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The concept of compositionality.
Our discussion has led us to one of the key concepts of contemporary semantic theory: compositionality.
The principle of compositionality
Compositionality is a simple and natural concept underlying most of the work in natural language semantics and in the semantics of programming languages. A common formulation of the principle of compositionality is the following: ``The meaning of a compound expression is a function of the meaning of its parts and of the syntactic rule by which they are combined.'' This at least requires that we have a notion of parts that form such an expression according to rules, and that each of these parts can be assigned a meaning.
Suppose we have some sort of theory of hierarchical syntactic structure providing us with a notion of parts of the expressions in our language. That is, the syntactic structures we would like to have should allow us to classify sentences into subparts, sub-subparts, and sub-sub-subparts, ..., and so on - ultimately into individual words (beyond that, it doesn't matter too much for our purposes what sort of syntactic theory it is). The syntactic structures provided by that theory then make it possible to tell an elegant story about where semantic representations come from.
Ultimately, semantic information flows from the lexicon, where each lexical item is associated with a representation. But how is this information combined? Here, we make use of the sentence hierarchy a syntactic analysis provides us with. Suppose the syntax tells us that some kind of sentential subpart (a VP, say) is decomposable into two sub-subparts (a TV and an NP, say). In that case it's our task to describe how the semantic representation of the VP subpart is to be built out of the representation of its two sub-subparts. If we succeed in doing this for all the grammatical constructions covered by the syntax, we'll have developed a compositional semantics for the corresponding language (or at least, for that fragment of the language covered by our syntactic analysis).
Why compositionality?
What is so nice about compositional semantics? First of all, the principle of compositionality is of great immediate appeal: The meaning of a sentence should somehow be derivable from that sentence itself. And two kinds of information that every sentence provides us with are its words and its structure. Moreover, in the course of compositional semantic construction, each of the syntactic subparts of a sentence gets assigned a meaning of its own at some point, and syntactic rules are correlated with actions on the semantic side. This may be seen as giving a semantic justification of syntactic structure.
More philosophically, the principle of compositionality offers an explanantion of how a human being can understand a possibly infinite number of sentences never heard before (namely by constructing their meaning from a finite set of rules and a finite set of known lexical meanings). Also, a compositional account of meaning suggests a plausible explanation of why we perceive a connection in meaning between sentences that share syntactic parts. Consider the sentences:
``John's father likes Mary.''
``John's father hates Mary.''
Clearly, these two sentences have meaning elements in common. Assuming a compostional semantics that is constructed part-by-part, along with the syntactic structure, this can be explained by pointing to their shared syntactic parts ``John's father'' and ``Mary''.
Discussion
Of course, both of these arguments aren't really coercive. Satisfactory explanantions for the described phenomena could probably also be given on the basis of semantic construction methods that would not neccesarily be counted as compositional. In fact, in spite of its apparent simplicity, the notion of compositionality raises a number of interesting issues, and is by no means uncontroversial.
On the one hand, if we do not further constrain the class of functions that may be used to combine the meanings of the parts of expressions, the principle doesn't really say very much any more. It can even be shown that any semantic construction method can be made compositional in some sense. On the other hand, it can be doubted that systematic semantics always have to be compositional. There is for instance the standard top-down algorithm for Discourse Representation Theory (DRT). It is a systematic approach to semantic construction (under any reasonable interpretation of the word systematic) but it does not work by assigning meaning representations to each of the syntactic constituents of a sentence separately. Therefore many semanticists have argued that it is not truly compositional.
Apart from this, the principle of compositionality doesn't come with a fixed domain. One way to read it is in terms of meaning representations, as stating that we should have a separate meaning representation for each single syntactic part of an expression, and that the representation of the whole expression should be formed from these separate representations. This requirement is also known as surface- or s-compositionality . As another extreme, it can be read as referring to truth-functional content, e.g. as stating that we should be able to say exactly for each syntactic part how it is to be interpreted with respect to a given first-order model (one also speaks of deep or d-compositionality ).
In the following, we shall stick to the first way of reading the principle of compositionality. In fact, we shall view it more or less as a practical design guideline than as a philosophical claim. This means that when constructing semantic representations, we will try to solve our problems locally and modularily, striving for a nice and clearly structured system. But occasionally, we shall also look at our semantic formalisms in view of d-compositionality.
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