Compositionality: Similarity versus Interpretability

The present paper studies the general implications of the principle of compo-sitionality for the organization of grammar. It will be argued that Janssen's (1986) requirement that syntax and semantics be similar algebras is too strong, and that the more liberal requirement that syntax be interpretable into semantics leads to a formalization that can be motivated and applied more easily, while it avoids the technical complications that encumber Janssen's formalization. Moreover, this alternative formalization even allows one to fur-ther`complete' the formal theory of compositionality, in that it is capable of clarifying the role played by model-theoretic interpretation and meaning postulates, two aspects that received little attention in Janssen (1986) and Montague (1970). In its most general form, the principle of compositionality states that`the meaning of an expression is a function of the meanings of its parts and of the way they are syntactically combined' (Partee 1984, p. 281). In other words: the meaning of an expression is determined completely by the meanings of its parts plus the information which syntactic rules have been used to build that expression out of those parts. The principle is also known as`Frege's principle', but this attribu-tion is at best a tribute according to Janssen (1986), who gives a formalization of the principle which is based on Montague's paper`Universal Grammar' (1970). Janssen's formalization and the framework deened in Montague (1970) are, roughly speaking, `diierent views of the same mathematical object' (Janssen 1986, Part 1, p. 91). The main diierence is that Janssen employs many-sorted algebras, whereas Montague uses one-sorted algebras (though with much additional structure). As a consequence of this, Janssen's approach has various advantages (see Hendriks 1993, p. 136). The term`many-sorted algebra' stems from Adj (1977). We deene the notion as follows: 1 h(A s) s2S ; (F) 2? i is a many-sorted algebra of signature (algebra) ii (a) S is a non-empty set (of sorts); (b) (A s) s2S is an indexed family of sets (A s is the carrier of s); (c) ? is a set (of operator indices); (d) (the type-assigning function) assigns to each 2 ? a pair hhs 1 (e) (F) 2? is an indexed family (of operators) such that if To cut a long story short, Janssen argues that the compositionality principle dictates the following: (a) the syntax is a-algebra A = h(A s) s2S ; (F) 2? i with 1 Our terminology deviates from Janssen (1986), where a …