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This paper proposes a compositional semantics for lexicalized tree adjoining grammars (LTAG). Tree-local multicompnent derivations allow seperation of semantiv contribution of a lexical item into one component contributing to the predicate argument structure and second a component contributing to scope semantics. Based on this idea a syntx-semantics interface is presented where the compositional semantics depends only on the derivation structure. It is shown that the derivation structure allows an appropriate amount of underspecification. This is illustrated by investigating underspecified representations for quantifier scpoe ambiguities and related phenomena such as adjunct scope and island constraints.