Among modern critics who believe in the attributed authorship of the Gospels, what are their arguments? According to the many online lectures by biblical scholars, not to mention entries in  Wikipedia and other online sites, whether the experts are on either the liberal side or the conservative side of the table, they all state the four Gospels were actually anonymous works and that someone appended the names of Matthew, Mark, Luke and John to them long after the anonymous writers wrote them.

The average person you talk with has no idea of this, and he or she wholly believe the actual eyewitnesses (viz., the apostles) wrote the Gospels.

I am interested in the views of critical scholars who believe the Gospels were written by the men whose names are appended to them. I am not interested in purely religious answers such as, "The Holy Spirit guided the church to ascribe authorship to Matthew, Mark, Luke, and John."  Would someone please 1) point  me to scholars in the field of historical criticism who believe the attributions are accurate; and more importantly 2) summarize their arguments for me?
This is part two of a two part answer.  See my previous post for general arguments.
Specific books
Matthew
External - On Matthew Papias writes, &quot;Matthew collected the oracles in the Hebrew language,  and each interpreted them as best he could.&quot;.  This is probably the most debated phrase in all of the patristic writings - the words translated as &quot;oracles&quot;, &quot;Hebrew language&quot;, and &quot;interpreted&quot; are all ambiguous.  One possible, natural reading is that Matthew wrote a Gospel in Aramaic and other people then translated from that.  Critiques use this interpretation of Papias' statement against his accuracy because we &quot;know&quot; our Matthew was written in Greek and doesn't show signs of being translated.4
However, there are several other possible translations - some critical scholars see the statement as a reference to the hypothetical Q document that lies behind portions of Luke and Matthew, which does show signs of Aramaic-isms.  Others have suggested Papias actually meant something like &quot;written in the Hebrew style&quot;.4
Whatever Papias meant exactly, we can be pretty sure he knew of a Greek Gospel of Matthew.  (Papias himself wrote in and spoke Greek and quotes from Matthew are known from works predating Papias.)  We also know that there was an Aramaic Gospel closely associated with the Gospel of Matthew (by contemporaries) that survived for several hundred years and was used by the church in Palestine.  Since this document has not survived to the present day, scholars can't judge what it was exactly.  Still, that means there are several possible ways Papias could be accurate within our knowledge - Matthew could have written a Gospel in Aramaic and also a version in Greek from scratch (i.e. he was bilingual so didn't need to translate); Matthew could have written a Q-like document in Aramaic that was translated and then expanded; Matthew could have written in Greek, but in the &quot;Hebrew style&quot; (i.e. that he Gospel was aimed towards Jews); and so on.  As such, dismissing Papais as inaccurate is unwarranted.
In any case, writing not long after Papias, Irenaeus also attests Matthean authorship and appears to be drawing upon an independent tradition to do so.  (He states when the gospel was authored, a subject Papias did not touch upon as far as we know.) Irenaeus knew Polycarp, who is thought to be a disciple of John the Apostle, so he likely had access to good information.  Matthean authorship is then continually attested to by church fathers for the next several hundred years.  Summing up the external evidence, Donald Guthrie writes:8

there is no conclusive reason for rejecting the strong external testimony regarding the authorship of Matthew

