Subject: Request: help with Colwell's quantitative analysis

From: "Perry L. Stepp" <[email-masked]>
Date: Thu, 1 May 1997 03:00:23 -0500

Help!  

I'm looking at Colwell and Tune's article on quantitative analysis ("Method
in Establishing Quantitative Relationships between Text-Types of New
Testament Manuscripts," 56ff in Colwell, *Studies in Methodology in Textual
Criticism of the New Testament*).  It occured to me, as I read the article,
that I'm not totally clear on part of the method.  I thought I understood
it, but now I'm wondering.

Colwell says: "the quantitative definition of a text-type is a group of
manuscripts that agree more than 70 per cent of the time and is separated
by a gap of about 10 per cent from its neighbors" (59).  I'm fine with the
70 percent agreement, but I'm not sure how to discern the 10 percent gap. 
Looking at the percentages, how does one use the 10 percent gap to
delineate one group from another?  What am I missing?

If anyone out there has a copy of Colwell's *Studies* and is willing to
explain the 10 percent gap (preferably via reference to Colwell's charts),
I'd *really* appreciate it.

Grace and peace, 

Perry L. Stepp

************************************************************
Pastor, DeSoto Christian Church, DeSoto TX
Ph.D. candidate in New Testament, Baylor University

"A system of morality which is based on relative 
emotional values is a mere illusion, a thoroughly vulgar
conception which has nothing sound in it and nothing
true."
                    Phaedo 69b
************************************************************

---

From: "Professor L.W. Hurtado" <[email-masked]>
Date: Thu, 1 May 1997 16:49:50 +000

> From:          "Perry L. Stepp" <[email-masked]>
> 

Colwell's numbers are inductively arrived.  Here's how it works.  
Question:  What does "text-type" agreement mean (i.e., when are mss 
to be thought of as primary members of the same "text-type"/group)?
Steps toward the answer:
1) Text-type agreement must be (a) based on *all* types of variation, 
not merely on variants against the TR etc., and (b) must evidence a 
sufficiently close whole text of a NT writing or chunk of it to 
constitute a likely real relationship.
2) We start by assuming that widely recognized witnesses of major 
text-types are rightly so regarded, and thus choosing such leading 
reps. we collate them all together listing all variants where any two 
or more of the whole selection of reps. agree, and counting the 
agreements at those 'variation units' of each possible pair of mss, 
then converting the count into percentages of the total number of 
variation-units.
3) when we do this (e.g., in Mark) Aleph & B come out agreeing at 
least 70% or much more, and considerly (at least by 10 % points) more 
than either of them agrees with any major witness of any other major 
textual group/type.
4) Thus, if we're going to take Aleph and B as an example of a 
"text-type" relationship, then their quantitative agreement becomes 
the measure for such things.
There's nothing transcendent about this.  It's purely a pragmatic way 
of trying to move the definition a bit along the lines of precision 
while trying to stay as close as we can to the empirical data.
 
L. W. Hurtado
University of Edinburgh,
New College
Mound Place 
Edinburgh, Scotland EH1 2LX
Phone: 0131-650-8920
Fax: 0131-650-6579
E-mail:  [email-masked]

---

From: "Robert B. Waltz" <[email-masked]>
Date: Thu, 1 May 1997 11:42:13 -0700

>I'd *really* appreciate it.

Let's start with the rates of agreement. I'm going to refer to Table I
(page 60 in Colwell's book) and give the data for B as Colwell gives it:

 75  TR
 61  p45
 69  p66
 73  p66**
 92  p75
 79  Aleph
 83  Aleph**
 73  A
 73  A**
 53  D
 77  W
 78  W**
 65  Theta
 81  Psi
 74  Omega
 67  Cr
 71  565

Now let's sort that in ascending order of agreement:

 53  D
 61  p45
 65  Theta
 67  Cr
 69  p66
 71  565
 73  A
 73  A**
 73  p66**
 74  Omega
 75  TR
 77  W
 78  W**
 79  Aleph
 81  Psi
 83  Aleph**
 92  p75

Now I'm going to add in the separations -- that is, the distance between
the percentage agreement between the current manuscript and the next one
in the list

 53  D         8
 61  p45       4
 65  Theta     2
 67  Cr        2
 69  p66       2
 71  565       2
 73  A         0
 73  A**       0
 73  p66**     1
 74  Omega     1
 75  TR        2
 77  W         1
 78  W**       1
 79  Aleph     2
 81  Psi       2
 83  Aleph**   9
 92  p75       -

Looking at these separations, there largest of these is 9 (between
Aleph** and p75). If that were 10%, instead of 9%, it would be a
10% gap and meet Colwell's definition. If we ignore Aleph**, though,
as a text that may never have existed, we get a separation of 11 points
(*not* percentage points) between p75 and Psi, so that at least
would would constitute a gap. (Does this mean that p75 and B form
a text-type all on their own? Well, not exactly.)

That's what Colwell and Tune mean by the "gap." Now for the flies
in the ointment. (This is where I reveal the truth about the defects
in Colwell and Tune, so all you people who think I can't get anything
right can go back to sleep now. We're going to do some mathematical
thinking here.)

I'm quoting my web page, from the Text-Types article, section
"Definition of a Text-type":

It was not until the mid-Twentieth century that E.C. Colwell offered the first 
balanced definition of a text-type.[*1] In one essay he gave a qualitative 
definition ("A Text-type is the largest group of sources which can be 
generally identified").[2] He adds the important qualification, "This 
definition is a definition of a text-type as a *group of manuscripts* 
[emphasis mine], not... a list of readings." Five years later, in an 
influential essay, Colwell went further. He attempted a quantitative 
definition. (Indeed, his method is frequently called the "quantitative method" 
-- a name that makes me cringe, since any statistical method is a 
"quantitative method.") His statement on the subject is perhaps the most-
quoted statement on genealogy since Hort's time: 

"This suggest that the quantitative definition of a text-type is a group of 
manuscripts that agree more than 70 per cent of the time and is separated by a 
gap of about ten percent from its neighbors."[3] 

Colwell deserves immense credit for offering this definition (as well as for 
his other methodological studies; he is perhaps the greatest worker in this 
field in the twentieth century). This definition has the advantages of being 
clear, precise, and usable. Unfortunately, in the author's experience, it does 
not work. (It strikes me as almost tragic that Colwell's most-frequently-cited 
comment on text-types is also one of the few that is not entirely correct. 
It's worth noting that he rarely if ever refers back to this criterion.)

There are two reasons for this. First, the percentage of agreements between 
manuscripts is entirely dependent on the sample. Second, the "gap" which 
Colwell refers to disappears when working with mixed manuscripts. Let us offer 
examples.[*4] 

To take the first point first, consider the relationship between B/03 and 
Aleph/01 in chapter 2 of Colossians. The two manuscripts agree in only two of 
the seven variations cited in GNT4, or 29%. If we take the 29 variants cited 
in NA27 (excluding conjectures), we find that they agree in 18 of 29, or 62%. 
If we turn to the Munster Institute's New Testament "Auf Papyrus," and examine 
the variants supported by two or more uncials (excluding orthographic 
variants), we find that the two agree in 32 of 47, or 68%. But if we turn to 
the editia minor of Tischendorf8, we find agreement in 19 of 32 non-
orthographic variants, or 59%. Even if we throw out the small GNT sample, we 
still have almost a ten percent variation between the three remaining sample 
sets, all of which form large and reasonable bases for comparison. Which one 
should we use in deciding whether B and Aleph belong to the same text-type? 
The 68% number, which places them on the fringe of qualifying? The 59% number, 
which isn't even close? Or something else? 

All told, Aleph and B have 25 disagreements in this chapter (though some are 
scribal errors, usually in Aleph). How do we decide how many variants to 
spread these 25 differences out over to determine if there is 70% agreement?

A thought-experiment about mixed minuscules should be sufficient to 
demonstrate the non-existence of the "gap." Suppose X is an unmixed 
manuscript, Y is copied from X with five percent Byzantine admixture, Z is 
copied from Y with another 5% admixture, and so on. It follows that X can 
never have a ten percent gap; that space is occupied by Y, Z, and so on down 
the line. If that is not proof enough, one can present a concrete example 
based on B in the Gospels. Using a large (990 reading) sample and 39 Greek 
manuscripts, I found two documents (2427 and p75) which, in their particular 
areas, agreed with B over 80% of the time. Below this was a gap -- but most 
manuscripts that are considered to belong with B (including Aleph, L, 33, and 
892) are on the far side of the gap![*5] The next-closest manuscript was 
Xi/040 in Luke, at 68%. From there down to the final manuscript in the list 
(D/05, with 30% overall agreement), there was no gap larger than eight 
percentage points (and even this gap would have been filled had I included the 
Coptic versions).

The median gap among non-Byzantine manuscripts was one, and even the 
arithmetic mean ("average") was under two. Colwell's "gaps" will simply not 
exist in large manuscript samples. 

There is also a problem with the conceptual model of the Colwell system. Take 
a manuscript like L/019 of the gospels. It has a significant Byzantine 
component -- large enough that it will likely fail Colwell's 70% criterion. 
But -- where it is non-Byzantine -- it stands very close to B/03, and is one 
of the closest allies of that manuscript. Should we not be able to recognize L 
as a degenerate relative of B, and use it on that basis? 

Some would propose to address the problem by adjusting the numbers. This may 
work in some cases, but cannot be guaranteed; any statistic will be dependent 
on its sample. It is possible that we could assign percentages if we could 
produce a "representative" list of variants -- but what is a "representative" 
variant reading?

