Literary Greek

Bart Ehrman repeatedly says the traditional authorship of the canonical Gospels must be false because they are written in sophisticated literary Greek whereas the disciples of Jesus were Aramaic-speaking peasants. He also judges 1 Peter to be pseudonymous for the same reason.

The way Ehrman frames the argument is false on the face of it.

i) According to traditional authorship, only one of the four Evangelists would even be a candidate for "an Aramaic-speaking peasant": John. Certainly that description doesn't fit Matthew, Mark, or Luke. 

ii) It's simplistic to say John was an Aramaic-speaking peasant. For one thing, he had entree with the high priest. That suggests he moved in higher social circles. He was well-connected. 

The next question is whether the Gospels are even written in sophisticated literary Greek. Keep in mind that this is only germane to Jewish authors. Since Luke was gentile, there's be no incongruity in his writing in literary Greek. 

I'm going to quote the analysis of Nigel Turner in A Grammar of the New Testament Greek; Volume IV: Style (T&T Clark, 1980). I'm just giving samples of his detailed analysis. 

Unlike Ehrman, Turner is a Greek scholar by specialization. That's his area of expertise. 

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Mark

Howard concurred with Lagrange that the Greek was translation Greek (11).

There is considerable evidence favoring influence of an exclusively Aramaic kind upon the style of Mark, but the case for the translation of documents is somewhat weakened by the fact that here in the same gospel are instances both of exclusive Aramaisms and exclusive Hebraisms side by side (15). 

Mark's style is conspicuously different from the Ptolemaic Papyri and closer to the LXX, following the order: article>noun>article>genitive (54 times). He never has the position which is common in non-Biblical Greek: article>article>genitive>noun (17). 

Some features of Markan style recall Latin constructions and vocabulary. That they are probably more frequent in Mark than in other NT texts, except the Pastoral epistles, may raise the question whether Mark was written in Italy in a kind of Greek that was influenced by Latin. However, supposing that his language is influenced in that way, we presume that it could have happened as well in the Roman provinces (29).

Matthew

On the whole, Matthew is not as Septuagintal in style as Luke (36).

It is sometimes assumed that Matthew writes Greek of a less Aramaic quality than Mark, and that he tends to soften the Semiticisms in general. That is not always true: we have found already many Semiticisms which may be attributed to Matthew independently of Mark.

If we examine the Markan sections of Matthew we shall find the contrary evidence, suggesting that Matthew has altered Mark to something more Semitic, conforming what we have already found…It would seem then that there is very little to choose between the relative Semitism of Mark's and Matthew's style (37).

Luke

Hebrew influence: This is far more extensive, and is not confined to the Infancy narrative (46).

The literal translation of Hebrew infinitive absolute comes into Biblical Greek from the LXX (47).

Physiognomical expressions: The large proportion of its occurrences are not in the Koine, but in Biblical literature, and the papyri instances are relatively slight when compared line by line with the LXX, Testament of Abraham, Testaments of the Twelve Patriarchs, Greek Enoch, Psalms of Solomon, and other works of this kind. There are 34 instances in Luke Acts, 31 in Revelation. In view of its place in Luke's own composition, it is not only a word of translation Greek but belongs to Jewish Greek (49).  

Semitic influence: This is vast, enabling the respective advocates of Aramaic and Hebraic sources to claim the features as Aramaic or Hebrew to suit their purpose (50).

And (or for) behold! An exclusively Biblical Septuagintal phrase, perhaps also from Aramaic, it is frequent in the LXX, and Luke and Paul probably obtained the expression from here. As it occurs in the possibly "free" Greek of the Testament of Solomon (seven times) and Testament of Abraham (ten times) it may be a feature of free Jewish Greek, derived perhaps from the translated books. It is scattered throughout Luke-Acts… (53).

John

The Shepherd of Hermas [has] the same kind of Greek, influenced by Jewish idiom and marked by an over-use of asyndeton, though to a less extent than John (70).

The place of the verb is important: in Luke and John it is so often in the primary position that it is no longer secular Greek (72). 

The Gospel vocabulary is limited to 1011 words, only 112 which are NT hapax. Many of these words are repeated, so that the vocabulary is only 6 1/2 percent of total word-use, almost the lowest in the NT (76).

We conclude that John's language throughout is characteristic of Jewish Greek, syntactically very simple, dignified but without the flexibility of the secular language, pointlessly varied in syntax and vocabulary… (78).

[Jewish Greek] appears in some free-Greek books of the LXX (e.g. Tobit), and some Jewish works as far away in time as the Testament of Abraham and the Testament of Solomon, which cannot be shown to be translations of Semitic originals. Ignorance of Greek as a cause of Jewish Greek, is altogether less probably than the influence of the Greek Bible through widely scattered synagogues, forming a new community language (78).

We must conclude that 1 Peter wears a veneer of good stylistic revision upon a basic draft of the same kind of Greek that is found elsewhere in the NT. It is tempting to ascribe the veneer to an amanuensis, not necessarily Silvanus (130).