Internal -  The strongest internal evidence of authorship comes from Matthew 9:9-13.  This passage in the Gospel of Matthew has a couple differences when compared to the parallel passages of Mark 2:13-17 and Luke 5:27-32.  Most notably, Matthew calls the tax collector &quot;Matthew&quot; while the other Gospels call him &quot;Levi&quot;.  Yet all three Gospels list Matthew, not Levi, as a disciple of Jesus in their lists of the 12 (Matthew 10:2-4; Mark 3:16-18; Luke 6:13-16).  Matthew's list, however, does have a difference - he adds the words &quot;the tax collector&quot; after Matthew's name.  D. A. Hagner argues that the most natural explanation is that Matthew is adding a self-deprecating identification to 10:3 and substituting his new (apostolic) name in 9:9-13, just as Peter (originally Simon) and Paul (originally Saul) do in their own works.9
In support of this conclusion is Matthew 9:10, where the author says &quot;the house&quot; instead of &quot;his house&quot; as the other Gospels.  This is the kind of subtle difference one would expect if Matthew was the author of the Gospel that bears his name.  The strength of this argument is underscored by the fact that some who reject Matthean authorship suggest this passage was the cause of the &quot;anonymous&quot; work becoming associated with Matthew.  However, Daniel B. Wallace argues that an admirer of Matthew would be unlikely to adopt the subtle deprecation of the apostle, and that only Matthean authorship adequately explains the data.10
Briefly on additional evidence, Donald Senior sees the literary style of Matthew as compatible with Matthean authorship.11  Wallace, Gundry, and others notes Matthew's frequent use of precise monetary terms and his use of unique parables with monetary themes as evidence that someone familiar with money wrote the Gospel.8, 10  C. F. D. Moule suggests several subtle self-references and E. J. Goodspeed suggests Matthew would have have been uniquely equipped among the disciples to take shorthand notes during Jesus' ministry.12, 13
Objections - The most common objection to Matthean authorship is that Matthew appears to rely on Mark for a significant portion of his Gospel.  A true disciple of Jesus would surely not rely on a non-disciple the argument goes.  In response, it first should be noted that while the most popular solution to the Synoptic problem, Markean priority is far from certain.
More crucially, it is simply an assumption, not a fact, that a disciple wouldn't use a non-disciple as a source.  Our knowledge of contemporary writing suggests the practice of reusing another's work was quite common.  Additionally, Paul quotes Greek philosophers in a couple of places, but no one suggests that means he viewed them as authoritative.  Finally, if Matthew knew (or at least thought) that the Gospel of Mark is based on the teaching of Peter (see the next section), there is not even a hint of a conflict.4
Mark
External - Mark has arguably the best external witness of any Gospel, starting with Papias who writes:

Mark having become the interpreter of Peter, wrote down accurately whatsoever he remembered... he neither heard the Lord nor accompanied Him. But afterwards, as I said, he accompanied Peter... [Mark] took special care, not to omit anything he had heard [from Peter], and not to put anything fictitious into the statements.6

This testimony, according to Papias, comes from the &quot;elder John,&quot; which probably is meant to refer to the apostle John.  According to Wallace, the most notable thing here is despite a strong desire to tie scripture to apostles, Papias (and later church fathers) do not attribute the Gospel of Mark to Peter.  If one was simply ascribing a name to an anonymous work, there would be no reason to choose a follower of Peter instead of Peter himself.  Indeed, if &quot;according to X&quot; merely meant &quot;according to the teaching of X&quot;, as some critics argue, then this Gospel would be called the &quot;Gospel according to Peter&quot;.  The fact that the fathers resisted the temptation to attribute the Gospel to Peter enhances their credibility not just here, but also on other authorship testimony.14
After Papias, Mark's authorship is directly attested to by Clement of Alexandria, Origen, Irenaeus, and The Anti-Marcionite prologue within the next hundred years.  Tertullian and Justin Martyr also indirectly support the testimony (i.e. in off-hand remarks in Biblical commentary).  Some of these accounts are, no doubt, dependent on each other.  However, there is also much independent testimony in these varied accounts.  John A.T. Robinson, who does not necessarily believe the accuracy of the tradition in regards to other Gospel writers, says that one should &quot;take seriously the tradition&quot; in regards to Mark and that Markean authorship for, at minimum, the first draft of Mark is probable.15  Guthrie considers that the external evidence is conclusive, writing there is no need to &quot;do little more than mention [it]&quot;.8
Internal - Since we don't know a great deal about Mark the person, internal evidence will naturally not be super strong.  However, Hengel argues that the simple Greek of Mark is completely compatible with a Christian raised in Jerusalem.16  A.B. Bruce suggests that the description of the young man in Mark 14:51-52 reads like an autobiographical insert, which agrees with the early church tradition that the Last Supper took place at Mark's parents house.17
More compelling, there is good evidence that the Gospel was written by someone close to Peter.  Scholars such as Michael F. Bird and Richard Bauckham have argued that Mark used a technique called inclusio, which is thought to have been used by secular historians of the period, to bracket eyewitness testimony.  Since Peter regularly appears at the start and end of such passages, Mark is thus implying that Peter is the eyewitness behind those particular accounts.18
Detailed word studies, such as the work of C. H. Turner, have shown that there is a pattern in the use of third person plurals in passages involving Peter that read as if an original first person plural has been substituted. This is a literary technique known as the &quot;plural-to-singular device.&quot;  Writing on this phenomenon, F.F. Bruce states &quot;the reader can receive from such passages 'a vivid impression of the testimony [of Peter] that lies behind the Gospel'&quot;.19
Objections - The most common objection to Markean authorship seems to be that Mark (allegedly) commits geographical errors, betraying that he is not a native of Palestine.  First, this seems like an incredibly weak reason to reject all the other evidence, in my opinion.  Surely a person can have less-than-perfect knowledge of the geography of their home region.  More importantly, such claims of inaccuracy do not stand up to scrutiny.4  One example will be sufficient to illustrate the point.  Mark 7:31 states:

Then Jesus left the vicinity of Tyre and went through Sidon, down to the Sea of Galilee and into the region of the Decapolis.