**** Footnotes: ****

1. Indeed, Colwell was one of the first to plead exclusively for the use of 
the word "text-type" in this context. See Colwell, Studies in Methodology, p. 
9.

2. Ibid.

3. Ernest C. Colwell and Ernest W. Tune, "Method in Establishing Quantitative 
Relationships between Text-Types of New Testament Manuscripts," reprinted in 
Studies in Methodology, p. 59.

4. Ironically, it was Colwell himself who first pointed out the defect in his 
method -- four years before he proposed his definition! In "Method in Locating 
a Newly-Discovered Manuscript" (Studies in Methodology, page 33), he wrote 
"Weak members of a Text-type may contain no more of the total content of a 
text-type than strong members of some other text-type may contain. The 
comparison in total agreements of one manuscript with another manuscript has 
little significance beyond that of confirmation, and then only if the 
agreement is large enough to be distinctive."

5. If someone objects that comparisons across the gospel corpus are not valid, 
let me simply add that I examined individual books, and even sections of 
books, and the results were the same within the margin for error. At times the 
leading manuscripts (especially W) shifted slightly, but the general picture 
never did. So I present overal statistics because they are simpler.

-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-

                        Robert B. Waltz
                     [email-masked]

Want more loudmouthed opinions about textual criticism?
Try my web page: http://www.skypoint.com/~waltzmn
(A site inspired by the Encyclopedia of NT Textual Criticism)

---

From: "Professor L.W. Hurtado" <[email-masked]>
Date: Fri, 2 May 1997 09:25:45 +000

A few rjoinder notes to Waltz's piece on Colwell and quantitative 
approaches.
1) Waltz cites figures from *Table 1* on p. 60 of Colwell, 
_Methodology_, which includes *singular* readings, which are (as 
Colwell explains in the essay) not useful for identifying 
*relationships*.  Thus, Table 2 on the same page refines the numbers 
by including only those variation units where the tested mss exhibit 
agreement of at least two of them.  There the separation becomes 
clearer.  This exclusion of singular readings is an important 
feature.  By definition, singular readings don't attest 
relationships
 2) One ofthe ways Colwell's method has been refined subsequently (by 
e.g., Fee and yours truly) is to insist that identification of mss 
relationships cannot safely be done in samples (and in the essay from 
which the tables come and in which Colwell sketches the basics of his 
method, he cites figures for John 11, a sample only), and that we 
have to take measurements (a) across a whole NT writing, and (b) 
chapter by chapter.  If we do this, we can detect "block-mixture" 
(as, e.g., Fee did in Aleph in John, and as I confirmed in W in 
Mark).  
Where the "mixture" is not block-mixture but simply the apparent 
influence of one textual tradition upon another or the drift of a 
textual tradition towards another, the percentages of agreement will 
suggest what is going on *if you include relevant witnesses from 
sufficient major groups to measure comparative agreements (as, e.g., 
I attempted with ref. to W in Mark, showing that as the W-P45-Fam 13 
tradition progresses it becomes more closely allied with the 
"Byzantine" type mss witnesses.

L. W. Hurtado
University of Edinburgh,
New College
Mound Place

---

From: "Robert B. Waltz" <[email-masked]>
Date: Fri, 2 May 1997 08:58:25 -0700

>feature.  By definition, singular readings don't attest 

OK, I'll concede that one. I just took the first table I saw.
That affects the list of manuscripts, but does not affect the
meaning. I thought of doing my example with fudged numbers. I
suppose I should have done so.

The list supplied *did* demonstrate one point, though: how
few gaps actually exist. Mixture (not block mixture, *mixture*)
fills most of them in.

>(as, e.g., Fee did in Aleph in John, and as I confirmed in W in 

True, but not relevant to the definition of the gap. :-)
 
>"Byzantine" type mss witnesses.

Agreed. However, I interpret this differently (not your results,
but the meaning of the situation).

To examine text-types in their entirety means that we must
examine *all* the witnesses, not just the "good" ones. This
includes the mixed witnesses. This means, almost certainly,
that there *will be* no gaps, even if a text-type exists.

Allow me to give an example. I admit that this is based on
a sample, not on the entire Gospel corpus, but it is based on
a 990 reading corpus, so it should be representative assuming
it is possible to take a representative sample in the Gospels.
(By no means an assured point, and that could invalidate all
studies -- including Hurtado's -- done so far.)

Below is a sorted list of agreements with Aleph in those 990
readings. I've omitted the papyri as incomplete and W as block-mixed.

 %   MS      "Gap"
 35  D       0
 35  E       0
 35  G       0
 35  N       0
 35  Omega   0
 35  U       1
 36  A       0
 36  Gamma   0
 36  K       0
 36  M       0
 36  Pi      1
 37  1424    0
 37  565     0
 37  700     0
 37  fam 13  1
 38  1010    0
 38  1506    1
 39  1071    0
 39  Theta   1
 40  X       3
 43  1342    0
 43  fam 1   2
 45  1241    4
 49  579     1
 50  33      2
 52  892     0
 52  C       7
 59  B       1
 60  L

Notice that the largest gap in this list is 7, and the median gap is --
ahem! -- *zero*. And that's based on fewer than thirty manuscripts and
no versions. Throw in the Coptic versions and our largest gap shrinks to
four. Throw in every Greek manuscript I studied (I admit this is bad
technique) and the gap shrinks to *three*. In other words, there *is no
gap.*

I would also note that the highest rate of agreement observed is 60%.
This obviously is much lower than Colwell's 70% threshold. This demonstrates
another point: *Statistics depend upon their sample.* Even if we use
some fixed rule such as "all variants supported by at least two manuscripts,"
you will still have variations in your sample depending on what manuscript
base you use.

You can't assign some number here and expect it to work in every case.

Please, people, study statistics before you use Colwell!

Let me also stress that I am not attacking Colwell. I think he was
the greatest textual thinker of his generation. My article on
text-types quotes him more than any other. It is simply his misfortune
that his one major mistake is the thing that so many scholars have
latched onto as if it were gospel. It is, in fact, a statement with
inadequate statistical foundations.

                        Robert B. Waltz
                     [email-masked]

---

From: "Professor L.W. Hurtado" <[email-masked]>
Date: Fri, 2 May 1997 16:15:49 +000

Well, a few more response are called for by Waltz's latest.
1) Sure, the number counts will depend upon the mss selected for the 
exercise.  That's why it's essential to the method to choose an 
adequate list of all relevant witnesses, relevant to determining 
which group mss belong to.  This doesn't affect the validity of the 
method.
2) I fail to see how my work is invalidated if samples are invalid.  
I didn't use samples, but collated across the whole of Mark.
3) I quite agree with Waltz (and said previously the same) that there 
is nothing "magical" (his term; mine was "transcendent") about 
Colwell's numbers.  Their beauty is that they were arrived at 
*inductively* by beginning with counting agreements of mss and seeing 
what happened.  So, with all due respect to Waltz on statistics, one 
really doesn't have to be a stats techie to use the method, and I've 
seen no stats argument showing otherwise.
4) I do want to know from Waltz the basis for his own numbers in his 
last posting.  Agreements of this or that ms with Aleph at 990 
variation units . . .?  What is the basis for determining the 
variation units?  Mere counts of agreements of this ms with that one 
mean nothing *unless the counts are set within some meaningful 
context*.  That is why in the Colwell method all counts are so 
carefully defined:  e.g., the agreement of any two mss is seen in the 
context of the agreement of all other possible pairs (hence the grids 
of percentages, not merely a single row of numbers without a 
context).

L. W. Hurtado
University of Edinburgh,
New College
Mound Place

---

From: "Robert B. Waltz" <[email-masked]>
Date: Fri, 2 May 1997 11:33:15 -0700

Here we go again. Please, people, read what I say, not what you
think I'm arguing against. Specifically to Hurtado: I am *not*
attacking your study. In fact, I have quoted it -- and even
derived an interesting idea from it (could the so-called
"Caesarean" text actually be a true branch of the "Western" text,
and D an abberation?). I am attacking the method that Colwell
described in one essay and, as best I can tell, *never used again*....

>which group mss belong to.  This doesn't affect the validity of the 

Of *course* it affects the validity of the method. How do you
know, before you've done the work, which manuscripts are
significant?

You cannot assume the solution.

>I didn't use samples, but collated across the whole of Mark.

Simply false. You collated across the whole of Mark for readings
supported by at least two of the group p45 Aleph A B D W Theta f13
565 TR. That *is* a sample. You may say that it includes at least
two witnesses from the Alexandrian, Byzantine, and "Caesarean"
text-types. Probably true, but you don't know that. (Robinson
will tell you that neither the TR nor A is a perfectly Byzantine
witness.) And where is the second "Western" witness?

You are once again assuming the solution. And you also ignore
the possibility of mixture (not block mixture, mixture). Both
the Alexandrian and Byzantine texts are represented by only
two witnesses. Suppose one is mixed for a particular reading.
Out it goes from your sample.

You have used a sampling technique, merely one that uses a
very large sample. A large enough sample, I confess, that it
probably makes up for its methodoligical bias -- but only if
sampling works. (It probably does -- but can you prove it to
me?)

Then consider the possibility of an undiscovered text-type.
With this sample, you cannot find it.

>seen no stats argument showing otherwise.

I agree, anyone can use the method. So what? That doesn't make it
valid. I am not criticising your use of the method. I am criticising
the method itself.

I'll give an analogy. I know how to drive an automobile, and I
assume Hurtado does also. Does that make me competent to design
an automobile? Or repair one? Hardly. Nor would I want to critique
its design.

