ARG-ST as a head feature

[Carl Pollard]

Hi again, Martin,

It occurred to me that in order to really compare the empirical
consequences of Roger's and my proposal with Bayer's or Bayer and'
Johnson's (beyond what I said in my last message), it would be
necessary to know exactly what their algebra of neutralization
(\wedge) and coordination (\vee) is. In Roger's and my proposal it is
the Levy lattice, whose exact structure is known (for example, it is
bounded distributive, self-dual, primes = coprimes) -- we can draw the
Hasse diagram.

So, what does Johnson and Bayer's coordination/neutralization lattice
look like?  Or, equivalently, dividing out by logical equivalence,
what is the Lindenbaum algebra of the <\wedge, \vee>- fragment?

Now Johnson and Bayer (I finally looked at it) are explicit that these
connectives are boolean (they even call them that) and obey the
standard propositional calculus.  (In particular, they make clear in a
footnote that they are NOT the additive connectives of linear logic.
The lattice models of those -- IL algebras -- are not even
distributive.)

So the Johnson-Bayer lattice is a boolean algebra. This is a clear
difference from the Levy lattice, which is definitely not not
boolean. Assuming this boolean algebra is finite (and therefore
complete and atomic), it is isomorphic to the powerset of its set of
atoms. So the question is: what are the atoms of this lattice?

To keep it simple, suppose we have a language that would be described
in conventional) terms as having three cases (nom, acc, gen), and
nothing else except intransitive verbs that select the case of their
subject.  In this case the Levy lattice for the nominal categories has
18 elements and nom, acc, gen are the primes (and also the
coprimes). Now assume further that the language has two noun
declensions, one that neutralizes nom-acc and another that neutralizes
acc-gen. Then you should be able to take a neutralized form from each
declension, coordinate them, and get something that would satisfy a
verb that will settle for nom or gen (but not acc). In the Levy
lattice this coordinate structure gets the category

 (1)    (nom \neut acc) \coord (nom \neut gen)

and the verb selects for

 (2)    (nom \coord gen).

Since (1) is in the principal ideal generated by (2), this
hypothetical prediction is borne out. In fact, a good way to describe
the Levy lattice is: it is just big enough to handle coordinations of
syncretizations of "basic" (= prime) values and no bigger (no bigger
because the distributive law lets you put every element into
coordinate/syncretic normal form).

Now let's try to look at the same example in the Bayer setup. I'm at a
disadvantage now, because I don't know which finite boolean algebra we
are working in. One thing is for sure, it cannot be the one whose
atoms are nom, acc, gen, because in that lattice you can't distinguish
(say) between nom-acc syncretism and non-gen syncretism -- they
all go to bottom. So in order to know what JB would predict about
this kind of hypothetical example, it is not really enough to know
that \wedge and \vee are boolean, we need to know WHICH boolean
algebra is at stake.

If anyone out there knows the answer to that question (Mark, Sam, are
you there?), please let us know! Once we know what lattice it is, we
can see whether it correctly handles hypothetical examples of this
kind.

[But we would also want to make sure that it did not have TOO
MANY categories that it didn't make distinctions that had no
linguistic significance. We already know the 8-element finite boolean
algebra is too small; which one should we try next? We could try
Pow(Pow({nom, acc, gen}, but that allows 258 distinct case values,
most of which I suspect would be difficult to justify on empirical
grounds.]

>From the preceding considerations I would conclude that unless the
algebra of Johnson and Bayer's connectives is elucidated more
(especially how the "basic" values are embedded into the lattice and
how the atoms are described in terms of them), our proposal at least
has the advantage over theirs that its empirical consequences are
determinate. (I hasten to add that this claim would be more persuasive
in the face of REAL examples rather than hypothetical ones.)

Carl