Illiterate men.

[FallentoReason]

My comment was directed at the book of Matthew. The other three stand on their own merit.

True. I shouldn't have extrapolated.

Well, if the Gospel of Matthew sits on shaky foundations, then doesn't that start to paint a new picture about the credibility of the other witnesses?

Let's look at the intentions that Luke had when he wrote his account:

Inasmuch as many have undertaken to compile a narrative of the things that have been accomplished among us, just as those who from the beginning were eyewitnesses and ministers of the word have delivered them to us, it seemed good to me also, having followed all things closely for some time past, to write an orderly account for you, most excellent Theophilus, that you may have certainly concerning the things you have been taught.

So in 1:1 he acknowledges that he was familiar with previous texts describing the events i.e. Mark. Now, this is what wikipedia says about this introduction:

The traditional view is that Luke, who was not an eye-witness of Jesus' ministry, wrote his gospel after gathering the best sources of information within his reach (Luke 1:1-4)

Which makes sense with the fact he had access to the Gospel of Mark. Conclusion? The author couldn't have been the Apostle Luke as he didn't witness anything but merely relayed on the information he gathered. So once again it is Church tradition that holds together the ideal possibility that we are reading what the Apostles wrote. Not very convincing..