I am critiquing the design of Colwell's automobile -- which, as
someone with mathematical training, I have at least some skill
to do (not much, but more, at least, than Colwell had). Hurtado
is saying that I can't critique the design because he (Hurtado)
knows how to drive the car!

>of percentages, not merely a single row of numbers without a 

I gave a sample to demonstrate a point. The point was demonstrated:
There was no 10% gap. You are the one who says that I can use any
set of manuscripts, and therefore any sampling method, I want.
Therefore my results are valid. Q.E.D.

And before anyone attacks that, I agree, the result I just stated
is not valid. But it demonstrates the importance of knowing what
one is sampling.

If one wishes to know the list of readings involved, it starts with
the variants found in the apparatus of GNT. I supplemented this with
a filter to find variants roughly every ten verses. (I say "roughly"
because I too required two witnesses to support each reading, and I
also had to take variants for which I could learn the readings of
all the witnesses.) If someone wants a list of readings, I will
offer them.

I concede that this is not a perfect method. It is biased toward
readings where B and Aleph divide, because those are the readings
that puzzled the editors of GNT. (This is one of the reasons
why I added my 400+ arbitrary readings.) This probably lowers
the overall list of agreements.

I simply state that there *is no* perfect method. And -- since
there is no method that adequately captures lists of variants --
we cannot treat the 70% number as magic. Nor can we count
on the gap in all cases, since there are always mixed manuscripts
floating around. The inclusion or exclusion of such a manuscript
(in Mark, L would be a good example) will determine whether or
not the gap exists.

I will repeat -- again -- my example from Colossians 2.

In this chapter, Aleph and B disagree 25 times. Spread over how many
readings? (Particularly since eight of those readings have no other
uncial support.)

If we take readings which are supported by two or more uncials,
then the two agree in  32 of 47, or 68%. But if we take readings
found in the Nestle apparatus, we get 62%. If we take Tischendorf's
minor eighth edition, we get 59%.

Can someone tell me which of those numbers is "right," and why?

And, BTW, does this mean B and Aleph don't belong to the same
text-type in Colossians 2? (I say yes, but not many people seem
to agree with me. And even I say it for other reasons....)

                        Robert B. Waltz
                     [email-masked]

---

From: "Professor L.W. Hurtado" <[email-masked]>
Date: Fri, 2 May 1997 18:11:57 +000

Waltz writes:
> think I'm arguing against. 

Excellent advice!  Let's watch the beams as well as the motes, Bob!

Waltz asks:  How do you
> know, before you've done the work, which manuscripts are
Bob.  We have to start somewhere right?  So, in Colwell's method, we 
start with what we *think* we know, and test it to see if our 
knowledge is correct.  So, we start with the important mss witnesses 
we *think* are reps of what we *think* are major groups.  Then we 
collate them & the counts will indicate whether they hang together in 
groups or not, how much they diverge etc., provided that (a) you have 
enough witnesses to detect what you're looking for (e.g., alignment 
with *major* groups, or whatever), and (b) provided you've used 
enough text and variation units to represent something.
> You cannot assume the solution.

Uh, right.  the method involves testing assumptions, Bob.
> witness.) And where is the second "Western" witness?

Bob, in TC "samples" normally mean sample bits, chapters, etc.  
Second, I collated *all* of Mark, noting *all* variations among *all* 
witnesses chosen, and counted *all* variation-units that were 
*meaningful* -- i.e., where it was possible to measure *agreements* 
of the witnesses.  
I don't know (nor does Robinson) what a "perfect" Byz witness would 
be.  I used what are considered sufficiently good representatives of 
both earlier and later stages of the Byz text-type.  And D is the 
only Greek witness in Mark--not something I can do anything about.  
In any case, I know of nobody who suggests that D in Mark is *not* a 
major "Wstern" witness.

> Out it goes from your sample.
Once again, Bob, we all make assumptions, and the aim is to test 
them, which is *precisely what the method aims to offer*.  Moreover, 
I don't "ignore mixture" (by which I presume you mean mss influenced 
by more than one tradition/tendency, etc.).  E.g., I point out that 
Fam 13 shows a drift toward the Byz text-type--precisely the 
"mixture" (your term) you claim I ignore.  I've pointed this out 
earlier today, but you seem stronger on making charges than paying 
attention!
As to your last sentence in the quote above, I'm not sure what you 
mean.  All readings/variants that are supported by any two of the 
reps chosen are used whatever their complexion ("mixed" or not).
> sampling works. (It probably does -- but can you prove it to
You're simply not using "sample" in any meaningful way, but I can't 
stop you if you want to do so.  And could you justify such pejorative 
language as "methodological bias"?
> With this sample, you cannot find it.

Uh, Bob.  I *did* make some discoveries.  (1) W & P45 are *not* part 
of the "Caesarean" text-type, and (2) do seem to form agreement 
sufficient to make them look like a somewhat distinctive group.  So, 
the method does in principle allow for what you deny.

> valid. I am not criticising your use of the method. I am criticising
Again, Bob, take your own advice about careful reading of others.  
You've plainly distorted my point, which was that the method doesn't 
involve fancy, techie stats stuff, just empirical measurements and 
testings of hypotheses.
You say:
> Hurtado

You've lost me again, Bob.  I didn't say you *couldn't* criticize 
anything.  I said your criticisms aren't impressive because they miss 
the marks.

I asked:
> >context*. 
> Therefore my results are valid. Q.E.D.

This is getting tiresome, Bob.  I did *not* say "any set of mss", for 
heaven's sake!  I said chosen mss thought to represent major groups.  
I asked for the things that might make your figures meaningful and 
you again distort my words and dodge the question.  Not good, Bob. 

Waltz goes on: 
> we cannot treat the 70% number as magic.
Who said "perfect" and what would that be, Bob?  The aim is a method 
that is appropriate to the question and that stands up to competent 
analysis & criticism.  Thus far, the refined Colwell method meets 
these criteria, but publish your results and lets see.  And no one 
said "magic", either.  Just pragmatic and verifiable.

> Nor can we count
 Yup.  There are mss that seem to lie between major groups . . . as 
the Colwell method will demonstrate, and as it will even allow one to 
portray quantitatively in a meaningful way by comparison with the 
quantitative agreements of mss of the group that are solid reps of 
it. If the "gap" is not very large, then the method shows  . . . just 
what you refer to.  What's the problem?

L. W. Hurtado
University of Edinburgh,
New College
Mound Place

---

From: "Robert B. Waltz" <[email-masked]>
Date: Fri, 2 May 1997 12:49:18 -0700

TCers --

Hurtado and I have gotten into what is obviously a very nasty debate.
He is either not understanding or refusing to read what I say, and
evidently he feels that I am doing the same. What is more, it appears
to me that we are re-hashing arguments that have been said many
times before.

I could rebut Hurtado's post, but I'm not going to convince
him. And he won't convince me. (After all, I'm right and he's
wrong. :-)

The question is, is anyone else listening? Is there any point
in attempting a rebuttal? I doubt it -- but if anyone wants
us to keep flailing away at our respective viewpoints, let me know.
Otherwise, I will spare my blood pressure (and perhaps yours)
and let this thread die.

                        Robert B. Waltz
                     [email-masked]

---

From: "James R. Adair" <[email-masked]>
Date: Fri, 2 May 1997 14:21:32 -0400 (EDT)

I personally have found the debate between Hurtado and Waltz to be
interesting, especially when matters of substance and theory are
emphasized and personal comments are not.  Let me ask the two antagonists
(and anyone else who wants to respond) about a few specific points that I
see coming out of this debate.

(1) Is the existence of a gap of about 10% between groups of witnesses
really a useful measurement?  I can see measuring levels of agreement
(whether 70% or some other number), but surely manuscripts can always be
chosen in such a way as to eliminate any substantial gap in levels of
agreement.  Maybe the gap could be redefined in a manner similar to this:
if one group of mss agrees with one another in 70% of readings, and
another group of mss likewise agrees among its members in 70% of _its_
readings, _and_ there is a gap of 10% between any two members of one
groups vs. the other, _then_ we can identify two different text-types.
Such a definition would seem to eliminate mixed mss from consideration
_when determining text-types_ (i.e., not entirely from consideration).

(2) The 70% agreement has often been questioned.  Other than the Byzantine
and Alexandrian witnesses, does any other group show a 70% level of
agreement anywhere in the NT?  What level of agreement is there among the
OL witnesses (eliminating D from the Western group)?  Bob, what level of
agreement is there in the groups you identify in Paul?

(3) What exactly do levels of agreement tell us?  They say that mss are
very similar, but is that really the same thing as saying that they are
genetically related (as a group)?  Does discovery of text-types tell us
more about where patterns of reading originated or more about where they
ended up?  One way to increase the level of agreement, of course, is to
correct a ms from another ms, so levels of agreement may not say much
about historical origin.  As an analogy, consider the English language.
Because of the Norman invasion of England, Modern English contains many,
many words of French origin.  Nevertheless, at its root, it is a Germanic
language.  Using statistical sampling, would we call Modern English a
member of the Germanic or the Romance text-type (or would we say it's
mixed?)?

