\section{Semantic versus Syntactic Restrictions}
\label{sec6}
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The model which we have developed is ideal for so--called 
{\it flat\/} languages. It assumes no phrase structure 
whatsoever and nevertheless gets the semantics right. 
However, two issues must be addressed: (1) what does it 
mean that the syntax has flat structure and (2) are the 
restrictions on phrase structure semantically or 
syntactically motivated? 

To see that there is a problem, let us start with the 
words. The words are composed from roots by adding the 
case suffixes. By the definition of merge we can only 
attach a suffix to something that is not a suffix. But 
this restriction is not enough to guarantee conventions 1 and 2 
to hold. For there is nothing that says that cases are 
suffixes rather than prefixes and nothing tells us that 
they can only be attached to a word rather than a complex 
constituent. So, the right result is only obtained if we 
assume in addition a grammar that allows only the right kind 
of parses. To see this, let us 
note that if the cases can be alternatively prefixes 
and suffixes, an expression of the form {\tt x1y} is 
ambiguous, because it can be parsed alternatively as 
({\tt x1}){\tt y} or as {\tt x}({\tt 1y}). In languages 
this is usually not so problematic with cases (since 
they are not separated by a word boundary) but if the 
language has pre-- as well as postpositions a certain 
amount of ambiguity may arise. Secondly, if the cases 
can be also phrasal affixes we will not generate the 
right analysis either. For example, notice that the 
following would be parsed as correct in Old Georgian:
%%
\\[2mm]
\begin{tabular}{ll}
(\thesection.1) & {\sf sarel manisajta}
\end{tabular}
\\[2mm]
%%
Moreover, it would get the same translation as (5.1):
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf sarel manisajta}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf name}'(\mbox{\sc inst}{^{\smallfrown}\circ}); \\
\mbox{\sf father}'(\mbox{\sc gen}{^{\smallfrown}\mbox{\sc % 
inst}{^{\smallfrown}\circ}}, \mbox{\sc gen}{^{\smallfrown}\mbox{\sc 
gen}{^{\smallfrown}\mbox{\sc inst}{^{\smallfrown}\circ}}}); \\
\mbox{\sf belong-to}'(\mbox{\sc inst}{^{\smallfrown}\circ}, 
\mbox{\sc gen}{^{\smallfrown}\mbox{\sc inst}{^{\smallfrown}\circ}}).
\\
\hline
\end{array}$$
%%
Namely, the instrumental can also be added after the 
constituent {\sf sarel manisa} (name of the father) 
has been formed. 

This demonstrates the need for a regimentation. A 
context--free grammar defining the correct parses can 
easily be given. Let {\tt R} denote the type of root, 
{\tt F} the type of a suffix, {\tt W} the type of a word, 
and {\tt C} the type of a complex unit. Then we assume the 
following rules: 
%%
$$\begin{array}{l@{\quad \pf \quad}l}
\mbox{\tt C} & \mbox{\tt W} \; \mid\; \mbox{\tt C}\ \mbox{\tt C} \\
\mbox{\tt W} & \mbox{\tt R} \; \mid \; \mbox{\tt W}\ \mbox{\tt F} \\
\end{array}$$
%%
(Here, $\mid$ is a sign of disjunction. The first line is 
equivalent to two rules, one being 
$\mbox{\tt C} \pf \mbox{\tt W}$ and the other 
$\mbox{\tt C}\pf \mbox{\tt C}\ \mbox{\tt C}$.)
Thus, a complex constituent can be a word, or it can be 
made from two complex constituents. These rules allow to 
parse a string of words in any way one likes, as long as the 
structure is binary branching. A word is either a stem or
a word followed by a suffix.  That this much regimentation is 
enough is a consequence of the following
%%
\begin{prop}
\label{free}
Let $A$, $B$ and $C$ be plain semantic structures, that is, 
semantic structures whose referent system consists of 
$\circ : \circ$ only. Then $A \oplus B = B \oplus A$ and 
$A \oplus (B \oplus C) = (A \oplus B) \oplus C$. 
\end{prop}
%%
The proof is easy and omitted. (Notice that we get identity, 
not just semantic equivalence here.) Suffice it to say 
that if $A$ and $B$ are plain then the merge is the Zeevat--merge: 
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf X}/} \\
\hline
\circ : \circ \\
\hline\hline
V_1 \\\hline
\Delta_1 \\\hline
\end{array}
\; \oplus \; 
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf Y}/} \\
\hline
\circ : \circ \\
\hline\hline
V_2 \\\hline
\Delta_2 \\\hline
\end{array}
\; = \; 
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf X Y}/} \\
\hline
\circ : \circ \\
\hline\hline
V_1 \cup V_2 \\\hline
\Delta_1 \cup \Delta_2 \\\hline
\end{array}$$
%%
So, given any string of symbols, we must first parse it 
according to the grammar above and then assign the meanings 
to the parts and compose them according to the obtained 
constituent analysis. Now, it is certainly possible to 
constrain the syntax to be, for example, right branching on 
the words. So, rather than the above grammar we might propose 
the following one:
%%
$$\begin{array}{l@{\quad \pf \quad}l}
\mbox{\tt C} & \mbox{\tt W} \; \mid\; \mbox{\tt W}\ \mbox{\tt C} \\
\mbox{\tt W} & \mbox{\tt R} \; \mid \; \mbox{\tt W}\ \mbox{\tt F} \\
\end{array}$$
%%
However, no one would propose such a structure for a
languages in absence of any evidence even though that would 
be consistent with the data (and the semantics just given). 
But why not? Apparently, in absence of anything that would 
tell us about the constituent structure we assume that 
it is free. This means that the right kind of grammar would 
be as follows:
%%
$$\begin{array}{l@{\quad \pf \quad}l}
\mbox{\tt C} & \mbox{\tt W}^+ \\
\mbox{\tt W} & \mbox{\tt R} \; \mid \; \mbox{\tt W}\ \mbox{\tt F} \\
\end{array}$$
%%
The first line is an abbreviation for a series of infinitely many 
rules. It means that {\tt C} may be replaced by any nonzero number of 
{\tt W}s. This is actually what has been proposed for some Australian 
languages (eg Warlpiri). Here, a consituent can be analysed 
as a string of words of any length. However, the semantics 
outlined so far does not allow for such an analysis on the 
semantic level. Rather, it asks for a binary branching 
structure. To reconcile the semantical analysis with the 
constituent structure we propose to call a substring a 
{\it syntactic constituent\/} if it forms a unit in any 
semantical parse that derives the desired meaning. This 
means that no sequence of words of length $> 1$ which is 
not the entire sentence is a constituent since we could always 
avoid forming that constituent in a semantic parse and 
nevertheless get the same result. In this way, the syntactic 
analysis follows from the semantical analysis in virtue of 
Proposition~\ref{free}. 

