The principle of compositionality is the claim that the meaning of a complex expression is determined by its structure and the meanings of its constituents. Normally the thesis is taken to be about some particular language; questions of structure and constituency are then settled by the syntax of that language. By extension, we can talk about compositionality in other representational systems—thoughts, traffic signs, musical notation, and so on—as long as they have their own syntax.
Varieties of Compositionality
The principle is not committed to a specific conception of syntax and semantics, which is why it can be employed in debates between proponents of different conceptions (see, by way of comparison, Partee 1984). Still, if we reject all constraints on either structure or meaning, compositionality becomes trivial. As T. M. V. Janssen (1986) has shown, we can turn any meaning function on a recursively enumerable set of expressions into a compositional one, as long as we can replace the syntactic operations with different ones. And as W. Zadrozny (1994) has shown, we can turn an arbitrary meaning function into a compositional one, as long as we replace the old meanings with new ones from which they are uniformly recoverable. But because the task of semantics is to identify a meaning assignment that respects both what our best syntax tells us about structure and what our best intuitions tell us about synonymy, these results do not show compositionality to be empirically empty (compare Kazmi and Pelletier 1998; Westerståhl 1998; Dever 1999).
Although hardly trivial, the principle as stated is rather weak. For example, consider a view, according to which the meaning of a declarative sentence s is the set of possible worlds where s is true. According to such a view, tautologies are synonymous, even though (because Rudolf presumably has some tautological beliefs and lacks others) sentences resulting from embedding tautologies in the context of "Rudolf believes that … " are not. Intuitively, this is a violation of compositionality (compare Carnap 1947, sec. 14). Still, the semantics is not in conflict with compositionality as stated, because tautologies might differ structurally or in the meaning of their constituents, which could explain how embedding them may yield nonsynonymous sentences. The strengthening of the principle that is incompatible with this view requires that the meaning of a complex expression be determined by the meanings of its immediate constituents and the syntactic way these constituents are combined. (e is an immediate constituent of e ′ if e is a constituent of e ′ and e ′ has no constituent of which e is a constituent.) Call the strengthened principle local compositionality and the original one global compositionality.
Compositionality rules out the existence of a pair of nonsynonymous complex expressions built up from synonymous constituents through identical syntactic operations within the same language. As the principle is usually construed, it says nothing about the possibility of such pair of complex expressions existing in distinct languages. Still, intuitively, if the Estonian sentence s 1 and the Aramaic sentence s 2 mean different things despite having identical syntactic structure and pairwise synonymous constituents, we should conclude that either Estonian or Aramaic is not compositional. (The same structure and the same meanings of constituents cannot determine more than one meaning.) If we want our principle of compositionality to yield this result, we need to strengthen it: we could demand, for example, that there be a single function for all possible human languages that maps the structure of a complex expression and the meanings of its constituents to the meaning of that complex expression (compare Szabó 2000, p. 500). Call this principle cross-linguistic compositionality and the original one language-bound compositionality.
So, there are at least four versions of the principle of compositionality: language-bound global, language-bound local, cross-linguistic global, and cross-linguistic local. The first is the weakest and it corresponds to how the principle is officially announced; the last is the strongest and it better captures what is typically taken for granted.
There are three well-known claims that are also occasionally referred to as compositionality principles. The first is the building principle, which states that the meaning of a complex expression is built up from the meanings of its constituents. This is a fairly strong claim, at least if we take the building metaphor seriously. For then the meanings of complex expressions must themselves be complex entities whose structure mirrors that of the sentence (compare Frege 1984 , p. 193; Frege 1979 , p. 255). The second is the rule-to-rule principle, according to which every syntactic rule corresponds to a semantic one that assigns meanings to the output of the syntactic rule on the basis of the meanings of its inputs. If we assume that an arbitrary function deserves to be called a rule, this is equivalent to language-bound local compositionality. The third is the principle of substitutivity, according to which if two expressions have the same meaning, then substitution of one for the other in a third expression does not change the meaning of the third expression. Assuming that the semantics is Husserlian —that substitution of synonyms at a single syntactic position within a larger expression never changes the meaningfulness of the larger expression (compare Husserl 1913, p. 318)—this is also equivalent to language-bound local compositionality. (For the equivalence results, see Hodges 2001, theorem 4. If we want to insist—plausibly—that semantic rules must be at least computable, the rule-to-rule principle is stronger than language-bound local compositionality. The assumption that the semantics is Husserlian is far from trivial—it entails, for example, that because "Jacques is likely to leave" is meaningful and "Jacques is probable to leave" is not; "likely" and "probable" are not synonyms [compare Gazdar et al. 1985, p. 32].)
