Interpretation of results
The results can be grouped together as follows:

Luke group: 012, 002, 112
Matthew group: 212, 211, 210
Mark group: 222,221,220,122,022*,121,020,021,120
Sayings group 1: 200, 201, 202
Sayings group 2: 102
Again, for category definitions, look

First, what can we state with mathematical certainty? We can state, with a high degree of certainty, that the vocabulary of some categories, is more related to other individual categories than it is related to the overall vocabulary in the synoptics. Possible problems with attributing these differences directly to authorship are discussed

We notice that categories where Mark could have been the original author all group together. For example we see 221, which could have been authored by either Matthew or Mark, grouped with 121 which could only have been originally authored by the author of Mark. This gives strong support to Markian priority.

Also, we notice that categories that Luke and Matthew share in common apart from Mark, i.e. 212, and 202, group together with categories associated with Matthew, like 200, and 211. This tends to support Matthew as the second gospel.

If for the moment we assume that the results directly indicate authorship, we can say the following: Neither the 2 source hypothesis
2SH, nor the Farrer Hypothesis FH, nor an intermediate solution, the 3 Source Hypothesis 3SH, can be excluded. The 3SH has Matthew use Mark and a smaller Q, and has Luke use Matthew, the smaller Q, and Mark. All the hypotheses can be viewed here.

For the 2SH the relation between 200 and 202 is a bit problematic, but it could be explained by significant parts of 200 sondergut Matthew, being part of Q. The 2SH would expect 212 to be roughly equally related to Luke and Matthew. The fact that we did not find significant evidence of the relation to Luke, does not mean that the relation does not exist. However, the study tells us that the minor agreements (212), are clearly Matthian in style, and that there is no clear distinction between sondergut Matthew (200), and material that is clearly Q, on the 2SH (202). The combination of these 2 results should make us suspect, that perhaps Luke did use Matthew.

At the very least Luke seems to have had among his sources, the following:
1) Markian material.
2) Markian material with a Matthian flavor seems likely, based on the fact that category 212 seems Matthian and not Lukian.
3) Double tradition material. And based on the fact that 202 groups with 201 and 200, and not with 102, we would have to say on the 2SH that Matthew was much more faithful to Q, than Luke was.
4) Based on the fact that 202 closely resembles 200, we would have to say that Q contained much of what we call sondergut Matthew, and that Luke ignored it.

So, we have that Luke had probably access to a Matthian looking triple tradition, access to a Matthian looking double tradition, and access to at least significant parts of what we know as sondergut Matthew. It is not a large step from here to say that Luke knew Matthew.

For the FH, the more problematic results are first that there are sayings groups at all, and secondly that 102 does not group together with the other categories associated with Luke, and that the category it came closest to grouping with was 202, and finally that 102 and 201 appear to be about equally distant from 202. The first problem can be explained by assuming that effects other than authorship are at work in separating the sayings categories from the others. For example different pronouns may be used in the sayings, which often feature Jesus speaking, than are used in the narrative sections. Also, because these categories are generally in different pericope, the difference in subject matter could lead to different vocabulary as well. One solution to the second problem for the FH, that 102 seems more saying-like than Luke-like, and does not group with 002, is to assume that 102 does represent another source. But on the FH this is an oral source, not a written one. On the other hand, the differance between 002 and 102 could be attributed to the difference between narrative and sayings, with 002 seen as being largely narrative. Finally, on the FH, we would expect to see 201 more related to 202 than we do. The fact that 201 and 102 are about equally distant from 202, suggests a sayings source. However, in saying this, I am interpreting the lack of a strong 201-202 relationship, and the fact that the study did not find a relationship, does not mean that one does not exist. All we can say here is that a saying source fits more naturally with the results, and may be more probable than not, but we can not eliminate the FH, based on a lack of a finding.

On the 3SH, 200, and 202 could both be part Matthew, and part Q. This would help explain why they seem to look like each other. Also, 102 would have a lot of Q in it, and thus would not tend to look much like the categories associated with Luke. Finally we would expect 212 to look like the Matthian categories on the 3SH, and not look much like the categories associated with Luke. Thus the 3SH naturally explains all the results that could be seen as potentially problematic for the 2SH, and the FH. For a more detailed discussion of this look

However, one of the main arguments for the saying source being written instead of oral is now gone. On the 2SH, it is argued that the order of Q in Matthew and Q in Luke are correlated. But, if we assume that Luke was using Matthew, on the 3SH, then there is no reason to believe that the sayings Luke does take from the saying source, are in an order that is even roughly correlated with the sayings in Matthew.

It is possible, maybe even probable, that the true solution is more complicated than the ones considered here. For example, there could have been more than one version of Mark or Matthew. The problem is that as the number of hypothetical documents grows, the number of possible solutions grows exponentially, and the data that can be used to separate them is spread that much more thinly. Personally, I doubt enough data exists to come to any conclusions beyond the basic outline of what happened, with any real confidence.

So, in conclusion, based on this study and other more traditional forms of evidence, not presented here, I believe the 3SH, or some variation of it is most likely the correct solution. The study also provides almost as much support for the FH, and I do not believe the 2SH can be eliminated by this study. However, solutions that do not involve Markian priority have another hurdle to overcome, based on these results, and I think it becomes more difficult to argue that Luke did not use Matthew at all.

A discussion of a number of different possible solutions to the synoptic problem can be viewed
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