After the discussion in the last post about the authorship of the Gospels, I've created a spreadsheet model, in Open Office format, to illustrate the probability calculus for whether the Gospels were written by Matthew, Mark, Luke, and John. This post probably won't make much sense if you haven't read the comments on the previous one; if you're not a mathy person just skip it. The spreadsheet file is at the bottom of this post.
It's just a model: I don't claim to have incorporated every possible effect that could be important. I'm just trying to illustrate how the short chains of oral transmission to Papias and Irenaeus can provide significant information. This is a dynamic spreadsheet, so if you change one box the rest of the numbers will change accordingly. The idea is that if you don't accept my numbers, you can change it yourself to see what happens, rather than just griping at me that it should be different.
I've used the following hypothetical likelihoods for each Gospel/early writer to be pseudonymous instead of genuine, from the perspective of a person who isn't sure whether or not to believe Christianity:
Matthew: .1
Mark: .01
Luke: .001
John: .03
Papias: .05
Irenaeus: .001
Eusebius: negligible
This is based on the names attached to the documents, as well as internal evidence and the fact that they were received as genuine by the Church, but not yet taking into account the testimony of Papias through John (concerning Matthew and Mark), as quoted by Eusebius, and Irenaeus disciple of Polycarp disciple of John (for all 4 gospels).
The odds for Matthew are higher than the others because that is the only one where there were significant arguments for pseudonymity, instead of just arguments that it could have happened. For me, the actual names written on the documents, and their acceptance by the church, are better evidence than any of the evidence against, hence the numbers above. Papias is more likely to be pseudonymous since other than the few fragments preserved by Eusebius, we have to rely on the judgement of the early church about this.
I haven't taken into account possible lack of independence between the pseudonymity of the 4 gospels. Although the Gospels do incorporate text from each other, they were almost certainly written by different individuals, so they aren't strongly dependent. Nevertheless, there's some probability dependence here in their later cultural acceptance by the Church. If you want to consider different odds for their dependence, you could put that in by hand in the "scenarios" section at the bottom (where e.g. "13" would mean the probability of the 1st and 3rd Gospels are pseudonymous, relative to the probability of all of them being genuine.)
In accordance with the arguments here, I've assigned odds of .001 per century until the first branch point for each document. I've assumed Eusebius had only one copy of Papias, and I just guessed 4 centuries for Irenaeus since I couldn't figure out the first branch point for him from what I could find online. I'm pessimistically assuming that if an textual corruption has occured, it makes the entire document unreliable. I'm also assuming that any given nongospel writer has a .01 chance of being totally unreliable due to e.g. deliberate deception.
Finally, I'm assuming that for any two people related by an oral testimonial link, there's a .05 chance that a given fact about authorship will be garbled. I'm assuming optimistically there was only one John and that Papias interviewed him directly, but this is balanced by not including any of the numerous other chains back to the apostles which Papias cites. I'm giving the garbling more odds than any other form of error, but unlike the other errors I'm assuming that because this is inadvertent, if one piece of data in the document is garbled, the rest are all unaffected.
(In order to make the math easier when considering multiple gospels, I had to break out the total errors, the garbling errors and the nongarbling errors into three separate rows).
I haven't included the possibility that we might be wrong about the chains of testimony themselves, but you're free to play around with inserting extra people or changing their dependence or such.
I got the following probability odds for the Gospels being pseudonymous:
Matthew: .007
Mark: .0007
Luke: .00017
John: .0036
And for different numbers of gospels being genuine, the probability price you pay is about 
for one Gospel being pseudonymous, 
for two, 
for three, and 
for all four. This is before considering prior probabilities.
Interdependence between the four Gospels will make these last figures smaller, but I think any reasonable model will have some significant suppression of probability there.
All right then, here it is. Enjoy!
UPDATE: Replaced incorrect "genuine" with "pseudonymous" above.
UPDATE 2: Fixed a bug in the spreadsheet. See the comments section below. This doesn't change things much for the numbers I provided, but might affect things if you change the input assumptions. See oldspreadsheet to look at the old version.
Most of my probability mass is in variations of "stuff got corrupted/exaggerated before it was written down." Conspiracies are less likely than honest mistakes (people misunderstanding what they saw/heard or failing to verify sources, etc) but still more likely than people rising from the dead.
Incidentally, to see if the math is correct, I tried changing the all sources deceptive/honest fields in the spreadsheet to .99/.01 and it doesn't seem to change the final numbers much, so either something is wrong or I don't understand the spreadsheet, probably the latter. ^^ (I expected that doing this ought to make the conclusion just as firm in the opposite direction.) (Note: .99 is not my true belief, nor is .01!)
