The synoptic problem, the triplelink model and statistics

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The synoptic problem, thetriple-link model and
statistics

• Hypotheses about the relationships among the
  synoptic gospels

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The synoptic gospels

• Matthew (Mt), Mark (Mk) and Luke (Lk)
• Written in the second half of the first century
  (in Greek)
• From oral and/or written sources
• Originally written on scrolls
• The earliest surviving manuscript fragments, from
  the late second and third centuries, are in codex
  form

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The Codex Sinaiticus
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Pericopes (sections)

• Triple tradition
• Double tradition (mainly Mt and Lk, e.g., Sermon
  on the Mount material)
• Single tradition (e.g., birth and infancy
  narratives of Mt and Lk)
• Variations in the order of pericopes

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The texts of the pericopes vary from gospel to
gospel

• Published synopses gospels laid out in parallel
  columns for comparison of pericope texts
• Note that there are variations in the text among
  the manuscripts and from one modern edition to
  the next of the New Testament in Greek

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The synoptic problem

• How do we account for the complex patterns of
  agreement and disagreement between the synoptic
  gospels?
• The Griesbach hypothesis?
• The two-source hypothesis (Q)?
• The Farrer theory?
• Other theories?

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The Augustinian hypothesis(so-called)
Mt
Lk
Mk
Augustine of Hippo (354-430)
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The Griesbach hypothesis
Mt
Mk
Lk
Johann Jakob Griesbach (1745-1812) Farmer (1964),

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The two-source hypothesis
Q is the hypothetical source that accounts for
the large quantity of Mt-Lk double tradition
material that is absent from Mk.
Mt
Q
Lk
Mk
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The Farrer theory
Mk
Lk
Mt
Farrer (1955), Goulder, Goodacre
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Mk
Mt
Lk
e.g. Hengel (2000)
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Counting verbal agreements

• The numbers of common occurrences in the same
  section of material of the same Greek word in the
  same grammatical form
• Data from Honoré (1968),Tyson and Longstaff
  (1978) used here
• Data aggregated over all the triple tradition and
  double tradition pericopes

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Counts of verbal agreements, aggregated over
triple and double tradition
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The triple-link model (Honoré)
A
C
z
x
y
B
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(A, B, C) any permutation of (Mt, Mk, Lk).
x the probability that a given word in Gospel A
is transmitted unaltered to Gospel B.
y the probability that a given word in Gospel B
is transmitted unaltered to Gospel C.
z the probability that a given word in Gospel A
is transmitted unaltered directly to Gospel C.
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A the event that a given word is in Gospel A
B the event that a given word is in Gospel B
C the event that a given word is in Gospel C
C1 the event that a given word is in Gospel C and
has been transmitted via Gospel B
C2 the event that a given word is in Gospel C and
has been transmitted directly from Gospel A
A the event that a given word is in Gospel A
According to the triple-link model, any word that
Gospel C has in common with either Gospel A or
Gospel B has been transmitted to Gospel C from
either Gospel A or Gospel B.
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x Pr(BA)
y Pr(C1B)
z Pr(C2A)
x is evaluated directly as a relative frequency,
the ratio of the number of words that are in both
Gospel A and Gospel B to the number of words that
are in Gospel A.
(Other conditional probabilities involving A, B
and C may be evaluated in a similar way.)
Additional assumptions have to be made in order
to evaluate y and z.
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Honorés three conditional independence
assumptions

• Given that a word is in Gospel A, the event that
  it is transmitted to Gospel B and the event that
  it is transmitted directly from Gospel A to
  Gospel C are independent.
• Given that a word is in Gospel B, the event that
  it is in Gospel A and the event that it is
  transmitted from Gospel B to Gospel C are
  independent.
• Given that a word is in Gospel A and Gospel B,
  the event C1 that it is transmitted from Gospel B
  to Gospel C and the event C2 that it is
  transmitted directly from Gospel A to Gospel C
  are independent.

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Using Honorés conditional independence
assumptions, we obtain the following formulae,
which we use to evaluate z and then y.
Pr(B?CA)
z
Pr(BA)
Pr(CB) - z Pr(AB)
y
1 - z Pr(AB)
We find also the formulae that were given by
Honoré
Pr(B?CA) xy xz - xyz
Pr(CA) xy z - xyz
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An additional criterionfor consideration
x gt max(y,z)
Lk-Mk-Mt and Mk-Lk-Mt would be ruled out
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Honorés assumptions modified

• Given that a word is in Gospel A, the event that
  it is transmitted to Gospel B and the event that
  it is transmitted directly from Gospel A to
  Gospel C are independent.
• Given that a word is in Gospel B, the event that
  it is in Gospel A and the event that it is
  transmitted from Gospel B to Gospel C are
  independent.
• The event C1 that a word is transmitted from
  Gospel B to Gospel C and the event C2 that it is
  transmitted directly from Gospel A to Gospel C
  are mutually exclusive.

