
Anyway, Brant gives a list of all 27 of the earliest manuscripts which have "page 1" or the end "page". Now in all 27 cases the title on the front page or at the end is either "Gospel according to (name)" (i.e. Matthew Mark Luke or John) or "According to (name)". The word "Gospel" appears in 18 titles and is missing in 9 titles. (The earliest of them all are Papyri 4 (Matthew), 62 (Matthew), 66 (John), & 75 (Luke & John)).
Then he quotes Bart Ehrman:
Because our surviving Greek documents provide such a wide variety of (different) titles for the Gospels [emboldening mine], textual scholars have long realized that their familiar names (e.g., "The Gospel according to Matthew") do not go back to a single "original" title, but were added later by scribes." (Note 16: Quoted from "Jesus: Apocalytic Prophet" by B Ehrman, 248-49n1.)
Brant Pitre continues:
Look back at the chart [of the 27 manuscripts] showing the titles of the earliest Greek manuscripts. Where is the "wide variety" of titles that he is talking about? The only significant difference is that in some of the later copies, the word "Gospel" is missing, probably because the title was abbreviated. In fact, it is precisely the familiar names Matthew, Mark, Luke, and John that are found in every single manuscript that we possess! According to the basic rules of textual criticism, then, if anything is original in the titles, it is the names of the authors. They are at least as original as any other part of the Gospels for which we have unanimous manuscript evidence.
Well I'm done, but to finish:
Because our surviving Greek documents provide such a wide variety of titles for the Gospels...
In the light of Brant's presentation of the evidence that statement is just mind-bogglingly misleading and scandalous.
That may not be the answer you were looking for, but I think it is beyond question the correct one. I recommend Brant's book to all, it isn't a slog to read in the slightest - it would make a great, slightly different, birthday or Christmas present, too.
The only way that precisely the same names could have been added in the early 2nd century to these four Gospels is if there was a huge "conspiracy" across the whole Roman Empire and beyond. Such a conspiracy would have required a central ecclesiastical authority to impose its will on all the Church. The nearest claim to such a central authority would have been the Bishop of Rome. So the best line of pursuit would be to ask the added question "What authority did the Bishop of Rome exercise over the whole worldwide Church in the first two centuries AD?"
For a whole world of Christians to see the appearance of names to previously anonymous works without any mention of it to successive generations would require a very excessive level of worldwide self-deception. It would have required a level of cultish control in the early church for which there is absolutely no evidence.
(For an idea of how cults manage to gain such a level of control over their adherents I suggest Steve Hassan's books, and his "BITE model" for starters... Behaviour Control..incl "keep 'em busy", Information Control.. incl. "tell 'em all info outside the group is devilish/unnecessary to pay any attention to", Thought Control, Emotional Control.. "Keep 'em feeling guilty they are not doing enough, & give 'em a fear of leaving the group, you will lose all hope of the real salvation".)
Furthermore, surely unbelieving parents would warn their children of the deceit to try to stop their children being persuaded by these works (i.e. the Gospels). And some of these warnings would have been made by scholars at the time, and some of these warnings would be extant today: there are no such warnings extant. So the conspiracy would have had to have extended to the seizure of such written warnings.
Obviously, there is no historical evidence of any such attempts of the suppression of material hostile to a Church's agenda, itself unsubstantiated, in the first two centuries.
In addition to all the above, the existence of
variations in the text of the Gospels shows the Gospels were not edited or controlled by any central authority. I am referring not only to significant variations such as the location or existence of John 7:53-8:11 in John's Gospel, but scores of trivial variations as well:
These variants prove there was no central committee deciding to add titles to all Gospel manuscripts in the second or third century.
It might interest the reader to compare the lack of historical evidence of any central authority in the early church capable of exercising such an enormous sway over the whole Church, and lack of any historical evidence of any conspiracy within the Church in the creation of the NT and Gospels of the Scriptures, with the situation in the Muslim world when the Quran we have today was being created:-