\section{Three Regimes}
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In semantical as well as syntactical terms the problem in 
understanding a sentence is to know which of the words is 
the head or functor and which of the others are which 
arguments of the functor. This problem can be looked at in 
two ways: as a {\it language choice\/} or as a {\it speaker 
choice}. If looked at as a speaker choice, we will talk of 
{\it strategies}. A {\it strategy\/} is a way to say clearly 
and unambiguously what one wants to say in a given language. 
If looked at as a language choice will talk of {\it regimes}. 
A {\it regime\/} is a fixed way of encoding relations in a 
language. Strategies and regimes are of course closely connected. 
Strategies depend on the regimes of the language. To allow for 
a formal definition of regimes, we shall fix a language of semantical 
structures, which should be unambiguous. We may take, for 
example, semantic derivations in Montague Grammar. The 
regime is then a recipe of `sugaring' a derivation into a 
legitimate string of the language with the corresponding 
meaning. Sugaring is a process that eliminates parts of 
structure and adds some functional elements in its place. 

Let us illustrate the problem with a thought experiment. Imagine a 
language, {\it Ektian}, with only three words, {\sf eg}, which means 
`cat', {\sf ag}, which means `dog', and the word {\sf um}, which means 
`to chase'. 
%%
\footnote{This is an adaptation of a language described by Franz 
Hohler in his {\it Wegwerfgeschichten}. It might be worth emphasizing 
why we engage in a thought experiment rather than taking real 
language examples. The reason is simply that a fictitious language 
can be made to function any way we please. So we can make it 
consistently SOV, for example. If we take real language example 
we always run the risk that one objects that such and such 
construction is actually not SOV, and that one finds marginal 
constructions of type VSO, and so on. This objections will 
however never matter for what we really want to say and so we 
have chosen for the thought experiment.}
%%
There are no determiners, so {\sf eg} may mean, 
depending on circumstances, either `a cat' or `the cat'. 
But this will hardly matter. Now consider the English sentence 
(\thesection.1) and its semantic translation. 
%
\\[2mm]
\begin{tabular}{ll}
(\thesection.1) & {\sf The cat chases the dog.} \\
   & 
$\mbox{\sf chase}'(x,y) \und \mbox{\sf cat}'(x) \und 
	\mbox{\sf dog}'(y)$
\end{tabular}
\\[2mm]
%
Suppose we want to translate this sentence into Ektian. We certainly
expect the words {\sf eg}, {\sf ag} and {\sf um} to occur exactly once. 
But, in what order will they appear? What if anything is added to these 
words? In particular, which additional information (order, markers, 
etc) does Ektian use to identify the subject and the object? There are 
three basic choices. 
%%
\begin{enumerate}
\item
Inferential or Null Regime. Subject and Object must be inferred 
from their meaning and the context. {\sl Examples.} Lusi. 
%%
\footnote{We take this example from \cite{palmer:roles}, who 
quotes Li and Thompson~\cite{lithompson:subject} for the claim 
that Lusi does not distinguish subject and object.}
%%
\item
Positional Regime. It is agreed that the elements follow 
in a certain order, for example subject first. 
{\sl Example.} English, Chinese. 
\item
Marking Regime. The words get markers that allow to 
disambiguate the sentence. 
%%
\begin{enumerate}
\item
Head Marking Regime. The head is marked for some quality of the 
subject and the object. We write as follows 
%%
$$N_1 \quad N_2 \quad n_1\mbox{--}n_2\mbox{\it --V}$$
%%
Here $n_1$ and $n_2$ are markers (which we write as prefixes, 
even though they may also be suffixes or clitics etc) 
that reflect a property of $N_1$ and $N_2$ (gender, class etc). 
We call these markers also {\it agreement markers}. Also, 
$n_1$ is the {\it reflex\/} of $N_1$ in $V$. Some 
agreement markers may be absent. Note that the order of subject 
and object marker must be fixed for this to disambiguate the 
sentence. {\sl Examples.} Mohawk, Bantu languages. 
%%
\item 
Dependent or Case Marking Regime. The head identifies subject by 
a subject marker and the object by an object marker. These markers 
are called {\it cases}. We write as follows. 
%%
$$N_1\mbox{--}\gamma_1 \quad N_2\mbox{--}\gamma_2 \quad V$$
%%
Note that $\gamma_1$ and $\gamma_2$ must be distinct in order 
for this to disambiguate the sentence. {\sl Example.} 
Tagalog.
\end{enumerate}
\end{enumerate}
%%
Let us illustrate this with Ektian. If it uses the inferential 
regime, then any permutation of the sequence {\sf eg ag um} 
means the same, all circumstances being equal. Furthermore, {\sf eg 
ag um} means both `the cat chases the dog' as well as 
`the dog chases the cat'. In the positional regime it is declared, 
for example, that the subject precedes the 
object (this is in fact all that is needed in the present case). 
Then the sentences are unique in meaning. Thus, {\sf eg ag um} means 
`the cat chases the dog', while {\sf ag eg um} means `the dog 
chases the cat'. When the marking regime is employed, we may 
either have cases, say {\sf -su} for subject and {\sf -ob} for 
object. Then any permutation of the sequence {\sf eg-su ag-ob 
um} will mean `the cat chases the dog', and any permutation 
of the sequence {\sf eg-ob ag-su um} will mean `the dog chases 
the cat'. If we finally choose the head marking regime, we 
shall have certain prefixes, say {\sf e-} for cats and {\sf a-} 
for dogs, and then any permutation of {\sf eg ag e-a-um} means 
`the cat chases the dog', while any permutation of 
{\sf eg am a-e-um} means `the dog chases the cat'. So much 
for an illustration of the basic regimes.