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Using the modified assumptions, we obtain the
followingformulae, which we use to evaluate z
and then y.
Pr(B?CA)
z
Pr(BA)
y Pr(CB) - z Pr(AB)
For z the formula is unchanged, but for y it is
simpler than before. We find also the following
simpler formulae
Pr(B?CA) x(y z)
Pr(CA) xy z
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Macro or micro level?
To find evidence for or against any particular
model, biblical scholars study in detail the
texts of individual pericopes.Should we attempt
to fit the triple-link model to
individualpericopes, specifically those from the
triple tradition, and to examine in detail how
the model works for each pericope? When we
attempt to do so, it turns out that there is
greatvariation from pericope to pericope in how
well the model fitsand in which permutation
A-B-C gives the best fit.
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The importance of oral tradition
Synoptic models tend to assume that the close
similarities between the gospels are to be
explained by the evangelists use of written
sources. In recent years there has been a
renewed focus on the role oforal performance in
the transmission of stories about Jesusand his
teaching, e.g., Dunn (2003, 2005), Bauckham
(2006).The gospel writers may have had the
texts of earliergospels available to them, but
they would also have beenfamiliar with
traditions transmitted by oral performance.
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A simple model based on Dunn
Mt
mainly oral source material
Mk
Lk
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Indications of oral tradition
Where the different versions of a pericope are
very similar to each other, word for word, this
suggests literary dependence. Where the
relationship is looser, this may suggest the use
of oral traditions. Using the Tyson and
Longstaff data, we study the 100 triple-tradition
pericopes.
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A measure of similarity
For a given pericope, letnA be the number of
words in Gospel A, nB the number of words in
Gospel B, nAB the number of words common to both
A and B. Define a measure of similarity by
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Exploratory analysis of similarities
For each of the 100 triple-tradition
pericopes, let x1 s(Mt,Mk), x2 s(Mt,Lk), x3
s(Mk,Lk). The mean vector of (x1, x2, x3) is
given by (0.440, 0.290, 0.331) and the
correlation matrix by
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Principal component analysis
Based upon the covariance matrix of x1
s(Mt,Mk), x2 s(Mt,Lk), x3 s(Mk,Lk), the
principal components are given by
PC1 0.56 x1 0.55 x2 0.62 x3 (57) PC2
0.82 x1 - 0.31 x2 - 0.47 x3 (29) PC3 - 0.07
x1 0.78 x2 - 0.63 x3 (14)
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Extreme scores on PC1
Pericopes with high scores on PC1, very similar
wording Mk 1330-32//Mt 2434-36//Lk
2132-33 (the Time of the Coming words of
Jesus) Mk 1328-29//Mt 2432-33//Lk 2129-31 (the
Parable of the Fig Tree words of
Jesus) Pericopes with low scores on PC1,
dissimilar wording Mk 135-39//Mt 423-25//Lk
442-44 (Jesus Ministry in Galilee essentially
narrative) Mk 37-12//Mt 1215-21//Lk 617-19 (a
Multitude Healed narrative)
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Two other highlighted pericopes
Pericope with low score on PC2 and PC3, Mk and Lk
very similar, Mt different Mk 1238-40//Mt
231-12//Lk 2045-47 (Beware of the Scribes
words of Jesus plus narrative
framework) Pericope with low score on PC2, high
score on PC3, Mt and Lk very similar, Mk
different Mk 811-13//Mt 1238-42//Lk
1129-32 (the Sign of Jonah words of Jesus
plus narrative framework)
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More nuanced ways of measuring agreement?
The counting of exact verbal agreements, counting
only words having exactly the same grammatical
form, is a simple, yet crude way of measuring
agreement between texts. We might consider
counting the same words even if they have a
different ending. We might consider counting
synonyms. We might take into account word order
or, more generally, in some way, the grammatical
and narrative structure of the text. (Tyson and
Longstaff give data on some aspects of this.)
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Computational questions
The data compiled by Honoré and by Tyson and
Longstaff were produced by examining the texts by
eye and counting the words. How easy would it be
nowadays to carry out such a process by computer
analysis of the texts? What other, more
sophisticated, methods of measuring textual
similarity are there available?