Only two of the canonical Gospels, Matthew and John, are alleged by tradition to have been written by eyewitnesses - but I will also address Mark and Luke.
First of all, I should say that none of the four canonical Gospels names its own author, none of them claim to be eywitness accounts or even to have spoken to eyewitness of Jesus. All are written in the third person and none of the authors tell us anything about themselves. All of the traditional ascriptions of authorship come from 2nd century tradition.
G.Mark
The first gospel written is Mark. Mark is not by tradition an eyewitness account but 2nd century tradition casts him as a secretary of the Apostle Peter who haphazardly wrote down everything Peter said in no particular order.
The basis for this tradition stems from a single claim by Papias who said (c. 130 CE) that he got the information from John the Presbyter (not to be confused with John the Apostle). That's it. That's the entire case for Mark as a secretary of Peter.
Now let's examine the credibility of this claim.
First, Mark does not say that he knew Peter, talked to Peter, ever met Peter or got any information from any eyewitness.
Secondly, the author is extremely hostile to Peter. Mark is a decidedly Pauline, anti-Jewish and anti-Petrine diatribe. Mark is very hostile to the apostles in general and to Peter in particular. He takes every opportunity to depict the apostles as being dense and not getting Jesus' true message (reflecting the tension between Pauline communities and the Jerusalem cult in the last half of the first century). More to the point (and this is important) Mark does not give Peter any redemption after his betrayal. Mark does not grant Peter and appearance from Jesus. Mark's Peter denies Jesus, runs away and that's it. Now why would a Petrine memoir not include a Petrine witness of the resurrection? Wouldn't that be the most important part? How does it make any sense to exclude it?
Thirdly, the book is quote obviously a literary construction and is manifestly not a transcription of oral anecdotes. The literary structure of Mark, both in its chiastic forms and its use of the Hebrew Bible as a allusory template or "hypertext" preclude the possibility of transcribed oral tradition. GMark is a carefully constructed literary work.
It should also be mentioned that Mark is a Greek composition which shows no signs of translation from Aramaic, the language of Peter and the language he would have dictated his memoirs in.
Fourth, Mark makes a number of errors regarding Palestininan geography and Jewish laws and customs which show that his information could not have been collected from a Palestinian Jew. Mark's passion, in particular, is so riddled with factual. historical and legal inaccuracies that it cannot be historical and cannot have come from an eyewitness. (I will address the specific errors in the section devoted to that subject)
Fifth, the book could not have been written during the lifetime of Peter. Mark knows about the destruction of the Temple which means that Peter was dead (at least by Christian tradition) when the book was written.
To summarize, the canonical Gospel of Mark is an anonymous book written outside of Palestine in a Gentile language to a Gentile audience sometime during or after the Jewish-Roman War. The author is hostile to Jews and to the apostles. He does not know Jewish laws or customs. He does not know the geography of Palestine. He does not like Peter. He never makes any claim to have known Peter or to have ever been to Palestine.
In 130 CE some guy said he heard from another guy that the author was a secretary of Peter's.
G.Matthew
Let's move on to Matthew. The Gospel of Matthew, by tradition, is attributed to the apostle of that name. Like Mark, this authorship tradition stems from Papias (it was also claimed by Irenaeus but he was probably parroting Papias). Papias clamed that, "Matthew composed the sayings [of Jesus] in the Hebrew language, and each one interpreted them as best he could." In Adv. Haer. 3.1.1.
If such a Logia ever existed, it is not Canonical Matthew. GMatt is not a sayings gospel for one thing and was not written in Hebrew for another. Furthermore, GMatt is largely dependent on Mark and (most probably)another written sayings tradition (in Greek, not Hebrew) called Q. Matt's dependence on Mark also puts its date somewhere around 80 CE (if not later) which is pushing the envelope for the plausibility of the author being a contemporary of Jesus. It's not impossible, of course, but this is an era when people generally didn't live much past forty or fifty years of age.
The bigger obstacle for apostolic authorship is that fact that Matthew copies so extensively from secondary sources. An eyewitness should not be expected to copy verbatim from a non-eyewitness.
There is also the fact that GMatt contains some of the more demonstrable fictions and signs of OT cannibalism but more on those aspects in their proper sections.
It also bears repeating that the author Matthew never claims to have been an apostle or a witness, never states his name and never claims to have known any other witnesses.
To sum up for Matthew:
Papias claims that an apostle named Matthew compiled a sayings Gospel in Hebrew.
The Canonical Gospel of Matthew is written in literary Greek and is not a sayings gospel. The author never claims to have been an apostle or an eyewitness. It relies heavily on secondary Greek sources as well as the Septuagint. Once again, an eyewitness would not rely on the accounts of non-witnesses to recount events that he had supposedly seen for himself. It was written at least 50 years after the alleged crucifixion. The author includes demonstrable fictions which can clearly be shown to have been derived from the Septuagint.
Papias' Logia, if it existed, has never been found.