In the upper left hand corner of the spreadsheet, you'll see that I have a space in which I assigned likelihood ratios to each of the 4 Gospels being pseudonymous, equal to (.1, .01, .001, .03), prior to the testimony of Papias and Irenaeus. These low odds contribute significantly to the final probabilities, however you should notice that the addition of the testimony of P & I does make a noticable change.
Also, I'm interpreting deceptiveness to imply that P & I provide zero evidence for traditional authorship, not that this would be evidence against traditional authorship. That's another reason why .99 and .01 aren't symmetric in the way you expected.
I just realized, however, that the spreadsheet does not give correct odds when the Gospel pseudonymity odds are taken to be comparable to 1. I was using the approximation where if the probability is p, the odds ratio is p:1. This approximation is OK when p is small (as it is for the numbers I provided), but if p isn't small one should use the odds ratio (1-p):p. I've created a revised version of the spreadsheet which fixes this; see the main post above. It also contains a new column which converts the odds ratios that N gospels are pseudonymous to a normalized probability distribution.
In the revised spreadsheet, you could go and change the odds of the Gospels being pseudonymous to (.5, .5, .5, .5)--i.e. no evidence either way until taking into account P & I. You'll see that the chance of all 4 gospels being pseudonymous is about .006, due to the testimony of P & I. On the other hand, if we use (.15, .15, .15, .15), the probability goes down to under
, with only about .003 odds that at least 2 gospels are pseudonymous. Even if we raise the odds of deliberate deception to .05 per author, you still pay a significant price for pseudonymity. Try it and see!
So even under assumptions much more cynical than mine, there still seems to be a few orders of magnitude of likelihood ratios which can be extracted from the situation. If we assume that the most efficient route for the skeptic involves taking the "liberal" view that the Gospels are mostly pseudonymous, then these liklihood ratios against pseudonymity convert directly into evidence for Christianity. This would be on top of any evidence for Christianity coming from "minimal facts" about the early Christian claims, which should be accepted even on the liberal view.
Hm, I'm just thinking out loud about if this is the correct thing to do or not. I can see several possible ways reality could be arranged. Suppose P & I literally knew nothing about the situation at all; then their answers ought to be completely uncorrelated with the truth, and they'd indeed provide zero evidence. But "deceptiveness" implies to me the situation where P & I do actually know stuff, but are writing down either exaggerations, falsehoods, or a subset of truth carefully chosen to cause us to believe an untruth. I'm not sure what to expect under this scenario, but probably anti-correlation (as I was originally thinking) isn't right.
More later when I look at the revised spreadsheet.
I looked at the revised spreadsheet. I'm still not 100% clear on what everything means, but it seems that when I plug in some priors that more realistically represent my state of knowledge, it does indeed change in ways I'd expect. When I plugged in numbers that seemed reasonable to me, it claimed it was most likely that two of the gospels were pseudonymous, which also seems reasonable to me. Yes, I'm much more cynical with my priors than you. :) However, if one looks here, one can see that I'm usually only slightly underconfident: http://predictionbook.com/users/lavalamp
You haven't posted your numbers, so I can only speak generally. But even under these more cynical assumptions, you seem to have concluded that (with some unknown probability ratio) probably about two of the gospels are genuine.
If two of the gospels are genuine, this makes it more likely that what they say is true. How much more likely, depends on a whole host of other questions, but it seems clear that there's some additional evidence for Christianity here. How much, I leave to you to decide yourself since I don't have much more time to get into this now.
Ideally, this series would have concluded with a post on this topic, since it needs to be addressed to complete the historical argument. It is true that some people do lie, but I think the gospels have several literary features which correlate with honest reporting. I might get to posting on this eventually, but I've just had a bunch of wisdom teeth removed, so I'm a bit down for the count right now.
FWIW, my position is that Luke seems likely to have been written by Luke or someone very good at pretending to be Luke. John was likely written by John, but with a decent chance of having been written on John's behalf, on account of its age. Matthew and Mark I'm more skeptical about. It's mostly a moot point to me, though, because I don't view Matthew, Mark, and Luke as independent, and John is so late I trust it much less (and it's also not completely independent).
I'm not so sure about that. I think I've said before, most of my probability mass is in things having gotten garbled/exaggerated/misunderstood/misinterpreted (g's scenario being one member of this set) before they got written down, and I'm honestly not sure what genuineness or lack thereof does to this scenario, I can think of arguments from both directions. I guess I think my numbers have been mostly under the assumption that the books are genuine enough, and therefore positive evidence of forgery would be evidence against Christianity, but additional evidence of genuineness doesn't help much-- effectively I've already updated on it. I think the date affects my confidence a lot more than the author, especially the "latest possible date" (I forget the fancy latin phrase).
Agree, if the gospels are lies they're well crafted ones; to repeat myself, I think it's more likely that the authors were honestly mistaken. Anyway, I hear wisdom teeth removals are no fun--so take it easy. I promise to ignore anything you write while on Vicodin. :)