This paper sets up a framework for Lexicalized Tree Adjoining Grammar (LTAG) semantics that brings together ideas from different recent approaches addressing some shortcomings of LTAG semantics based on the derivation tree. The approach assigns underspecified semantic representations and semantic feature structure descriptions to elementary trees. Semantic computation is guided by the derivation tree and consists of adding feature value equations to the descriptions. A rigorous formal definition of the framework is given. Then, within this framework, an analysis is proposed that accounts for the different scopal properties of quantificational NPs (including nested NPs), adverbs, raising verbs and attitude verbs. Furthermore, by integrating situation variables in the semantics, different situation binding possibilities are derived for different types of quantificational elements.