In this article we argue that there is a large class of expressions, typified by plastic flower, stuffed animal and kosher bacon , that have a unique semantics combining compositional, idiomatic and decompositional interpretation. These expressions are compositional because their constituents contribute their meanings to the meanings of the wholes; they are idiomatic because their interpretation involves assigning dictionary entries to nonterminal elements in their syntactic structure; and they are decompositional because their meanings have proper parts that are not the meanings of any of their syntactic constituents. We argue that extensionalist semantics, on which the meaning of an expression is a function from domains to extensions in those domains, cannot provide an adequate account of the semantics of these expressions, and that supplementation with a theory of pragmatic interpretation does not improve the situation. We show how our account explains the intensionality and the productivity of these expressions.