Since R. Montague (1974), it has been customary to capture compositionality formally as the existence of a homomorphism between a syntactic and a semantic algebra. Let the syntactic algebra be a partial algebra E =〈E, (F γ)γ∈Γ〉, where E is the set of (simple and complex) expressions and every F γ is a syntactic operation on E with a fixed arity. Let m be a meaning assignment function from E to M, the set of meanings. Let F be a k -ary syntactic operation on E ; then m is F -compositional if there is a k -ary partial function G on M such that whenever F (e 1, …, e k) is defined,
m (F (e 1, …, e k))=G (m (e 1), …, m (e k)).
Finally, let m be compositional just in case m is F compositional for every operation of the syntactic algebra. Whenever m is compositional, it induces the semantic algebra M =〈M, (G γ)γ∈Γ〉 on M and it is a homomorphism between E and M (compare Westerståhl 1998). (For details and formal results, see Janssen 1986, 1997; Hodges 2001.) As stated, this captures language-bound local compositionality.
Arguments for Compositionality
The argument most frequently used to support the compositionality of natural languages is the argument from productivity. It goes back (at least) to Frege, who claimed that "the possibility of our understanding sentences which we have never heard before rests evidently on this, that we can construct the sense of a sentence out of parts that correspond to words" (Frege 1980 [1914?], p. 79). The argument is an inference to the best explanation, which can be expanded and rephrased without assuming that meanings are Fregean senses as follows. Because speakers of a language can understand a complex expression e that they have not previously encountered, it must be that they (perhaps tacitly) know something on the basis of which they can figure out, without any additional information, what e means. If this is so, something they already know must determine what e means. But this knowledge cannot plausibly be in general anything but knowledge of the structure of e and knowledge of the meanings of the primitive constituents of e.
If successful, the argument from productivity establishes global language-bound compositionality. To show that a language is locally and/or cross-linguistically compositional requires detailed empirical investigation. As an argument for global language-bound compositionality, it can be criticized on the ground that although we clearly do understand some complex expressions we have never heard before, it is not self-evident that we could in principle understand all complex expressions in this manner. In fact, it is hard to see how the sort of general considerations mentioned by the argument from productivity could rule out the existence of isolated exceptions to compositionality. (Isolated putative exceptions are often declared to be idioms. Criteria for being an idiom are controversial [compare Nunberg, Sag, and Wasow 1994].)
Besides productivity, two other features of our language comprehension are cited in support of compo-sitionality. One is unboundedness : although we are fi-nite beings, we have the capacity to understand eachof an infinitely large set of complex expressions. (An ex-ample from Platts 1979, p. 47: "The horse behind Pegasus is bald," "The horse behind the horse behind Pegasus is bald," "The horse behind the horse behind the horse behind Pegasus is bald," and so on.) 47. From unboundedeness, productivity follows (assuming that finite beings cannot encounter infinitely many expressions), and thus this is not really an independent consideration. The other feature of language comprehension that supports compositionality is systematicity : that there are definite and predictable patterns among the sentences we understand. (For example, anyone who understands "The rug is under the chair" can understand "The chair is under the rug" and vice versa.) Because productivity does not follow from systematicity the argument from systematicity provides independent support for compositionality.
In fact, systematicity supports a stronger principle. The standard explanation for why understanding "black dog" and "white cat" is sufficient for understanding "black cat" and "white dog" is that we can decompose the meanings of complex expressions into the meanings of their constituents and then compose these into meanings of other complex expressions. The best explanation for the possibility of our ability to compose the meanings of complex expressions from the meanings of their constituents is supposed to be compositionality. By parity of reasoning, the best explanation for the possibility to decompose the meanings of complex expressions into the meanings of their constituents is inverse compositionality : that the meaning of any complex expression determines the meanings of its lexical constituents (as well as its syntactic structure) (compare Fodor and Lepore 2002, p. 59; Pagin 2003, p. 292). Compositionality and its inverse yield the view that the meanings of complex expressions can be viewed as having a structure isomorphic to the syntactic structures of those expressions, which in turn, may capture the idea behind the metaphor of the building principle.
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