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Truth and Meaning
allows for in this respect is an admission of ambiguity of the expression involved. If one wishes discourse factors to contribute to meaning, the notion of mean- ing must be enriched in order to incorporate such factors (see Sect. 3 below).
2.5 AmbiguitiesareLexicalorDerivational
Ambiguities may simply be lexical. If they are not, the principle of compositionality of meaning as applied in Montague Grammar allows for only one alternative source: a difference in the derivational history. This has certain consequences for the theory. Suppose, for example, that one takes the sentence, 'Every Belgian speaks two languages.' to be ambiguous between a reading in which there are two languages that are spoken by every Belgian, and one in which, merely, every Belgian is bilingual. The Montagovian com- positionality principle now requires that the ambi- guity resides either in the words or in the derivational history of this sentence. The former option being less likely, the syntax will have to be such as to reflect at least two different derivational histories of this sentence. Semantic decisions are thus seen to impose conditions on the syntax. In Montague Grammar, syntax and semantics, though distinct, are thus forced to remain in step with each other.
3. Representations of Meanings
Meanings are, in Montague Grammar and in almost all semantic theories, considered to be model-theoretic entities, such as truth values, sets of a certain kind, or functions of a certain type. Usually these abstract entities are represented by means of an expression in some logical language (e.g., Vx[man(x) -»mortal(x)]). These representations are not themselves meanings, and should not be confused with them. Differences in representation do not always constitute a difference in meaning: ^ Aq>and <pA ty express the same mean- ing. And logically equivalent expressions are equally good as representations for some meaning. A semantic theory cannot be based on accidental properties of meaning representations, since then it would be a the- ory about representations and not the meanings them- selves.
As has been shown above, there is for each syntactic rule a corresponding semantic rule that says how the meaning of the compound expression depends on the meanings of the parts. Since these meanings are rep- resented by logical expressions, it seems natural to represent such functions by means of an operation on logical expressions. This operation has to represent an operation on meanings and therefore it should not make use of accidental properties of the meaning rep- resentation. For instance, the operation 'enclose the formula between brackets and write a negation sign in front of it' is acceptable since it corresponds with the semantic operation of negation. But the operation 'negate the second conjunct of the formula' is not
acceptable since it does not correspond with an oper- ation on meanings. This is evident from the fact that it has different effects on the two equivalent meaning representations <pA*!/ and i^ A<p.
The above discussion, however, only makes sense if one takes meanings to be abstract entities. In some theories such as discourse representation theory (Kamp 1981, henceforth DRT) meaning rep- resentations are an essential ingredient of the semantic theory. This claim of DRT is linked with the pos- tulation of the psychological relevance of their rep- resentations. In such a theory, operations on representations of meaning are, of course, acceptable. For such operations the issue of compositionality can be raised as well, but it becomes a different issue.
Thus two extremes have been met: no use of rep- resentations and total dependency on them. There are also theories that lie somewhere in between; for instance, some theories for anaphora essentially use the indices of variables (for a discussion see Landman and Moerdijk 1983).
4. TheStatusofCompositionality
The principle of compositionality implies no restric- tions on the rules of syntax. We know that a rich variety of rules is needed. A rule is needed, for exam- ple, to introduce the verb do in the negative sentence Mary does not cry, and one to ensure the correct morphological form does, or cries in Mary cries. In general, rules may perform permutations, insertions and deletions, and have therefore the same power as Turing machines or Chomsky type-0 grammars. The principle implies no restrictions either on the nature of meanings or on the operations that can be per- formed on them. It can be proven that, in such a system, any sentence can be assigned any meaning in a compositional way (see Janssen 1986a; 1997). Compositionality, therefore, implies no restriction on the final results that can be obtained. Without additional empirical constraints compositionality has no empirical content. For this reason the principle of compositionality in Montague Grammar is not con- sidered to be a claim about natural language, but merely a methodological or heuristic principle, i.e., a
criterion for evaluating theories (not all theories that have been proposed are compositional).
As such it has proved its value. Proposals that do not satisfy the compositionality criterion often turn out to be inadequate precisely at the points where they infringe the principle. Janssen (1986b; 1997) discusses several proposals from the literature that do not obey compositionality, in the sense sketched in Sect. 3. It is shown that those proposals have unacceptable logical consequences, and that by reformulating the under- lying ideas in a compositional way, they gain in gen- erality and empirical adequacy.
Now a more elaborated illustration of the heuristic value of the principle will be considered. It has to
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