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Lambda expressions are evaluated "hierarchically"--we resolve functions in the daughter node before we resolve functions in the mother node. In a given constituency, a sister node may define a variable for a lambda expression. But if there are three sister nodes, there will be ambiguity as to which order the variables get assigned. Does this mean that syntax trees must be binary branching in order for lambda calculus to work?

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    Another excellent question! – Tim Osborne Feb 15 '15 at 10:12
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Applicable to what? Meaning assembly? For this, λ-calculus is used only in categorical grammars, which are binary-branching by definition.

In syntactic frameworks with a context-free backbone, glue semantics (based on linear logic) is used at times, but it has problems, too.

Meaning assembly via composition rules is best done using unification. If one uses Davidsonian (or neo-Davidsonian, i.e., Parsonsian) logical forms, every phrase (including preterminals) is associated with an LF fragment and an individual. If you have a ternary (or, in general, a nonbinary) rule, the individual of a subordinated phrase is unified with a variable in the LF fragment of the head. For example, the predicate of "give" is quaternary and the corresponding rule is VP -> V NP NP. Then the individual associated with V (the eventuality) is unified with that of VP. The individual associated with the indirect object is unified with the fourth argument of the predicate of the verb, etc. The subject variable remains open until it's unified later by another rule. Since (neo-)Davidsonian formulae are existentially closed conjunctions of literals, when the parsing is completed we take all literals used during parsing to be conjuncts and add a quantifier for each variable occurring in the LF.

The above procedure can be informally described as a "relaxed" lambda calculus, but rather than replace variables we unify them. It can also be used in dependency grammars if we add unifications to ID rules.

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