Some critics have interpreted this to mean that Jesus went through Sidon in order to get to the Sea of Galilee, which would be the wrong direction.  However, it need not mean more than that He went to Sidon and then to the Sea of Galilee.  Even ignoring that simple solution, Douglas Edwards notes:20

even the Jesus movement's travel from Tyre to Sidon to the Decapolis depicted in Mark... is, in fact, quite plausible. Josephus notes that ... a dispute regarding boundaries arose between Sidon and Damascus, a city of the Decapolis. It is therefore conceivable that the movement headed east toward Damascus and then south through the region of the Decapolis, following major roads linking Damascus with either Caesarea Philippi or Hippos.

Luke
External - Among the Gospels, Luke's is the one where the attribution is most likely to be original (i.e., that is was on the autograph).   M. Dibelius argues that a work addressed to an individual (Theophilus) would normally also contain the name of the person it came from (on an external tag).  Furthermore, it seems clear that the author of Luke intended his work to be widely distributed, which would normally require a title or name attached to the work.21
Lukan authorship is attested to by Justin Martyr, Irenaeus, the Muratorian Fragment, Origen, the Anti-Marcionite Prologue, and Tertullian by the end of second century.  Perhaps most notably, the heretic Marcion accepted Lukan authorship while rejecting the other three Gospels as &quot;too Jewish&quot;.  Despite this, the Church continued to accept the Gospel and continued to attribute it to a relatively unknown figure.  There are no dissenting voices among the church fathers, despite numerous possible Pauline figures to pick from (i.e. even if we assume the Church knew that the Gospel of Luke was associated with Paul, there is no compelling reason why Luke was unanimously chosen.)  Summing up this evidence, Darrell L. Bock concludes:22

Such unanimity, when numerous Pauline companion candidates exist, argues for the veracity of this identification.

Internal - A discussion about the internal authorship of Luke will naturally start with Acts, where the evidence is strong.  (Nearly all scholars believe Acts and Luke were written by the same person, regardless of their opinion of the author's identity.)  Even a casual reading of Acts will reveal two things - after the first few chapters, Paul receives most of the writer's attention and many of the later passages use the first person plural (collectively called the &quot;we passages&quot;).
The most natural reason for the increasingly heavy emphasis on Paul as Acts progresses is that the author was a follower of Paul and thus had more detailed information about Paul than other early church leaders.  The most natural explanation for the use of &quot;we&quot;, especially since it does not occur throughout Acts, but instead only in places, is that the author is among the group of people referred to in such passages.  Examining all the known companions/part-time companions of Paul, Guthrie finds that the only other person who seems likely to have been present on all occasions is Luke.8  Andreas J. Köstenberger, L. Scott Kellum, and Charles Quarles have additional cited reasons, apart from the &quot;we passages&quot;, why each of these people is an unlikely author of (Luke-)Acts.23
To reduce the force of this conclusion, some critics have suggested that the author had access to the travel diary of a companion of Paul.  However, the &quot;we passages&quot; are written in the same style as the rest of the work, meaning that the author would have had to rewrite these passages in his own style while choosing not to change the pronoun.  As Carson and Moo correctly conclude, &quot;this is highly improbable&quot;.4
Objections - The chief objections to Lukan authorship are a couple of apparent discrepancies between the person of Paul in his own writings and in Acts and supposed differences between the theology of Paul and Luke.
On the apparent historical discrepancies Wallace notes that, even if they are true discrepancies, this doesn't really argue against Lukan authorship - a companion could mis-remember a detail just as easily as an outsider could be misinformed.24  Indeed, if the author was only pretending to be a follower of Paul, one would think he would make sure to match the details to that which was known from the letters of Paul (those that think Luke was not the author almost universally date the work late, by which time Paul's letters were surely circulating.)  Furthermore, Carson and Moo as well as Guthrie have examined the alleged discrepancies and found no differences that cannot easily be explained by different writers (neither of which was intending to write a comprehensive history) choosing to emphasize different details of the same events.4, 8
In regards to theology, the differences, to my mind, have been greatly exaggerated - casual readers usually don't detect any difference.  Furthermore, differences really only tell us that Paul himself did not write Luke-Acts.  That a student could have a slightly different theology than his teacher is hardly surprising.  That said, it will be worthwhile to examine one such difference: the use of the OT Law.  In Acts, Paul follows the Law by, for example, having Timothy circumcised (16:3) and participating in temple  purification rites (21:17-26).  In his own letters, Paul recommends against circumcision (Galatians 5:2-4) and ritual restrictions (1 Corintians 8-10).  This appears to be a serious contradiction, but only via a caricature of Paul.  In reality, Paul's view of the Law is much more nuanced.  He says he is willing to follow the Law when it is beneficial to do so (1 Corinthians 9:19-22) and says circumcision or uncircumcision is a meaningless difference (Galatians 6:15).  For these reasons,  Richard N. Longenecker concludes that there are no material difference between the theology of Luke and Paul.25
John
Finally, we turn to John, the only book where there would be a clear motive for the attributed authorship - John was one of the most important apostles.  Naturally, this does not itself argue against the authorship, but, perhaps, means we should demand strong evidence to uphold it.  That, in my opinion, is exactly what we find.
External - The strongest testimony in favor of Johannine authorship for the Gospel of John comes from Irenaeus.  In a letter to Florinus, Irenaeus vividly recalls learning from Polycarp in his youth.  He states that Polycarp (c. 70-156) was a follower of John and otherwise received testimony from eyewitnesses.4  Thus, Irenaeus' testimony on things related to John comes from a very good authority.  On the Gospel of John, he writes:26