However, still we must account for the fact that the case suffixes 
cannot be added to complex constituents, only to words. Up to this 
point this was treated as  a syntactic restriction. (In fact, it 
is most likely a morphological restriction, but for the 
present argumentation this difference does not matter.) There 
is however a way to make that follow from the `semantics'. 
First, we shall assume that a lexical item can state 
explicitly that it wants a complement to one side rather 
than another. So, rather than writing $\circ : \mbox{\sc %
gen}{^{\smallfrown}\circ}$ into the header of the genitive case, 
we write $\auf \circ : \mbox{\sc gen}{^{\smallfrown}\circ} : 
\triangleright\zu$. (The brackets are inserted for readability 
only.) This means that the item in question seeks a complement 
to its left and will substitute $\circ$ by 
$\mbox{\sc gen}{^{\smallfrown}\circ}$. This will ensure that 
the cases are always suffixes. Finally, 
we introduce another symbol, $\ast$, in place of $\circ$. 
Its combinatorics is the same as that of $\circ$. It serves 
mainly as a flag. Now lexical entries come in the form 
shown to the left rather than that shown to the right.
%%
$$\begin{array}{|c|}
\hline
\ast : \ast \\
\hline\hline
V \\\hline
\Delta
\\\hline
\end{array}
\qquad\qquad
\begin{array}{|c|}
\hline
\circ : \circ \\
\hline\hline
V \\\hline
\Delta
\\\hline
\end{array}$$ 
%%
We shall agree that composition is only defined if the 
two referent systems have the form $\ast : \ast$, or 
both $\circ : \circ$, or if one has a statement 
of the form $\ast : \alpha{^{\smallfrown}\circ}$ 
($\circ : \alpha{^{\smallfrown}\ast}$) and the other 
of the form $\ast : \ast$ ($\circ : \circ$). This 
serves to make the merge well--defined. Case suffixes 
have the form 
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sc gen}/} \\
\hline
\auf \ast : \mbox{\sc gen}{^{\smallfrown}\ast} : \triangleright \zu
\\
\hline\hline
\varnothing 
\\\hline
\varnothing 
\\\hline
\end{array}$$
%%
Finally, we shall assume that the word boundary marker 
($\#$) has the following semantics:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\#/} \\
\hline
\auf \ast : \circ : \triangleright \zu
\\
\hline\hline
\varnothing 
\\\hline
\varnothing 
\\\hline
\end{array}$$
%%
The idea behind this proposal is quite simple. As long as we 
keep stacking case suffixes, we are still at the level of a word. 
This is expressed by the fact that the stack variable has the 
form $\ast$. The word is completed when the word boundary 
marker is added to the right. Then the stack variable changes 
from $\ast$ to $\circ$ and allows the word to compose with other 
full words, but not with uncompleted words. 