(4) The word "sample" is being used differently by Hurtado and Waltz, it
seems to me.  Hurtado says, "Bob, in TC 'samples' normally mean sample
bits, chapters, etc.  Second, I collated *all* of Mark, noting *all*
variations among *all* witnesses chosen, and counted *all* variation-units
that were *meaningful* -- i.e., where it was possible to measure
*agreements* of the witnesses."  It appears to me that Larry is using
the word "sample" in a way commonly used in TC circles, whereas Bob is 
using a more formal mathematical (statistical) definition.  I think that
TCers will have to bow to math in this case.  Even if "samples" normally
mean "sample bits, chapters, etc." in TC circles, it is clear that this
terminology is weak and should be changed to agree with the more rigorous
math definition, especially when TCers are dealing with statistics.
Despite the "alls" of Hurtado's study, there is a sample: in the choice of
manuscripts.  It may be a good sample, but it's still a sample.

I'd be interested to see what Larry and Bob (and others) have to say about
these points. 

Jimmy Adair
Manager of Information Technology Services, Scholars Press
    and
Managing Editor of TELA, the Scholars Press World Wide Web Site
---------------> http://scholar.cc.emory.edu <-----------------

---

From: "Robert B. Waltz" <[email-masked]>
Date: Fri, 2 May 1997 15:30:33 -0700

On Fri, 2 May 1997, "James R. Adair" <[email-masked]>
wrote:

>emphasized and personal comments are not.

I must apologise for my part in that. I will admit that the recent
unpleasantness about Academic Imperialism made me rather thin-skinned,
since I am one who does not have university credentials in TC.

>Let me ask the two antagonists

Nice to see something I can answer easily.

>agreement.

That is my contention. Surprisingly, I do think there is value
in the concept of the "gap" -- but I don't believe it can be applied
rigidly. In a large enough sample set, a gap of only a few percentage
points can be significant.

Also, I think it more useful to apply the gap to classified agreements
than to overall percentages of agreement. I offer as an example 630
in the Paulines. In my overall table of agreements, its closest relative
is L (77%). That's because 630 has suffered heavy Byzantine correction.

However, if we look at 630's handful of near-singular readings, we
find that it shares no fewer than 6 with 1739. Other than 1739 and
its obvious relative 1881, the next highest rate of near-singulars
is 3 with B and 2 with p46. So *there* we see a gap of significance.

By combining these facts, we come to the conclusion (borne out
by other data) that 630 is a heavily mixed manuscript, more Byzantine
than anything else, but with a strong infusion of family 1739
readings (in fact, family 1739 readings outnumber Byzantine readings
in Romans-Galatians; after that, 630 is effectively purely Byzantine).

>Maybe the gap could be redefined in a manner similar to this:

Personally I see a lot of merit in this idea. It makes analysis a lot
harder, but the concept is good.

It does leave one problem: It leaves us stuck if we have to investigate
a text-type which no longer exists in pure form in any manuscript.
Which, sadly, puts us back where we started in my last argument with
Hurtado: the advocates of the "Caesarean" text explicitly defined it
in terms of readings mined from mixed manuscripts.

Still, if you'll allow me to fudge those numbers a little, depending
on the sample, I could live with this idea.

>agreement is there in the groups you identify in Paul?

I'll give as much data here as I can type in half an hour. (BTW -- if
some of the numbers don't agree, blame the fact that I'm retyping all
this. If there's an inconsistency, tell me and I'll check it.)

*************************************

Gospels: If we omit D, and people are willing to live with my samples,
the "European" Latins (a, b, ff2) seem to agree about 70-80% of the
time. (By comparison, they agree with D about 60-65% of the time,
and with D about 35% of the time. They agree with k about 55% of the
time.)

*************************************

I have no good data for Acts.

*************************************

I'm going to annotate the data for Paul. :-)

*************************************

p46/B type:
p46: Closest relative is B, 62% of the time. Next closest is 1739, 56% of
  the time. p46 and B also have 34 near-singular agreements, as opposed to
  14 between p46 and Aleph, 21 between p46 and D, 16 between p46 and 1739.
B: Closest relative is actually the Sahidic, 68% of the time. After that,
  it's about a wash between 1739 (64%) and p46 (62%). Aleph is at 57%,
  D at 42%. But p46 remains the clear winner in near-singulars (34, to
  26 for the Sahidic and 18 for 1739).

Alexandrian text:
Aleph: Closest relative is A, 76%. Outside Romans (where 33 is Byzantine),
  33 takes the prize at about 77%. C agrees 76% , 1506 72%, the
  Bohairic 73%. The fragmentary I gets the prize where it exists, 80%.
  For comparison, B agrees with Aleph 57%, D agrees 47%, 1739 agrees 63%.
A: Closest is C at 77%, followed by Aleph at 76. With B, 53%; with D,
  47%; with 1739, 62%
C: Closest is A, 77%, followed by Aleph, 76%. With B, 51%; with D,
  48%; with 1739, 63%.
33: Closest is Aleph, 69%. A C follow at 65% (exclude Romans and all
  these numbers go up by about 7-8 points). With B, 54%; with D, 48%;
  with 1739, 61%.

"Western" text:
D: Closest relative , not surprisingly, is d, 84%. Next are F G, 72%,
  then the other Old Latins in the 60-70% range. With B, 42%; with
  Aleph, 47%; with 1739, 47%.
G: Other than, of course, F f, the closest relatives are D d, 72%.
  With B, 41%; with Aleph, 46%; with 1739, 47%.

As a typical example of the Byzantine text, we note that K L
  agree 91% of the time.

Family 1739:
0243 and 1739 agree 96% of the time. (I hope soon to study this
  relationship in detail.)
6 agrees with 1739 64% of the time (compared to 69% with L), but
  has 19 near-singular agreements. (Next on the near-singular list,
  other than 424** with 18 and 1881 with 12, is B with 10)
The corrections in 424 go with 1739 91% of the time, and of 424**'s
  26 near-singulars, 20, or 77%, are shared by 1739. (This is
  extraordinary, since in the typical manuscript at least a third
  of the near-singulars are errors.)
1881 agrees with 1739 77% of the time.
1739 itself agrees with p46 56% of the time, with B 64% of the time,
  with Aleph 63% of the time, with D 47% of the time.

*************************************

Now to the Catholics. Here I'll just make a table.

      p72   Aleph A     B     C     K     33    323   614   1241  1739  2495
p72   -     38    43    66    45    21    41    47    39    48    51    39  
Aleph 38    -     60    48    54    37    57    50    45    51    60    46
A     43    60    -     44    57    35    79    55    52    60    66    50
B     66    48    44    -     49    20    46    44    32    48    56    36
C     45    54    57    49    -     42    56    60    51    57    70    51
K     21    37    35    20    42    -     38    47    52    38    34    48
33    41    57    79    46    56    38    -     59    58    59    67    54
323   47    50    55    44    60    47    59    -     50    71    78    50
614   39    45    52    32    51    52    58    50    -     47    52    82
1241  48    51    60    48    57    38    59    71    47    -     80    49
1739  51    60    66    56    70    34    67    78    52    80    -     50
2495  39    46    50    36    51    48    54    50    82    49    50    -

*************************************

To summarize (and I'm using some additional data not offered here):

In the Gospels, we find that the "Western" group falls on the ragged 
edge of the 70% rule -- so close to the edge that the sample might
tip things one way or the other. There is a huge "gap" between
"Western" and non-Western, though.

In Paul, the Alexandrian, "Western," and Family 1739 texts all
meet the 70% rule for their stronger members, though some of the
weaker ones fail. p46 and B *don't* make it, though they are clearly
akin. In Paul, for my sample, 60% might be a better threshold.
BTW -- if you observe, 1739 seems to agree with all three of
the other text-types more than they agree with each other.
This is the basis for my belief that it forms its own text-type.

In the Catholics, we find 80% agreement between the members of
family 2138 (614 630 1505 1611 1799 2138 2412 2495, etc.), with
subgroups breaking the 90% mark. The members of family 1739
(C 323 945 1241 1881 2298) gnerally exceed this mark with respect
to 1739 though not to each other. There is a group within the
Alexandrian text (A Psi 33 81 436 bo etc.) which exceeds the 70%
threshold, but neither Aleph nor p72/B are part of it. There
probably needs to be work done on the Alexandrian text in the
Catholics. (It may well be that p72/B are a separate text-type.)

N.B.
One thing that I think is worth noting is that, in the Catholics,
1241 agrees with 1739 80% of the time, and C agrees with 1739 70%
of the time. But 1241 and C agree only 57% of the time. Interesting,
huh? This ought to tell us something (I'm not sure what yet, but
it's one of those reasons why I'm concerned with the statistical
nature of variants).

>genetically related (as a group)?

To give a great answer, I say, "Yes and no." That is, a high rate of
agreement means that they have something in common -- but it may
be just a bunch of Byzantine mixture.

In my view, a manuscript (call it X) that is 80% Byzantine and
20% something else has value only in the 20% that's something
else. If we want to examine the Byzantine text, we have plenty
of purely Byzantine manuscripts to look at. For simplicity,
let's say that X had a Byzantine ancestor B and a less-Byzantine
ancestor O. Now it may be that a lot of X's Byzantine readings
came from O; we can't tell.

But it doesn't matter. I will freely concede to Hurtado, Colwell
and Tune, et al that X is more closely related to B and the
Byzantine text than to O and something else. However, I don't
see that it matters. Since we don't care about the Byzantine
text (given how many good examples we have), our interest is
in the small relationship with O.

>Does discovery of text-types tell us
>about historical origin.

Entirely agreed, from where I sit. See above.

>As an analogy, consider the English language.

Depends on the sampling method. :-)

If we sampled grammatically, or based on words in common use, we'd
say it's a Germanic language. If we look at a dictionary, where
obscure Latin-derived words predominate, we'd say it's a Romance
language.

In my view, the Grammar is simplified Germanic, the vocabulary is
mixed. I don't know what one would call the result.

