\section{A Comparison}
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We have proposed in the previous sections two alternative ideal 
languages: Polish Notation and the Ideal Case Marking Language. 
We have presented them both in their pure abstract instantiation. 
Both enjoy unique readability and with each we can associate 
a formal semantics. While Polish Notation lends itself easily 
to a functional interpretation, the case marking languages
are better treated using a relational interpretation. 
%%
\footnote{Without going into much detail, we shall explain 
the basis of this remark. It is known that in predicate 
logic, a theory formulated in an arbitrary signature can 
be replaced by a theory that uses only relations. Namely, 
each $n$--ary function symbol $f$ is replaced by an 
$n+1$--ary relation symbol $R$. An equation $f(x_0, \ldots, 
x_{n-1}) \doteq y$ is replaced by the formula 
$R(x_0, \ldots, x_{n-1},y)$. Terms are eliminated in a 
stepwise construction. As it turns out, the semantics we are 
proposing here for word marking languages resembles exactly 
the result of turning a functional signature (that of Montague 
Grammar) into a relational one.}
%%
In this 
section we will discuss in some depth the similarities and 
differences between the notations. 

A first difference that is worth noting is that Polish Notation 
neither allows for word order variation nor for argument elision. 
While this does not seem to be problematic for computer languages, 
natural languages are different in this respect. Speakers like 
to drop all material in a sentence that is given by the context. 
Although languages resist this (you cannot simply omit what you 
want) there is generally some freedom. Specifically, those languages 
with rich case marking do allow for argument elision. Australian 
languages are generally very liberal in this respect. In Martuthunira 
one is allowed to drop any constituent or word whatsoever 
(see Section~\ref{sec9}, and \cite{dench:martuthunira}). 

Now we are in a certain dilemma: we have shown that case marking 
languages allow for free word order and free omission, but why 
is it that rich case marking does not lead to random word order? 
There are two kinds of answers. The first answer is historical: 
if a language starts with poor morphology, it has strict word 
order. If it acquires richer morphology, there is no need to 
free the word order in tandem. Languages do not generally strive 
for word order freedom (see Lehmann~\cite{lehmann:factors} on this 
point). But here again there is the question why this should be 
so. This brings us to the second answer: the order of thought. 
If a language is free in its word order this practically means 
that speakers can use a word when they need or want to. That in 
turn means that they can map the order of thought almost directly 
onto the linear order. If that is so, it is unlikely, for example, 
that NPs or subordinate sentences become discontinuous just because 
there is the possibility for that. However, omission of argument is 
much different. We just choose not to say something, that's all. 
There is no need to adapt the mapping from thought to speech in any 
complicated way. 

Both of these models have wider application. In fact, languages 
do admit ambiguity and the most favoured language in absence 
of morphology is SVO. Montague has originally proposed his analysis 
for English. The ambiguities that formally arise are actually now 
welcome because they reflect a basic trait of the language under 
analysis. In this vein, the analysis can be extended to languages 
of type SVO, OVS, SOV and VOS, and if one allows the verb to form 
a constituent with the subject, also to the remaining types 
OSV and VSO. (See \cite{keenan:binding} for arguments that in 
such languages the verb forms a constituent with its subject.)

