THEORY: How many parameters, constraints, or "types" are there?

Hello, the list.

I don't really expect anyone to be able to give an exact answer to that
question ("How many parameters, constraints, or 'types' are there?"),
although someone may surprise me.

I have read that there are between 4000 and 8000 natlangs now; and I have
read (though I can't find it anymore) that, at the estimated period of
greatest linguistic diversity (either 15000 B.P. or 15000 B.C.E., I forget
which), there were estimated to be about 15,000 natlangs.

Now, if "parameters" have exactly two values each, and they are _logically_
independent -- that is, any parameter can assume either value, no matter
what values the others have -- then, with 12 parameters we get 4096
combinations, with 13 parameters we get 8192 combinations, and with 14
parameters we get 16,384 combinations.
So if there are 15 or more independent binary parameters, there are more
language "types" than there have ever been languages at any given time.

If, OTOH, "parameters" have exactly three values each, then with 7
parameters we get 2187 combinations, with 8 parameters 6561 combinations,
and with 9 parameters 19,683 combinations.
So if there are 9 or more independent ternary parameters, there are more
language "types" than there have ever been languages at any given time.

Suppose we require the parameters to be only pairwise-independent; that is,
for any two different parameters, each one of the two can assume any value,
regardless of the value assumed by the other one?  In effect, then, any set
of three parameters, if they're binary, will yield between 5 and 8
combinations; so we might expect at least 3125 combinations out of 15
parameters and at least 15,625 combinations out of 20 parameters.

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Leaving Principles&Parameters aside for the moment, let's look at
Optimality Theory.

In Optimality Theory, what counts is what order the constraints are in --
that is, which of each pair of constraints each language regards as the
higher-priority constraint.

Any two constraints can occur in either order.

Assume, for example, three constraints, called, for lack of creativity,
ConstraintA, ConstraintB, and ConstraintC.

There may be extant natlangs attesting ConstraintA >> ConstraintB;
and there may be extant natlangs attesting ConstraintA >> ConstraintC;
and there may be extant natlangs attesting ConstraintB >> ConstraintA;
and there may be extant natlangs attesting ConstraintB >> ConstraintC;
and there may be extant natlangs attesting ConstraintC >> ConstraintA;
and there may be extant natlangs attesting ConstraintC >> ConstraintB.

Now, whether ConstraintA >> ConstraintB
or ConstraintB >> Constraint A,
we may have natlangs extant attesting
both ConstraintA >> ConstraintC
and ConstraintC >> ConstraintA.

That is, the three questions
1. ConstraintA >> ConstraintB, or ConstraintB >> ConstraintA?
2. ConstraintA >> ConstraintC, or ConstraintC >> ConstraintA?
3. ConstraintB >> ConstraintC, or ConstraintC >> ConstraintB?

may be pairwise-independent; for each two of them, there may be natlangs
attesting all four combinations of answers.

Indeed, O.T. would expect that there are; that is, O.T. would expect that
each two of the questions are, indeed, _logically_ (if not statistically)
independent.

However, the three questions can't possibly be an independent set-of-three;
by O.T.,
if ConstraintA >> ConstraintB and ConstraintB >> ConstraintC,
then ConstraintA >> ConstraintC.

In O.T., the number of different "types" of languages
will be N! ("enn factorial"), where N is the number of constraints.

But 7! = 5040, and 8! = 40320.

That means, with just eight (8) constraints, there would be more "types" of
languages, than there have ever been (estimated to be) contemporaneously-
existing languages.

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So, what about it?  Any P&P-ers want to guess that there are at most 15
Parameters? Any OT-ers want to guess that there are at most 8 constraints?

For that matter, what would you guess the top 10 parameters or the top 5
constraints are?

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Thanks to anyone who answers.

Tom H.C. in MI

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