>manuscripts.  It may be a good sample, but it's still a sample.

Thanks for pointing out the difference in language. You're right, I was
using the mathematical sense -- and not even thinking of other meanings.

Naturally, I vote for the mathematical use. :-)

>I'd be interested to see what Larry and Bob (and others) have to say about

I hope this clarifies things. If not, well, I'll try again next week
some time. :-)

Thanks for both the conciliatory words and the curiosity.

                        Robert B. Waltz
                     [email-masked]

---

From: "Robert B. Waltz" <[email-masked]>
Date: Sun, 4 May 1997 10:57:58 -0700

TCers --

In the last week or so, I've gotten several requests for information on
the use of statistics in TC. In one sense, that's good; it means I've
been getting you thinking. But I haven't been much help in my replies.
Given that the questions keep coming in, I think I have to try to give
something of an overview of *why*.

The basic question that has been asked is, "What is a good mathematical
reference for TCers?" As far as I know, there is none. If there were,
I would be quoting it, rather than stumbling along looking for methods
of my own.

Now I'll admit that there could be a great reference out there somewhere.
But I doubt it. And the reason that I doubt it is that we haven't even laid
all the groundwork.

To give an example: In the last few decades, we've done a great deal to
define what constitutes a "variant." Epp and Fee are among the big names
here. They've given us a nomenclature both for the variants themsleves
and for their nature. What they haven't given us is a rigid system for
determining the scope of variants. And the latter is vital.

Consider this: In Larry Richards's statistical analysis of the Catholic
Epistles, he goes to great lengths to make all his variants binary
(that is, to have *two and only two* readings). As I recall (and I'm
not checking this), he had only four ternary variants (those with
three readings) in his whole set of a hundred and some.

By contrast, about 25% of the variants in the apparatus of UBS/GNT
are ternary (or of even higher order).

Does this matter? You bet it does! If all variants are binary, then
in general we can say that, for any variant, one reading is Byzantine
and one reading is non-Byzantine. This means that, if two manuscripts
both have the non-Byzantine reading here, then they automatically
agree. In other words, agreement in non-Byzantine readings is meaningless;
every manuscript -- even one that is purely Byzantine -- will have 100%
agreement with every other in non-Byzantine readings.

But it's worth remembering that Lake, Streeter, et al *defined* the
"Caesarean" text in terms of the non-Byzantine readings. If all variants
are binary, then *all* manuscripts become "Caesarean" under this definition.
Whereas, if many of our variants are ternary, there are two non-Byzantine
readings, and agreement in non-Byzantines becomes a meaningful statistic.

Is anyone prepared to state that the existence or non-existence of a
text-type depends entirely on how many readings we allow at a variant?
Seems pretty silly to me....

If we are to make mathematical analysis of TC a truly respectable
discipline, we must create a greater degree of rigour. Now I know that
not everyone wants to do this (witness my exchange with Ehrman about
the Comprehensive Profile Method. I say that profiles should be created
by mechanical means. He says that human beings should create them.
There is no "right" answer here; it's just that one way is rigourous
and the other is not). All I am saying is that such rigour is required
from proper mathematical analysis. Results must be repeatable!

Another problem with recommending a math text or reference to TCers
is the level of math required. I don't know about your colleges,
but my school offered two statistics courses -- one, called "Statistics,"
was taught by the economics department, and had no prerequisites.
The math department offering was called "Probability and Statistics,"
and had as prerequisites multi-variable calculus and linear algebra.

Now I'll tell you the truth: the Economics version of the course was
more "useful." They taught you a lot more things. In Probs & Stats,
we didn't even learn such basic tools as chi-square testing. Instead,
we studied possible outcomes, probability distributions -- and
where and when one can apply various methods. The Economics class
never taught that. And it's a real lack. Those economic students
frequently applied tests and formulas to results where they simply
didn't apply. But I frankly don't know how to *tell* people how
to apply the tests until they've reached the level of mathematical
maturity implied by four semesters of calculus, linear algebra, etc.
(There may be a way; I just don't know it. :-)

To show how much work there is to be done, I will point out that,
in the six or so years since I first started applying statistics
to textual criticism, I have refined my methods five times. And
I *still* don't have a method that lets me plug in a collation
and have the computer read off the text-type of the resulting
manuscript. I have a lot more data than I used to, and it's all
very pretty, but it isn't rigorous. (And, I fear, I'm about used
up. I'm 35, and mathematicians stop having ideas at 30. So somebody
else will likely have to take up the torch.)

This brings up another point. I may, at some point, have said that
Colwell and Tune and/or Hurtado are wrong in their definition of a
text-type. I should not say that. Their methods are faithful to
their definition.

But note that word: "Definition." They have *postulated* the
definition of a text-type (70% agreement, 10% gap). That is, they
have said that "this is what we mean by a text-type."

That is a legitimate mathematical method. They could, for all
it matters mathematically, define text-types in terms of the
color of ink used in the manuscript. This is logically consistent.
It's just that the results wouldn't match what we instinctively
think of as a text-type.

The informal definition of a text-type is "the loosest grouping
of manuscripts between which some degree of relationship can be
discerned." Now I say that, due to mixture, this definition
contradicts the Colwell/Tune definition. Allow me enough manuscripts,
and I will show you a set in which there are no 10% gaps.

That is the reason why I did, and do, reject the Colwell definition.
I've spent the last several years trying to find a definition which
more or less conforms with what we already knew, without shutting
us off from new discoveries. And I am the first to admit that I
have failed. I believe I have constructed an accurate picture of
the text-types in Paul and the Catholics -- but I also admit
that this is based on my informal analysis of the statistics
I produce, not on the statistics themselves.

The Colwell/Tune definition carries with it two seemingly
unnoticed corollaries:

1. A text-type can only exist if it has a pure representative
2. A sufficiently mixed text cannot belong to a text-type.

I personally reject this. By this definition, the Alexandrian
text did not exist in the Gospels until B and Aleph became known!

I would postulate something different:

1. Text-types exist.
2. Text-types need not have a pure representative.

On that basis I have stumbled around (and stumbled is the right word!)
looking for a mathematical rule that will let me prove the existence
of text-types.

I hope that is a decent summary of the situation. We can't just go
out and call on mathematicians to help us out. If we ever want TC
to be a rigorous subject (and I say *if*, because some do *not*
want it to become one), we need to get our house in order. Get
our definitions straight, study the nature of variants so that
we know what tests we can apply. If we do that, then maybe finally
we can write the textbook that everybody wants.

All right, the voice yapping in the wilderness will now stop yapping
for a bit. :-)

                        Robert B. Waltz
                     [email-masked]

---

From: Bart Ehrman <[email-masked]>
Date: Sun, 4 May 1997 13:25:08 -0400 (EDT)

> from proper mathematical analysis. Results must be repeatable!

   Mr. Waltz: is it still the case that you haven't actually read any of
my stuff?  

    I completely agree that our models have to be mathematically based,
repeatable, and rigourous.  I thought that was why I developed the
comprehensive profile method in the first place.  If you have suggestions
about the method itself, I really would like to hear them, since there are
several doctoral dissertations being written right now that are using it,
and refinements could prove quite useful (Vincent Broman's comments have
been extremely helpful; I have to say that I prefer his approach of
offering suggestions of a method that he has read and understand to
your approach of criticizing a method you haven't even examined;  but
maybe it's a matter of taste). 

   With respect to our disagreement, it is not over either rigor or
repeatability.  It's over how one *decides* what constitutes a variant or
not and how one goes about using these variants to classify manuscripts. 
The points you're raising over that issue are precisely what is at stake,
in part because they show that the decision cannot be made mechanically
until you, a subjective being, actually *make* these decisions.  I.e.,
it's wrong to say that I think human beings "should" make decisions; I
think humans *have* to make them.  There is no choice about the matter. 
The question about whether they are rigourous or not in doing so is
altogether different. (BTW, your point that binary agreements always
create a Byzantine vs. a non-Byzantine reading of course is obviously
false, so I wonder what you're thinking of.  Could you tell me, by the
way, if you've evaluated yet the standard criticism of the earlier methods
of establishin a Caesarean text by considering only those variants that
differ from the Byzantine text?  To most of us, this criticism is
absolutely valid, and based precisely on a rigorous set of
considerations.) 

-- Bart D. Ehrman, University of North Carolina at Chapel Hill

---

From: "Robert B. Waltz" <[email-masked]>
Date: Sun, 4 May 1997 15:00:26 -0700

>   Mr. Waltz: is it still the case that you haven't actually read any of

To answer the question: Yes, I have read various things you have written.
But to answer the *implied* question: No, I haven't gotten to the article
on the Comprehensive Profile method. I admit that this is something I
need to do -- but it hardly affects my discussion of the Colwell/Tune
definition.

[ ... ]

>think humans *have* to make them.  There is no choice about the matter.

In part I agree. But human beings are always subjective, and should
be kept *out* of the process as long as possible. (Sez I.) So I
prefer to use the computer to do every bit of the work it possibly
can. There may be methodological bias in my programming -- but at
least that bias will affect all manuscripts equally.

>altogether different.

I will try to be charitable and assume that we are using "rigour" in
different ways. So what in the world do you mean if you do not mean
something that can be done absolutely mechanically? (This excludes,
of course, the meaning in mathematical logic -- but even I don't
see any way to apply mathematical logic to TC.)

>(BTW, your point that binary agreements always
>false, so I wonder what you're thinking of.

What? Assume that, at some point of variation, there are two readings,
A and B. That is what I mean by a binary variant.

