λ-Grammars and the Syntax-Semantics Interface ∗

type syntactic dimensions semantic dimension s νt st np νt e n νt e(st) Table 2: Concretizations of abstract types used in this paper. The terms that our signs consist of are typed, but it is expedient to type the signs themselves as well. Types for signs will be called abstract types. Abstract types in this paper are built up from ground types s, np and n with the help of implication, and thus have forms such as np s, n((np s)s), etc. A restriction on signs is that a sign of abstract type A should have a term of type A in its i-th dimension. The values of the function . for ground types can be chosen on a per grammar basis and in this paper are as in Table 2. For complex types, the rule is that (AB) = AB. This means, for example, that np(np s) = np(np s) = (νt)((νt)νt) and that np(np s) = e(e(st)). As a consequence, (2c) should be of type np(np s). Similarly, (2a) and (2b) can be taken to be of type np, (3a) and (3b) are of types np s and s respectively, etc. In general, if M has abstract type AB and N abstract type A, then the pointwise application M(N) is defined and has type B. Abstraction can also be lifted to the level of signs. Supposing that the variables in our logic have some fixed ordering and that the number of dimensions of the grammar under consideration is n, we define the kth n-dimensional variable ξ of abstract type A as the sequence of variables 〈ξ1, . . . , ξn〉, where each ξi is the k-th variable of type A . The pointwise abstraction λξM is then defined as 〈λξ1M1, . . . , λξnMn〉. A definition of pointwise substitution is left to the reader. With the definitions of pointwise application, pointwise abstraction, and n-dimensional variable in place, we can consider complex terms built up with these constructions. (7a), for example, is the pointwise application of (6b) to the pointwise composition of (6a) and (2c). Here ζ is of type np. (7a) can be expanded to (7b), where each dimension of a lexical sign is denoted with the help of an appropriate subscript (e.g. (6b)1 is λT.T ([a woman])). The terms here can be reduced and the result is as in (7c), a sign coupling the c-description in its first dimension to one of its possible readings. The other reading is obtained from (7d), which reduces to (7e). (7) a. (6b)(λζ.(6a)((2c)(ζ)))