G.Luke
The traditional author of Luke-Acts is supposedly a physician and travelling companion of Paul named Luke. Neither Luke nor Paul is a witness of Jesus even by tradition so I suppose I could stop right there but I think I'll take the time to point out that even the tradition which does exist is dubious. First of all, the author of Luke-Acts never claims to have known Paul. The earliest known claim for this tradition comes from Irenaeus in the late 2nd century who probably based his conclusion on the "we passages" from Acts as well as a stray mention of someone named Luke in Philemon (the name turns up in a couple of the non-authentic Pauline letters as well but the authentic corpus onle mentions the name once in passing).
There is no reason whatever to suppose that the Luke mentioned by Paul has aything to do with either GLuke or Acts.
The "we" passages in Acts are those passages during which the narrative voice changes from third person to first person plural. This is the source of the supposition that the author of Luke-Acts was a companion of Paul's but Vernon Robbins has shown that this was merely a Greek literary device for describing sea voyages.
Furthermore, Luke knew Josephus, which puts that gospel into the mid 90's CE at a bare minimum and probably later. This means that Paul had been dead 30 years before Luke-Acts was written. It is highly unlikely, then, that the book was written by a companion of Paul and there is absolutely no reason to connect the "Luke" who is so casually mentioned by Paul in one letter to the composition of Luke-Acts.
Furthermore, Luke is dependent on both Mark and Q which (contrary to some Christian folklore) means that Luke had no access to first hand accounts from other witneses.
There are also historical inaccuracies in Luke as well as contradictions with other Gospels which I will get to in time.
So, to sum up Luke, it is an anonymous gospel whose author makes no claim to first hand knowledge and no claim to knowledge even of Paul. It was written more than a half century after the crucifixion, is dependent on secondary sources and contains numerous historical errors and contradictions with the other gospels.
The fable of a physician named Luke who travelled with Paul comes from a claim made 150 years after the crucifixion and is corroborated by nothing in the text itself.
G.John
By tradition, the GJohn is written by the apostle of that name and is also identified as the mysterious "Beloved Disciple" mentioned within the text. This tradition, like Luke, stems from a late 2nd century claim by Irenaeus (who is known to have confused John the Apostle with another John, called 'the Presbyter" and may have been doing so again).
As with the other canonical Gospels, the author of GJohn does not identify himself or claim to be a witness (The seeming self-identification in 21:24 is a later redaction to the book, not part of the orginal manuscript and did not name the author "John" in any case. It is also not really a first person singular assertion, ("I wrote this") but a first person plural avowel that "we know" these were the words of a disciple (without naming the disciple).
Looking at the text of GJohn, we can see that any claim to the book as an eyewitness account does not hold water. First of all there is the very late date (c. 100 CE if not later) which puts it at the absolute edge of any plausible lifespan for a contemporary of Jesus. It also shows a heavy Hellenistic influence, both in its literary style and its theology. How does an illiterate Palestinian fisherman suddenly become proficient in stylized literary Greek and become aware of Alexandrian Jewish-Greek concepts like the Logos?
GJohn is also arguably the most anti-Jewish work. It goes beyond being just a polemic against the Pharisees or the priests and becomes a full on indictment of all Jewish people. Kind of weird since the author (like Jesus) was allegedly a Jew.
GJohn contains some of the longest, most otherwordly and most implausible speeches for Jesus. The length of the discourses in itself mitigates against their historicity simply by virtue of the implausibilty of those speeches surviving verbatim for 70 or more years in the memory of this fisherman (and nowhere else. These discourses are found nowhere else in early Christian literature). They do not have the short and sweet anecdotal quality of the Q pericopes which are easy to remember and transmit through oral tradition.
GJohn also shows layered authorship. It is not the contiguous work of a single author but the result of multiple redactions by multiple hands.
What is really the nail in the coffin, though, is that GJohn anachronistically retrojects the expulsion of Christians from Jewish synagogues (an event which occurred c. 85-95 CE) to within the life of Jesus. An eyewitness could not have made this mistake.
To sum up for John, it is an early 2nd century book which is heavily Hellenistic in its language and theology. It is markedly anti-Jewish, it contains speeches for Jesus which are not only incompatible with the character of Jesus as he is presented in the synoptics (not to mention that it simply strains all credulity that a 1st century Jewish audience would tolerate a guy claiming he was God) but simply cannot be credibly defended as authentic transcriptions of speeches remembered verbatim for 70 years by an illiterate Palestinian fisherman (and by nobody else) and then translated into Greek by that same fisherman. It contains contradictions with the synoptics which I will get to in time. It shows muliple hands of authorship and it contains an anachronism so glaring that it is a fatal blow to any consideration of eyewitness testimony.
Its traditional authorship stems from a single unreliable claim by Irenaeus (a guy who couldn't keep his "Johns" straight) around 180 CE.
Conclusion
None of the Gospels were written by any eye-witness to any historical Jesus.
