Cogitate on refers to the act of thinking deeply, considering or reflecting on an issue or concept. The antonym for cogitate on may be to ignore, to neglect, to dismiss or to avoid. Ignoring implies paying no attention or giving no thoughts to the subject matter while neglecting involves failing to give proper care, attention or consideration. Dismissing, on the other hand, means dismissing an idea or concept as unimportant or unworthy of consideration. Avoiding implies that one is choosing not to think or consider something, perhaps because of fear or uncertainty.