Emergence is a distinguishing feature in systems, especially when complexity grows with the number of components, interactions, and connectivity. There is immense interest in emergence, and a plethora of definitions from philosophy to sciences. Despite this, there is a lack of consensus on the definition of emergence and this hinders the development of a formal approach to understand and predict emergent behavior in multi-agent systems. This paper proposes a grammarbased set-theoretic approach to formalize and verify the existence and extent of emergence without prior knowledge or definition of emergent properties. Our approach is based on weak (basic) emergence that is both generated and autonomous from the underlying agents. In contrast with current work, our approach has two main advantages. By focusing only on system interactions of interest and feasible combinations of individual agent behavior, state-space explosion is reduced. In formalizing emergence, our extended grammar is designed to model agents of diverse types, mobile agents, and open systems. Theoretical and experimental studies using the boids model demonstrate the complexity of our formal approach.