John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia.

On this evidence, Guthrie remarks:8

There can be no doubt, therefore, that Irenaeus accepted John the apostle as author of the Gospel ... on the basis of Polycarp’s testimony.

Further second century testimony to Johannine authorship is found in Clement of Alexandria, Tertullian, Origen ,Theophilus of Antioch, the Muratorian Canon, and the Anti-Marcionite Prologue.  The last of which is especially interesting because it references Papias as its source.  (The relevant portion of Papias' work hasn't survived [very little of his work has], but there is no particular reason to doubt that he wrote that John was the author of the Gospel of John.)  Since Papias himself was a follower of John, Carson and Moo argue that his testimony on the authorship is especially strong evidence.4
In addition to the usual Patristic testimony, there is strong evidence for the attribution of the Gospel of John in manuscript evidence.  Wallace points out that there are two manuscripts of John - P66 and P75 - which date to the end of the second century and attest to Johannine authorship.  Textual analysis shows the two manuscripts are not closely related and that their common ancestor must precede them by at least three generations.  Furthermore, another early manuscript, B, shows signs of being more primitive than P75.  The earliest known New Testament manuscript, P52, also comes from John.  It doesn't contain the start of the Gospel, so can't itself attest authorship, but does definitely prove the Gospel was written before the year 100.  Together, the documentary evidence suggests that the Gospel of John was quickly accepted by the church and widely distributed, and that the attribution was (at minimum) very early.27
Internal - The internal evidence for Johannine authorship is also very strong.  Indeed, some critics argue it is &quot;too strong&quot; to be trusted.  The most famous argument for Johannine authorship is B. F. Westcott's &quot;concentric proofs&quot;, expanded by Leon Morris:28

The author was a Jew
The author was a Jew in Palestine
The author was an eyewitness
The author was an apostle
The author was the apostle John

