We present a model of the syntax-semantics interface for Tree-Adjoining Grammars (TAGs). It is based on the encoding of TAGs within the framework of Abstract Categorial Grammars (ACGs). This encoding relies on a functional interpretation of the substitution and adjunction operations of TAGs. In ACGs, the abstract terms representing derivation trees are full-fledged objects of the grammar. These terms are mapped onto logical formulas representing the semantic interpretation of natural expressions that TAGs can analyze. Because of the reversibility properties of ACGs, this provides a way to parse and generate with the same TAG encoded grammar. We propose several analyses, including long distance dependencies, quantification, control and raising verbs, and subordinate clauses. We also show how this encoding easily extends to other phenomena such as idioms or scope ambiguities. All the lexical data for theses modellings are provided and can be run with the ACG toolkit, a software package dedicated to the development of ACGs that can use these grammars both for parsing and generation.