The case marking regime actually splits into two further subregimes. 
The first is called {\it group marking\/} and the second the 
{\it word marking regime}. Group marking languages use only 
one item of a case marker, which appears either peripherally 
or on the head. (For example, assume that Ektian has a word 
{\sf bul}, which means `blue'. Then {\sf bul-su eg-su} 
means `blue cat' in the nominative, if Ektian is word 
marking, and {\sf bul eg-su} if it is group marking.) 
Examples of group marking languages are Japanese, Turkish, 
and Hungarian. In word marking languages, the case marker is 
iterated on every single word of the constituent. This definition 
is actually somewhat inexact, and we shall return to this question 
below. Examples are Finnish, German, Latin and Russian. 

Some notes are in order. First, languages usually use a  
mixture of these regimes. Indo--European languages usually 
have subject verb agreement in addition to case marking (if 
present). Furthermore, as we shall see, in languages the systems 
are never pure enough to guarantee full disambiguation (for example, 
in German there is no distinction between nominative and 
accusative in feminine and neuter nouns as well as all plural 
nouns). In that case some other regime jumps in. Second, 
a marker is to be understood in an abstract sense. For 
example, intonation is also added as a cue for disambiguation. 
In the German case, the subject--first preference jumps in as soon 
as the cases fail to distinguish nominative from accusative. 
This can be overruled by adding intonation. Also, the context 
may help (then we are in Regime 1). In what is to follow, 
we will exclude both possibilities. Third, the markers can 
be suffixes or words or clitics. To keep matters simple, 
we often do as if they are part of their host word, neither 
clitics nor separate words. However, this will hardly make 
a difference.

Before we go on, we note some direct consequences. The 
head marking regime and the dependent marking regime 
are very distinct in the following sense. If the head 
marking regime is chosen, the reflex $n_1$ of $N_1$ 
depends on the actual choice of $N_1$. This can be a 
semantic property or what we call a {\it ritualized\/} 
property. Here by {\it ritualized\/} we mean roughly 
{\it must be learnt by heart}. These two, semantic and 
ritualized head marking, are not sharply distinguished. 
Gender systems in Indo--European languages are semantically 
motivated but to a large extent ritualized. 
If the reflex is semantic it can be inferred from the 
meaning of $N_1$. It may well be that the reflex depends 
on the phonetic form of $N_1$, but we ignore that choice 
as it makes no difference in principle. A neutral term 
in place of {\it semantic\/} is {\it inferential}. 
In contrast to head marking, dependent marking languages 
need to record in the lexicon what the cases of the 
actants must to be. Again, this can be inferential 
(i.~e. mainly semantic) or ritualized. The first 
option is realized in Manipuri (a Tibeto--Burman language, 
see \cite{dixon:ergativity}) and the second in Indo--European 
languages. Mixtures do occur. In Finnish, the case of 
the direct object varies between accusative and partitive 
depending on the meaning of the sentence and whether the 
object is entirely affected by the action etc. 

