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In the tradition of Model Theoretic Syntax, we propose a logical approach to the description of grammars. We combine in one formalism several tools that are used throughout computer science for their power of abstraction: logic and lambda-calculus. We propose then a high-level formalism for describing mildly context sensitive grammars and their semantic interpretation. As we rely on the correspondence between logic and finite state automata, our method combines conciseness to effectivity. We illustrate our approach with a simple linguistic model of several interleaved linguistic phenomena involving extraction. The level of abstraction provided by logic and λ-calculus allows us not only to use this linguistic model for several languages, namely English, German and Dutch, but also for semantic interpretation.