\section{Introduction}
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The dominant paradigm in formal semantics ever since its 
beginnings in the 70ies has been Montague Semantics. 
In Montague Semantics, the meaning of a sentence is 
computed on the basis of its structural description. 
Although Montague Semantics is potentially very powerful, 
it cannot explain the role of agreement and case marking. 
From its point of view, both should be redundant, since 
we have structure to give us all the information we need. 
We have argued elsewhere (see \cite{kracht:agreement}) 
that this is the wrong route to take. If we assume that 
there is not enough structure to begin with, 
there is no reason to assume that agreement or case 
marking is redundant. In this paper, we shall explore 
the potential of case marking as a substitute for syntactic 
structure. We shall show that in the presence of case 
marking of a specific kind, no syntactic structure whatsoever 
is needed. The basic mechanism needed to make this work is 
called {\it suffixaufnahme\/} (see \cite{plank:suffixaufnahme} 
for the history of that term). We shall develop an idealized 
model for such languages, called ideal case marking languages. 
Although this model is mainly of theoretical significance, 
it may have far reaching consequences in the study of formal 
semantics, some of which we shall indicate at the end of 
this paper. 

Cases are difficult for language theory because of their 
dual nature. On the one hand they do service in syntax 
by marking arguments for their syntactic status and on 
the other hand cases also have a meaning, although to 
a varying degree. Core cases tend to be more syntactic, 
while other cases (eg locative and instrumental cases) 
have a clearly definable semantic meaning, which make 
them more semantic in nature. The degree to which a 
given case in a given language is semantic or syntactic 
differs from one construction to another, so it is by 
no means fixed. One and the same case 
(eg the allative in Finnish) can on one occasion be 
completely semantic (in which case it is typically an 
adverbial, indicating the movement of some object 
towards the allative marked NP) and the next time a 
purely syntactic one (being selected by some verb, for 
example).

While the semantic side of cases is surely interesting, 
we shall deal in this paper exclusively with the 
syntactical side of cases. We consider cases in their 
role as devices to keep track of the syntactic status 
of an argument. Our investigation will be held theory 
neutral. This is why we have chosen to speak of the 
combinatorics of cases. We shall investigate in what 
ways cases can help to keep track of arguments. As we 
shall show, the more elaborate the case marking system, 
the freer the word order is allowed to be. This approach 
has hardly been prominent in the last decades. All syntactic 
theories we know of study the mechanics of cases independently 
of any word order phenomenon (a notable exception is the 
dissertation by Nordlinger \cite{nordlinger:case}). 
Moreover, we find in \cite{baker:polysynthesis} the 
remarkable statement that head--marking languages tend 
to have free word order, while case marking languages 
do not. As we shall show in the beginning of this paper, 
there are purely combinatorial arguments that this is 
unlikely. Several empirical arguments against this 
view will also be adduced. 

The paper is organized as follows. We begin with an 
overview of language strategies to encode syntactic 
relations. It will emerge that there are two systems 
which yield unambiguous languages: Polish Notation
and case marking. We shall subsequently develop a 
semantics for case marking languages that is flexible 
enough for both types of case marking: group and word 
marking languages. We will then compare this semantics to 
Montague Semantics. Finally, we will pick up some 
remaining issues on the relationship between grammatical 
restrictions and semantical restrictions. 

