Sum: the 'only six' argument

Sat Sep 9 19:05:04 UTC 2000

 Larry Trask wrote:

> I will therefore content myself with reporting that no one
> who has so far replied has expressed any great sympathy with
> any version of the 'only six' argument, and several people
> have been openly hostile.
>
> These negative responses don't surprise me at all.  I am
> certainly not sympathetic to the 'only six' argument.  It's
> just that I keep coming across claims of this sort every now
> and again, and I was beginning to wonder if a significant
> number of historical linguists were embracing such arguments.
> Apparently not.

Wait just a second there. I may have sounded negative myself. But when I
thought about it a little more, I realized that there is a legitimate
and  interesting argument there, and it ought to be in historical
linguistics textbooks if it isn't. ( I don't know if this is the
argument you have seen, but I'd be interested to know if it IS any
textbooks?).

Basically, IF one has set up the question properly and IF one has
carried out the comparison with discipline and honesty (big ifs, of
course), then a very small number of examples of a single sound
correspondence is sufficient to demonstrate a historical (not
necessarily a genetic) relationship beyond any reasonable doubt.
Practically speaking, given the sample sizes we usually work with and
the way that phonological systems are set up, in most cases, the
necessary number is indeed around six or not much more. This isn't
anything for anyone to be hostile to (or sympathetic to, for that
matter); it simply follows necessarily from the logic of probability.
I'll explain, but first a clarification.

When you ask about numerical criteria for a genetic relationship, you
are asking (at least) two separate questions. Most of the respondents
addressed the second question-- what are the criteria for determing if
two historically related languages are related genetically-- as opposed
to being related by contact or borrowing, or by being in a
lexifier-creole relationship. Some respondents addressed the question of
what criteria are relevant for subclassifying genetically related
languages. As far as I can see (and as most of the respondents said),
numerical criteria simply are not relevant for making these kinds of
judgements. It's the nature of the similarities or commonalities, not
the number of them that count. In any case probability theory doesn't
come into play because in all these cases we have already ruled out
coincidence as an explanation.

But when approaching unclassified languages or languages which haven't
been compared to each other before, the first question we have to ask is
whether these languages have something in common which cannot be due to
chance or coincidence. Numerical criteria and probability theory are the
most reliable means for making judgements of this type.

 Here's how you end up with only six: First the average expected number
of chance matches between any two consonants in any two languages (that
is the expected number of times the consonants will appear in the same
position in a word with the same meaning) is the frequency of the first
consonant in its language times the frequency of the second consonant in
its language times the number of word pairs available for comparison.
Thus if ten percent of the words start with /t/ in one language and ten
percent of the words in the other language start with /b/ then in a
hundred word sample, there should be (by chance) one case where the
translation of a word starting with /t/ in the first language starts
with  /b/ in the second.  In a 1000 word sample there should be about
ten such cases. One rough guide to frequency of a consonant is simply 1
over the number of consonants in the inventory. So if you have twenty
consonants the average frequency of each consonant is 1/20 or .05. If
you have a Macintosh with a graph calculator try entering this formula
1/x^2*n100 (one over x squared times n times 100). This gives you the
expected number of correspondences, in a sample with n100 word pairs, of
two languages both with x number of consonants, evenly distributed. You
can see from this that as long as the average size of the consonant
inventory is greater than 10 (or put another way, where no consonant
occupies more than ten percent of the word positions being compared) the
expected number of chance matches in a 100 wd sample is between 1 and 0.
That is in a 100 word sample you expect that each consonant (in initial
position) in one language will match up with each consonant in the other
in one word or not at all. In a 1000 word sample the expected chance
avgs. are not all that much higher-- basically if the average size of
the consonant inventories is 14 (or the avg. frequency no more thant
1/14), you only expect to get 5 chance correspondences, though below 14
the expected number starts to climb dramatically. (At 5 the expected
number is 40).

 The next question is how far above the average do we have to get before
coincidence becomes an absurdly unlikely explanation. There is a formula
for this, but I won't go through it since this post has gotten long. But
here is one example: In the case where two langauges both have 20
consonants evenly distributed (or more realistically in comparing two
consonants in two languages both of which have a frequency of 5% in the
word-position being compared in their respective languages), the
probabilty of finding more than 5 correspondences (i.e. 6 or more) in a
100 wd. sample is 0.000000356, or roughly 1 in 2.8 million. (The chance
of finding 5 or more is roughly 1 in 163,000.) So in this set of
circumstances "6 or more" (i.e a single correspondence set occuring in a
given position-- say word-initially-- in 6 or more words) should be
pretty well conclusive for demonstrating a non-chance and hence almost
certainly historical (genetic or contact) relationship.

 I think that working all this out mathematically is interesting and
important for compartive linguistics for two reasons. First it means
that if you apply the comparison strictly (allow only one-to-one word
comparisons, and one-to-one phoneme comparisons) you can get more
knowledge from less information-- you can potentially demonstrate a
relationship with much less data than comparativists have traditionally
thought necessary. This is important to me, because I work in
Afroasiatic, where the perpetual concern is exactly how to get more
knowledge with less information (few old texts for most langauges).

    But the other side of this is that the mathematics makes it
perfectly clear that if you relax the semantic and phonemic criteria far
enough, you quickly come to a point where the expected number of chance
correspondences becomes so high, that it becomes practically impossible
to mount an effective demonstration of a relationship. The relevant
parameters are number of comparisons and frequency of consonants. If you
allow for comparison of each word with a wide range of semantically
close words you multiply the number of comparisons and effectively
increase the sample size. (A pair of 1000 wd-lists with one-to-one
matching is the same mathematically as two 100 wd. lists with each word
compared with 10 words in the other language-- both give 1000 pairs or
trials).  Going back to the previous example with frequency of 5% for
each consonant the number of matches you need to get to the 1 in a
million or better range for different samples sizes are: 200-8, 500-10,
1000-14, 2000-19. In other words although the average number of expected
chance correspondences increases geometrically with sample size, the
number needed for reasonable certainty of non-chance goes up at a higher
rate. If you are considering each word in a 1000 wd list against 20 or
30 semantically close words, the effective sample size-- and hence the
number of matches needed to demonstrate a non-chance relationship--
becomes gigantic. (I don't have a calculator powerful enough to
calculate it though, sorry.) Similarly If you allow many-to-many phoneme
matchings, you effectively increase the frequency. If you compare two
systems of 15 consonants at 3 points of articulation one-to-one the
chance of a match is on average 1/15 squared. The expected number of
chance matches in a 1000 word sample is between 4 and 5 (4.44)--
reasonable. The chance of matching any two consonants at the same point
of articulation  is 1/3 squared. In a 1000 wd. sample the expected
number of chance matches is 111-- a big jump.

Thus with very loose criteria, the comparatist is in the paradoxical
position of having to prove the existence of hundreds of "bad" (random)
correspondences in order to have any confidence of having found in any
good ones (ones which actually reflect language history). And if there
really are any good correspondences, the problem of how to pick them out
from all the random "noise" which is certain to be there is daunting.

>

  -- -----------------------------------------------------------
Robert R. Ratcliffe
Associate Professor, Arabic and Linguistics,
Dept. of Linguistics and Information Science
Tokyo University of Foreign Studies
Asahi-machi 3-11-1,
Fuchu-shi, Tokyo
183-8534 Japan

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