Let us close with a remark on the first regime. In such a 
language, (\thesection.2) (and any permutation thereof) is 
ambiguous between (\thesection.2a) and (\thesection.2b). 
%%
\\[2mm]
\begin{tabular}{ll}
(\thesection.2) & {\sf eg ag um.}  \\
(\thesection.2a) & {\it The dog chases the cat.} \\
(\thesection.2b) & {\it The cat chases the dog.} 
\end{tabular}
\\[2mm]
%%
This can lead to the following regime. It is fixed that the 
sentence means only one of the above. Typically, arguments 
are ranked (for agentivity) and the higher ranked argument 
takes the role of subject. 
%%
\footnote{The diligent reader may notice that nothing is said 
in the case when they have equal rank. Languages have developed 
means to avoids this. In Plains Cree, for example, third person 
arguments are assigned two possible discourse functions: proximate or 
remote. The proximate argument is ranked higher than the remote 
argument. It is not allowed to have more than one proximate 
argument.}
%%
To generate the other meaning some other sentence must be 
used. An obvious possibility is to have a marker on the verb 
(let it be {\sf inv}) that exchanges the role of subject and 
object. So, if (\thesection.2) expresses (\thesection.2a), 
(\thesection.3) will express (\thesection.2b), and if 
(\thesection.2) expresses (\thesection.2b) then (\thesection.3) 
will express (\thesection.2a). 
%%
\\[2mm]
\begin{tabular}{ll}
(\thesection.3) & {\sf eg ag um-inv.} 
\end{tabular}
\\[2mm]
%%
This leads to a regime known as {\it inverse marking}. An example 
is Plains Cree (see \cite{vanvalinlapolla:syntax}).
In principle the problem could be solved also by marking an 
argument as being `demoted'. So, assume that the suffix {\sf dem} 
means `is demoted' (which is attached to the subject and says that 
it becomes the new object). Then if (\thesection.2) expresses 
(\thesection.2a), (\thesection.4) now expresses (\thesection.2b), 
and if (\thesection.2) expresses (\thesection.2a) then 
(\thesection.5) expresses (\thesection.2b).
%%
\\[2mm]
\begin{tabular}{ll}
(\thesection.4) & {\sf ag-dem eg um.} \\
(\thesection.5) & {\sf ag eg-dem um.}
\end{tabular}
\\[2mm]
%%
(A note of clarity. If regimes other than the Positional 
Regime are employed, the sentence above may in principle 
be replaced by any permutation of its words with identical 
meaning.) 

The list above is surely not complete. We could in principle 
invent a new word {\sf per} (for permuted) which functions 
just like {\sf inv}, which however can appear anywhere 
in the sentence. (See also the discussion surrounding 
the reflexes, whether they are separate words.) We know of 
no language where this option is chosen. There are obvious 
shortcomings of this regime, though it is of course not 
prima facie excluded. 

Some of the regimes discussed earlier have some gaps in them. 
Here to contain a {\it gap\/} means that there exist sentences 
which are semantically ambiguous if that regime is used alone. 
We shall use the term {\it ambiguous\/} in a rather strict sense 
here: the sentence is counted as ambiguous if it can have two 
meanings, judging from the mere string alone. Though it may mean 
one thing in one context and another in a different context, it 
has two meanings, if the context is not specified. This allows us 
to measure the disambiguating potential of the regime alone. The 
higher the disambiguating potential of the regime, the less context 
dependent the language is. We begin with the inferential regime. 
Here matters are clear: any sentence where a verb has two arguments 
is ambiguous. 

Now consider head--marking. Suppose that there are 
two NPs with identical reflex. Then it is not possible to decide 
which of the two is subject and which of the two is the object.
We summarize this in the following observation. 
%%
\begin{quote}
{\sl Observation.}
The Inferential Regime has gaps if there exists different 
arguments. The Head Marking Regime has gaps if there exist 
argument with identical reflex on the head.  
\end{quote}
%%
The conditions of the theorem are always satisfied. There 
exist transitive verbs in any language we know of. Furthermore, 
head marking languages typically use some classification 
system that distiguishes a few types of things. These are 
gender or class systems. A more elaborate case are noun 
incorporation languages. In such languages, the object 
agreement is realized by a noun of the language rather than 
an agreement marker of the usual sort. If we would make our language 
into a noun incorporation language, we get the following 
sentences: 
%%
\\[2mm]
\begin{tabular}{ll}
(\thesection.6) & {\sf eg ag-um} \\
		& the cat dog-chase \\
		& {\it the cat chases the dog} \\
(\thesection.7) & {\sf ag eg-um} \\
		& the dog cat-chase \\
		& {\it the dog chase the cat}
\end{tabular}
\\[2mm]
%%
However, even this is not enough even though it avoids many 
of the abovementioned problems. It so happens in these languages 
that there is no incorporation when there is talk of people with 
names. So, we do not say in such languages {\sf John Paul-hit}, rather 
we say {\sf John Paul hit} (or {\sf John hit Paul}, depending on the 
wrd order type). So, in these examples there is no incorporation. 
%%%
\footnote{One wonders if there can be a formal proof that 
incorporation is insufficient. We are quite confident that this 
is so, but it needs a precise definition of incorporation to begin 
with.}
%%%
Typically, this does not mean that there is no marking at all. Mohawk, 
for example, also has a gender system. But, as we have seen, gender 
systems have gaps. This concludes the discussion of incorporation. 

What about the other two regimes? We will see shortly that in both 
cases there exist idealized languages which have no gaps, 
that is, in which every sentence is unambiguous! For languages 
with word order regimes this has been shown already in the 20ies. 
This is what we shall now turn to.