If this is true, then there are only two possibilities: Either the
Byzantine text is split (which can happen, of course, but is not
especially common), or one of the readings is Byzantine and the
other is not. For simplicity, let's say that reading B is Byzantine
and A is not. Together so far?

Now take two manuscripts, X and Y. Since, *by definition of a binary
variant*, they can only have reading A or reading B here, there are
four possible results:

1. X reads A, Y reads A -- Comment: This is the only case where both are
                                    Non-Byzantine. And X and Y agree.
2. X reads A, Y reads B -- Comment: Y is Byzantine, so this is not a
                                    case where both are non-Byzantine.
3. X reads B, Y reads A -- Comment: X is Byzantine, so this is not a
                                    case where both are non-Byzantine.
4. X reads B, Y reads B -- Comment: Both Byzantine, so this is not a
                                    case where both are non-Byzantine.

In other words, *if all readinngs are binary*, then wherever two manuscripts
are non-Byzantine, they agree.

Q.E.D.

This does not admit of argument. If you wish to prove this statement untrue,
you must attack the premise that all readings are binary. I do not say that
all readings are binary, only that, if they are, then agreement in
non-Byzantines is meaningless.

Or were we disagreeing about what "agreement in non-Byzantines" means?

>Could you tell me, by the

I'm not sure I understand the question. Are you asking, "have I examined
the Lake/Blake/New/Streeter system of defining the 'Caesarean' text in
terms of its divergences from the Byzantine text?"?

If so, the answer is that I have not repeated the experiment that they
undertook, so I do not know if the "Caesarean" text exists. I have suspicions,
but no final answer.

However, I have no objection, in theory, to their method. They applied it
incorrectly (the TR is *not* the Byzantine text), and of course they
applied it circularly. That means that they needed better application of
their method, not a better method.

Let me try asking the question from the other direction: Do you insist
that a text-type can only exist if there is a pure representative of the
type? Because I will readily concede that there are no surviving pure
representatives of the "Caesarean" text. All of them, from Theta on
down, are mixed (if they form a text at all). But if you cannot examine
manuscripts at the points where they are free of Byzantine mixture,
how can you examine that lost text?

***

Let me make another observation on this exchange. Those who are
neutrals (lucky you :-) will observe that I keep being attacked
at *the same points.* Usually it's where I have questioned a common
assumption, but not always.

The one thing that is universal is that we are debating over
postulates. Colwell, or Ehrman, or Hurtado, are offering postulates.
(You don't think Colwell postulated his definition? Consider how
he "derived" it. He looked at one small set of data, offered
some numbers -- and by some amazing process those numbers have
become gospel, accepted without proof by very many TCers.)

So I would ask a question: What is wrong with trying alternate
postulates? I can't give a TC analogy -- but I can state one
thing with certainty: Mathematics has always gotten richer when
it has tried using different postulates.

                        Robert B. Waltz
                     [email-masked]

---

From: "Professor L.W. Hurtado" <[email-masked]>
Date: Mon, 5 May 1997 17:16:02 +000

As Jimmy Adair and a couple of others think they found some value in 
the exchanges  Bob Waltz and I had late last week, and as substantive 
matters continue to be on the table, I hope I shall be permitted 
patience for another contribution.  (And Bob, I never saw our 
exchanges as shouting matches, at least on my end.)
Adair highlights some interesting matters.  Also, in his response to 
Adair and in his subsequent posting Waltz makes a number of 
statements that require some clarification in my view, merely to be 
able to understand what he means before I can evaluate them.  Let me 
try to be concise and itemize matters.
1.) It bears remembering what Colwell (and Tune) were up to in the 
two or three relevant essays in Colwell's _Methodology_ collection in 
which the % matters come up.  Colwell posits that the only 
helpful definition of "text-Type" is "a group of manuscripts, not  . 
. .  a list of readings" (Colwell, _Methodology_, p. 9).  That is 
essential to understand the quantitative method he tried to develop.  
We're trying to group mss so that we can then proceed toward a 
history of the NT text.  In order to do this, we can apply the 
Colwell-Tune quantitative approach, and in this approach they offered 
as a *quantitative* definition of a text-type (i.e., group of mss), 
70% or more agreement in point of variation where any two of the 
leading witnesses of all known major groups share a variant across a 
significant body of text (in my view the collation has to be book by 
book across whole books).
As I've stated the reason for the 70% figure is simply that when they 
did this exercise in their sample, and when Fee carried out his 
exercise in John and when I did mine in Mark, we found that the major 
reps. of  Hort's "Neutral" type and of the "Byzantine" type agreed at 
this level and their agreement was separated from agreement with 
major reps of other text types by 10% or more.  These figures might 
well be modified, if one changes the nature of the experiment to ask 
other questions, but that would say nothing about the usefulness of 
the figures *for the purpose at hand*--viz. to group mss into 
text-types and to then see where and whether  new mss fit with any of 
these groups.
Thus, when Waltz says that we can introduce enough mss to make the 
"gap" of agreement much smaller, this is misleading.  One might well 
introduce mss that are "weak" members of a text-type/group and 
the agreement of these mss with the text-type of which they are 
weak members would *not* be separated as distinctly from their level 
of agreement with members of other text-types.  But the validity 
of the 10% "gap" and the 70% agreement observation is that these 
figures are functional quantitative definitions of the behavior of 
*major/good reps. of text-types when collated with major/good reps. 
of all other known text-types*.  Thus, that weaker members of the 
text-type might not exhibit this behavior is very interesting, almost 
predictable, but not relevant for the validity of the definition of 
strong members of a text-type, at least a text-type like the Neutral 
and Byzantine ones.  
2.) There has been some confusion over the word "sample".  Given that 
we cannot collate all known Greek mss, we have to chose well certain 
representative mss for any such exercise.  I suppose one could call 
such reps "samples", but "samples" in at least some social science 
exercises can involve *random* sampling, and when you're choosing 
particular mss because they are already thought to be leading reps. 
of groups this is hardly "sampling".
Moreover, when the purpose is to measure the agreements of mss (a) 
where the mss collated vary, and (b) where their variations enable us 
to say anything about agreements with one another, then if we count 
all such places how is this a "sample"?  I suppose one could think of 
such variation units as "samples" in the sense that they are not 
every word/phrase in the NT, but only places where the text varies.  
But, again, "sample" implies a portion of the relevant data.  But if 
all the relevant data is counted, how is it only a sample?
Waltz cites critically L. Richardson's study of the Johannines, and I 
too have strong reservations about R's approach, which, N.B. is *not* 
the sort of application of quantitative analysis that Fee and I have 
practiced and advocated.
 3) With respect (sincere, though my patience 
wears thin at times), Waltz makes several statements that either seem 
to me incorrect or else not comprehensible to me.--He allgeses as 
what he calls corollaries of the Colwell-Tune method (a) that a 
text-type can exist only if it has a pure representative, and (b) 
that a sufficiently mixed ms can't belong to a text-type.  Both are 
incorrect allegations.  To repeat:  per Colwell, a text-type is a 
group of mss that can be shown to belong together in some objective 
way (i.e., open for others to examine and test).  No pure rep is 
required, only that a group of mss exhibit sufficiently strong 
agreement to indicate that they are a group.  Mixed mss may show up 
as weaker members of such a group, or a group of "mixed" mss. might 
exhibit sufficient agreement to form a group.  I really don't see 
where Waltz gets these corollaries, for the studies done using and 
adapting the Colwell-Tune approach demonstrate my statements.--Waltz 
alleges that Fee & Epp haven't given us "a rigid system for 
determining the scope of variants", and Waltz regards this as 
"vital".  I'd really like Waltz to specify what in the impressively 
cogent discussions in chaps. 3-4 of the recent Epp/Fee _Studies in 
The Theory & MEthod_ volume he takes issue with, and where in these 
discussions he finds mis-steps or omissions.  E.g., the "recommended 
definitions & limitations" on Epp/Fee pp. 57-61 are fairly explicit 
and precise.  Can you improve on these paragraphs, Mr. Waltz?  If so, 
please specify.--Waltz criticizes the Colwell-Tune quantitative 
definition (70%, etc.) as contrary to "what we instinctively think of 
as a text-type".  "Instinctively"???  Given that some have used 
text-type to = a "pattern of readings" and others (my preference) an 
identifiable major group of mss distinguisable from other major 
groups, where does instinct come in?--Mr. Waltz quite commendably 
wants "mathematical analysis" in NT TC to be "a truly respectable 
discipline" and calls for "a greater degree of rigour".  I therefore 
fail to understand how it is an advance to list mere numbers of 
agreements of two mss in "non-Byzantine" readings.  And I simply 
don't know what his tables of numbers and percentages mean.  Mere 
numbers and percentages mean nothing without a context.  Percentages 
of what?  Numbers of what? Mr. Waltz earlier mentioned counting 
variantion units from N/A27, but I didn't catch the details.  Could 
you please give us a simple statement of what it is you do, as 
scientifically clear as possible?  Merely stating alleged results 
without the scientific notation of process is near useless.
--What do you mean by seeking to "apply the gap to classified 
agreements"?  
--What do you mean by "near-singular readings"?
--How is it rigorous or meaningful to make a judgment about mss 
relationships based on a handfull of readings (as in Waltz's list of 
6 agreements of 630 with this or that ms)?

