Page 127 - Encyclopedia of Philosophy of Language

do with discourse pronouns. Let us consider first the relevant phenomena on the basis of the following two discourses:
A man walks in the park. He whistles. (1)
only for the phenomena that are treated in DRT, but for other phenomena as well (see Groenendijk and Stokhof 1991). Thus it is seen that the attempt to stay compositional led in a certain direction. Since this approach has its intrinsic value, it is not surprising that, independently of any compositionality require- ment, it is proposed by other authors as well, e.g., Seuren 1985. For more discussion, see van Eyck and Kamp 1997.
5. Formalization of the Principle
As the rules of grammar specify the structural parts of an expression, they can be regarded as operators taking input expressions and delivering output expressions. Mathematically speaking, this means that they are considered operators in an algebra of expressions, turning the grammar into an algebra. The notion 'part of an expression' is defined as 'being the input to a rule forming that expression.' As has been shown above, the meaning of an expression is deter- mined by its derivational history. This means, math- ematically speaking, that meaning assignment is a function that is defined on the elements of the term algebra over the grammar for the source language.
The principle of compositionality also describes the character of meaning assignment. Suppose an expression is obtained by application of operation f to arguments a, , . . . , an , hence with the derivational history f(a,,...,an). Then the meaning in a model should be obtained from the meanings of its parts, hence by application of an operator g (corresponding withf)tothemeaningsofT(f(a],...,an)).IfT denotes the meaning function, then:
The meaning assignment is thus seen to be a homo- morphism.
These considerations produce the following for- malization of the principle: A compositional meaning assignment of language A is obtained by designing an algebra (A,Fy as syntax for A, an algebra (B, (?) as a semantic model, and by letting the meaning assign- ment be a homomorphism from the term algebra over Ato<£,G>.
Since there is no restriction on the operators, any language that can be generated by some algorithm (i.e., any recursively enumerable language) can be described by a compositional grammar. Furthermore, it can be shown that every meaning assignment to sentences can be put in the form of a homomorphism. If some logic is used to represent meanings, <A, F> is in fact translated into expressions from a logical algebra <C, H>. This algebra is then homo- morphically interpreted in the semantic algebra <B,G>. The requirement that meaning assignment be compositional is guaranteed if one translates into polynomials over <C, H>. This model is introduced in Montague (1970). For an introduction see Halvorsen
Not all men do not walk in the park. He whistles.
(2)
In (1), the pronoun he in the second sentence is inter- preted as anaphorically linked to the term a man in the first sentence. This is not possible in (2), where he has to refer to a third party. The meanings of (1) and (2) are, therefore, different. And since the second sentences of (1) and (2) are identical, we must look to the first sentence to find the meaning difference:
These two sentences are logically equivalent. If mean- ing is taken to be identical to truth-conditions, as is customary, then (3) and (4) have the same meaning. But so do the second sentences of (1) and (2). There- fore, if the meaning of each discourse, (1) and (2), were a function of the meaning of the constituent sentences, (1) and (2) would have identical meanings, which is contrary to fact. This phenomenon thus seems to provide an argument against the assumption of compositionality in discourses.
In discourse representation theory, meaning rep- resentations constitute an essential level. There, (3) and (4) are assigned different representations. The two negation signs in the representation of (4) trigger different interpretation strategies for the discourse. This is one of the ways in which the difference between DRT and compositional grammars becomes evident.
Nevertheless, a compositional treatment for this kind of phenomenon is quite feasible and, in fact, the principle itself points to a solution. Since the two discourses (1) and (2) have different meanings, and their second sentences are identical, the difference. must reside in their first sentences, i.e., (3) and (4). And since (3) and (4) have identical truth-conditions, a richer notion of meaning is required if the principle of compositionality is to be saved for discourses. Truth-conditions of sentences (which involve possible worlds and assignments to free variables) are just one aspect of meaning. Another aspect is that the pre- ceding discourse has a bearing on the interpretation of a sentence (and especially of the so-called discourse pronouns). Moreover the sentence itself extends this discourse and thus has a bearing on sentences that follow it. Thus a notion of meaning is required which takes into account the semantic contribution that a sentence makes to a discourse. Sentences (3) and (4) make different contributions to the meaning of the discourse, especially concerning the interpretation of later discourse pronouns. These ideas have led to dynamic predicate logic and dynamic Montague Grammar, a compositional theory that accounts not
A man walks in the park.
(3) Not all men do not walk in the park. (4)
Compositionality of Meaning
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