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We propose formal means for synthesizing switching protocols that determine the sequence in which the modes of a switched system are activated to satisfy certain high-level specifications in linear temporal logic (LTL). The synthesized protocols are robust against exogenous disturbances on the continuous dynamics and can react to possibly adversarial events (both external and internal). Finite-state approximations that abstract the behavior of the underlying continuous dynamics are defined using finite transition systems. Such approximations allow us to transform the continuous switching synthesis problem into a discrete synthesis problem in the form of a two-player game between the system and the environment, where the winning conditions represent the high-level temporal logic specifications. Restricting to an expressive subclass of LTL formulas, these temporal logic games are amenable to solutions with polynomial-time complexity. By construction, existence of a discrete switching strategy for the discrete synthesis problem guarantees the existence of a switching protocol that can be implemented at the continuous level to ensure the correctness of the nonlinear switched system and to react to the environment at run time.