Finally, as I've stated earlier (and demonstrated in my study of the 
Codex W)  quantitative analysis is only the *first step* in grouping 
mss and in analysing their relationships.  It has to be complemented 
with analysis of the specific readings shared by mss.  No one has 
suggested that I know of that quantitative agreement alone is more 
than a rough measurement and one with heuristic value.
L. W. Hurtado
University of Edinburgh,
New College
Mound Place 
Edinburgh, Scotland EH1 2LX
Phone: 0131-650-8920
Fax: 0131-650-6579
E-mail:  [email-masked]

---

From: "Robert B. Waltz" <[email-masked]>
Date: Mon, 5 May 1997 13:43:50 -0700

On Mon, 5 May 1997, "Professor L.W. Hurtado" <[email-masked]>
wrote, in part:

>. .  a list of readings" (Colwell, _Methodology_, p. 9).

If you read my web page, you'll note that I quote that statement too.

Although this, too, is something that might be worthy of discussion.
Is a text-type a group of readings or a group of manuscripts?
If it is a group of manuscripts, how do we determine which manuscripts
belong except by looking at some list of readings? And if it is a
list of readings, how do we choose the readings except by looking
at the manuscripts containing them?

I wonder, therefore, if we can separate manuscripts and readings.
(I'm not being dogmatic here; I just don't know. Thoughts, anyone?)

>That is 
>history of the NT text.

With which goal I hearily agree -- probably more heartily than most
on this list, since I don't trust internal evidence if I don't have
to.

>In order to do this, we can apply the 
>significant body of text

Notice that I said in my last post that this is a consistent method.
I just don't think it produces quite the results we want. Also, it
really *does* assume the solution. It's all very well to say that
we need to start somewhere. Believe me, I know the need! My
database defaults to comparing limited lists of manuscripts.

But we need to make our assumptions as few as possible. If we just
keep *assuming* that this witness or that is "the best," we can
never tell if something else is better.

>(in my view the collation has to be book by 

I agree in part with that last. That is, one has to be alert for
block mixture.

However, complete collation of all manuscripts is simply not possible
at this time. So if we are to study all manuscripts, we must use a
sampling technique. (Note that I am not saying this is *good*; I'm
saying this is *necessary.*) This means that we may not have enough
data points, especially for short books, to produce detailed results
book-by-book. To get more statistical accuracy, we need to check
book-by-book for block mixture, and then -- if possible -- study
results for an entire corpus. Only that way can we involve enough
readings to be certain our results are reliable.

>text-types and to then see where and whether  new mss fit with any of 

I'm not saying that you didn't achieve these results. I'm saying the
value you find depends on the sample. In your system, based on variants
found in two or more witnesses in a group of less than a dozen, the
addition of even one more manuscript, depending on its nature, could
significantly alter your results.

>of all other known text-types*.

Assume that to be true. It very well may be. It still leaves you
unable to assess mixed manuscripts. And I would claim that there
are several known text-types that exist *only* in mixed manuscripts.
The "Caesarean" text type, if it exists, is an example. The p46/B
type of Paul, if *it* exists, is an example.

And there is at least one undisputable example: Family 2138 in the
Catholics. This is a very large family (I've personally gathered
statistics for 614 630 1505 1611 1799 2138 2412 2495, and there
is evidence that there are many more.) On average, these manuscripts
agree with each other in about 70%-80% of the cases. But if we look
at the reading found in the majority of these manuscripts, they
all agree with that 80%-90% of the time. What's more, in my spot
checks, *every time they deviated,* it was in the direction of the
Byzantine reading. In other words, here is a group where the members
meet the Colwell definition, but even so can be shown to be slightly
mixed. It doesn't take much to see that only a little more Byzantine
mixture would have knocked the group below the Colwell threshold.
Does it cease to become a text-type just because the members are
a little more mixed? I don't see it that way.

>Thus, that weaker members of the 
>predictable,

More than predictable. A logical necessity.

>but not relevant for the validity of the definition of 

But this is my key objection: I don't care about using *only* the
strong members of a text-type. Sure, it's nice where they are
available. But I refuse to write off the weak ones.

>representative mss for any such exercise.

I'm afraid I don't accept that, either. If you haven't studied all
manuscripts, how do you know which are representative? (This strikes
me as too obvious for words.)

Better to sample a large number of manuscripts than to know only
a handful in detail.

I'll give an example. There is a family in Paul consisting of 330, 451,
and 2492. (Incidentally, Maurice Robinson confirmed the relationship
between 330 and 451 in Acts.) This group seems to have an interesting
and independent text (though it's mostly Byzantine, and I don't know
if there was ever a pure non-Byzantine type behind it). If we insisted
on looking only at well-known, pure witnesses, we would never know that
this group exists.

>I suppose one could call 

"Sample" as I have been using it applies not to manuscripts but
to readings. And sampling need not be random; a "sample" is just
a bunch of stuff. In certain instances one prefers a random sample.
However, I would point out that often one does *not* want a random
sample. Without going into details (all the more so since I don't
remember them), consider all the political polls that call up lists
of people carefully selected to meet certain characteristics.

Now this may sound like I'm agreeing with Hurtado and saying that one
should use selected manuscripts. I'm not. Remember, I use "sample" to
apply to readings. We may choose to select readings carefully (although
frankly I tried in my work on the gospels to select them randomly).
But I would strongly argue that we cannot selectively choose manuscripts.
When a pollster calls John Smith, it's because the pollster known that
John Smith is 35-50 years old, has two years of college, has two
children, etc. When we leave, say, manuscript 223 out of our calculations,
is it because we know its text? Much more likely it's because we *don't*
know its text. In other words, we are creating a biased sample without
even knowing the biases.

I have elsewhere said some rather unkind things about the Alands'
"manuscript categories" and "thousand readings." I still don't think
it's an adequate classification tool. But at least it lets us separate
the bulk of Byzantine manuscripts from the ones which are something
(it hardly matters *what*) else. I would maintain that this is where
we have to start.

[ ... ]

>the sort of application of quantitative analysis that Fee and I have 

Agreed. It's a modified Claremont approach. And, actually, I agree
with many of Richards' results. I find the same three families
non-Byzantine families he does (though I use different and, I think,
more informative names: "Alexandrian," "family 1739," "family 2138").
My complaints with him are that his collations are inaccurate, that
"Mixed" is not a text-type, and that his three so-called Alexandrian
groups are actually separate text-types (surely this is obvious
in the case of Family 2138!)

>incorrect allegations.

Let's be clear: Not the Colwell-Tune *method*, the Colwell-Tune
*definition.*

I'm going to demonstrate my point very simply. I've done this before;
I hope (sorry, but I'm going to be acid here) you'll read it this
time.

Let's start with a pure manuscript of a text-type. Call it M.
Assume, without loss of generality (to put that in layman's terms,
the way I construct this example will work for any number of readings),
that there are six points of variation we might study. Number
them 1, 2, 3, 4, 5, 6 (how's that for original). At each of these
points let "A" represent the reading of M. Our alternate reading,
which we may consider to be the reading of the Byzantine text, is "B."

Make two copies of M; call them N and P. Initially, both N and
P have the same profile:

   N    P
  ---  ---
1. A    A
2. A    A
3. A    A
4. A    A
5. A    A
6. A    A

Now let's randomly mix three Byzantine readings into N and P. Let's
say that N receives Byzantine readings at 1, 3, and 4; P gets them
at 2, 3, and 5. So now our profile is:

  N**  P**
  ---  ---
1. B    A
2. A    B
3. B    B
4. B    A
5. A    B
6. A    A

Now copy N** (call the result Q) and P** (call that R). Then destroy
M, N, and P.

If we compare Q and R, they agree only twice: at 3 (where both are
Byzantine) and at 6 (where both have the original reading).

Now I say that Q and R still represent -- albeit badly -- the original
text-type of M. But they *do not* meet the Colwell definition, and
no manuscripts mixed in this manner can.

Try it yourself, with as many readings and as many Byzantine corruptions
as you like. If you don't want to do the math yourself, the article
on mathematics on my web page (see the section labelled "Probability")
will show you the expected degree of agreement between manuscripts
which have suffered various degrees of mixture.

[ ... ]

>Waltz 
>discussions he finds mis-steps or omissions.

You completely missed the point of what I was saying. What I was
out is that the way we set variants determines what statistical
analysis we can use. I showed why, too.

I pointed out that Epp & Fee made great progress in this area.
But their work was not related to statistical analysis, merely
to classification. We need to add the statistical element -- e.g.
do we insist on binary readings, etc.

Those who think I am denigrating Epp would be advised to read all
the various postings in which I have quoted Epp's opinions *against*
the orthodox views of textual critics.

[ ... ]

>Waltz criticizes the Colwell-Tune quantitative 
>groups, where does instinct come in?

Plenty of people around here have said "I know a text-type when I
see one." Perhaps I mis-spoke; maybe some people on this list
don't have a feeling for text-types. The "instinctive" definition
is "the highest [loosest] degree of relationships between manuscripts
that we can perceive."

A lot of people have perceived the "Caesarean" text. Zuntz perceived
the p46/B text. Neither meets the Colwell definition. That must say
something.

[ ... ]

>And I simply 

I *never once* mentioned "variantion units from N/A27" with respect
to my most recent table (for the Catholics). I did say the numbers
were precentages of agreement. I may not have said in the last post
that these were based on the variation units in GNT3 and T&T. But
I have stated that in the past. I've also put most of the details
on my web page.  

>--What do you mean by seeking to "apply the gap to classified 

That depends on what system of classifications one uses. Ehrman
has one, I have another -- and we disapprove of each others'
methods. So I can't answer that categorically. But the idea
is that, for any given manuscript and any given statistic,
*other than overall agreements*, there will probably be a gap
how close it stands between its immediate relatives and how
it stands with respect to non-relatives. The size and nature
of that gap, however, are dependent on the statistic, the
manuscript, and the control manuscripts.
 
