These are not very common words, but people who use them—especially lawyers—tend to mix them up. “Militate” is usually followed by “against” in a phrase that means “works against”: “His enthusiasm for spectacular collisions militates against his becoming a really effective air traffic controller.”
“Mitigate” means almost the opposite: to make easier, to moderate. “His pain at leaving was mitigated by her passionate kiss.” It should not be followed by “against.”
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