ion’ have proven useful in compositional semantic analyses. λ-conversion is the formal operation accompanying the semantic operation of functional application. It combines the meaning of two expressions one of which serves as argument to the other. λabstraction is the opposite process. It allows for a decomposition of the (known) meaning of a complex structure into the meaning of its parts. λ-abstraction filters out the semantic contribution of those parts of the structure whose meaning is known, yielding the meaning for those parts whose meaning was previously unknown. At the same time, application of λ-abstraction turns saturated expressions (in Frege’s terminology) into unsaturated expressions, i.e. into functions, which demand an argument. An example from Frege (1891) may serve to illustrate this point. Consider the DP in (134). (134) the capital of the German Empire In Frege’s time, the DP in (134) denoted the city of Berlin. Frege argues that this meaning can be derived on the base of the meaning of its parts. He suggests to decompose the DP into two parts of which we know the meaning of one (the German Empire). He further proposes to analyse the meaning of the remaining part of the DP, the capital of, as a function which takes countries as input and maps them onto their respective capital. The functional nature of the meaning of the capital of is illustrated in (135). (135a) shows the functional nature of the expression. Choosing different countries as argument to this function yields different values (cf. 135b-c). (135) a. [[the capital of]] (x) = y b. [[the capital of]] (germany’) = berlin’ c. [[the capital of]] (finland’) = helsinki’ The functional nature of the meaning of the capital of can be derived by applying λabstraction to the meaning of the DP the capital of the German Empire in (134). By factoring out the known meaning of German Empire, we arrive at the semantic representation for [[the capital of]] in (136). (136) [[the capital of]] = λx . the unique city which is the capital of x. The expression on the right-hand side of the equation represents a function and reads as ‘a function from individuals x (of type ) into the unique city which is the capital of x’. By convention, expressions to the left of the dot, stand for the arguments of the function. Expressions to the right of the dot stand for the function values. These are, typically, an INTERPRETING JEWEILS 217 individual (as in the case at hand), a truth value (represented by the truth conditions), or another function. What is of general interest here is that λ-abstraction is a process that creates unsatured expressions from saturated ones. As such, it can be employed in semantic derivations in order to ensure interpretability. Bittner (1994:69) postulates the rule of λ-abstraction as a cross-linguistic universal. (137) Bittner’s Rule of λ-Abstraction (simplified version): Let α have have a translation [[α]], and let the index ‘i’ be the index of either α or a sister of α, and let [[α]] contain a variable u with index ‘i’. Then λui.[[α]] is a translation of α. Stated in simple terms, the rule applies by adding a λ-operator to the denotation of α, thus binding a variable which has an index identical to an index on either α or α’s sister. Applying λ-abstraction results in ‘opening up’ an argument position in the denotation of α. Thus, λ-abstraction turns saturated expressions which contain a free variable into functions which can apply to arguments of the same type as the variable. On this view, λabstraction enters the semantic derivation in order to ensure the interpretability of otherwise uninterpretable configurations. A typical non-interpretable configuration obtains when two syntactic sisters are of the ‘wrong’ semantic type, i.e. when they cannot be combined with each other. Such situations are generally grouped under the label of ‘type mismatch’. Type mismatch arises in three cases: (i) both sisters denote saturated expressions such that none can take the other as its argument; (ii) both sisters are functions, but none is of the right kind to serve as argument to the other; (iii) one sister denotes a function, and the other a saturated expression, but of the wrong kind. Summing up, λ-abstraction applies in order to ensure interpretability in cases of type mismatch. Since it is meant to apply universally, (137) is stated in a very general form. For the purposes at hand, I will use a more restricted version of (137), which allows for a composition of the meaning of jeweils-DP and its syntactic sister, and which can be shown to apply elsewhere in the grammars of German and English. The restricted λ-rule is given in (138): (138) Index-Triggered λ-Abstraction: If the semantic types of a proposition-denoting expression α and its syntactic sister β do not match, and if [[α]] contains a free variable ui that shares an index ‘i’ with β, λ-abstraction in [[α]] over index ‘i’ is licensed, and λui.[[α]] is a value for α. 54 E.g., (137) is defined in such a way that it can also account for the semantic effects of syntactic head movement, which shall not concern us here. 55 At first sight, the formulation of (138) creates a problem for compositionality. The meaning of the node consisting of α and β cannot be derived by looking at the meanings of α and β alone. For λ-abstraction to be licensed, the co-indexed free variable must be visible at the stage of the derivation where α and β combine, which it is not. As a result, the semantic derivation apparently looks into the meaning of α, of which the variable forms a part, a process not in line with compositionality (and which seems to lead Bittner (1994) to assume a mechanism of variable storage). However, Dekker (1993:165f.) shows that the apparent problem for compositionality can be overcome by working with partial assignment functions. See also Dekker 1998:332, fn.5 on the special case of the assignment function assigning the empty set. (thanks to Ede Zimmermann, p.c., for pointing this out)
Stored only when both papers are in the corpus — 65 citations exist upstream.