We motivate the need for a new notion of observability for systems over finite alphabets and propose three new notions of output observability, thereby shifting our attention to the problem of state estimation for output prediction. We then consider a class of systems over finite alphabets with linear internal dynamics, finite-valued control inputs, and finitely quantized outputs. We derive a set of sufficient conditions and a set of necessary conditions for these systems to be output observable, propose an algorithmic procedure to verify one of these conditions, and construct finite memory output observers when certain conditions are met.