Semantic theories of natural language associate meanings with utterances by providing meanings for lexical items and rules for determining the meaning of larger units given the meanings of their parts. Traditionally, meanings are combined via function composition, which works well when constituent structure trees are used to guide semantic composition. More recently, the functional structure of LFG has been used to provide the syntactic information necessary for constraining derivations of meaning in a cross-linguistically uniform format. It has been difficult, however, to reconcile this approach with the combination of meanings by function composition. In contrast to compositional approaches, we present a deductive approach to assembling meanings, based on reasoning with constraints, which meshes well with the unordered nature of information in the functional structure. Our use of linear logic as a 'glue' for assembling meanings also allows for a coherent treatment of modification as well as of the LFG requirements of completeness and coherence.