The semantics for case marking languages works also for other 
types of languages. In particular, it can be used for languages 
that are consistently peripherally group marking. In these languages, 
the case marker is put either in front or at the end of the entire 
phrase. If we want to do this, nothing needs to be changed, except 
that the case marker will now be a phrasal affix rather than a 
word affix. To illustrate this, we shall compute the semantics 
of the phrase `good doctor'. 
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sc nom}/} \\
\hline
\circ : \mbox{\sc nom}{^{\smallfrown}\circ} \\
\hline\hline
\varnothing \\\hline
\varnothing \\\hline
\end{array} 
\qquad
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf good}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf doctor}'(\circ) \\
\hline
\end{array}
\qquad
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf doctor}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf doctor}'(\circ) \\
\hline
\end{array}$$
%%
We may either compose both {\sf good} and {\sf doctor} with 
the nominative and then compose the result, or we may first 
compose {\sf good} and {\sf doctor} and then compose with the 
nominative. In both cases the result is the same: 
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf good doctor}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf good}'(\mbox{\sc nom}{^{\smallfrown}\circ}); \\
\mbox{\sf doctor}'(\mbox{\sc nom}{^{\smallfrown}\circ}). \\
\hline
\end{array}$$
%%
We shall also prove this in a rather formal manner: 
%%
\begin{prop}
Let $A$ and $B$ be plain, and let $C$ be a case marker. Then 
$(A \oplus B) \oplus C = (A \oplus C) \oplus (A \oplus C)$. 
\end{prop}
%%
\proofbeg
Let $A = [\circ : \circ, [V : \Delta]]$, 
$B = [\circ : \circ, [V', \Delta']]$ and 
$C = [\circ : \mbox{\sc N}{^{\smallfrown}\circ}, [\varnothing : 
\varnothing]]$. Denote by $\sigma$ the substitution 
$\circ \mapsto \mbox{\sc N}^{\smallfrown}\circ$. 
Then $A \oplus B = [\circ : \circ, [V \cup V', 
[\Delta \cup \Delta']]$ and so $(A \oplus B) \oplus C = 
[\circ : \circ, [V^{\sigma} \cup V'^{\sigma}, \Delta^{\sigma} 
\cup \Delta'^{\sigma}]]$. On the other hand, $A \oplus C = 
[\circ : \circ, [V^{\sigma}, \Delta^{\sigma}]]$ and 
$B \oplus C = [\circ : \circ, [V'^{\sigma}, \Delta'^{\sigma}]]$ 
and so $(A \oplus C) \oplus (B \oplus C) = 
[\circ : \circ, [V^{\sigma} \cup V'^{\sigma}, 
\Delta^{\sigma} \cup \Delta'^{\sigma}]]$. This shows the 
claim.
\proofend

In languages which have phrasal affixes, case suffixes may 
actually end up in a sequence, giving the appearance of 
suffixaufnahme. The following is found in Sumerian (see 
\cite{plank:suffixaufnahme}).
%%
\\[2mm]
\begin{tabular}{llll}
(\thesection.1) & {\sf \'e} & {\sf lugal-ak} \\
     & house & king-{\sc gen} \\
     & \multicolumn{3}{l}{\it house of the king} \\
(\thesection.2) & {\sf \'e} & {\sf lugal-ak-a} \\
     & house & king-{\sc gen}-{\sc loc} \\
     & \multicolumn{3}{l}{\it in the house of the king} \\
(\thesection.3) & {\sf \'e} & {\sf \v{s}e\v{s}} & 
        {\sf lugal-ak-ak-a} \\
     & house & brother & king-{\sc gen}-{\sc gen}-{\sc loc} \\
     & \multicolumn{3}{l}{\it in the house of the brother of 
        the king} 
\end{tabular}
\\[2mm]
%%
Here, the case is a phrasal suffix, and the genitive complement 
also follows the head noun. Hence the last example is to be 
bracketed as follows. 
%%
$$(\mbox{\sf \'e} \ (\mbox{\sf \v{s}e\v{s}} \ 
(\mbox{\sf lugal})\mbox{\sf -ak})\mbox{\sf -ak})\mbox{\sf -a}$$
%%
This is admittedly a rare example, though we find in the 
abovementioned source also examples from Late Elamite 
and Kanuri (a Nilo--Saharan language). 

Most languages are in between these types. Latin has case 
agreement, but no stacking of cases. These languages are 
intermediate cases, since they are neither entirely word 
marking (because then they would look like Kayardild or 
Martuthunira) nor are they group marking. Nevertheless, 
if the morphological and syntactic restrictions are properly 
implemented, the semantics works for these examples, too. 
There is, however, a condition. If for example, we wish to 
give a semantics for the genetive that parallels that of Old 
Georgian, we must assume that a genetive specifier agrees 
in case with the head noun. We must therefore posit an 
invisible case marker for the specifier. Likewise, we must 
assume that relative clauses, in fact all clauses, are case 
marked. 