>--What do you mean by "near-singular readings"?

The technical definition is "a reading is near-singular if it is
supported by no more than x manuscripts of those being examined."
The number x will, of course, depend on the number of manuscripts
examined. I've been working with 30-50 manuscripts, and typically
use x=5 or x=6.

>6 agreements of 630 with this or that ms)?

Oh muses, I call on you to make people read what I say!

I used a sample of 550+ readings. In this set, 630 was found to
have a certain number of near-singular readings (19, I believe).
Of these 19, 6 agreed with 1739. No other manuscript had even
close to that many near-singular agreements.

***

My apologies to all of you caught in the crossfire. I hope this
will be my last reply to Hurtado on this subject. If others of
you have questions, feel free to ask. I hope I will be able to
answer -- but I can't spend time like this repeating myself.
Sadly, I have a real job too....

                        Robert B. Waltz
                     [email-masked]

---

From: "Professor L.W. Hurtado" <[email-masked]>
Date: Tue, 6 May 1997 12:38:46 +000

Thanks to Robt. Waltz for his response to my previous posting.  I too 
hope that my previous postings have been sufficiently clear to make 
it possible to move on to other things now, with only the following 
brief clarifications:
Waltz writes:
> If it [a text-type] is a group of manuscripts, how do we determine
> which manuscripts
If the question we seek to answer is whether a given selection of mss 
constitutes an identifiable group such as a "text-type", then we 
collate them across a large body of text counting their behavior at 
all points of variation, and then asssessing the count in comparison 
to the behavior of other manuscripts that can afford us what we think 
might be a valid analogy and/or contrast.  Thus, we compare counts 
with counts, agreement of mss with agreement of other mss.  It's 
*overall significantly high levels of agreements in variants* that is 
the crucial indicator of a text-group.  Analysis of readings comes 
later, and enables us to say something about the *nature/quality* of 
the kind of readings "preferred" by a given group, and about the 
"mixture" or other factors that have influenced the text of the 
group.
(And, by the way, "mixture", by which Waltz seems to assume always 
influence of one kind of  ms upon another, is only *one* way that a 
ms might acquire kinds of readings.  Scribes can quite independently 
have produced the same variants, e.g., harmonizations, stylistic 
improvements, etc., because they had same/similar tastes and 
concerns.  This is why we have to go by *overall high numbers of 
agreements* and not merely small numbers of shared readings.)

> really *does* assume the solution.
 
> never tell if something else is better.
But, as Waltz will surely know, the essence of scientific work is to 
assume/hypothesize something *which we then devise a means of 
testing*.  The Colwell-Tune type approach offers a means of testing 
whether mss are what we might have assumed.  There's nothing wrong 
in assuming something so long as (a) you're aware of the assumption 
you're making, and (b) can devise a means of testing the assumption.
Thus, if we assume that Codex W has sufficiently strong agreement 
with Theta and 565 to make it a "pre-Caesarean" witness, and then we 
conduct the exercise in Mark I conducted in my published study, the 
Colwell-Tune approach indicates that W does not in fact show anything 
approaching strong enough agreement to indicate any particular 
relationship twixt W and Theta in Mark.  And when you then follow 
this up (as I did) with detailed analysis of the *particular 
readings* W and Theta do share, you find that even the *kinds* of 
shared readings are less likely to be the result of any direct 
relationship of the two mss. 
> significantly alter your results.
For what it's worth, for the published version of the study I did add 
another ms (565) to the original list (in the PhD thesis version), 
and the results were strengthened!  I have always wanted since then 
to have the chance to run, say, L and 33 in Mark through this 
collation, to see if we can say anything more precise about how and 
whether they are in fact (as often thought) "weaker" 
Alexandrian/Neutrals in Mark, and if so what other affiliations they 
might have and/or what kind(s) of readings they may prefer.  I do 
doubt that this would alter the evident & strong relationships show 
among primary Neutrals/Alexandrians and primary Byzantines.  So, I 
rather doubt that it would "significantly alter results" in fact.  
But it might contribute to our grasp of the mss added.

Waltz states (again and again!) that my approach:
>  . . . still leaves you
> The "Caesarean" text type, if it exists, is an example.
But I *did* assess "mixed manuscipts" in Mark in my study, for 
example.  I characterized in some precision and detail the "mixed" 
nature of the text of W and P45, and even Family 13, etc.  Not by 
mere counting, but by then following up the counting with analysis of 
readings.  The full method Fee and I have advocated involves *both* 
quantitative study *and* readings-analysis, the combination of which 
permits *identification* of "mixture (by quantitative study) *and 
characterization* of "mixture" (by analysis of readings identified in 
the countings).  I've staed this more than once but Waltz seems not 
to have noticed.  I hope this will be the last time necessary.
> available. But I refuse to write off the weak ones.

Nor does anyone else "care about using *only* the strong members of a 
text-type" and urge us "to write off the weak ones".  One tries to 
select good/strong reps of a text type *for the initial purposes of 
seeing if a new ms belongs to this or that group*.  One might also 
use/select deliberately "mixed" or "weaker" members of a text-type to 
see if a new ms accords more closely with them or with the stronger 
members.  No one's writing off anything.  Where does this notion come 
from?

> me as too obvious for words.)
And I find it "too obvious for words" that Waltz is mistaken again.  
Once more:  We *have* been studying mss for quite a while, and these 
studies may have been accurate or less than accurate in conclusions.  
So, we take what's been done and *TEST* the previous conclusions by 
some approach/method that will help tell us something.  Which is what 
the Colwell-Tune approach was designed to assist.  Which is what Fee 
did in John (correcting previous identification of Aleph, for 
example), which is what I did in Mark (correcting previous 
identification of W, P45, Fam. 13, etc.).

> analysis we can use. I showed why, too.
No, Waltz hasn't shown what he claims in his last sentence.  And he 
hasn't specifically shown what's wrong/missing in the fairly clear 
definition of variants, variation-units, etc. that Epp offers.

> do we insist on binary readings, etc.

I've not seen from Waltz anything specifically as to what 
"statistical analysis" offers us.

> the p46/B text. Neither meets the Colwell definition. That must say
Yup.  It probably means that "Caesarean" text-type has to be 
re-thought or else has to be demonstrated as being anything like what 
we otherwise expect of agreement of mss alleged to belong to the same 
text-type.  I'm satisfied that Theta & 565 show this kind of 
agreement (or approach it).  So there's something in Mark of the 
period of these mss, and the Colwell-Tune approach helps us to see it 
in some objective terms.

> examined. I've been working with 30-50 manuscripts, and typically
I take it that "technical definition" here must be Waltz's, as I've 
never even come across the term before in nearly 30 yrs. of NT 
text-critical work.  I've seen "sub-singular", but not this one.  But 
then maybe I've missed it somewhere.

So, for my part as well, I hope this final posting is final on the 
subject for me and has been helpful in clarifying some matters for 
others.
 
L. W. Hurtado
University of Edinburgh,
New College
Mound Place

---

From: "Robert B. Waltz" <[email-masked]>
Date: Tue, 6 May 1997 09:41:46 -0700

I promised not to continue the debate with Hurtado, and I won't. If I
haven't made my opinions clear by now, I probably am unable to do so.
(E.g. I can't stop using mathematical language, which I gather
confuses certain people.)

But there is one point that I thought I should clarify, since it may
be that I *haven't* discussed this before.

>then maybe I've missed it somewhere.

The definition is indeed mine. As far as I know, I am the first
person to give a precise definition to the term "near-singular"
(although it seems to me that I saw it used somewhere).

It will be observed that a near-singular reading is not the same as
subsingular reading (although all sub-singulars are near-singular,
the reverse is not true). Subsingulars are usually defined as
readings having support from only one significant document;
a near-singular can have several strong supporters. The point
is simply that it represents only a small fraction of the
tradition.

It will be observed that the concept only has meaning when one
is studying a fairly large number of manuscripts (I wouldn't use
it on a set of less than 25 mss).

The goal of the concept is to find the characteristic readings
of a text-type, without the inherent biases of having a human
being look for them. It also has the mechanical advantage that
such a near-singular profile can be computed on the fly. (To
compute such a profile, on my slow database on my slow machine,
takes about three minutes per 100 readings -- and that's while
computing four other statistics as well. Even if the results
are arguably not as good as Ehrman's profiles, I can probably
find them for a hundred manuscripts in the time it takes a
human being to do *one*.)

The concept is largely equivalent to any previous system of
classified agreements, going all the way back to Hutton's triple
readings. However, it avoids Hutton's problem of assuming the
solution. (And Hutton *clearly* assumed the solution; his system
acknowledged only Alexandrian, Byzantine, and "Western" types.)

It should be noted that near-singular agreements in and of themselves
are not sufficient to classify manuscripts; we also need overall
rates of agreement, and (sez I) at least one other statistic as
well. (I prefer more). The reason for this should be obvious:
While a distinctive manuscript like B will have many near-singular
readings (over 100 in my set of 990 readings in the Gospels),
many of the Byzantine manuscripts have none at all -- or if they
do have one or two, they are often the result of error.

Rates of agreement in near-singular readings are often low -- closely
related manuscripts may agree no more than 30-40% of the time. But
unrelated manuscripts will often have no near-singular agreements at
all. So the statistic can be meaningful, and it is a good place to
apply the concept of the "gap."

Or so say I....

Bob Waltz
[email-masked]