The first two points were once disputed, but the discovery of the Qumran literature (Dead Sea Scrolls), showing John's themes were well within the norm for contemporary Judaism have convinced nearly all scholars of the accuracy of the first point.  The second point is no longer disputed either, as archaeological finds have demonstrated the author of John had an intimate knowledge of Palestine.4, 19
On the third point, there are numerous passages that suggest an eyewitness report.  For example, 1:14 says &quot;we have seen his glory&quot;.  Careful analysis of this passage reveals not only the first person plural, but also a verb (θεάομαι) that normally signifies a physical examination.27  On the forth point, the author seems to have intimate knowledge of what happened in Jesus' inner circle. For example, 4:27 (&quot;Just then his disciples came back&quot;, ESV) and 6:19 (&quot;When they had rowed about three or four miles&quot;).  In favor of the fifth point is the author's use of &quot;John&quot; to describe John the Baptist.  This suggests that the author was either completely unaware of the apostle John, an unlikely possibility, or expected his readers to naturally distinguish between the two Johns.  The most natural explanation for this is that the writer is John and that his audience would know that he uses &quot;the disciple who Jesus loved&quot; when he talks about himself.28
Objections - The chief objections to Johannine authorship are the use of &quot;beloved disciple&quot; moniker and the high theology of John's Gospel.  On the first point, the idea is that while an admirer of John may give him such a title, no Christian author, especially not a humble apostle, would give themselves such a title.  On this objection, Carson and Moo state point blank that it &quot;should be abandoned&quot; as it doesn't hold up to scrutiny.  When a New Testament writer says God/Jesus loves them &quot;it is never to suggest that other believers are not loved or are somehow loved less.&quot;  (See, for example, Galatians 2:20 and Ephesians 3:14-21.)  For them, the moniker is not a mark of arrogance, but rather a mark of &quot;brokenness&quot;.  That is, John sees himself as needing Jesus' love.4
&quot;Theological development&quot; is notoriously subjective - who is to say how long it &quot;must&quot; have taken for a high Christology to develop?  To claim such a framework can be developed a priori and then the New Testament fit into that framework, and that this then proves a late date for a given work, is nothing more than begging the question.  When the idea was first developed in the early 1800s, the proponents of the technique were convinced the Gospel of John must have been written late in the second or early in the third century because of connections to ideas popular at that time.  The manuscript discoveries since then (cited above) have proven this date utterly impossible, yet some scholars still hold to such subjective methodologies - the time frame is collapsed to fit the external realities, but the idea that John reflects &quot;development&quot; of theology remains.
Even if we allow such considerations, it is not a problem for Johannine authorship.  According to the church fathers, John lived into the last decade of the first century.  One might doubt their testimony, but certainly it would be foolish to say it is impossible that John lived that long, and the Gospel could hardly have been written any later than the 90s based on the hard documentary evidence.  Furthermore, Robinson turns the high theology argument completely on its head.  He asks, is it really plausible the writer of what many throughout church history have described as one of the greatest pieces of theology ever written would be completely forgotten?  He says to make such a conclusion shows &quot;an indifference to evidence (or rather to the lack of it)&quot;, for there is, in fact, no concrete evidence for any author other than John.  Remarking on the unexplained oddities that various different hypothetical authors create, he concludes:29

I find it much easier to believe ... that this disciple himself 'wrote these things', and that this certificate, given in his presence, is true [a reference to John 21:24]... It is these self-created aporiai, or perplexities, in Johannine studies which seem to me so much more baffling that the breaks and discontinuities at which critics balk...  In fact ironically it is the lack of final reduction to which the evidence most powerfully points.

Conclusion
The most obvious conclusion is that I failed in my goal to keep this answer to a reasonable length. :) In truth, though, I've only covered a fraction of the evidence and have barely touched on the arguments in support of the evidence...
The external authorship for the Gospels is very strong, stronger than almost any secular work of antiquity.  If what we were talking about was not religious in nature, I doubt anyone would reject the attributed authors.  It is only by deciding that the nature of the works means they would have been anonymous tracts that developed over time that the abundance of early witnesses for authorship are brushed aside.  However, this decision is entirely a priori, lacking in hard evidence to support it.  Despite an abundance of textual variations in other places, there are no extant copies of the Gospels that have different authors attached; nor is there any conflicting witness about authorship among the church fathers.
Furthermore, in the cases of Luke and Mark, there is no plausible reason why such figures would be picked as authors of anonymous works; all of the evidence suggests the early church desired apostolic authorship for its scriptures, yet two of the four Gospels bear the name of non-apostles.  If the church fathers were truly making things up, we would expect variation in their attribution and would expect them to pick &quot;bigger&quot; names.  How the universal church came to accept the attributed authors of widely circulated &quot;anonymous&quot; works, without any dispute or variation, would be quite a mystery.  Thus, lightly tossing aside the Gospel attributions is unwarranted - we must have solid internal evidence to consider the proposition.
Looking at the internal evidence, there is nothing to seriously challenge the authorial attributions.  There are a few minor oddities, sure, but nothing that is not easily explained by a careful examination of the evidence.  And in most cases, there is internal evidence that can best be explained by the traditional author.  In total, the internal evidence affirms the traditional authors much more than it argues against them, as each Gospel's characteristics are just as we'd expect if the attributions were accurate.  By its nature, such evidence can't be conclusive, but certainly provides no reason to doubt the testimony of the church fathers, and some reasons to believe them.
Combining the external and internal evidence, I and the two dozen or so authors I cited, conclude that there is sufficient reason to trust the &quot;according to X&quot; titles that appear on our Gospels.  In their quest for the &quot;real story&quot; behind the New Testament, some scholars have been too quick to dismiss what to a historian working on secular history would be quite compelling evidence.  Most likely, the Gospels were written by the people who the Church has always believed wrote them.

See part one for list of citations