In this talk, we examine a fundamental question posed some time ago by Birkhoff and von Neumann: What are the possible structures of measurement in a theory of Physics? We find striking connections with the conceptual framework of First Order Logic, and apply classification theorems for the automorphism groups of first order relational structures to argue for the uniqueness of Hilbert space as the phase space of a theory with probabilistic transitions between atomic states. We discuss possible applications of these observations to Algebraic Quantum Field Theory.