The concept of "strong generative capacity" (SGC) of a linguistic formalism was introduced by Chomsky in the early sixties in order to characterize descriptive capacity. However, the original definition proposed by Chomsky turned out to be unusable, especially when one wished to compare the SGC of different types of formalisms. This book provides for the first time a rigorous and useful characterization of SGC, defining it as the model theoretic semantics of linguistic formalism. The book begins by reviewing classical definitions of weak and strong generative capacity. An in-depth discussion of how abstract interpretation domains are defined in theory-neutral set-theoretical terms is presented. The book also explains how interpretation domains are defined for labeled constituency, dependency, endocentricity, and linking, and applied to the analysis of a range of linguistic formalisms, among which are context-free grammars, dependency grammars, X-bar grammars, tree-adjoining grammars, transformational grammars and categorial grammars.