As happens most days that the earth is spinning around the sun I have been thinking about the criteria of authenticity again. In particular I've been thinking about what they can't do. And what they can't do is a lot. They can't tell us anything that isn't already in the sources.
Collingwood has a great example that is quite apposite here. Imagine I see a sailboat in the water. I look away, then back again a few minutes later. I see the sailboat still in the water, but in a different location. Now, all the criteria of authenticity can ask is what whether I actually saw the sailboat. Was my apprehension genuine? What they cannot do, because they do not ask this sort of question, is ask how it is that the sailboat is in two places. The answer, obvious without any particularly sophisticated procedure, is of course that the sailboat moved along the water.
Consider now an example from the gospels. Nowhere is Jesus said to have engaged in efforts to recruit followers in Bethany/Bethphage, just outside Jerusalem. Yet all the gospels suggest that he has followers in that region before his final journey to Jerusalem. This is clearest in John's Gospel of course, when we encounter the family of Mary, Martha, and Lazarus. This is intimated in the Synoptics, when Jesus knows where in Bethany/Bethphage his followers can find the animal(s) that he needs for his triumphal entry into the city and tells them what to say if challenged. The criteria at best can tell us that these passages describe events that really happened. They cannot tell us what we must reasonably infer, namely that prior to his first visit to Bethany/Bethphage either Jesus or others operating on his behalf must have worked to establish the connections that we see throughout the passages in what we might call a "Bethany/Bethphage complex."
Note that the above example does not depend upon the authenticity of any particular passage in this complex. It only requires that from the existence of this complex we can reasonably infer that Jesus had supporters in and around Bethany/Bethphage. Luke might in fact evince some awareness of such earlier contacts; cf. his account of the anointing, wherein the episode of anointing, which in the other gospels occurs in Bethany/Bethphage near the final entry into Jerusalem and in the other Synoptics at the house of a certain Simon the Leper, occurs earlier in Jesus's ministry at the house of a Pharisee named Simon. The similarities are sufficient to think that they are all drawing upon the same tradition, and Luke's decision to locate it earlier than the rest is quite tantalizing (does he know of the Johannine tradition surrounding Mary, Martha, and Lazarus, and thus that Jesus was in Bethany/Bethphage earlier? Does he omit the name of the location of the anointing in order to make the account better fit his structure?).
But I digress. The above does however bring us to a further difficulty with the criteria of authenticity: they cannot easily handle discordant accounts. Take the sending of the Twelve and the Seventy(-Two). Matthew, Mark, and Luke have Jesus send out twelve missionaries to the villages of Israel; Luke has a second account in which Jesus sends out seventy or seventy-two. The criteria can only determine which of these putative events happened. When they come to Luke then the question is "Did Jesus send twelve, seventy(-two), both, or none?" They cannot easily reckon with the possibility that the sending accounts are entirely schematic, that perhaps Jesus sent out different followers at different times as he either thought it necessary or as he thought them ready. Perhaps there was never one sending or two of a specific number but a whole bunch of sendings, in dribs and drabs, and the Synoptic Gospels merely telescope this into one or two accounts. Note that I am not arguing that this is the case; I am merely arguing that it is a quite reasonable possibility but one that the criteria cannot apprehend.
Anyways, yeah, the upshot is that I don't think that the criteria work.